Universidade Federal do Espírito Santo Centro de Ciências Exatas Departamento de Matemática Dissertação de Mestrado em Matemática

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊s♣ír✐t♦ ❙❛♥t♦

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

  

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛

▼✉❧t✐♣❧✐❝✐❞❛❞❡ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s

♣♦s✐t✐✈❛s ♣❛r❛ ✉♠❛ ❡q✉❛çã♦ ❡❧í♣t✐❝❛ q✉❛s✐❧✐♥❡❛r

  

❏♦sé ❈❛r❧♦s ❞❡ ❖❧✐✈❡✐r❛ ❏✉♥✐♦r

❛ ❛

  ❖r✐❡♥t❛❞♦r❛✿ Pr♦❢ ✳ ❉r ✳ ▼❛❣❞❛ ❙♦❛r❡s ❳❛✈✐❡r

  

❆❣r❛❞❡❝✐♠❡♥t♦s

  ✲ ❉❡ ✉♠❛ ❢♦r♠❛ ♠✉✐t♦ ❡s♣❡❝✐❛❧✱ à ❉❡✉s✱ q✉❡ t❡♠ s✐❞♦ ✉♠ ót✐♠♦ ♣❛✐ ❡ t❡♠ ♠❡ ❝♦♥❝❡❞✐❞♦ ❣r❛♥❞❡s ♦♣♦rt✉♥✐❞❛❞❡s✳ ❙❡ ❝❤❡❣✉❡✐ ❛té ❛q✉✐✱ ✐ss♦ ❞❡✈♦ à ❊❧❡✳ ✲ ❆♦ ♠❡✉ ♣❛✐ ✭♠❛✐s ❞♦ q✉❡ ❡♠ ♠❡♠ór✐❛✮✱ ❏♦sé ❈❛r❧♦s✱ ❡ à ♠✐♥❤❛ ♠ã❡✱ ▼❛r✐❛ ❞❡ ▲♦✉r❞❡s✱ ♣❡❧❛ ❡❞✉❝❛çã♦✱ ♣❡❧♦ ❢♦rt❡ ✐♥❝❡♥t✐✈♦ ❡ ♣♦r t♦❞♦s ♣r❡❝✐♦s♦s ❝♦♥s❡❧❤♦s✳ ➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♣❡❧♦ ❛❝♦❧❤✐♠❡♥t♦ ❞✉r❛♥t❡ ❛ ❣r❛❞✉❛çã♦ ❡ ♦ ♠❡str❛❞♦ ❡ ♣♦r s❡ ♣r❡♦❝✉♣❛r❡♠ t❛♥t♦ ❝♦♠✐❣♦ ♥❡ss❡ ♣❡rí♦❞♦✳ ❉❡st❛❝♦ ♠✐♥❤❛ ❛✈ó✱ ◆❛s❝✐r❡♠❛✱ ❝✉❥♦ ❝❛r✐♥❤♦ ❡ ❝✉✐❞❛❞♦ ❝♦♠ ♠✐♥❤❛ ❢♦r♠❛çã♦ ♠❡ ❞❡r❛♠ ❢♦rç❛s ♣r❛ ❝♦♥t✐♥✉❛r✳ ✲ ➚ t♦❞♦s ♦s ❛♠✐❣♦s ❞❛ ✐❣r❡❥❛✱ ❞❡♥tr❡ ♦s q✉❛✐s ❞❡st❛❝♦ ▼❛✉rí❧✐♦ ❡ ●❛❜r✐❡❧❛✱ ♣♦r ♠❡ ♣r♦✲ ♣♦r❝✐♦♥❛r❡♠ ❞✐❛s ✐♥❞❡s❝r✐tí✈❡✐s q✉❡ s❡♠♣r❡ ❣✉❛r❞❛r❡✐ ❝♦♠✐❣♦✳ ❆♦s ❛♠✐❣♦s ❞♦ ♣♦❦❡r✱ ♣❡❧♦s ♠♦♠❡♥t♦s ú♥✐❝♦s ❞❡ ❞❡s❝♦♥tr❛çã♦✳ ➚s ❛♠✐❣❛s ▼✐❝❤❡❧❧❡ ❡ ❏❛q✉❡❧✐♥❡✱ ♣♦r t❡r❡♠ t♦r♥❛❞♦ ❛ ❣r❛❞✉❛çã♦ ✉♠ ♣❡rí♦❞♦ ♠❛✐s ❛❣r❛❞á✈❡❧ ❡ ❞✐✈❡rt✐❞♦✳ ✲ ❆♦s ♣r♦❢❡ss♦r❡s ▼❛r❝❡❧♦ ❋❡r♥❛♥❞❡s ❋✉rt❛❞♦ ❡ ❏♦ã♦ P❛❜❧♦ P✐♥❤❡✐r♦ ❞❛ ❙✐❧✈❛✱ ♣♦r t❡✲ r❡♠ ❛❝❡✐t♦ ♣❛rt✐❝✐♣❛r ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ❞❡st❡ tr❛❜❛❧❤♦✳ ✲ ➚ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ▼❛❣❞❛ ❙♦❛r❡s ❳❛✈✐❡r✱ ♣♦r s✉❛ ❡♥♦r♠❡ ♣❛❝✐ê♥❝✐❛ ❡♠ r❡s♣♦♥❞❡r ♠✐♥❤❛s ✐♥ú♠❡r❛s ♣❡r❣✉♥t❛s✱ ♣♦r s✉❛ ❞❡❞✐❝❛çã♦ ❛ ❡st❡ tr❛❜❛❧❤♦ ❡ ♣❡❧❛ ❡①♣❡r✐ê♥❝✐❛ q✉❡ ♣✉❞❡ ❛❞q✉✐r✐r ❝♦♠ s❡✉s ❡♥s✐♥❛♠❡♥t♦s✳ ✲ ➚ ❈❛♣❡s✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳

  

❘❡s✉♠♦

  ◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛♠♦s r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s ♣♦✲

  N

  s✐t✐✈❛s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❡♠ R ❡♥✈♦❧✈❡♥❞♦ ♦ ♦♣❡r❛❞♦r p✲❧❛♣❧❛❝✐❛♥♦ ❝♦♠ 2 ≤ p < N ✱ ✉♠❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ ❞♦ t✐♣♦ ♣♦tê♥❝✐❛ ❝♦♠ ❡①♣♦❡♥t❡ q s✉❜❝rít✐❝♦✱ ✉♠ ♣❛✲ râ♠❡tr♦ λ ♣♦s✐t✐✈♦ ❡ ✉♠ ♣♦t❡♥❝✐❛❧ a(x) s❛t✐s❢❛③❡♥❞♦ ❝❡rt❛s ❤✐♣ót❡s❡s✳ ❚❛❧ ♣r♦❜❧❡♠❛ ❢♦✐

  ✐♥✐❝✐❛❧♠❡♥t❡ ❡st✉❞❛❞♦ ♣♦r ❇❛rts❝❤ ❡ ❲❛♥❣ ❡♠ ❬✺❪ ♥♦ ❝❛s♦ ❞♦ ♦♣❡r❛❞♦r ❧❛♣❧❛❝✐❛♥♦ ✭p = 2✮✳ ❆♣r❡s❡♥t❛♠♦s ❛s ✈❡rsõ❡s ❞♦s r❡s✉❧t❛❞♦s ❞❡ ❬✺❪ ♣❛r❛ ♦ ❝❛s♦ ❞♦ p✲❧❛♣❧❛❝✐❛♥♦✱ ❞❡♠♦♥str❛❞❛s ♣♦r ❋✉rt❛❞♦ ❡♠ ❬✶✼✱ ✶✽❪✳

  

❆❜str❛❝t

  ■♥ t❤✐s ✇♦r❦✱ ✇❡ st✉❞② r❡s✉❧ts ♦♥ ❡①✐st❡♥❝❡ ❛♥❞ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s

  N

  ❢♦r ❛ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥ ✐♥ R ✐♥✈♦❧✈✐♥❣ t❤❡ p✲❧❛♣❧❛❝✐❛♥ ♦♣❡r❛t♦r ✇✐t❤ 2 ≤ p < N✱ ❛ s✉❜❝r✐t✐❝❛❧ ♥♦♥❧✐♥❡❛r✐t②✱ ❛ ♣♦s✐t✐✈❡ ♣❛r❛♠❡t❡r λ ❛♥❞ ❛ ♣♦t❡♥❝✐❛❧ a(x) s❛t✐s❢②✐♥❣ s♦♠❡ ❤②♣♦t❤❡s❡s✳ ❙✉❝❤ ♣r♦❜❧❡♠ ✇❛s ✜rst st✉❞✐❡❞ ❜② ❇❛rts❝❤ ❛♥❞ ❲❛♥❣ ❬✺❪ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❧❛♣❧❛❝✐❛♥ ♦♣❡r❛t♦r ✭p = 2✮✳ ❲❡ ♣r❡s❡♥t ✈❡rs✐♦♥s ♦❢ t❤❡ r❡s✉❧ts ♦❢ ❬✺❪ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ p

  ✲❧❛♣❧❛❝✐❛♥✱ ✇❤✐❝❤ ✇❡r❡ ❞❡♠♦♥str❛t❡❞ ❜② ❋✉rt❛❞♦ ❬✶✼✱ ✶✽❪✳

  ❙✉♠ár✐♦

  ■♥tr♦❞✉çã♦ ✶

  ✶ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✺

  ✶✳✶ ❈♦♥❥✉♥t♦s ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ❋✉♥❝✐♦♥❛❧ r❡str✐t♦ ❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ❆ ❝❛t❡❣♦r✐❛ ❞❡ ▲❥✉st❡r♥✐❦✲❙❝❤♥✐r❡❧♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✹ ❘❡s✉❧t❛❞♦s ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❡ ❝♦♠♣❛❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✺ ❖ ▲❡♠❛ ❞❡ ❇ré③✐s✲▲✐❡❜ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✻ ❯♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✼ ❚❡♦r❡♠❛ ❞❡ ❡①t❡♥sã♦ ❞❡ ❉✉❣✉♥❞❥✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

  ✷ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✷✵

  ✷✳✶ ❈♦♥s✐❞❡r❛çõ❡s ✐♥✐❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷ ❆ ❝♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✸ ❙♦❧✉çõ❡s ♣♦s✐t✐✈❛s ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✹ ❈♦♥❝❡♥tr❛çã♦ ❞❛s s♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

  ✸ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✺✷

  ■♥tr♦❞✉çã♦

  ❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛✱ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❡ ❝♦♥❝❡♥✲ tr❛çã♦ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ❞❛ s❡❣✉✐♥t❡ ❝❧❛ss❡ ❞❡ ❡q✉❛çõ❡s q✉❛s✐❧✐♥❡❛r❡s ❞❡ ❙❝❤rö❞✐♥❣❡r 

  p−2 q−2 N

   −∆ u + (λa(x) + 1)|u| u = |u| u, x ∈ R ,

  p

  (S )

  λ,q 1,p N

   u ∈ W (R ),

  ∗ ∗ p−2

  = N p/(N − p) u = ∇u)

  p

  ❡♠ q✉❡ 2 ≤ p < N, p < q < p ✱ ♦♥❞❡ p ✱ ∆ ❞✐✈(|∇u| é ♦ ♦♣❡r❛❞♦r p✲❧❛♣❧❛❝✐❛♥♦ ❡ λ é ✉♠ ♣❛râ♠❡tr♦ ♣♦s✐t✐✈♦✳ ❱❛♠♦s ❝♦♥s✐❞❡r❛r a ✉♠❛ ❢✉♥çã♦ s❛t✐s❢❛③❡♥❞♦

  −1 N

  2

  , R) (0) ✭A

  

1 ✮ a ∈ C(R é ♥ã♦ ♥❡❣❛t✐✈❛✱ Ω = ✐♥t a é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞❡ ❝❧❛ss❡ C

−1

  (0) ❡ Ω = a ❀

  > 0

  2

  ✭A ✮ ❡①✐st❡ M t❛❧ q✉❡

  N

  L({x ∈ R : a(x) ≤ M }) < ∞ ✱

  

N

  ♦♥❞❡ L é ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❡♠ R ✳ )

  ❯♠ ❞♦s ♠♦t✐✈♦s ❞❡ ❡st✉❞❛r♠♦s s♦❧✉çõ❡s ❞❡ (S λ,q é q✉❡✱ ♣❛r❛ λ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❉✐r✐❝❤❧❡t

  (

  p−2 q−2

  −∆

  p u + |u| u = |u| u

  ❡♠ Ω, (D )

  q

  u = 0 ❡♠ ∂Ω, s❡ t♦r♥❛ ✉♠ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛ ❧✐♠✐t❡✳

  )

  q

  ❊♠ ❬✻❪✱ ❇❡♥❝✐ ❡ ❈❡r❛♠✐ ❡st✉❞❛r❛♠ ♦ ♣r♦❜❧❡♠❛ (D ♥♦ ❝❛s♦ ❞♦ ♦♣❡r❛❞♦r ❧❛♣❧❛❝✐❛♥♦✳

  q )

  ❊❧❡s ♣r♦✈❛r❛♠ q✉❡ (D ❝♦♠ p = 2 ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ❝❛t(Ω) s♦❧✉çõ❡s ♣❛r❛ q ♣ró①✐♠♦ ❞❡

  ■♥tr♦❞✉çã♦ ✷

  ∗

  2 ✳ ❆q✉✐✱ ❝❛t(Ω) ❞❡♥♦t❛ ❛ ❝❛t❡❣♦r✐❛ ❞❡ ▲❥✉st❡r♥✐❦✲❙❝❤♥✐r❡❧♠❛♥♥ ❞♦ ❝♦♥❥✉♥t♦ Ω✳ P♦st❡✲ r✐♦r♠❡♥t❡✱ ♦✉tr♦s ❛rt✐❣♦s ❢♦r❛♠ ♣✉❜❧✐❝❛❞♦s tr❛t❛♥❞♦ ❞❛ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ♣❛r❛

  (D )

  q

  ❝♦♠ p = 2 ❡♠ ❢✉♥çã♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞❡ Ω✱ t❛♥t♦ ♥♦ ❝❛s♦ s✉❜❝rít✐❝♦ ❬✽✱ ✶✶✱ ✼❪ q✉❛♥t♦ ♥♦ ❝❛s♦ ❝rít✐❝♦ ❬✷✽✱ ✷✷❪✳ ❖ ❝❛s♦ q✉❛s✐❧✐♥❡❛r (2 < p < N) ❢♦✐ ❝♦♥s✐❞❡r❛❞♦ ♣♦r ❆❧✈❡s ❡ ❉✐♥❣ ❡♠ ❬✷❪ ❡ ❋✉rt❛❞♦ ❡♠ ❬✶✻❪✳

  ) ❖ ♣r♦❜❧❡♠❛ (S λ,q ❢♦✐ ❡st✉❞❛❞♦ ♣♦r ❇❛rts❝❤ ❡ ❲❛♥❣ ❡♠ ❬✺❪ ♥♦ ❝❛s♦ s❡♠✐❧✐♥❡❛r p = 2✳

  ❊❧❡s ♣r♦✈❛r❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ♣❛r❛ λ s✉✜❝✐❡♥t❡✲ ♠❡♥t❡ ❣r❛♥❞❡✳ ▼❛✐s ❛✐♥❞❛✱ q✉❛♥❞♦ λ → ∞✱ ❡ss❛s s♦❧✉çõ❡s s❡ ❝♦♥❝❡♥tr❛♠ ❡♠ ✉♠❛ s♦❧✉çã♦

  )

  

q

  ♣♦s✐t✐✈❛ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ❞♦ ♣r♦❜❧❡♠❛ (D ✳ ❚❛♠❜é♠✱ s✉♣♦♥❞♦ Ω ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ )

  λ,q

  ❡ ❛♣♦✐❛❞♦s ♥♦ r❡s✉❧t❛❞♦ ❛♥t❡r✐♦r ❞❡ ❇❡♥❝✐ ❡ ❈❡r❛♠✐✱ ♦s ❛✉t♦r❡s ♣r♦✈❛r❛♠ q✉❡ (S ❝♦♠

  ∗

  p = 2 ♣♦ss✉✐ ❝❛t(Ω) s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ♣❛r❛ λ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ❡ q ♣ró①✐♠♦ ❞❡ 2 ✳

  ❊♠ ❬✶✼✱ ✶✽❪✱ ❋✉rt❛❞♦ ❡st✉❞♦✉ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿ 

  p−2 q−2 N

  −∆

  p u + (λa(x) + 1)|u| u = |u| u ,

   ❡♠ R

   

  N

  u(τ x) = −u(x) ,

  ♣❛r❛ t♦❞♦ x ∈ R  

  1,p N

   u ∈ W (R ),

  ∗ N

  ❝♦♠ λ > 0✱ 2 ≤ p < N, p < q < p ✱ τ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ♦rt♦❣♦♥❛❧ ❞❡ R ❡♠

  N

2 R

  = Id ), (A )

  1

  2

  s❛t✐s❢❛③❡♥❞♦ τ 6= Id ❡ τ ✳ ❖ ♣♦t❡♥❝✐❛❧ a s❛t✐s❢❛③ (A ❡ é ✐♥✈❛r✐❛♥t❡

  N

  ♣♦r τ✱ ✐st♦ é✱ a(τx) = a(x) ♣❛r❛ t♦❞♦ x ∈ R ✳ ❯t✐❧✐③❛♥❞♦ ♠ét♦❞♦s ✈❛r✐❛❝✐♦♥❛✐s ❡❧❡ ♦❜t❡✈❡✱ ♣❛r❛ λ ❣r❛♥❞❡✱ r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s q✉❡ ♠✉❞❛♠ ❞❡ s✐♥❛❧ ❡①❛t❛♠❡♥t❡ ✉♠❛ ✈❡③✱ ❛❧é♠ ❞❛ r❡❧❛çã♦ ❡♥tr❡ ♦ ♥ú♠❡r♦ ❞❡ss❛s s♦❧✉çõ❡s ❝♦♠ ❛ t♦♣♦❧♦❣✐❛ ❡q✉✐✈❛r✐❛♥t❡ ❞♦ ❝♦♥❥✉♥t♦ Ω ♦♥❞❡ ♦ ♣♦t❡♥❝✐❛❧ s❡ ❛♥✉❧❛✱ q✉❛♥❞♦ q é ♣ró①✐♠♦ ❞❡

  ∗

  p ✳ ❆❞❛♣t❛♥❞♦ ❛s ✐❞❡✐❛s ❞❛s ❞❡♠♦♥str❛çõ❡s ❞❡ss❡s r❡s✉❧t❛❞♦s ❞❡ s♦❧✉çõ❡s ♥♦❞❛✐s ♣❛r❛ ♦

  ❝❛s♦ s❡♠ ❤✐♣ót❡s❡ ❞❡ s✐♠❡tr✐❛✱ ♦ ❛✉t♦r ♣ô❞❡ ❡st❡♥❞❡r ♦s r❡s✉❧t❛❞♦s ❞❡ ❬✺❪ ♣❛r❛ ♦ ❝❛s♦ q✉❛s✐❧✐♥❡❛r✳ ◆♦ss♦ ♦❜❥❡t✐✈♦ ❛q✉✐ é ❡st✉❞❛r ♦s r❡s✉❧t❛❞♦s ❞❡ ❇❛rts❝❤ ❡ ❲❛♥❣ ❬✺❪ ❡st❡♥❞✐❞♦s ♣❛r❛

  ♦ ❝❛s♦ 2 ≤ p < N✱ ❞❡♠♦♥str❛❞♦s ♣♦r ❋✉rt❛❞♦ ❡♠ ❬✶✼✱ ✶✽❪✳ ❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡♠ q✉❡ tr❛❜❛❧❤❛r❡♠♦s é Z

  1,p N p E = u ∈ W (R ) : a(x)|u| < ∞ .

  N R λ = (E, k · k λ )

  P❛r❛ λ ≥ 0✱ ❞❡✜♥✐♠♦s E ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ E ♠✉♥✐❞♦ ❞❛ ♥♦r♠❛

  1/p

  Z

  p p kuk = (|∇u| + (λa(x) + 1)|u| ) . λ

  N R

  ■♥tr♦❞✉çã♦ ✸

  

λ,q λ,q : E → R

  ❆ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ✭S ✮ t❡♠♦s ♦ ❢✉♥❝✐♦♥❛❧ eI ❞❡✜♥✐❞♦ ♣♦r Z Z

  1

  1

  p p q

  e I (u) = (|∇u| + (λa(x) + 1)|u| ) − |u| .

  λ,q N N

  p R q R )

  λ λ,q

  ❉✐③❡♠♦s q✉❡ u ∈ E é ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ (S q✉❛♥❞♦ Z Z Z

  

p−2 p−2 q−2

  |∇u| ∇u · ∇φ + (λa(x) + 1)|u| uφ − |u| uφ = 0,

  N N N R R R λ λ,q )

  ♣❛r❛ t♦❞❛ ❢✉♥çã♦ φ ∈ E ✳ ❯♠❛ s♦❧✉çã♦ u ❞❡ (S é ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ q✉❛♥❞♦ n o e e

  I (u) = inf I (v) : v ) .

  λ,q λ,q λ,q

  é ✉♠❛ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ❞❡ (S ◆♦ ❈❛♣ít✉❧♦ ✷✱ ❛♣r❡s❡♥t❛♠♦s ♦s r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s

  )

  λ,q

  ♣♦s✐t✐✈❛s ❞❡ (S ♣❛r❛ λ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳ ❙ã♦ ❡❧❡s✿ ) ) = Λ (q)

  ❚❡♦r❡♠❛ ❆ ❙✉♣♦♥❤❛ q✉❡ (A

  1 ❡ (A 2 s❡❥❛♠ ✈á❧✐❞❛s✳ ❊♥tã♦ ❡①✐st❡ Λ t❛❧ q✉❡✱ λ,q

  ♣❛r❛ t♦❞♦ λ ≥ Λ ✱ ♦ ♣r♦❜❧❡♠❛ ✭S ✮ t❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛✳

  ) ⊂ R → ∞ )

  n n n

  ❚❡♦r❡♠❛ ❇ ❙❡❥❛♠ (λ t❛❧ q✉❡ λ ❡ (u ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ) (u )

  λ n ,q λ n ,q n

  ❞♦ ♣r♦❜❧❡♠❛ (S t❛❧ q✉❡ eI é ❧✐♠✐t❛❞♦✳ ❊♥tã♦✱ ❛ ♠❡♥♦s ❞❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱

  1,p N

  u → u (R ) )

  n ❢♦rt❡ ❡♠ W ❝♦♠ u s❡♥❞♦ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❞♦ ♣r♦❜❧❡♠❛ (D q ✳

  → ∞

  n ) ⊂ R n n )

  ❈♦r♦❧ár✐♦ ❈ ❙❡❥❛♠ (λ ❝♦♠ λ ❡ (u ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s

  1,p N

  ) ) (R )

  λ n ,q n

  ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ❞♦ ♣r♦❜❧❡♠❛ (S ✳ ❊♥tã♦ (u ❝♦♥✈❡r❣❡ ❡♠ W ❛♦ ❧♦♥❣♦ ❞❡ )

  q

  ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ♣❛r❛ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ❞❡ (D ✳

  −1

  (0) ◆♦ ❈❛♣ít✉❧♦ ✸✱ ❡st✉❞❛♠♦s ❛ r❡❧❛çã♦ ❡♥tr❡ ❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω = ✐♥t a ❡ ♦

  ) ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ❞❡ (S λ,q ✳ ▼❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡♠♦♥str❛✲ çã♦ ❞♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✳

  1 ) 2 )

  ❚❡♦r❡♠❛ ❉ ❙✉♣♦♥❤❛ q✉❡ (A ❡ (A s❡❥❛♠ ✈á❧✐❞❛s ❡ q✉❡ Ω s❡❥❛ ❧✐♠✐t❛❞♦✳ ❊♥tã♦ ❡①✐st❡

  ∗ ∗

  q ∈ (p, p ) , p )

  ❝♦♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡✱ ♣❛r❛ ❝❛❞❛ q ∈ (q ✱ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ Λ(q) > 0 t❛❧ )

  λ,q

  q✉❡✱ ♣❛r❛ t♦❞♦ λ ≥ Λ(q)✱ ♦ ♣r♦❜❧❡♠❛ (S t❡♠ ♣❡❧♦ ♠❡♥♦s ❝❛t(Ω) s♦❧✉çõ❡s ♣♦s✐t✐✈❛s✳ ❊♠❜♦r❛ ❛s ❞❡♠♦♥str❛çõ❡s ❞♦s r❡s✉❧t❛❞♦s ❛❝✐♠❛ s✐❣❛♠ ❛q✉❡❧❛s ❞❡ ❇❛rts❝❤ ❡ ❲❛♥❣✱

  ❡①✐st❡ ✉♠❛ ♠❛✐♦r ❞✐✜❝✉❧❞❛❞❡ té❝♥✐❝❛ ❞❡✈✐❞❛ à ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ ❞♦ ♦♣❡r❛❞♦r p✲❧❛♣❧❛❝✐❛♥♦ ❡ ♣❡❧♦ ❢❛t♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❢✉♥çõ❡s ❡♠ q✉❡ s❡ ❞❡✈❡ tr❛❜❛❧❤❛r ♥ã♦ s❡r ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✳ ❉❡♥tr❡ ♦s r❡s✉❧t❛❞♦s q✉❡ ❛✉①✐❧✐❛♠ ❛ tr❛♥s♣♦r ❛ ❞✐✜❝✉❧❞❛❞❡ ✐♥tr♦❞✉③✐❞❛ ♣❡❧♦ p✲❧❛♣❧❛❝✐❛♥♦✱

  ■♥tr♦❞✉çã♦ ✹

  ✉♠❛ ✈❡rsã♦ ❞❡ ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❡ ❝♦♠♣❛❝✐❞❛❞❡ ✭❬✸✸❪✱ ▲❡♠❛ ✶✳✹✵✮ ❞❡♠♦♥s✲ tr❛❞❛ ♣♦r ❋✉rt❛❞♦ ✭❬✶✽❪✱ ▲❡♠❛ ✷✳✻✮ ❡ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛s s❡q✉ê♥❝✐❛s ♠✐♥✐♠✐③❛♥t❡s ♣❛r❛ S

  ∗ 1,p p

  (Ω) ֒→ L (Ω) ✭❬✶✽❪✱ ▲❡♠❛ ❆✳✶✶✮✱ ♦♥❞❡ S é ❛ ♠❡❧❤♦r ❝♦♥st❛♥t❡ ❞❛ ✐♠❡rsã♦ ❞❡ ❙♦❜♦❧❡✈ W ✳

  ❈❛♣ít✉❧♦ ✶ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s

  ◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s r❡s✉❧t❛❞♦s ❡ ❞❡✜♥✐çõ❡s q✉❡ ✉t✐❧✐③❛♠♦s ♥♦ ❞❡❝♦rr❡r ❞❡st❡ tr❛❜❛❧❤♦✳

  ✶✳✶ ❈♦♥❥✉♥t♦s ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s

  ❉❡✜♥✐çã♦ ✶✳✶ ❙❡❥❛♠ X ❡ Y ❞♦✐s ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳ ❯♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ ❛s ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s f, g : X → Y é ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ H : [0, 1] × X → Y t❛❧ q✉❡ ♣❛r❛ t♦❞♦ x ∈ X

  ✱ t❡♠✲s❡ H(0, x) = f(x) ❡ H(1, x) = g(x)✳ ❊s❝r❡✈❡✲s❡ f ≃ g ♣❛r❛ ✐♥❞✐❝❛r q✉❡ ❡①✐st❡ ✉♠❛ ❤♦♠♦t♦♣✐❛ H ❡♥tr❡ f ❡ g✳ ❉❡✜♥✐çã♦ ✶✳✷ ❙❡❥❛♠ X ❡ Y ❞♦✐s ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳ ❉✐③❡♠♦s q✉❡ X ❡ Y sã♦ ❤♦♠♦t♦✲ ♣✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s s❡ ❡①✐st❡♠ ❛♣❧✐❝❛çõ❡s ❝♦♥tí♥✉❛s f : X → Y ❡ g : Y → X t❛✐s q✉❡ f ◦ g ≃ Id

  Y

  X X Y

  ❡ g ◦ f ≃ Id ✱ ♦♥❞❡ Id é ❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ ❞❡ X ❡ Id ❞❡ Y ✳

  N

  P❛r❛ Ω ⊂ R ❡ r > 0 ❞❡✜♥✐♠♦s

  −

  • N

  Ω = {x ∈ R : = {x ∈ Ω :

  

r ❞✐st(x, Ω) < r} ❡ Ω r ❞✐st(x, ∂Ω) ≥ r}✳

  ◆❡st❛ s❡çã♦✱ ♥♦ss♦ ♦❜❥❡t✐✈♦ é ♣r♦✈❛r ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✱ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❛ ❞❡♠♦♥s✲ tr❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❉✳

  N

  2

  , r > 0

  1

  2

  ▲❡♠❛ ✶✳✸ ❙❡❥❛ Ω ⊂ R ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❞❡ ❝❧❛ss❡ C ✳ ❊♥tã♦ ♣❛r❛ r

  ❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✻

  ❆ ❞❡♠♦♥str❛çã♦ q✉❡ ❞❛♠♦s ❛q✉✐ ❡♥❝♦♥tr❛✲s❡ ❡♠ ❬✶✸❪ ✭Pr♦♣♦s✐çã♦ ✷✳✸✳✸✮✳ ❆♥t❡s✱ ❛♣r❡✲ s❡♥t❛♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ ✉♠ r❡s✉❧t❛❞♦ s♦❜r❡ s✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s q✉❡ s❡rã♦ ♥❡❝❡s✲ sár✐♦s✳

  m N k

  ⊂ R ❉❡✜♥✐çã♦ ✶✳✹ ❙❡❥❛ M = M ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ ❞✐♠❡♥sã♦ m ❡ ❝❧❛ss❡ C ✱ k ≥ 1✳

  m N

  → U ⊂ R ❙❡❥❛ ϕ : U ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ❝♦♠ U ✉♠ ❛❜❡rt♦✱ U ⊂ R ❡ p = ϕ(x) ∈ M ♣❛r❛ ❛❧❣✉♠ x ∈ U ✳ ❖ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❛ M ♥♦ ♣♦♥t♦ p é ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ m

  ′ m T M = ϕ (x) · R . p N

  ❉✐③❡♠♦s q✉❡ ✉♠ ✈❡t♦r u ∈ R é ♥♦r♠❛❧ ❛ s✉♣❡r❢í❝✐❡ M ♥♦ ♣♦♥t♦ p q✉❛♥❞♦ u ❢♦r ♣❡r♣❡♥❞✐❝✉❧❛r ❛ t♦❞♦s ♦s ✈❡t♦r❡s t❛♥❣❡♥t❡s ❛ M ♥♦ ♣♦♥t♦ p✱ ✐st♦ é✱ q✉❛♥❞♦ s❡ t✐✈❡r u · v = 0 M

  p

  ♣❛r❛ t♦❞♦ v ∈ T ✳ ■♥❞✐❝❛♠♦s ♦ ❝♦♥❥✉♥t♦ ❞♦s ✈❡t♦r❡s ♥♦r♠❛✐s ❛ M ♥♦ ♣♦♥t♦ p M

  ♣♦r ν p ✳ ❉✐③❡♠♦s q✉❡ ♦ s❡❣♠❡♥t♦ [p, a] = {p + t(a − p) : 0 ≤ t ≤ 1} é ♥♦r♠❛❧ ❛ M ♥♦ ♣♦♥t♦ p

  p M

  s❡ p ∈ M ❡ v = a − p ∈ ν ✳

  ⊥

  (p; ε) ❉❛❞♦ ε > 0✱ ❛ ❜♦❧❛ ♥♦r♠❛❧ B é ❛ r❡✉♥✐ã♦ ❞♦s s❡❣♠❡♥t♦s ♥♦r♠❛✐s ❛ M ♥♦ ♣♦♥t♦ p

  ✱ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ♠❡♥♦r q✉❡ ε✳ ❉✐③❡♠♦s q✉❡ ε é ✉♠ r❛✐♦ ♥♦r♠❛❧ ❛❞♠✐ssí✈❡❧ ♣❛r❛ ✉♠ s✉❜❝♦♥❥✉♥t♦ X ⊂ M q✉❛♥❞♦✱ ❞❛❞♦s q✉❛✐sq✉❡r ❞♦✐s s❡❣♠❡♥t♦s [p, a] ❡ [q, b]✱ ♥♦r♠❛✐s ❛ M✱ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ♠❡♥♦r q✉❡ ε✱ ❝♦♠ p 6= q ∈ X✱ t❡♠✲s❡ [p, a] ∩ [q, b] = ∅✳

  ❊♥✉♥❝✐❛♠♦s ❛❣♦r❛ ♦ t❡♦r❡♠❛ ❞❛ ✈✐③✐♥❤❛♥ç❛ t✉❜✉❧❛r ♣❛r❛ s✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s ❝✉❥❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷✹❪✳

  m N k

  ⊂ R ❚❡♦r❡♠❛ ✶✳✺ ❙❡❥❛ M = M ✉♠❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛ ❞❡ ❝❧❛ss❡ C ✱ k ≥ 2✳ ❊♥tã♦✿

  ✭✐✮ ❊①✐st❡ ε > 0✱ r❛✐♦ ♥♦r♠❛❧ ❛❞♠✐ssí✈❡❧ ♣❛r❛ M✳

  ⊥ ε (M ) = ∪ p∈M B (p; ε)

  ✭✐✐✮ ❆ r❡✉♥✐ã♦ V ❞♦s s❡❣♠❡♥t♦s ♥♦r♠❛✐s ❛ M ❞❡ ❝♦♠♣r✐♠❡♥t♦

  N

  ♠❡♥♦r q✉❡ ε é ✉♠ ❛❜❡rt♦ ❞❡ R ❝❤❛♠❛❞♦ ❛ ✈✐③✐♥❤❛♥ç❛ t✉❜✉❧❛r ❞❡ M ❞❡ r❛✐♦ ε✳ (M ) → M (M )

  ε ε

  ✭✐✐✐✮ ❆ ❛♣❧✐❝❛çã♦ η : V ✱ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣♦♥t♦ q ∈ V ♦ ♣é ❞♦ ú♥✐❝♦

  k−1

  s❡❣♠❡♥t♦ ♥♦r♠❛❧ q✉❡ ♦ ❝♦♥té♠✱ é ❞❡ ❝❧❛ss❡ C ✳ ❉❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✶✳✸✿ P♦r ❤✐♣ót❡s❡✱ ∂Ω é ✉♠❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛ ❞❡ ❝❧❛ss❡

  ❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✼

  −

  • r

  ⊂ Ω g : Ω → Ω ❛ ♣r♦❥❡çã♦ ❞❛❞❛ ♣♦r

  1 r

  1

  (

  −

  1

  x, , s❡ x ∈ Ω r

  g(x) =

  −

  π(x) − r η(x), ,

  1

  s❡ x ∈ Ω \ Ω r

  1

  ♦♥❞❡ η(x) é ♦ ✈❡t♦r ✉♥✐tár✐♦ ♥❛ ❞✐r❡çã♦ ❞♦ ú♥✐❝♦ s❡❣♠❡♥t♦ ♥♦r♠❛❧ q✉❡ ❝♦♥té♠ x✳ ❉❡✜♥✐♠♦s

  • → Ω t❛♠❜é♠ f : Ω r ♣♦r

  1

  ( g(x), s❡ x ∈ Ω, f (x) =

  • \ Ω.

  x − r

  1 η(x),

  s❡ x ∈ Ω

  r

  1 Ω

  ❖❜s❡r✈❛♠♦s q✉❡ f ❡ g ❡stã♦ ❜❡♠ ❞❡✜♥✐❞❛s ❡ sã♦ ❝♦♥tí♥✉❛s✳ ❱❛♠♦s ♠♦str❛r q✉❡ f◦g ≃ Id : [0, 1] × Ω → Ω

  • 1

  ❡ g◦f ≃ Id ✳ P❛r❛ ✐ss♦✱ t♦♠❛♠♦s H ❞❛❞❛ ♣♦r

  Ω r1 H (t, x) = (1 − t)f ◦g(x) + tx = (1 − t)g(x) + tx.

  1 − − −

  ⊂ Ω

  1 (t, x) = x ∈ Ω 1 (t, x)

  ❙❡ x ∈ Ω r ✱ H r ✳ ❙❡ x ∈ Ω \ Ω r ✱ H ❡stá ❝♦♥t✐❞♦ ♥♦ s❡❣♠❡♥t♦

  1

  1

  1

  [x, g(x)] ⊂ [π(x), g(x)] (∂Ω)

  ε

  ❞❡ V ✳ ❉❡s❞❡ q✉❡ ❡ss❡ s❡❣♠❡♥t♦ ♥ã♦ ❝♦♥té♠ ♦✉tr♦ ♣♦♥t♦ ❞❡ (t, x) ⊂ Ω

  1

  1

  ❢r♦♥t❡✐r❛ ❛❧é♠ ❞❡ π(x)✱ s❡❣✉❡ q✉❡ H ✳ ▲♦❣♦ H ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡ é ✉♠❛

  : [0, 1] × Ω → Ω

  Ω

  2

  ❤♦♠♦t♦♣✐❛ ❡♥tr❡ f ◦g ❡ Id ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ❞❡✜♥✐♠♦s H

  r 1 r 1 ♣♦r H (t, x) = (1 − t)g◦f (x) + tx.

  2

  ❖❜s❡r✈❛♠♦s q✉❡ ( g(x), s❡ x ∈ Ω, g◦f (x) =

  • \ Ω.

  π(x) − r η(x),

  1 s❡ x ∈ Ω r

  1

  (t, x) (∂Ω)

  ▲♦❣♦ H

  2 ❡stá ❝♦♥t✐❞♦ ♥♦ s❡❣♠❡♥t♦ ♥♦r♠❛❧ ❞❡ V ε q✉❡ ❝♦♥té♠ x✱ s❡❣♠❡♥t♦ ❡ss❡

  2

  • r

  ❝♦♥t✐❞♦ ❡♠ Ω ✳ ❆ss✐♠ H é ✉♠❛ ❤♦♠♦t♦♣✐❛ ❡♥tr❡ g◦f ❡ Id ✳ ❉❛í Ω é ❤♦♠♦t♦♣✐❝❛♠❡♥t❡

  1 Ω r

  1 r1 −

  ❡q✉✐✈❛❧❡♥t❡ ❛ Ω✳ ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ♠♦str❛✲s❡ q✉❡ Ω r t❛♠❜é♠ ♦ é✳

  2 ✶✳✷ ❋✉♥❝✐♦♥❛❧ r❡str✐t♦ ❛ ✉♠❛ ✈❛r✐❡❞❛❞❡

  1

  (X, R) ❈♦♥s✐❞❡r❡ (X, k · k) ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱ ψ ∈ C ✱

  ′

  V = {v ∈ X : ψ(v) = 0} (v) 6= 0 ❡ ψ ✱ ♣❛r❛ t♦❞♦ v ∈ V ✳

  ❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✽

  ′

  T z V = {y ∈ X : hψ (z), yi = 0}

  ′

  V (z) ♦✉ s❡❥❛✱ T z é ♦ ♥ú❝❧❡♦ ❞♦ ❢✉♥❝✐♦♥❛❧ ψ ✳

  1

  (X, R) := ϕ ❉❡✜♥✐çã♦ ✶✳✼ ❙❡❥❛ ϕ ∈ C ✳ ❆ ♥♦r♠❛ ❞❛ ❞❡r✐✈❛❞❛ ❞❡ ϕ|

  V V ❡♠ v ∈ V é

  ❞❡✜♥✐❞❛ ♣♦r

  ′ ′ kϕ (v)k = sup {hϕ (v), yi : y ∈ T V e kyk = 1} . v

  V ∗

  ▲❡♠❛ ✶✳✽ ❙❡❥❛♠ f ❡ g ❢✉♥❝✐♦♥❛✐s ❧✐♥❡❛r❡s ❡♠ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❋✳ ❙❡ N(f) ⊂ N(g) ❡♥tã♦ g ≡ kf ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ k ∈ R✱ ♦♥❞❡ N(f) ❡ N(g) ❞❡♥♦t❛♠ ♦ ♥ú❝❧❡♦ ❞❛s ❢✉♥çõ❡s f ❡ g✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡♠♦♥str❛çã♦✿ ❙❡ f ≡ 0 ❡♥tã♦ g ≡ 0 ❡ ♦ ❧❡♠❛ ❡stá ♣r♦✈❛❞♦ ♣❛r❛ q✉❛❧q✉❡r ❡s❝♦❧❤❛ ❞❡ g(v) k ∈ R

  ✳ ❈❛s♦ f 6= 0✱ ❡①✐st❡ v ∈ F t❛❧ q✉❡ f(v) 6= 0✳ ❈♦♥s✐❞❡r❡ k = ❡ ♦ ❢✉♥❝✐♦♥❛❧ f (v) ❧✐♥❡❛r h(x) = g(x) − kf(x) ♣❛r❛ x ∈ F ✳ ❱❛♠♦s ♠♦str❛r q✉❡ h ≡ 0 ❡✱ ❝♦♠ ✐ss♦✱ ❝♦♥❝❧✉✐r ❛ f (w) ❞❡♠♦s♥tr❛çã♦ ❞♦ ❧❡♠❛✳ P❛r❛ t❛♥t♦✱ s❡❥❛ w ∈ F ✳ ❊s❝♦❧❤❡♥❞♦ s = s(w) = ✈❡♠♦s q✉❡ f (v) w = sv + u

  ♦♥❞❡ u = w − sv ∈ N(f)✳ ❈♦♠♦ N(f) ⊂ N(g) t❡♠♦s h(u) = 0 ❡✱ ♣♦rt❛♥t♦✱ h(w) = sh(v) + h(u) = 0 ✳ ▲♦❣♦✱ h ≡ 0✳

  ❖ ❧❡♠❛ ❛ s❡❣✉✐r ❝♦rr❡s♣♦♥❞❡ à Pr♦♣♦s✐çã♦ ✺✳✶✷ ❞❡ ❬✸✸❪✱ ❝✉❥❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ t❛♠✲ ❜é♠ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✶✸❪✳ ▲❡♠❛ ✶✳✾ ❙❡❥❛ u ∈ V ✳ ❊♥tã♦

  ′ ′ ′ ′

  kϕ (u)k = min kϕ (u) − tψ (u)k

  ∗ V ✳ X t∈R

  ❉❡♠♦♥str❛çã♦✿ P❛r❛ t ∈ R✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s ❉❡✜♥✐çõ❡s ✶✳✼ ❡ ✶✳✻✱

  ′ ′

  kϕ (u)k = sup{hϕ (u), yi : y ∈ T

  V

  ∗ u V ❡ kyk = 1} ′ ′

  = sup {hϕ (u) − tψ (u), yi : y ∈ T

  V

  u

  ❡ kyk = 1} ✭✶✳✶✮

  ′ ′

  ≤ sup{hϕ (u) − tψ (u), yi : y ∈ X ❡ kyk = 1}

  ′ ′ ′

  ❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✾

  P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❍❛♥❤✲❇❛♥❛❝❤✱ ❡①✐st❡ Φ : X → R ❧✐♥❡❛r ❝♦♥tí♥✉❛ t❛❧ q✉❡

  ′ ′ ′

  Φ | = ϕ (u) = kϕ (u)k

  T u

  V X ∗ V ❡ kΦk V ′ ′

  (u)) = T V ⊂ N (ϕ (u) − Φ) ∈ R

  u

  ❈♦♠♦ N(ψ ✱ ♣❡❧♦ ▲❡♠❛ ✶✳✽✱ ❡①✐st❡ t t❛❧ q✉❡

  ′ ′ ϕ (u) − Φ = t ψ (u).

  ❆ss✐♠✱

  ′ ′ ′ ′ ′

  kϕ (u) − t ψ (u)k = kΦk = kϕ (u)k ∗ .

  X X

  V

  ❈♦♠❜✐♥❛♥❞♦ ❝♦♠ ✭✶✳✶✮✱ ❝♦♥❝❧✉í♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛✳

  ✶✳✸ ❆ ❝❛t❡❣♦r✐❛ ❞❡ ▲❥✉st❡r♥✐❦✲❙❝❤♥✐r❡❧♠❛♥♥

  ◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❛ ❝❛t❡❣♦r✐❛ ❞❡ ▲✉st❡r♥✐❦✲❙❝❤♥✐r❡❧♠❛♥♥ ❜❡♠ ❝♦♠♦ ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ✉s❛❞❛s ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❉✳ P❛r❛ ✉♠ ♠❡❧❤♦r ❡st✉❞♦ s♦❜r❡ ♦ ❛ss✉♥t♦✱ ✈❡❥❛ ❬✷✾✱ ✸✸✱ ✶✷✱ ✸❪ ❡ t❛♠❜é♠ ❬✶✸❪✳ ❉❡✜♥✐çã♦ ✶✳✶✵ ❉✐③❡♠♦s q✉❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ A ❞❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ❝♦♥trát✐❧ ❡♠ X q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ h : [0; 1] × A → X t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ x ∈ A✱ h(0, x) = x ∈ X

  ❡ h(1, x) = x ♣❛r❛ ❛❧❣✉♠ x ✳ ❊q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ ❡①✐st❡ ✉♠❛ ❤♦♠♦t♦✲ ♣✐❛ ❡♥tr❡ ❛ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡ ❞❡ A ❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥st❛♥t❡✳ ❚❛❧ h é ❝❤❛♠❛❞❛ ❞❡ ❞❡❢♦r♠❛çã♦ ❞❡ A ❡♠ X✳ ❉❡✜♥✐çã♦ ✶✳✶✶ ❙❡❥❛ A ⊂ X✱ ♦♥❞❡ X é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ❆ ❝❛t❡❣♦r✐❛ ❞❡ A ❡♠ X (A)

  X

  ✱ q✉❡ ❞❡♥♦t❛♠♦s ♣♦r ❝❛t ✱ é ♦ ♠❡♥♦r ✐♥t❡✐r♦ k t❛❧ q✉❡ A ♣♦❞❡ s❡r ❝♦❜❡rt♦ ♣♦r k

  X (A) =

  s✉❜❝♦♥❥✉♥t♦s ❢❡❝❤❛❞♦s ❡ ❝♦♥trát❡✐s ❡♠ X✳ ❙❡ ♥ã♦ ❡①✐st✐r t❛❧ ✐♥t❡✐r♦✱ ❞✐③❡♠♦s q✉❡ ❝❛t

  • ∞ (∅) = 0 (X)

  X X

  ✳ ❆❧é♠ ❞✐ss♦✱ cat ❡ r❡♣r❡s❡♥t❛♠♦s ❝❛t ♣♦r ❝❛t(X)✳

  N −1

N N

R R

  (S ) = cat (B

  1 (0)) = 1

  ❈♦♠♦ ❡①❡♠♣❧♦s✱ t❡♠♦s q✉❡ cat ✳ ❊①❡♠♣❧♦s ❞❡ ❝♦♥❥✉♥t♦s

  N N +1 N N

  ⊂ R (S ) = 2 q✉❡ t❡♠ ❝❛t❡❣♦r✐❛ ♠❛✐♦r q✉❡ ✶ é ❛ ❡s❢❡r❛ N✲❞✐♠❡♥s✐♦♥❛❧ S ❝✉❥❛ cat S

  N N N

  (T ) = N + 1

  T

  ❡ ♦ t♦r♦ N✲❞✐♠❡♥s✐♥♦♥❛❧ T ❝✉❥❛ cat ✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡ss❡s ❢❛t♦s ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷✾❪✳ Pr♦♣♦s✐çã♦ ✶✳✶✷ ❙❡❥❛♠ A ❡ B s✉❜❝♦♥❥✉♥t♦s ❞❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X✳ ❆ ❝❛t❡❣♦r✐❛

  ❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✵

  X (A) ≤ cat X (B);

  ✭✐✮ ❙❡ A ⊂ B✱ ❡♥tã♦ cat (A ∪ B) ≤ cat (A) + cat (B)

  X X

  X

  ✭✐✐✮ cat ❀

  X (B) ≤ cat B (B)

  ✭✐✐✐✮ ❙❡ B é ❢❡❝❤❛❞♦ ❡♠ X✱ ❡♥tã♦ cat ❀ ✭✐✈✮ ❙❡❥❛♠ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ♠♦❞❡❧❛❞❛ ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❡ K ✉♠ s✉❜❝♦♥❥✉♥t♦

  (K) < +∞

  M

  ❝♦♠♣❛❝t♦ ❞❡ M✳ ❊♥tã♦✱ cat ❡ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ❞❡ K t❛❧ q✉❡ cat (U ) = cat (K)

  M M ❀

  (X) = cat (Y )

  X Y

  ✭✈✮ ❙❡ X é ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ Y ✱ ❡♥tã♦ cat ✳

  1

  ❚❡♦r❡♠❛ ✶✳✶✸ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱ M ⊂ X ✉♠❛ C ✲✈❛r✐❡❞❛❞❡ ❡ I ∈

1 C (X, R)

  c

  ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡ ❡♠ M✳ ❙✉♣♦♥❤❛ q✉❡ I s❛t✐s❢❛③ (P S)

  d

  = {u ∈ M : I(u) ≤ d} ♣❛r❛ t♦❞♦ c ≤ d ❡ ❝♦♥s✐❞❡r❡ I ✳ ❊♥tã♦ ♦ ❢✉♥❝✐♦♥❛❧ I r❡str✐t♦ à M

  d

  ) t❡♠ ♣❡❧♦ ♠❡♥♦s cat(I ♣♦♥t♦s ❝rít✐❝♦s u t❛✐s q✉❡ I(u) ≤ d✳

  ✶✳✹ ❘❡s✉❧t❛❞♦s ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❡ ❝♦♠♣❛❝✐❞❛❞❡

  ❊♥✉♥❝✐❛♠♦s ❛q✉✐ r❡s✉❧t❛❞♦s ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❡ ❝♦♠♣❛❝✐❞❛❞❡ q✉❡ s❡rã♦ út❡✐s ♥♦ ❈❛♣í✲ t✉❧♦ ✸✳ ❆♥t❡s✱ ❞❛♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ ✉♠ r❡s✉❧t❛❞♦ ❞❛ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛ q✉❡ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬✶✺❪✳

  N N

  ❉❡✜♥✐çã♦ ✶✳✶✹ ❙❡❥❛ µ ✉♠❛ ♠❡❞✐❞❛ ❞❡ ❇♦r❡❧ ❡♠ R ❡ B ⊂ R ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ ❇♦r❡❧✳ µ é r❡❣✉❧❛r ❡①t❡r✐♦r ❡♠ B s❡ µ(B) = inf {µ(U) : U ⊃ B, U é ❛❜❡rt♦} ❡ µ é r❡❣✉❧❛r ✐♥t❡r✐♦r ❡♠ B s❡ µ(B) = sup {µ(K) : K ⊂ B, K é ❝♦♠♣❛❝t♦}✳ ❉✐③❡♠♦s q✉❡ µ é r❡❣✉❧❛r s❡ µ é r❡❣✉❧❛r ❡①t❡r✐♦r ❡ r❡❣✉❧❛r ✐♥t❡r✐♦r ❡♠ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s ❞❡ ❇♦r❡❧✳

  N

  ❉❡✜♥✐çã♦ ✶✳✶✺ ❯♠❛ ♠❡❞✐❞❛ ❞❡ ❘❛❞♦♥ ❡♠ R é ✉♠❛ ♠❡❞✐❞❛ ❞❡ ❇♦r❡❧ q✉❡ é ✜♥✐t❛ ❡♠ ❝♦♥❥✉♥t♦s ❝♦♠♣❛❝t♦s✱ r❡❣✉❧❛r ❡①t❡r✐♦r ❡♠ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s ❞❡ ❇♦r❡❧ ❡ r❡❣✉❧❛r ✐♥t❡r✐♦r ❡♠ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s✳

  N N

  ) ❉❡♥♦t❛♠♦s ♣♦r M(R ♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❞❛s ♠❡❞✐❞❛s ❞❡ ❘❛❞♦♥ s♦❜r❡ R ❡q✉✐♣❛❞♦ ❝♦♠ ❛ ♥♦r♠❛

  Z

  N |ω| = sup φ(x)dω (R ), |φ| ≤ 1 .

  ∞ N : φ ∈ C

  R N N

  ) ⊂ M(R ) )

  n

  ❉✐③❡♠♦s q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ (ω ❝♦♥✈❡r❣❡ ❢r❛❝❛♠❡♥t❡ ♣❛r❛ ω ❡♠ M(R s❡ Z Z

  ❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✶

  N

  1 N n ) ⊂ L (R , µ)

  Pr♦♣♦s✐çã♦ ✶✳✶✻ ❙❡❥❛ µ ✉♠❛ ♠❡❞✐❞❛ ♣♦s✐t✐✈❛ ❞❡ ❘❛❞♦♥ ❡♠ R ❡ (f ❝♦♠ Z

  |f |dµ ≤ C

  n

  ♣❛r❛ t♦❞♦ n ∈ N ❡ ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ C✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ♠❡❞✐❞❛

  N R

  |dµ ⇀ µ

  n = |f n

  ♣♦s✐t✐✈❛ ❞❡ ❘❛❞♦♥ µ t❛❧ q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ µ ❢r❛❝❛♠❡♥t❡ ♥♦ s❡♥t✐❞♦ ❞❛s ♠❡❞✐❞❛s✳ ❉❡s❞❡ ♦s tr❛❜❛❧❤♦s ❞❡ P✳▲✳ ▲✐♦♥s ❡♠ ❬✷✻✱ ✷✼❪✱ ♦ ♠ét♦❞♦ ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ ❝♦♠♣❛❝✐✲

  ❞❛❞❡ t❡♠ s✐❞♦ ❧❛r❣❛♠❡♥t❡ ✉t✐❧✐③❛❞♦ ♣♦r ✈ár✐♦s ❛✉t♦r❡s ♣❛r❛ ❝♦♠♣❡♥s❛r ♣r♦❜❧❡♠❛s ❞❡ ❢❛❧t❛ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡✳ ❖ ❧❡♠❛ q✉❡ ❛♣r❡s❡♥t❛♠♦s ❛ s❡❣✉✐r ❝♦rr❡s♣♦♥❞❡ ❛♦ ▲❡♠❛ ✷✳✻ ❡♠ ❬✶✽❪✳

  ∗

  ≡ 2 ❊❧❡ ❢♦✐ ♣r♦✈❛❞♦ ♥♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r p = 2✱ q n ♣♦r ❲✐❧❧❡♠ ❡♠ ❬✸✸❪ ♥♦ ▲❡♠❛ ✶✳✹✵✳ ❆ ✈❡rsã♦ ❞❡♠♦♥str❛❞❛ ♣♦r ❋✉rt❛❞♦✱ ❜❡♠ ❝♦♠♦ s✉❛ ❞❡♠♦♥str❛çã♦✱ ❢♦✐ ✐♥s♣✐r❛❞❛ ♥❡ss❡ ú❧t✐♠♦ tr❛❜❛❧❤♦ ❡ t❛♠❜é♠ ♣❡❧♦ ❞❡ ❙♠❡ts ❡♠ ❬✸✶❪ ✭✈❡❥❛ ▲❡♠❛ ✷✳✶ ❡ ❖❜s❡r✈❛çã♦ ✷✳✷✮✱ ♦♥❞❡ ♦ ❛✉t♦r

  ∗

  ≡ p

  n

  ❝♦♥s✐❞❡r❛ ♦ ❝❛s♦ 1 < p < N✱ q ❡ ♣❡r♠✐t❡ ♦ ❛♣❛r❡❝✐♠❡♥t♦ ❞❡ ✉♠ ♣♦♥t❡♥❝✐❛❧ V q✉❡ ♣♦❞❡ s❡r s✐♥❣✉❧❛r✳

  ∗ ∗

  ) ⊂ [p, p ] → p

  n n

  ▲❡♠❛ ✶✳✶✼ ❙❡❥❛ (q ✉♠❛ s❡q✉ê♥❝✐❛ ♥ã♦✲❞❡❝r❡s❝❡♥t❡ t❛❧ q✉❡ q ✳ ❙❡❥❛

  1,p N

  (u ) ⊂ W (R )

  n

  s❛t✐s❢❛③❡♥❞♦

  1,p N

  u ⇀ u f racamente em D (R ),

  n p N

  |∇(u − u)| ⇀ ω f racamente em M(R ),

  n n q N

  |u − u| ⇀ ν f racamente em M(R ),

  n N

  u n (x) → u(x) q.t.p. x ∈ R ,

  N

  ∇u (x) → ∇u(x) q.t.p. x ∈ R ,

  n

  ❡ ❞❡✜♥❛ Z Z

  p q n ω ∞ = lim lim sup |∇u | , ν ∞ = lim lim sup |u | . n n R→∞ R→∞

n→∞ n→∞

|x|>R |x|>R

  ❊♥tã♦

  

p/p∗ −1

  |ν| ≤ S |ω|,

  p p

  lim sup |u | N = |∇u| N + |ω| + ω ,

  n ∞ p,R p,R n→∞

  ❡

  ∗ n

  ❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✷

  ∗ p/p −1

  |ω| = S

  ❆❧é♠ ❞♦ ♠❛✐s✱ s❡ u = 0 ❡ |ν| ✱ ❡♥tã♦ ❝❛❞❛ ✉♠❛ ❞❛s ♠❡❞✐❞❛s ω ❡ ν s❡ ❝♦♥❝❡♥tr❛ ❡♠ ✉♠ ú♥✐❝♦ ♣♦♥t♦✳

  ❆❜❛✐①♦ ❡♥✉♥❝✐❛♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈♦♥❝❡♥tr❛çã♦✲❈♦♠♣❛❝✐❞❛❞❡ ❞❡✈✐❞♦ ❛ ▲✐♦♥s ✭❬✷✼❪✱ ▲❡♠❛ ■✳✶✮ ❝✉❥❛ ❞❡♠♦♥str❛çã♦ t❛♠❜é♠ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✸✷❪ ✭▲❡♠❛ ✹✳✽✮✳

  1,p N

  ) ⊂ D (R )

  n

  ▲❡♠❛ ✶✳✶✽ ❙❡❥❛ (u ✉♠❛ s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣✐♥❞♦ ❢r❛❝❛♠❡♥t❡ ♣❛r❛ u ❡♠

  1,p N N

  D (R ) )

  ✳ ❊♥tã♦ ❡①✐st❡♠ ❞✉❛s ♠❡❞✐❞❛s ✜♥✐t❛s ♥ã♦✲♥❡❣❛t✐✈❛s µ, ν ∈ M(R ✱ ✉♠ ❝♦♥✲

  N

  )

  j j∈J

  ❥✉♥t♦ ♥♦ ♠á①✐♠♦ ❡♥✉♠❡rá✈❡❧ ❏✱ ✉♠❛ ❢❛♠✐❧í❛ (x ❞❡ ♣♦♥t♦s ❞✐st✐♥t♦s ❡♠ R ❡ ❞✉❛s ) , (ν ) s❡q✉ê♥❝✐❛s (µ j j∈J j j∈J ❝♦♥t✐❞❛s ❡♠ (0, ∞) t❛✐s q✉❡

  X

  ∗ p

  ν + ν = |u| δ ,

  j x j j∈J

  X

  

p

  µ ≥ |∇u| + µ j δ x j ,

  j∈J

p/p

  ≥ Sν µ j ,

  j ∗ ∗

p/p

  P

  N 1/p

  ≥ ν < ∞ )

  ♣❛r❛ t♦❞♦ j ∈ J✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ✳ ❆❧é♠ ❞♦ ♠❛✐s✱ s❡ ν(R

  j∈J j p N 1/p −p/q N

  Sµ(R ) = γ C µ ∈ R

  x

  ❡♥tã♦ ν = γδ ♣❛r❛ ❛❧❣✉♠ x ❡ γ ≥ 0✳ ❯t✐❧✐③❛♥❞♦ ♦ ▲❡♠❛ ✶✳✶✽ ❡ ✐❞❡✐❛s ❞❡ ❬✸✵❪✱ ❋✉rt❛❞♦ ❬✶✽❪ ❞❡♠♦♥str♦✉ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡✲

  ❞❛❞❡ ❞❛s s❡q✉ê♥❝✐❛s ♠✐♥✐♠✐③❛♥t❡s ♣❛r❛ S✳

  ∗

  R

  1,p p p

  ) ⊂ W (Ω) |v | dx = 1 k → S

  n n n

  ▲❡♠❛ ✶✳✶✾ ❙❡❥❛ (v t❛❧ q✉❡ ❡ kv ✳ ❊♥tã♦ ❡①✐st❡ v ∈

  

1,p 1,p

  W (Ω) ⇀ v (Ω) (x) →

  n n

  t❛❧ q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ v ❢r❛❝❛♠❡♥t❡ ❡♠ W ❡ ∇v ∇v(x) q✳t✳♣✳ ❡♠ Ω✳

  ✶✳✺ ❖ ▲❡♠❛ ❞❡ ❇ré③✐s✲▲✐❡❜

  ◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛♠♦s ♦ ▲❡♠❛ ❞❡ ❇ré③✐s✲▲✐❡❜ ❬✶✵❪ ❡ t❛♠❜é♠ ✉♠❛ ✈❡rsã♦ ✈❡t♦r✐❛❧ ❞♦ ♠❡s♠♦✱ ❞❡✈✐❞♦ ❛ ❬✶❪✳ ❆ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ❞❡ ❇ré③✐s✲▲✐❡❜ q✉❡ ❞❛♠♦s ❛q✉✐ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷✶❪ ✭▲❡♠❛ ✹✳✻✮✳

  N

  )

  n n∈N

  ▲❡♠❛ ✶✳✷✵ ✭❇ré③✐s✲▲✐❡❜✮ ❙❡❥❛♠ 1 ≤ p < ∞✱ D ⊂ R ❛❜❡rt♦ ❡ (f ✉♠❛ s❡q✉ê♥❝✐❛

  p

  (D) ❧✐♠✐t❛❞❛ ❞❡ ❢✉♥çõ❡s ❡♠ L q✉❡ ❝♦♥✈❡r❣❡ ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡ D ♣❛r❛ f✳ ❊♥tã♦

  p

  f ∈ L (D) ❡

  ❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✸

  p p,D (D)

  ♦♥❞❡ | · | ❞❡♥♦t❛ ❛ ♥♦r♠❛ ✉s✉❛❧ ❡♠ L ✳

  p

  | ≤ C ❉❡♠♦♥str❛çã♦✿ P♦r ❤✐♣ót❡s❡✱ ❡①✐st❡ C > 0 t❛❧ q✉❡ |f n ✱ ♣❛r❛ t♦❞♦ n ∈ N✳ P❡❧♦

  p,D

  ▲❡♠❛ ❞❡ ❋❛t♦✉✱ Z Z Z

  

p p p

|f | = lim inf |f | ≤ lim inf |f | ≤ C. n n

n→∞ n→∞

  D D D p

  (D) ■ss♦ ♠♦str❛ q✉❡ f ∈ L ✳

  ε > 0

  ❙❡❥❛ ε > 0✳ ❆✜r♠❛♠♦s q✉❡ ❡①✐st❡ C q✉❡ ❞❡♣❡♥❞❡ ❞❡ ε ❡ ❞❡ p t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ s ∈ R ✱ s❡ t❡♠

  

p p p

− |s| − 1 + C .

  ε

  ✭✶✳✷✮ |s + 1| ≤ ε|s|

  p p

  |s + 1| − |s| − 1 = 0 > 0

  ε

  ❈♦♠ ❡❢❡✐t♦✱ ❝♦♠♦ lim ✱ ❡①✐st❡ A t❛❧ q✉❡

  p |s|→∞

  |s|

  p p p − |s| − 1 , .

  ε

  s❡♠♣r❡ q✉❡ |s| > A ✭✶✳✸✮ |s + 1| ≤ ε|s|

  ε

  P❛r❛ |s| ≤ A t❡♠♦s

  p p p p p p p p p p

  − |s| − 1 |s| + 2 + |s| + 1 ≤ 2 A + 2 + A + 1. |s + 1| ≤ 2 ε ε

  p p p p

  = 2 A + 2 + A + 1 > 0

  ε

  ❉❡✜♥✐♥❞♦ C ε ε ✱ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✸✮ ♠♦str❛♠

  p

  ♥♦ss❛ ❛✜r♠❛çã♦✳ ❉❛❞♦ b ∈ R✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✷✮ ♣♦r |b| ♦❜t❡♠♦s

  p p p p p

  − |sb| − |b| + C |b| ,

  ε

  ♣❛r❛ t♦❞♦ s ∈ R, |sb + b| ≤ ε|sb|

  ❞♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ R t❡♠✲s❡

  p p p p p

  − |a| − |b| |b| + C ε .

  ✭✶✳✹✮ |a + b| ≤ ε|a|

  P❛r❛ ❝❛❞❛ n ∈ N✱ ❝♦♥s✐❞❡r❡

  • p p p p

  u = | − |f − f | − |f | = (u − ε|f − f | ) ,

  n n n n n n

  |f ❡ Z Z

  • = max{u, 0}

  u → 0

  n

  ♦♥❞❡ u ♣❛r❛ q✉❛❧q✉❡r ❢✉♥çã♦ r❡❛❧ u✳ ❱❛♠♦s ♠♦str❛r q✉❡ ✳ P❛r❛

  D

  − f

  n

  ✐ss♦✱ ❛♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✹✮ ❝♦♠ a = f ❡ b = f ♦❜t❡♠♦s

  p p

  u − ε|f − f | ≤ C |f |

  n n ε

  ❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✹

  ❡✱ ♣♦rt❛♥t♦✱

  p p 0 ≤ Z = max {(u − ε|f − f | ), 0} ≤ C |f | .

n n n ε

  → 0

  n (x) → f (x) n

  ❈♦♠♦ f q✳t✳♣✳ ❡♠ D✱ t❡♠♦s q✉❡ Z q✳t✳♣✳ ❡♠ D✳ ❙❡❣✉❡ ❞♦ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ❉♦♠✐♥❛❞❛ ❞❡ ▲❡❜❡s❣✉❡ ✭❚✳❈✳❉✳▲✳✮ q✉❡

  Z Z → 0

  n

  q✉❛♥❞♦ n → ∞. ✭✶✳✺✮

  D

  ❚❡♠♦s t❛♠❜é♠ q✉❡

  • p + p p p +

  u = u = (ε|f − f | + u − ε|f − f | ) ≤ ε|f − f | + (u − ε|f − f | )

  n n n n n n n n p = ε|f − f | + Z . n n p p

  − f | ≤ M

  n ) (D) n

  P♦r ❤✐♣ót❡s❡✱ (f é ❧✐♠✐t❛❞❛ ❡♠ L ❡✱ ♣♦rt❛♥t♦✱ ❡①✐st❡ M > 0 t❛❧ q✉❡ |f p ✳ ❉❛í ❡ ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱ s❡❣✉❡ q✉❡

  Z Z Z

  p

  − f | ≤ εM + u + n = ε|f n Z n Z n .

  p D D D

  Z u → 0

  n

  ❋❛③❡♥❞♦ n → ∞ ❡ ε → 0✱ ♣♦r ✭✶✳✺✮ ♦❜t❡♠♦s ✳ ❈♦♠♦

  D

  Z

  p p p

  |f | − |f − f | − |f | ≤ u ,

  

n n n

p,D p,D p,D D

  ♦ ❧❡♠❛ ❡stá ♣r♦✈❛❞♦✳ ❖❜s❡r✈❛çã♦ ✶✳✷✶ ❆❞❛♣t❛♥❞♦ ❧✐❣❡✐r❛♠❡♥t❡ ❛ ❞❡♠♦♥str❛çã♦ ❛❝✐♠❛✱ é ♣♦ssí✈❡❧ ♠♦str❛r q✉❡✱ ♣❛r❛ 1 < p < ∞✱

  Z Z Z

  p p p

  |∇(u − u)| |∇u | − |∇u|

  n = n + o(1)

  ✭✶✳✻✮

  D D D p

  ) |) (D) q✉❛♥❞♦ n → ∞✱ s❡♠♣r❡ q✉❡ (u n é ✉♠❛ s❡q✉ê♥❝✐❛ t❛❧ q✉❡ (|∇u n é ❧✐♠✐t❛❞❛ ❡♠ L ❡ ∇u

  n (x)

  ❝♦♥✈❡r❣❡ ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡ D ♣❛r❛ ∇u(x)✳

  p

1 N

  ∈ L (R )

  ❉❡ ❢❛t♦✱ ♣❡❧♦ ▲❡♠❛ ❞❡ ❋❛t♦✉✱ |∇u| ✳ ❚❛♠❜é♠ ♣♦❞❡♠♦s ♠♦str❛r q✉❡✱ ♣❛r❛ ε > 0 > 0

  ε

  ❞❛❞♦✱ ❡①✐st❡ C q✉❡ ❞❡♣❡♥❞❡ ❞❡ ε ❡ ❞❡ p t❛❧ q✉❡

  p p p p p N

  − |A| − |B| |B| + C , .

  ε ♣❛r❛ t♦❞♦s A, B ∈ R ✭✶✳✼✮

  |A + B| ≤ ε|A|

  ❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✺

  N p N

  , x ∈ R ❈♦♠ ❡❢❡✐t♦✱ s❡❥❛♠ A, B ∈ R ❡ ❝♦♥s✐❞❡r❡ F (x) = |x| ✳ P❡❧♦ ❚❡♦r❡♠❛ ❋✉♥❞❛✲ ♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦✱

  Z

1 Z

  1

  d |F (A + B) − F (A)| = ∇F (A + tB) · B dt

  F (A + tB) dt = dt

  ✭✶✳✽✮

  

N N

  Z

1 Z

  X ∂F

  1 X

  p−1 ≤ (A + tB) | dt ≤ p|A + tB| |B | dt. i i

  |B ∂x i

  

i=1 i=1

  P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ d(ε) > 0 t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ 0 ≤ t ≤ 1✱

  p−1 p p p p p p |A + tB| |B | ≤ ε|A + tB| + d(ε)|B | ≤ ε2 |A| + (2 ε + d(ε)) |B| . i i

  ❙✉❜st✐t✉✐♥❞♦ ❡♠ ✭✶✳✽✮ ♦❜t❡♠♦s ❝♦♥st❛♥t❡s C, k(ε) > 0 t❛✐s q✉❡

  p p p p − |A| + k(ε)|B| .

  |A + B| ≤ εC|A| − ∇u

  n

  ❉❛í s❡❣✉❡ ✭✶✳✼✮✳ ❆❣♦r❛ ❡s❝♦❧❤❛ A = ∇u ❡ B = ∇u✳ ❊♥tã♦

  p p p p p

  | − |∇u − ∇u| − |∇u| − ∇u| |∇u|

  

n n n + C ε

  |∇u ≤ ε|∇u

  N p p p

  = |∇u | − |∇u − ∇u| − |∇u|

  n n n ✱ ❛

  ♣❛r❛ t♦❞♦ x ∈ R ❡ n ∈ N✳ ❈♦♥s✐❞❡r❛♥❞♦ eu ✈❡r✐✜❝❛çã♦ ❞❡ ✭✶✳✻✮ s❡❣✉❡ ♦s ♠❡s♠♦s ♣❛ss♦s ❞❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛ ❛♥t❡r✐♦r✳

  ❆♣r❡s❡♥t❛♠♦s ❛❜❛✐①♦ ✉♠❛ ✈❡rsã♦ ✈❡t♦r✐❛❧ ❞♦ ❧❡♠❛ ❞❡ ❇ré③✐s✲▲✐❡❜ q✉❡ ❢♦✐ ♣r♦✈❛❞❛ ♣♦r ❆❧✈❡s ❡♠ ❬✶❪✳

  s−2 K

  y ▲❡♠❛ ✶✳✷✷ ❙❡❥❛♠ K ≥ 1✱ s ≥ 2 ❡ A(y) = |y| ✱ ♣❛r❛ y ∈ R ✳ ❈♦♥s✐❞❡r❡ ✉♠❛

  N K s N K

  : R → R ) ⊂ (L (R )) → 0

  n n n

  s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s ✈❡t♦r✐❛✐s η t❛❧ q✉❡ (η ❡ η q✳t✳♣✳

  N s N K

  |

  n (L (R ))

  ❡♠ R ✳ ❊♥tã♦✱ s❡ |η é ❧✐♠✐t❛❞♦✱ t❡♠♦s Z

  s/(s−1)

  lim |A(η ) + A(ω) − A(η + ω)| = 0

  n n

  ✳

  n→∞ N R s N K

  (R )) ♣❛r❛ ❝❛❞❛ ω ∈ (L ✜①❛❞♦✳

  K N

  (y) (x)

  i i

  ❉❡♠♦♥str❛çã♦✿ P❛r❛ y ∈ R ❡ x ∈ R ❡ 1 ≤ i ≤ K✱ s❡❥❛♠ A ❡ ω ❛ i✲és✐♠❛ ❝♦♠♣♦♥❡♥t❡ ❞♦s ✈❡t♦r❡s A(y) ❡ ω(x)✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦

  Z Z

  1

  1

  d

  ❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✻

  K

  ❖❜s❡r✈❡ q✉❡✱ ♣❛r❛ ❝❛❞❛ y ∈ R ❝♦♠ y 6= 0 ❡ 1 ≤ j ≤ K✱ (

  s−4

  |y| ∂A (y) (s − 2)y i y j ,

  

i s❡ j 6= i,

  =

  s−2 2 s−4

  ∂y j |y| + (s − 2)y |y| , i s❡ j = i.

  | ≤ |y|

  j

  ❈♦♠♦ |y ✱ s❡❣✉❡ q✉❡ ∂A (y)

  i s−2

  . ≤ (s − 1)|y|

  ∂y

  j

  > 0

  1

  ▲♦❣♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C t❛❧ q✉❡ Z

1 Z

  1 s−2

  |A |∇A |ω| |η

  i (η n + ω) − A i (η n )| ≤ |ω| i (η n + tω)|dt ≤ C 1 n + tw| dt s−2

  ≤ C |ω|(|η | + |ω|) .

  1 n

  , C > 0 ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❡①✐st❡♠ ❝♦♥st❛♥t❡ C

  2 3 t❛✐s q✉❡ s−1 s−2

  |A (η + ω) − A (η )| ≤ C |ω| + C |ω||η | .

  i n i n

  2 3 n

  P❛r❛ s = 2✱ ♦ ❧❡♠❛ é ✈❡r❞❛❞❡✐r♦✳ P❛r❛ s > 2✱ 0 < ε < 1 ✜①❛❞♦ ❡ θ = s − 2✱ ✉s❛♥❞♦ ❛

  s−1

  > 1 ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣ ❝♦♠ ❡①♣♦❡♥t❡s ❡ s − 1 ♥❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱ ♦❜t❡♠♦s

  s−2

  C

  3 θ s−2 s−1

  |A (η + ω) − A (η + )| ≤ C |ω| |ω|ε |η |

  i n i n 2 n θ

  ε

  s−

  3 s−1 θ s−1 s−1

  2

  1 C

  ≤ C |ω| |ω| + ε |η |

  • s−

  2 n θ

  ε ▲♦❣♦

  s−1 s−1

  |A (η + ω) − A (η )| ≤ b C |ω| + ε|η | ,

  i n i n ε n

  C

3 C ε = C

  • 2 > 0 ε > 0

  ♦♥❞❡ b ✳ P♦rt❛♥t♦ ❡①✐st❡ C t❛❧ q✉❡

  s−2

  ε

  s−1 s−1 |A(η + ω) − A(η )| ≤ C |ω| + ε|η | . n n ε n ✭✶✳✾✮ N

  : R → R

  ε,n

  ❈♦♥s✐❞❡r❡ ❛❣♦r❛ ❛ ❢✉♥çã♦ G ❞❡✜♥✐❞❛ ♣♦r

  s−1 G ε,n (x) = max{|A(η n + ω) − A(η n ) − A(ω)|(x) − ε|η n (x)| , 0}.

  ❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✼

  N n (x) → 0 ε,n

  P♦r ❤✐♣ót❡s❡✱ η q✳t✳♣✳ ❡♠ R ✳ ❉❛í✱ ❝♦♠♦ A é ❝♦♥tí♥✉❛✱ s❡❣✉❡ ❞❛ ❞❡✜♥✐çã♦ G q✉❡

  N lim G (x) = 0, . ε,n

  q✳t✳♣✳ ❡♠ R

  n→∞

  ❉❡ ✭✶✳✾✮ t❡♠♦s

  s−1 s/(s−1) N 0 ≤ G (x) ≤ (C + 1)|ω| ∈ L (R ). ε,n ε

  ▲♦❣♦ ♣♦❞❡♠♦s ✉s❛r ♦ ❚✳❈✳❉✳▲✳ ♣❛r❛ ❝♦♥❝❧✉✐r q✉❡ Z

  s/(s−1)

  |G lim (x)| dx = 0.

  ε,n

  ✭✶✳✶✵✮

  n→∞ N R ε,n

  ❚❛♠❜é♠ ❞❛ ❞❡✜♥✐çã♦ ❞❡ G ✱ t❡♠♦s

  s−1

  |A(η | n + ω) − A(η n ) − A(ω)| ≤ ε|η n + G ε,n .

  s/(s−1)

  < ε > 0

  5

  ❈♦♠♦ 0 < ε < 1✱ t❡♠♦s ε ❡ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ✐♠♣❧✐❝❛ q✉❡ ❡①✐st❡ C t❛❧ q✉❡

  s/(s−1) s s/(s−1) |A(η + ω) − A(η ) − A(ω)| ≤ C ε|η | + C |G | . n n 5 n 5 ε,n s N K n ) (R )) 6 > 0

  P♦r ✭✶✳✶✵✮ ❡ ♣❡❧♦ ❢❛t♦ ❞❡ (η s❡r ❧✐♠✐t❛❞❛ ❡♠ (L ✱ ♦❜t❡♠♦s C t❛❧ q✉❡ Z

  Z

  s/(s−1) s lim sup |A(η + ω) − A(η ) − A(ω)| dx ≤ C ε lim sup |η | dx ≤ C ε. n n 5 n

  6 N N n→∞ R n→∞ R

  ❋❛③❡♥❞♦ ε → 0✱ s❡❣✉❡ q✉❡ Z

  s/(s−1)

  0 ≤ lim inf |A(η + ω) − A(η ) − A(ω)| dx

  

n n

n→∞ N R

  Z

  s/(s−1)

  ≤ lim sup |A(η + ω) − A(η ) − A(ω)| dx ≤ 0,

  

n n

N R n→∞

  ♦ q✉❡ ♠♦str❛ ♦ ❧❡♠❛✳

  ✶✳✻ ❯♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ p

  ❆q✉✐ ❛♣r❡s❡♥t❛♠♦s ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠ ❡s♣❛ç♦s L ❝✉❥❛ ❞❡♠♦♥str❛çã♦ ❡♥❝♦♥tr❛✲s❡ ❡♠ ❬✷✶❪ ✭✈❡❥❛ ▲❡♠❛ ✹✳✽✮✳

  N

  ▲❡♠❛ ✶✳✷✸ ❙❡❥❛♠ Ω ⊂ R ✉♠ ❛❜❡rt♦ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❧✐♠✐t❛❞♦✱ 1 < p < ∞ ❡

  p

  ❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✽

  p

  f n ⇀ f (Ω) ✳ ❊♥tã♦ f ❢r❛❝❛♠❡♥t❡ ❡♠ L ✳

  p p

  (Ω) ) (Ω) ❉❡♠♦♥str❛çã♦✿ P❡❧♦ ▲❡♠❛ ❞❡ ❋❛t♦✉✱ f ∈ L ✳ ❈♦♠♦ (f n é ❧✐♠✐t❛❞❛ ❡♠ L ❡

  p p

  L (Ω) (Ω) n ⇀ g é r❡✢❡①✐✈♦✱ ❡①✐st❡ g ∈ L t❛❧ q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ f ❢r❛❝❛♠❡♥t❡

  p

  (Ω) ❡♠ L ✳ ❱❛♠♦s ♠♦str❛r q✉❡ g(x) = f(x) q✳t✳♣✳ ❡♠ Ω✳ ❈♦♥s✐❞❡r❛♠♦s ♣❛r❛ j ∈ N ♦ ❝♦♥❥✉♥t♦

  Ω := {x ∈ Ω : |f (x) − f (x)| ≤ 1,

j n ♣❛r❛ t♦❞♦ n ≥ j} .

  ∞

  ∈ C (Ω ) (x) → f (x)

  j j n

  ❋✐①❛❞♦ j ≥ 1✱ s❡❥❛ ϕ ✳ P♦r ❤✐♣ót❡s❡✱ f q✳t✳♣✳ ❡♠ Ω✳ P❛r❛ n ≥ j✱ t❡♠✲s❡ |f (x)ϕ (x)| ≤ C|f (x)| ≤ C(|f (x) − f (x)| + |f (x)|)

  n j n n

  ≤ C + C|f (x)| := h(x) := ⊂ Ω

  j j j j

  ♣❛r❛ t♦❞♦ x ∈ K s✉♣♣ ϕ ❡ ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ C > 0✳ ❈♦♠♦ K é ❝♦♠♣❛❝t♦

  p

  1

  (Ω) (K )

  ❡ f ∈ L ✱ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ h ∈ L j ✳ P❡❧♦ ❚✳❈✳❉✳▲✳✱

  Z Z Z Z lim f (x)ϕ (x) = lim f (x)ϕ (x) = f (x)ϕ (x) = f (x)ϕ (x).

  

n j n j j j

  ✭✶✳✶✶✮

  n→∞ n→∞ Ω K j K j Ω ′

  R

  p p

  ∈ L (Ω) (Ω) → R f ϕ

  j j

  ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ ϕ ✱ T : L ❞❡✜♥✐❞❛ ♣♦r T (f) = é ❧✐♥❡❛r

  Ω p

  ⇀ g (Ω) ) → T (g) ❝♦♥tí♥✉❛✳ P❡❧❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❢r❛❝❛ f n ❡♠ L s❡❣✉❡ q✉❡ T (f n ✱ ✐st♦ é✱

  Z Z lim f n (x)ϕ j (x) = g(x)ϕ j (x).

  ✭✶✳✶✷✮

  n→∞ Ω Ω

  ❈♦♠❜✐♥❛♥❞♦ ✭✶✳✶✶✮ ❡ ✭✶✳✶✷✮ ♦❜t❡♠♦s Z

  ∞

  ∈ C (g(x) − f (x))ϕ (x) = 0, (Ω ).

  j ♣❛r❛ t♦❞❛ ϕ j j Ω j

  P❡❧♦ ▲❡♠❛ ✹✳✷✹ ❡♠ ❬✾❪✱ g(x) = f (x) q✳t✳♣✳ ❡♠ Ω j ♣❛r❛ t♦❞♦ j ∈ N.

  ⊂ Ω

  j j

  ❉❛í✱ ♣❛r❛ ❝❛❞❛ j ∈ N✱ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ Z ❞❡ ♠❡❞✐❞❛ ♥✉❧❛ t❛❧ q✉❡ g(x) = f(x)

  ∞ ∞

  \ Z Ω Z ♣❛r❛ t♦❞♦ x ∈ Ω j j ✳ ❈♦♥s✐❞❡r❡ A = ∪ j ❡ N = ∪ j ✳ ❉❛❞♦ x ∈ A \ N t❡♠✲s❡

  j=1 j=1

  \ Z x ∈ Ω j j ♣❛r❛ ❛❧❣✉♠ j ∈ N ❡✱ ♣♦rt❛♥t♦✱ g(x) = f(x)✳ ▼♦str❛♠♦s ❝♦♠ ✐ss♦ q✉❡ g(x) = f (x)

  ♣❛r❛ t♦❞♦ x ∈ A \ N. ✭✶✳✶✸✮

  ❈❛♣ít✉❧♦ ✶✳ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✾

  n (x) → f (x)

  P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ f q✳t✳♣✳ ❡♠ Ω✱ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ Z ⊂ Ω ❞❡ ♠❡❞✐❞❛ (x) → f (x)

  n

  ♥✉❧❛ t❛❧ q✉❡ f s❡❥❛ q✉❛❧ ❢♦r x ∈ Ω \ Z✳ ■ss♦ ❞✐③ q✉❡ ♣❛r❛ ❝❛❞❛ x ∈ Ω \ Z✱ ∈ N

  ❡①✐st❡ j t❛❧ q✉❡ |f

n (x) − f (x)| ≤ 1 .

s❡♠♣r❡ q✉❡ n ≥ j

  ⊂ A ❆ss✐♠ x ∈ Ω j ✳ ❈♦♠ ✐ss♦✱ Ω \ Z ⊂ A ❡✱ ❞❛í✱ Ω \ (Z ∪ N) ⊂ A \ N✳ ❙❡❣✉❡ ❞❡ ✭✶✳✶✸✮ q✉❡ g(x) = f (x) ♣❛r❛ t♦❞♦ x ∈ Ω \ (Z ∪ N).

  ❈♦♠♦ Z ∪ N t❡♠ ♠❡❞✐❞❛ ♥✉❧❛✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳

  ✶✳✼ ❚❡♦r❡♠❛ ❞❡ ❡①t❡♥sã♦ ❞❡ ❉✉❣✉♥❞❥✐

  ❆ s❡❣✉✐r ❡♥✉♥❝✐❛♠♦s ✉♠ t❡♦r❡♠❛ q✉❡ tr❛t❛ ❞❛ ❡①t❡♥sã♦ ❞❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s✱ ❞❡✈✐❞♦ ❛ ❉✉❣✉♥❞❥✐✳ ❙✉❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✶✹❪ ✭❚❡♦r❡♠❛ ✻✳✶✮✳ ❉❡✜♥✐çã♦ ✶✳✷✹ ❙❡❥❛ L ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✳ ❊♥tã♦ L é ✉♠ ❡s♣❛ç♦ ❛✜♠ ❞♦ t✐♣♦ ♠ s❡✱ ♣❛r❛ ❝❛❞❛ ❡s♣❛ç♦ ♠étr✐❝♦ X ❡ ♣❛r❛ t♦❞❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ f : X → L✱ ✈❛❧❡ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ ♣❛r❛ t♦❞♦ x ∈ X ❡ t♦❞❛ ✈✐③✐♥❤❛♥ç❛ W ⊃ f(x)✱ ❡①✐st❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ⊃ x

  ❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦ C ⊂ L t❛❧ q✉❡ f(U) ⊂ C ⊂ W ✳ ❚❡♦r❡♠❛ ✶✳✷✺ ✭❉✉❣✉♥❞❥✐✮ ❙❡❥❛♠ X ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✱ A ⊂ X ✉♠ s✉❜❝♦♥❥✉♥t♦ ❢❡✲ ❝❤❛❞♦ ❡ L ✉♠ ❡s♣❛ç♦ ❛✜♠ ❞♦ t✐♣♦ m✳ ❊♥tã♦✱ ❝❛❞❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ f : A → L ♣♦ss✉✐ ✉♠❛ ❡①t❡♥sã♦ ❝♦♥tí♥✉❛ F : X → L✳

  ❈❛♣ít✉❧♦ ✷ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✷✳✶ ❈♦♥s✐❞❡r❛çõ❡s ✐♥✐❝✐❛✐s

  ❊st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❡st✉❞❛r ❛ ❡①✐stê♥❝✐❛ ❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦

  

  p−2 q−2 N

   −∆ u + (λa(x) + 1)|u| u = |u| u, x ∈ R ,

  p

  (S )

  λ,q 1,p N

   u ∈ W (R ),

  ∗ p−2

  u = u|∇u)

  p

  ❡♠ q✉❡ 2 ≤ p < N, p < q < p ✱ ∆ ❞✐✈(|∇ é ♦ ♦♣❡r❛❞♦r p✲▲❛♣❧❛❝✐❛♥♦ ❡ λ é ✉♠ ♣❛râ♠❡tr♦ ♣♦s✐t✐✈♦✳ ❱❛♠♦s ❝♦♥s✐❞❡r❛r a ✉♠❛ ❢✉♥çã♦ s❛t✐s❢❛③❡♥❞♦

  N −1

  , R) (0)

  1

  ✭A ✮ a ∈ C(R é ♥ã♦ ♥❡❣❛t✐✈❛✱ Ω = ✐♥t a é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❝♦♠ ❢r♦♥t❡✐r❛

  −1

  (0) s✉❛✈❡ ❡ Ω = a ❀ > 0

  ✭A

  2 ✮ ❡①✐st❡ M t❛❧ q✉❡ N

  L({x ∈ R }) < ∞ : a(x) ≤ M

  ✱

  

N

  ♦♥❞❡ L é ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❡♠ R ✳ ❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥❛t✉r❛❧ ❡♠ q✉❡ tr❛❜❛❧❤❛♠♦s é

  Z

  1,p N p E = u ∈ W (R ) : a(x)|u| < ∞ .

  N R

  = (E, k · k )

  λ λ

  P❛r❛ λ ≥ 0✱ ❞❡✜♥✐♠♦s E ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ E ♠✉♥✐❞♦ ❞❛ ♥♦r♠❛

  1/p

  Z

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✷✶

  1,p N

  (R ) ❉✉r❛♥t❡ t♦❞♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❞❡♥♦t❛♠♦s ♣♦r k·k ❛ ♥♦r♠❛ ❡♠ W ❞❛❞❛ ♣♦r

  Z 1/p

  p p kuk = (|∇u| + |u| ) .

  N R N s s,D (D)

  ❉❛❞♦s 1 ≤ s < ∞ ❡ D ⊂ R ✱ ✐♥❞✐❝❛♠♦s ♣♦r | · | ❛ ✉s✉❛❧ ❞❡L ✳ P❛r❛ s✐♠♣❧✐✜❝❛r✱

  N N s

  q✉❛♥❞♦ D = R ✱ ❡s❝r❡✈❡♠♦s |u| ❛♦ ✐♥✈és ❞❡ |u| s,R ✳ ) : E → R

  λ,q λ,q λ

  ❖ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛ (S é eI ❞❡✜♥✐❞♦ ♣♦r Z Z

  1

  1

  1

  1

  p p q p q

  e I (u) = (|∇u| + (λa(x) + 1)|u| ) − |u| = kuk − |u| .

  λ,q q λ

  N N

  p q p q

  R R

  1

  ∈ C (E , R)

  λ,q λ

  ➱ ♣♦ssí✈❡❧ ♠♦str❛r ✭✈❡❥❛ Pr♦♣♦s✐çã♦ ✶✳✶✷ ❡♠ ❬✸✸❪✮ q✉❡ eI ❝♦♠ Z

  Z

  

′ p−2 p−2 q−2

  he I (u), vi = |∇u| ∇u · ∇v + (λa(x) + 1)|u| uv − |u| uv,

  λ,q N

  N R R λ λ λ,q )

  ♣❛r❛ t♦❞♦ u, v ∈ E ✳ ❉✐③❡♠♦s q✉❡ u ∈ E é s♦❧✉çã♦ ❢r❛❝❛ ❞❡ (S q✉❛♥❞♦

  ′

  he I (u), vi = 0, .

  λ

λ,q ♣❛r❛ t♦❞♦ v ∈ E

  )

  λ,q

  ❯♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ (S é ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ q✉❛♥❞♦ n o e e

  I λ,q (u) = inf I λ,q (v) : v λ,q ) .

  é ✉♠❛ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ❞❡ (S

  λ,q

  ❉❡✜♥✐♠♦s ❛ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ ♣❛r❛ ♦ ❢✉♥❝✐♦♥❛❧ eI ♣♦r n o

  p q ′

  e N = u ∈ E \ {0} : he I (u), ui = 0 = u ∈ E \ {0} : kuk = |u| .

  λ,q λ λ,q λ λ q

  N ) N

  ❖❜s❡r✈❛♠♦s q✉❡ e λ,q ❝♦♥té♠ t♦❞❛s ❛s s♦❧✉çõ❡s ❢r❛❝❛s ❞❡ (S λ,q ✳ ◗✉❛♥❞♦ u ∈ e λ,q t❡♠✲s❡

  1

  1

  p

  e I (u) = − kuk ,

  λ,q λ

  p q N

  λ,q λ,q

  ❡✱ ♣♦rt❛♥t♦✱ ♦ ❢✉♥❝✐♦♥❛❧ eI é ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡ ❡♠ e ✳ ❆ss✐♠✱ é ✜♥✐t♦ ♦ ♥ú♠❡r♦ e = inf I (u).

  λ,q λ,q

  ec

  

N

u∈ e λ,q

  ◆❡st❡ ❝❛♣ít✉❧♦✱ ✐♥✐❝✐❛❧♠❡♥t❡ ♠♦str❛♠♦s q✉❡ ♣❛r❛ t♦❞♦ λ > 0 s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ (S )

  λ,q

  ♣♦ss✉✐ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛✳ P♦st❡r✐♦r♠❡♥t❡✱ ✈❛♠♦s ♠♦str❛r

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✷✷

  λ,q )

  q✉❡ ❛s s♦❧✉çõ❡s ❞❡ (S s❡ ❝♦♥❝❡♥tr❛♠ ♥✉♠❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ (

  p−2 q−2

  −∆

  p u + |u| u = |u| u

  ❡♠ Ω, (D )

  q

  u = 0 ❡♠ ∂Ω.

  1,p

  J : W (Ω) → R

  q,Ω

  P❛r❛ ❡st❡ ♣r♦❜❧❡♠❛✱ ♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ é e ❞❛❞♦ ♣♦r Z Z

  1

  1

  p p q

  e J (u) = (|∇u| + |u| ) − |u| .

  q,Ω

  p q

  Ω Ω 1,p

  1 J ∈ C (W (Ω), R) q,Ω

  ❚❡♠♦s q✉❡ e ❡ ❛ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ ♣❛r❛ ❡st❡ ❢✉♥❝✐♦♥❛❧ é ♦ ❝♦♥❥✉♥t♦ n o

  1,p 1,p p q ′

  M f = u ∈ W (Ω) \ {0} : h e J (u), ui = 0 = u ∈ W (Ω) \ {0} : kuk = |u| ,

  q,Ω q,Ω Ω q,Ω 1,p

  

N

  (R )

  Ω

  ♦♥❞❡ kuk ❞❡♥♦t❛ ❛ ♥♦r♠❛ ❞❡ u ❡♠ W ❞❛❞❛ ♣♦r Z 1/p

  p p

  kuk = (|∇u| + |u| ) .

  Ω Ω λ,q

  ❯♠ ❝á❧❝✉❧♦ ❛♥á❧♦❣♦ àq✉❡❧❡ ❢❡✐t♦ ♣❛r❛ eI ♠♦str❛ q✉❡ é ✜♥✐t♦ ♦ ❡❧❡♠❡♥t♦ e m q,Ω = inf J q,Ω (u). e

  

M

u∈ f q, Ω

  ) ❈♦♠♦ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ❞❡ (S λ,q ✱ ✈❛♠♦s tr❛❜❛❧❤❛r ❝♦♠ ✉♠

  ♣r♦❜❧❡♠❛ ✉♠ ♣♦✉❝♦ ♠♦❞✐✜❝❛❞♦✳ ❆ s❛❜❡r✱ ♣❛r❛ λ ≥ 0✱ ❝♦♥s✐❞❡r❛♠♦s ♦ ♣r♦❜❧❡♠❛ 

  p−2 q−1 N

  

  • −∆ u + (λa(x) + 1)|u| u = (u ) , x ∈ R ,

  p

  • (S )

  λ,q 1,p N

   u ∈ W (R ) : E → R

  λ,q λ

  ❡ s❡✉ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ I ❞❡✜♥✐❞♦ ♣♦r Z Z

  1

  1

  p p q +

  I (u) = (|∇u| + (λa(x) + 1)|u| ) − (u ) ,

  λ,q N N

  p R q R

  • 1

  1

  = max{u, 0} ∈ C (E , R)

  λ,q λ

  ♦♥❞❡ u ✳ ❉❛ ♠❡s♠❛ ♠❛♥❡✐r❛ q✉❡ s❡ ♠♦str❛ q✉❡ eI ♠♦str❛✲s❡

  ∈ C (E , R)

  λ,q λ

  t❛♠❜é♠ q✉❡ I ✳

  λ,q

  ❆ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ ♣❛r❛ ♦ ❢✉♥❝✐♦♥❛❧ I é ♦ ❝♦♥❥✉♥t♦

  • ′ p q N = u ∈ E \ {0} : hI (u), ui = 0 = u ∈ E \ {0} : kuk = |u | .

  λ,q λ λ λ,q q λ

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✷✸

  λ,q

  ❆ss✐♠ ❝♦♠♦ ec é ✜♥✐t♦✱ ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ♦ ♥ú♠❡r♦ r❡❛❧ c = inf I (u).

  λ,q λ,q u∈N

λ,q

  • )

  λ

  ❉✐③❡♠♦s q✉❡ u ∈ E é ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ (S q✉❛♥❞♦

  λ,q

  Z Z Z

  • p−2 p−2 q−1

  |∇u| ∇u · ∇φ + (λa(x) + 1)|u| uφ − (u ) φ = 0,

  N N N R R R

  • )

  λ

  ♣❛r❛ t♦❞❛ ❢✉♥çã♦ φ ∈ E ✳ ❯♠❛ s♦❧✉çã♦ u ❞❡ (S é ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ q✉❛♥❞♦

  λ,q

  • I λ,q (u) = inf

  I λ,q (v) : v ) .

  é ✉♠❛ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ❞❡ (S

  λ,q

  • )

  ▼♦str❛r❡♠♦s q✉❡ s♦❧✉çõ❡s ❞❡ (S λ,q sã♦ ♣♦s✐t✐✈❛s ❡ s❡ ❝♦♥❝❡♥tr❛♠ ♥✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❞❡

  (

  • p−2 q−1

  −∆ u + |u| u = (u )

  p

  ❡♠ Ω,

  • (D )

  q

  u = 0 ❡♠ ∂Ω.

  1,p

  : W (Ω) → R

  q,Ω

  ❖ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛ ❡ss❡ ♣r♦❜❧❡♠❛ é J ❞❡✜♥✐❞♦ ♣♦r Z Z

  1

  1

  p p + q J q,Ω (u) = (|∇u| + |u| ) − (u ) .

  p q

  Ω Ω 1,p

  1

  ∈ C (W (Ω), R)

  ❊♥tã♦✱ J q,Ω ❡ ❛ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ ♣❛r❛ ❡st❡ ❢✉♥❝✐♦♥❛❧ é ♦ ❝♦♥❥✉♥t♦

  1,p 1,p p q

  • ′ M = u ∈ W (Ω) \ {0} : hJ (u), ui = 0 = u ∈ W (Ω) \ {0} : kuk = |u | .

  q,Ω q,Ω Ω q,Ω

  ❉❡✜♥✐♠♦s m = inf J (u).

  

q,Ω q,Ω

u∈M q, Ω

λ

  ❋✐♥❛❧✐③❛♥❞♦ ❡st❛ s❡çã♦✱ ❢❛③❡♠♦s ❛❧❣✉♠❛s ❝♦♥s✐❞❡r❛çõ❡s s♦❜r❡ ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ E ✳

  λ

  ▲❡♠❛ ✷✳✶ P❛r❛ λ ≥ 0✱ E é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ r❡✢❡①✐✈♦✳ )

  ) ❉❡♠♦♥str❛çã♦✿ ❙❡ (u n é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠ E λ ✱ ❡♥tã♦ (u n é ✉♠❛ s❡q✉ê♥❝✐❛

  1,p N 1,p N 1,p N

  (R ) (R ) (R ) ❞❡ ❈❛✉❝❤② ❡♠ W ✳ ❉❛í✱ ❝♦♠♦ W é ❝♦♠♣❧❡t♦✱ ❡①✐st❡ u ∈ W t❛❧

  1,p N p N

  → u (R ) → u (R )

  n n

  q✉❡ u ❡♠ W ❡✱ ♣♦rt❛♥t♦✱ u ❡♠ L ✳ ❆ss✐♠✱ ♣❛ss❛♥❞♦ ❛ ✉♠❛

  N

  (x) → u(x)

  n

  s✉❜s❡q✉ê♥❝✐❛✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ u q✳t✳♣✳ ❡♠ R ✳ ❱❛♠♦s ♠♦str❛r q✉❡ u → u

  n λ

  ❡♠ E ✱ ✐st♦ é✱ Z Z

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✷✹

  1,p N

  → u

  n (R )

  ❈♦♠♦ u ❡♠ W ✱ r❡st❛✲♥♦s ♠♦str❛r q✉❡ Z

  p a(x)|u − u| → 0.

n

  N R

  ∈ N )

  ❉❛❞♦ ε > 0✱ ❝♦♠♦ (u n é ❞❡ ❈❛✉❝❤② ❡♠ E λ ✱ ❡①✐st❡ n t❛❧ q✉❡✱ ♣❛r❛ n, m ≥ n ✱ Z Z

  p p 1/p 1/p p

  − a(x) − u | − u k a(x) u n u m = a(x)|u n m < ku n m < ε.

  λ N N R R

  1/p p N p N

  u ) (R ) (R )

  n

  ▲♦❣♦✱ ❛ s❡q✉ê♥❝✐❛ (a(x) é ❞❡ ❈❛✉❝❤② ❡♠ L ❡✱ ♣♦rt❛♥t♦✱ ❡①✐st❡ v ∈ L t❛❧

  1/p p N 1/p

  u → v (R ) u (x)

  n n

  q✉❡ a(x) ❡♠ L ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ a(x)

  N 1/p 1/p

  u (x) → a(x) u(x) ❝♦♥✈❡r❣❡ ♣❛r❛ v(x) q✳t✳♣✳ ❡♠ R ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ a(x) n q✳t✳♣✳

  N 1/p

  u λ ❡♠ R ✱ ✈❡♠♦s q✉❡ v = a(x) ✳ ■ss♦ ♠♦str❛ q✉❡ E é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳

  λ a = (E, k · k a )

  P❛r❛ ♠♦str❛r q✉❡ E é r❡✢❡①✐✈♦✱ ❞❡✜♥❛ E ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ E ♠✉♥✐❞♦ ❞❛ ♥♦r♠❛

  1/p

  Z

  p

  kuk a = (λa(x) + 1)|u| .

  N R a

  ❆✜r♠❛♠♦s q✉❡ E é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦✳ ❈♦♠ ❡❢❡✐t♦✱ ✈❛♠♦s ♠♦str❛r ❛♥t❡s q✉❡ ✈❛❧❡

  a

  ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♠ E ✿

  p p

  u + v u − v

  1

  p p ≤ (kuk + kvk ), .

  • a a

  ♣❛r❛ t♦❞♦ u, v ∈ E a ✭✷✳✶✮

  2

  2

  2

  a a

  P❛r❛ t❛♥t♦✱ ✉t✐❧✐③❛♠♦s ❛ ♣r✐♠❡✐r❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❧❛r❦s♦♥ ✭✈❡❥❛ ❬✾❪✱ ❚❡♦r❡♠❛ ✹✳✶✵❪✮✱ q✉❡ ❞✐③ q✉❡ ♣❛r❛ a, b ∈ R ✈❛❧❡

  p p

  a + b a − b

  1

  p p

  ≤ (|a| + |b| ). +

  2

  2

  2 ❉❡ss❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ s❡❣✉❡ q✉❡

  p p p p

  Z u + v u − v u + v u − v (λa(x) + 1) + = +

  N

  2

2 R

  2

  2

  a a

  Z

  1

  1

  p p p p

  ≤ (λa(x) + 1)(|u| + |v| ) = (kuk + kvk ),

  a a N

2 R

  2

  a

  ♦ q✉❡ ♠♦str❛ ✭✷✳✶✮✳ ❆❣♦r❛✱ ♣❛r❛ ✈❡r q✉❡ E é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦✱ s❡❥❛♠ ε > 0✱ u, v ∈ E ≤ 1 ≤ 1 > ε

  a a a a

  ❝♦♠ kuk ✱ kvk ❡ ku − vk ✳ ❆ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✶✮ ✐♠♣❧✐❝❛ q✉❡

  p p

  u + v ε

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✷✺ u + v

  

ε

p 1/p

  ≤ 1−δ ) ]

  a

  ❡✱ ❡♥tã♦✱ ✱ ♦♥❞❡ δ = 1−[1−( ✳ P♦rt❛♥t♦ E é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦

  

2

  2

  a p N N

  → E × (L

  a λ a (R ))

  ❡✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✸✳✸✶ ❡♠ ❬✾❪✱ E é r❡✢❡①✐✈♦✳ ❈♦♥s✐❞❡r❡ ❛❣♦r❛ T : E ❞❡✜♥✐❞❛ ♣♦r T (u) = (u, ∇u)✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ T ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡ é ❧✐♥❡❛r✳ ❖❜s❡r✈❡ q✉❡

  1/p p p N N p

  × (L (R )) = kuk |v|

  • p p N N

  a ×

  ♦ ❡s♣❛ç♦ ♣r♦❞✉t♦ E ♠✉♥✐❞♦ ❞❛ ♥♦r♠❛ k(u, v)k a é ✉♠

  (R )) ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ r❡✢❡①✐✈♦✱ ✉♠❛ ✈❡③ q✉❡ E a ❡ (L ♦ sã♦✳ ❚❡♠♦s q✉❡ T é ❧✐♥❡❛r

  × = kuk λ λ

  ❡ kT (u)k ♣❛r❛ t♦❞❛ ❢✉♥çã♦ u ∈ E ✱ ✐st♦ é✱ T é ✉♠❛ ✐s♦♠❡tr✐❛ ❧✐♥❡❛r✳ ❉❛í ❡ ❞♦

  p N N

  ) × (L (R ))

  λ λ a

  ❢❛t♦ ❞❡ E s❡r ❝♦♠♣❧❡t♦✱ T (E é ✉♠ s✉❜❡s♣❛ç♦ ❢❡❝❤❛❞♦ ❞❡ E ✳ ▲♦❣♦✱ ♣❡❧❛ )

  λ

  Pr♦♣♦s✐çã♦ ✸✳✸✵ ❡♠ ❬✾❪✱ T (E é r❡✢❡①✐✈♦✳ ❊ ❝♦♠♦ T é ✉♠❛ ✐s♦♠❡tr✐❛ ❧✐♥❡❛r s♦❜r❡❥❡t✐✈❛ )

  λ λ λ

  ❡♥tr❡ E ❡ T (E ✱ s❡❣✉❡ q✉❡ ✭✈❡❥❛ ♥♦t❛ ❡♠ ❬✾❪✱ ♣á❣✐♥❛ ✼✶✮ E é r❡✢❡①✐✈♦✳ ■ss♦ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛✳

  ) Pr♦♣♦s✐çã♦ ✷✳✷ ❙❡❥❛♠ λ ≥ 0 ❡ (u n ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❡♠ E λ ✳ ❊♥tã♦ ❡①✐st❡ u ∈ E λ t❛❧ q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱

  n ⇀ u λ

  ✭✐✮ u ❢r❛❝❛♠❡♥t❡ ❡♠ E ❀

  

s N ∗

  → u (R )

  n

  ✭✐✐✮ u ❢♦rt❡♠❡♥t❡ ❡♠ L loc ✱ ♣❛r❛ t♦❞♦ p ≤ s < p ❀

  N

  (x) → u(x)

  n

  ✭✐✐✐✮ u q✳t✳♣✳ ❡♠ R ✳ ⇀ u

  λ λ n

  ❉❡♠♦♥str❛çã♦✿ P❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ E é r❡✢❡①✐✈♦✳ ▲♦❣♦✱ ❡①✐st❡ u ∈ E t❛❧ q✉❡ u

  1,p N 1,p N

  ֒→ W (R ) (R ) ֒→ ❢r❛❝❛♠❡♥t❡ ❡♠ E λ ✳ ■ss♦ ♠♦str❛ (i)✳ ❈♦♠♦ E λ ❝♦♥t✐♥✉❛♠❡♥t❡ ❡ W

  s N ∗ s N

  L (R )

  λ ֒→ L (R )

  ❝♦♠♣❛❝t❛♠❡♥t❡ ♣❛r❛ p ≤ s < p ✱ s❡❣✉❡ q✉❡ E ❝♦♠♣❛❝t❛♠❡♥t❡✳

  loc loc s N

  → u (R )

  n

  ❆ss✐♠ u ❡♠ L ✱ ♦ q✉❡ ♠♦str❛ (ii)✳ P❛r❛ ♣r♦✈❛r (iii)✱ ✉s❛♠♦s ✉♠ ♣r♦❝❡ss♦

  loc N s

  = {x ∈ R : |x| < r} → u (B )

  r n

  1

  ❞✐❛❣♦♥❛❧✳ ❉❛❞♦ r > 0✱ s❡❥❛ B ✳ ❈♦♠♦ u ❡♠ L ✱ ⊂ N )

  1 n n∈N

  1

  ❡①✐st❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦ N t❛❧ q✉❡ ❛ s✉❜s❡q✉ê♥❝✐❛ (u ❝♦♥✈❡r❣❡ ♣❛r❛

  s

  u ) (B )

  ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡ B

  1 ✳ ❈♦♠♦ (u n n∈N 1 ❝♦♥✈❡r❣❡ ♣❛r❛ u ❡♠ L 2 ✱ ♦❜t❡♠♦s

  ⊂ N

  n ) n∈N

  2 1 n ) n∈N

  ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ (u

  2 ❝♦♠ N t❛❧ q✉❡ (u 2 ❝♦♥✈❡r❣❡ ♣❛r❛ u ❡♠ q✉❛s❡

  2

  t♦❞♦ ♣♦♥t♦ ❞❡ B ✳ Pr♦❝❡❞❡♥❞♦ ❞❡ss❛ ♠❛♥❡✐r❛✱ ♦❜t❡♠♦s ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s ❞❡ í♥❞✐❝❡s N

  ⊂ N ⊂ N )

  k+1 k n n∈N k k

  t❛✐s q✉❡ (u ❝♦♥✈❡r❣❡ ♣❛r❛ u ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡ B ✳ ❈♦♥s✐❞❡r❡

  ∗ ∗ ∗ ∗ ∗

  N

  ∗

  = {n , n , . . . , n , . . .} ⊂ N )

  ❝♦♠ n ♦ k✲és✐♠♦ ❡❧❡♠❡♥t♦ ❞❡ N k ✳ ❆ss✐♠✱ (u n n∈N é✱ ❛

  1 2 k k n ) n∈N

  ♣❛rt✐r ❞♦ s❡✉ k✲és✐♠♦ ❡❧❡♠❡♥t♦✱ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❞❡ (u k ❡✱ ♣♦rt❛♥t♦✱ ❝♦♥✈❡r❣❡ ♣❛r❛ u ⊂ B

  k k k

  ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡ B ✳ ❚❡♠♦s q✉❡✱ ♣❛r❛ ❝❛❞❛ k ∈ N✱ ❡①✐st❡ Z ❞❡ ♠❡❞✐❞❛

  ∞

  (x)) \Z Z

  n n∈N k k k m

  ♥✉❧❛ t❛❧ q✉❡ (u ❝♦♥✈❡r❣❡ ♣❛r❛ u(x) s❡❥❛ q✉❛❧ ❢♦r x ∈ B ✳ ❚♦♠❡ Z = ∪ m=1 ✳

  N

  \ Z \ Z

  k k

  ❊♥tã♦ Z t❡♠ ♠❡❞✐❞❛ ♥✉❧❛ ❡✱ ♣❛r❛ t♦❞♦ x ∈ R t❡♠♦s x ∈ B ♣❛r❛ ❛❧❣✉♠ k ∈ N

  ∗

  (x)) ❡ (u n n∈N ❝♦♥✈❡r❣❡ ♣❛r❛ u(x)✳ ■ss♦ ♠♦str❛ (iii)✳

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✷✻

  ✷✳✷ ❆ ❝♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ❞❡ P❛❧❛✐s✲❙♠❛❧❡

  1

  (F, R) n ) ◗✉❛♥❞♦ F é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡ I ∈ C ✱ ❞✐③❡♠♦s q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ (u

  c

  ❡♠ F é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ♣❛r❛ I ♥♦ ♥í✈❡❧ c✱ q✉❡ ❞❡♥♦t❛♠♦s ♣♦r (P S) ✱

  ′ ′

  ) → c (u ) → 0

  n n

  q✉❛♥❞♦ I(u ❡ I ❡♠ F q✉❛♥❞♦ n → ∞✳ ❉✐③❡♠♦s q✉❡ I s❛t✐s❢❛③ ❛

  c

  ❝♦♥❞✐çã♦ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ♥♦ ♥í✈❡❧ c q✉❛♥❞♦ t♦❞❛ s❡q✉ê♥❝✐❛ (P S) ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ ❡♠ F ✳

  ◆❡st❛ s❡çã♦✱ ❡st❛❜❡❧❡❝❡♠♦s ✉♠❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ♣❛r❛ ♦ ❢✉♥❝✐♦♥❛❧ I λ,q ✳ ▼♦s✲ tr❛♠♦s q✉❡ ❛ ❝♦♥❞✐çã♦ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ✈❛❧❡ ❛❜❛✐①♦ ❞❡ ✉♠ ❝❡rt♦ ♥í✈❡❧✱ ❞❡s❞❡ q✉❡ ♦ ♣❛râ♠❡✲ tr♦ λ s❡❥❛ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳ Pr❡❝✐s❛♠❡♥t❡✱ ✈❛♠♦s ❞❡♠♦♥str❛r ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✳

  > 0 = Λ (q) > 0

  1 λ,q

  Pr♦♣♦s✐çã♦ ✷✳✸ P❛r❛ t♦❞♦ C ❞❛❞♦✱ ❡①✐st❡ Λ t❛❧ q✉❡ I s❛t✐s❢❛③ ❛

  1

  ❝♦♥❞✐çã♦ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ♥♦ ♥í✈❡❧ c ♣❛r❛ t♦❞♦ c ≤ C ❡ λ ≥ Λ ✳ P❛r❛ ♣r♦✈❛r ❡st❡ r❡s✉❧t❛❞♦✱ ✈❛♠♦s ♣r❡❝✐s❛r ❞❡ ✈ár✐♦s ❧❡♠❛s ❛✉①✐❧✐❛r❡s✳

  ) ⊂ E

  n λ c λ,q

  ▲❡♠❛ ✷✳✹ ❙❡❥❛ (u ✉♠❛ s❡q✉ê♥❝✐❛ ✭P❙✮ ♣❛r❛ I ✳ ❊♥tã♦ )

  n λ

  ✭✐✮ (u é ❧✐♠✐t❛❞❛ ❡♠ E ✱ pq

  p

  • q

  ku k = lim |u | = c.

  n

  ✭✐✐✮ lim λ n q

  n→∞ n→∞

  q − p ∈ N

  c λ,q

  ❉❡♠♦♥str❛çã♦✿ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ s❡q✉ê♥❝✐❛ (P S) ♣❛r❛ I ✱ ❞❛❞♦ ε > 0✱ ❡①✐st❡ n t❛❧ q✉❡

  1

  ′ ′

  I (u ) ≤ c + ε kI (u )k ≤ ε,

  λ,q n n E

  ❡ λ,q

  λ

  q s❡♠♣r❡ q✉❡ n ≥ n ✳ ❖❜s❡r✈❡ q✉❡✱ ♣❛r❛ t♦❞♦ n ∈ N, Z Z

  1

  1

  1

  ′ p p q +

  hI i = | | I (u ) − (u ), u (|∇u + (λa(x) + 1)|u ) − (u )

  λ,q n n n n n λ,q n

  N N

  q p R q R Z Z

  1

  p p q +

  − (|∇u | + (λa(x) + 1)|u | ) − (u )

  n n n N N

  q R R

  1

  1

  p = − ku k . n ✭✷✳✷✮ λ

  p q

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✷✼ ❆ss✐♠✱ ♣❛r❛ n ≥ n ✱

  1

  1

  1

  p ′

  − ku k = I (u ) − hI (u ), u i ≤ c + ε + εku k .

  n λ,q n n n n λ λ λ,q

  p q q )

  ❙❡♥❞♦ 2 ≤ p < q✱ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ✐♠♣❧✐❝❛ q✉❡ (u n é ❧✐♠✐t❛❞❛ ❡♠ E λ ✳ ■ss♦ ♠♦str❛ (i)✳ )

  n c λ,q

  P❛r❛ ♣r♦✈❛r (ii) ♦❜s❡r✈❛♠♦s q✉❡✱ s❡♥❞♦ (u ✉♠❛ s❡q✉ê♥❝✐❛ (P S) ♣❛r❛ I ❧✐♠✐t❛❞❛✱ t❡♠♦s

  1

  ′

  lim hI (u ), u i = 0

  n n λ,q n→∞

  q ❡✱ ❛ss✐♠✱ ❞❡ (2.2) s❡❣✉❡ q✉❡

  1

  1

  1

  ′ p

  hI i − ku k c = lim I λ,q (u n ) − (u n ), u n = lim n .

  ✭✷✳✸✮

  λ,q λ n→∞ n→∞

  q p q ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❛ (2.2) ♦❜t❡♠♦s

  1

  1

  1

  ′ + q

  I (u ) − hI (u ), u i = − |u |

  λ,q n n n λ,q n q

  p p q ❡✱ ♣♦rt❛♥t♦✱

  1

  1

  1

  ′ q +

  c = lim I (u ) − hI (u ), u i = lim − |u | .

  λ,q n n n ✭✷✳✹✮ λ,q n q n→∞ n→∞

  q p q ❉❡ (2.3) ❡ (2.4) s❡❣✉❡ (ii)✳

  ) ⊂ E > 0

  ▲❡♠❛ ✷✳✺ ❙❡❥❛ (u n λ ✉♠❛ s❡q✉ê♥❝✐❛ ✭P❙✮ c ♣❛r❛ I λ,q ✳ ❊①✐st❡ C ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ λ t❛❧ q✉❡ s❡ c 6= 0 ❡♥tã♦ c ≥ C ✳

  1,p N q N

  (R ) (R )

  1 > 0

  ❉❡♠♦♥str❛çã♦✿ P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ✐♠❡rsã♦ ❞❡ W ❡♠ L ✱ ❡①✐st❡ C t❛❧ q✉❡

  q

  • q q q |u | ≤ |u| ≤ C kuk ≤ C kuk .

  

1

  1 q q λ ✭✷✳✺✮ λ

  ❙❡❣✉❡ ❞❛í q✉❡✱ ♣❛r❛ u ∈ E ✱

  

′ p p q

  • q

  hI − |u | ≥ kuk − C kuk (u), ui = kuk 1 .

  

λ,q λ q λ λ

  1 p q p

  1 p−q

  − C kuk ≥ kuk ≤ (2C ) := δ

  1 λ

  1

  ❖❜s❡r✈❡ q✉❡ kuk λ λ λ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ kuk ✳ P♦rt❛♥t♦

  2

  1

  p ′

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✷✽ (q − p)

  p

  · = δ > 0

  ❱❛♠♦s ♠♦str❛r q✉❡ ❛ t❡s❡ ❞♦ ❧❡♠❛ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ C ✳ ❙✉♣♦♥❞♦ q✉❡ pq

  

p p

  c < C ku k < δ ∈ N k < δ

  n n λ

  ✱ ♦ ▲❡♠❛ 2.4 (ii) ❞✐③ q✉❡ lim ✳ ▲♦❣♦ ❡①✐st❡ n t❛❧ q✉❡ ku

  λ n→∞

  ) ⊂ E

  n λ c λ,q

  ♣❛r❛ t♦❞♦ n ≥ n ✳ ❙❡♥❞♦ (u ✉♠❛ s❡q✉ê♥❝✐❛ (P S) ♣❛r❛ I ✱ ❞❛❞♦ ❛r❜✐tr❛r✐❛♠❡♥t❡

  ′

  ε > 0 ∈ N (u )k ≤ ε =

  1 n E

  1

  2

  ✱ ❡①✐st❡ n t❛❧ q✉❡ kI λ,q ♣❛r❛ t♦❞♦ n ≥ n ✳ ❚♦♠❛♥❞♦ n

  

λ

  } , n

  ♠❛①{n

  1 ✱ s❡❣✉❡ ❞❡ (2.6) q✉❡ ♣❛r❛ n ≥ n

2 ✱

  1

  p ′ ′ ′

  ku k ≤ hI (u ), u i ≤ kI (u )k ku k ≤ εku k ,

  n n n n E n λ n λ λ λ,q λ,q λ

  2 ♦✉ s❡❥❛✱

  1 p− 1 ku k ≤ (2ε) , . n λ

  2

  s❡♠♣r❡ q✉❡ n ≥ n

  • k → 0

  | → 0

  n λ q

  ■ss♦ ♠♦str❛ q✉❡ ku q✉❛♥❞♦ n → ∞✳ ❙❡❣✉❡ ❞❡ (2.5) q✉❡ |u n q✉❛♥❞♦ n → ∞✳

  1

  1

  p

  • q

  (u ) = ku k − |u | → 0

  λ,q n n

  ❆ss✐♠✱ I

  n q ✱ ✐st♦ é✱ c = 0✳ ❖ ❧❡♠❛ ❡stá ♣r♦✈❛❞♦✳ λ

  p q > 0 , R > 0 ) ⊂ E

  1 ε ε n λ

  ▲❡♠❛ ✷✳✻ ❙❡❥❛ C ✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ ε > 0✱ ❡①✐st❡♠ Λ t❛✐s q✉❡✱ s❡ (u

  c λ,q 1 ε

  é ✉♠❛ s❡q✉ê♥❝✐❛ ✭P❙✮ ♣❛r❛ I ❝♦♠ c ≤ C ❡ λ ≥ Λ ✱ ✈❛❧❡ Z

  q lim sup |u | ≤ ε. n

c

n→∞ B

  

c N

  = x ∈ R : |x| > R

  ε

  ♦♥❞❡ B R ε ✳ P❛r❛ ♣r♦✈❛r ❡ss❡ ❧❡♠❛✱ ✉s❛♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

  N

  ▲❡♠❛ ✷✳✼ ❙❡❥❛ A ⊂ R ✉♠ ❝♦♥❥✉♥t♦ ♠❡♥s✉rá✈❡❧ t❛❧ q✉❡ L(A) < ∞✳ ❊♥tã♦

  c lim L(A ∩ B ) = 0. R R→∞ c

  ) ∪ (A ∩ B )

  n

  ❉❡♠♦♥str❛çã♦✿ P❛r❛ n ∈ N✱ t❡♠♦s A = (A ∩ B n ✱ s❡♥❞♦ ❛ ✉♥✐ã♦ ❞✐s❥✉♥t❛✳ ❆ss✐♠

  c L(A) = L(A ∩ B ) + L(A ∩ B ).

n

n ✭✷✳✼✮

  ∞

  [ (A ∩ B ) )

  n n

  ❈❧❛r❛♠❡♥t❡ A = ❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❝♦♥❥✉♥t♦s (A ∩ B é ❝r❡s❝❡♥t❡✱ ✐st♦ é✱

  n=1

  A ∩ B ⊂ A ∩ B ⊂ · · ·

  1 2 ✳ P♦rt❛♥t♦✱ ♣❡❧♦ ▲❡♠❛ ✸✳✹ ❡♠ ❬✹❪✱

  !

  ∞

  [ L(A) = L (A ∩ B ) = lim L(A ∩ B ).

  n n

  ✭✷✳✽✮

  n→∞

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✷✾ ⊂ A

  n n ) < ∞

  P♦r ❤✐♣ót❡s❡✱ L(A) < ∞✳ ❈♦♠♦ A ∩ B t❡♠♦s q✉❡ L(A ∩ B ♣❛r❛ t♦❞♦ n ∈ N✳ ❈♦♠ ✐ss♦✱ ✉s❛♥❞♦ (2.7) ♦❜t❡♠♦s

  c

  L(A) − L(A ∩ B ) = L(A ∩ B )

  

n

n c

  L(A ∩ B ) = 0 ∈ N ❉❡ (2.8) s❡❣✉❡ q✉❡ lim n ✳ ❆ss✐♠✱ ❞❛❞♦ ε > 0✱ ❡①✐st❡ n t❛❧ q✉❡

  n→∞ c

  L(A ∩ B ) < ε = n > 0 = n ♣❛r❛ t♦❞♦ n ≥ n ✳ ❚♦♠❡ R ✳ ❙❡ R > R ❡♥tã♦

  n c c c c c

  (A ∩ B ) ⊂ (A ∩ B ) ) ≤ L(A ∩ B ) ≤ L(A ∩ B ) < ε ✳ ■ss♦ ✐♠♣❧✐❝❛ q✉❡ L(A ∩ B ♣❛r❛

  R n R R n

  t♦❞♦ R > R ✳ ■ss♦ ♠♦str❛ ♦ ❧❡♠❛✳ ❉❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✷✳✻✿ ❈♦♥s✐❞❡r❡ ♣❛r❛ R > 0 ♦s ❝♦♥❥✉♥t♦s

  N

  C(R) = {x ∈ R ; |x| > R, a(x) > M } ❡

  N D(R) = {x ∈ R ; |x| > R, a(x) ≤ M }. c N

  ∩ B }

  M M = {x ∈ R : a(x) ≤ M

  ❖❜s❡r✈❛♠♦s q✉❡ D(R) = A ✱ ♦♥❞❡ A ✳ P❡❧❛ ❤✐♣ót❡s❡

  R

  (A ) ) < ∞

2 M

  ✱ L(A ✳ ❙❡❣✉❡ ❞♦ ▲❡♠❛ ✷✳✼ q✉❡ L(D(R)) = 0. lim

  ✭✷✳✾✮

  R→∞

  P❛ss❛♥❞♦ ❛ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱ s❡ ♥❡❝❡ssár✐♦✱ ✉s❛♥❞♦ ♦ ▲❡♠❛ 2.4 (ii) ❡ ♦ ❢❛t♦ ❞❡ q✉❡ a(x) > M ❡♠ C(R)✱ ♦❜t❡♠♦s

  Z Z

  1

  p p

  |u | |

  n = (λM + 1)|u n

  λM + 1

C(R) C(R)

  Z

  1

  p

  < (λa(x) + 1)|u |

  n

  λM + 1

C(R)

  Z

  1

  p p

  ≤ (|∇u | + (λa(x) + 1)|u | )

  n n

  λM + 1

C(R)

  1

  

p

  ≤ ku k

  n λ

  λM + 1 1 pq ≤ · (c + 1)

  λM + 1 q − p

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✸✵ 1 pq ≤ ·

  (C

  1 + 1)

  ✭✷✳✶✵✮ λM + 1 q − p

  ♣❛r❛ t♦❞♦ n ∈ N✳ k ≤ σ ∈ E

  n ) λ n λ n λ ֒→

  P❡❧♦ ▲❡♠❛ 2.4 (i)✱ (u é ❧✐♠✐t❛❞❛ ❡♠ E ✱ ❞✐❣❛♠♦s ku ✳ ❈♦♠♦ u N

  N ∗ 1,p N s

  W (R ) ֒→ L (R ) ] ❝♦♥t✐♥✉❛♠❡♥t❡ ♣❛r❛ t♦❞♦ s ∈ [p, p ✱ ❡s❝♦❧❤❡♥❞♦ 1 < r < ✱

  N − p

  N p r ∗

  | ∈ L (R )

  n

  ✈❡♠♦s q✉❡ |u ♣❛r❛ t♦❞♦ n ∈ N✱ ✉♠❛ ✈❡③ q✉❡ p < pr < pN/(N − p) = p ✳ P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r ❡ ♣❡❧❛ ✐♠❡rsã♦ ❛❝✐♠❛ ♦❜t❡♠♦s ✉♠❛ ❝♦♥st❛♥t❡ C > 0 t❛❧ q✉❡

  ′

  Z 1/r Z 1/r Z

  ′ p pr p 1/r

  |u | ≤ |u | 1 ≤ |u | L(D(R))

  n n n pr

D(R) D(R) D(R)

  ✭✷✳✶✶✮

  ′ ′ p 1/r p 1/r

  ≤ Cku k L(D(R)) ≤ Cσ L(D(R)) .

  n λ

  Z

  N q s s c

  |u | (R ) ⊂ L (B )

  n λ

  ❱❛♠♦s ❡st✐♠❛r ✳ ❉❡✈✐❞♦ ❛ ✐♠❡rsã♦ ❝♦♥tí♥✉❛ ❞❡ E ❡♠ L R ♣❛r❛

  c B R

  

∗ p c p c ∗

  R R

  ] ∈ L (B )∩L (B ) ] t♦❞♦ s ∈ [p, p ✱ ✈❡♠♦s q✉❡ u n ✳ ❈♦♠♦ q ∈ [p, p s❡❣✉❡ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡

  ❞❡ ✐♥t❡r♣♦❧❛çã♦ q✉❡ ❡①✐st❡ α ∈ [0, 1] t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ n ∈ N✱

  q(1−α) q qα c ∗ c c

  |u | ≤ |u | |u |

  n n n q,B p ,B p,B R R R q(1−α) qα

  ∗ c

  ≤ |u | |u |

  n n p p,B

R

q(1−α) qα c

  ≤ Cku k |u |

  n n λ p,B R qα q(1−α)

c

  ≤ Cσ ku k

  n p,B

R

  Z qα/p Z

  p p q(1−α)

  = Cσ |u | dx + |u | dx

  n n C(R) D(R)

  ❯s❛♥❞♦ (2.10)✱ (2.11) ❡ ✭✷✳✾✮✱ ♣❛r❛ λ ❡ R s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡s ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ s❡ t♦r♥❛ tã♦ ♣❡q✉❡♥❛ q✉❛♥t♦ s❡ q✉❡✐r❛✳ ■ss♦ ♠♦str❛ ♦ ❧❡♠❛✳

  ′

  ) ⊂ E (u ) → 0

  n λ n

  ▲❡♠❛ ✷✳✽ ❙❡❥❛♠ λ ≥ 0 ❡ (u ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ t❛❧ q✉❡ I q✉❛♥❞♦

  λ,q

  n → ∞ ✳ ❊♥tã♦✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱

  N

  ∇u (x) → ∇u(x) ,

  n

  q✳t✳♣✳ ❡♠ R

  ′

  ∂u ∂u

  n p−2 p−2 p N

  • |∇u|
  • |∇u n
  • |∇u|
  • |∇u
  • |∇u|

  n

  ) t❛❧ q✉❡ 0 ≤ ψ ≤ 1✱ ψ ≡ 1 ❡♠ B

  N

  (R

  ∞

  ψ ∈ C

  : |x| ≤ r ✳ P❛r❛ ✈❡r✐✜❝❛r ✐ss♦✱ ❞❛❞♦s r, ε > 0 q✉❛✐sq✉❡r✱ t♦♠❛♠♦s

  r = x ∈ R N

  → 0 q✉❛♥❞♦ n → ∞, ♦♥❞❡ B

  n

  P

  B r

  ❆✜r♠❛♠♦s q✉❡✱ ♣❛r❛ t♦❞♦ r > 0✱ Z

  | − |∇u|) ♣♦ss✉❡♠ ♦ ♠❡s♠♦ s✐♥❛❧✳

  ) ❡ (|∇u

  ❡ ψ ≡ 0 ❡♠ R

  p−1

  − |∇u|

  p−1

  |

  n

  ✉♠❛ ✈❡③ q✉❡ ♦s ❢❛t♦r❡s (|∇u

  | − |∇u|) ≥ 0,

  n

  )(|∇u

  p−1

  − |∇u|

  p−1

  |

  n

  r

  N

  n

  R N

  n )).

  (∇u · ∇(u − u

  p−2

  ψ|∇u|

  R N

  − u) + Z

  n

  · ∇(u

  n

  ∇u

  p−2

  |

  n

  ψ|∇u

  Z

  \B

  ψ =

  

n

  P

  R N

  ψ ≤ Z

  n

  P

  B r

  = Z

  n

  P

  B r

  ✳ ❊♥tã♦✱ 0 ≤ Z

  r+ε

  | − |∇u|) = (|∇u

  (|∇u

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✸✶ ❉❡♠♦♥str❛çã♦✿ P❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✱ ❡①✐st❡ u ∈ E

  ❉❡✜♥✐♠♦s✱ ♣❛r❛ ❝❛❞❛ n ∈ N ❡ x ∈ R

  n (|∇u| p−2

  − ∇u · ∇u

  p

  p

  P n = |∇u n |

  ❡ n ∈ N✳ ❉❡ ❢❛t♦✱ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛rt③✱

  N

  (x) ≥ 0 ♣❛r❛ t♦❞♦ x ∈ R

  ∇u(x)) · ∇(u n (x) − u(x)). ❖❜s❡r✈❛♠♦s q✉❡ P n

  n (x) − |∇u(x)| p−2

  ∇u

  p−2

  ✱ P n (x) = (|∇u n (x)|

  N

  N .

  p−2

  (x) → u(x) q.t.p. em R

  n

  , u

  ∗

  ) ♣❛r❛ t♦❞♦ p ≤ s < p

  

N

  (R

  s loc

  → u ❡♠ L

  n

  u

  λ ,

  t❛❧ q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ u n ⇀ u ❢r❛❝❛♠❡♥t❡ ❡♠ E

  λ

  |

  ) ≥ |∇u

  p−1

  − |∇u|

  | − |∇u|) − |∇u|

  n

  (|∇u

  p−1

  |

  n

  |∇u| = |∇u

  p−1

  |

  n

  | − |∇u

  n

  |∇u

  p−1

  p

  n

  p

  |

  n

  ) = |∇u

  p−2

  |

  n

  p−2

  |(|∇u|

  n

  − |∇u||∇u

  p

  p

  |

  ✭✷✳✶✷✮

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✸✷

  ∞ N

  − u)ψ) ⊂ E

  n ) ⊂ E λ (R ) n λ

  ❈♦♠♦ (u é ❧✐♠✐t❛❞❛ ❡ ψ ∈ C t❡♠♦s ((u ❧✐♠✐t❛❞❛✳ P♦r

  ′

  (u ) → 0

  n

  ❤✐♣ót❡s❡✱ I q✉❛♥❞♦ n → ∞ ❡ ❛ss✐♠✱

  λ,q ′

  o(1) = hI (u ), (u − u)ψi

  n n λ,q

  Z Z

  p−2 p−2

  = |∇u | ∇u · ∇((u − u)ψ) + (λa(x) + 1)|u | u (u − u)ψ

  n n n n n n N N R R

  Z

  • q−1

  − (u ) (u − u)ψ

  n n N R

  Z Z

  p−2 p−2

  = ψ|∇u | ∇u · ∇(u − u) + |∇u | (∇u · ∇ψ)(u − u)

  n n n n n n N N R R

  Z Z

  • p−2 q−1

  (λa(x) + 1)|u | u (u − u)ψ − (u ) + (u − u)ψ

  n n n n n N N R R

  q✉❛♥❞♦ n → ∞✳ ❉❛í✱ Z

  p−2

  ψ|∇u | ∇u · ∇(u − u) = I (n) + I (n) + I (n) + o(1)

  n n n

  1

  2

  3 N R

  q✉❛♥❞♦ n → ∞✳ ❙✉❜st✐t✉✐♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ❡♠ ✭2.12✮ ♦❜t❡♠♦s✿ Z 0 ≤ P ≤ I (n) + I (n) + I (n) + I (n) + o(1)

  n

  1

  

2

  3

  4

  ✭✷✳✶✸✮

  B r

  q✉❛♥❞♦ n → ∞✱ ♦♥❞❡ Z

  

p−2

  |∇u | · ∇ψ)(u − u),

  I

  1 (n) = − n (∇u n n N R

  Z

  p−2

  | − u) ψ,

  I

  2 (n) = − (λa(x) + 1)|u n u n (u n N R

  Z

  • + q−1

  I (n) = (u ) (u − u) ψ,

  3 n n N R

  Z

  p−2 I (n) = ψ|∇u| ∇u · ∇(u − u ).

4 n

  N R

  I (n) = 0

  i

  ❱❛♠♦s ♠♦str❛r q✉❡✱ ♣❛r❛ 1 ≤ i ≤ 4✱ t❡♠✲s❡ lim ✳ ❈♦♠ ❡❢❡✐t♦✱ ❝♦♠♦ ψ ∈

  n→∞ ∞ N N

  C (R ) ∈ R

  1

  1

  ✱ ♣♦❞❡♠♦s s✉♣♦r |∇ψ| ≤ M ❡♠ R ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ M ✳ ❉❛í✱ ❞❛

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✸✸

  N

  \ B

  r+ε

  ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛rt③ ❡ ❞♦ ❢❛t♦ ❞❡ ψ ≡ 0 ❡♠ R s❡❣✉❡ q✉❡ Z

  p−2

  |I (n)| ≤ |∇u | |∇u ||∇ψ||u − u|

  1 n n n B r

  • ε

  Z

  p−1

  ≤ M |∇u | |u − u|

  1 n n B r

  • ε

  k ≤ M

  n λ

  P♦r ❤✐♣ót❡s❡✱ ku ♣❛r❛ t♦❞♦ n ∈ N ❡ ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ M > 0✳ P♦r ✐ss♦✱

  ′ p

  → u (B )

  n r+ε

  ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r ❝♦♠ ❡①♣♦❡♥t❡s p ❡ p ❡ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ u ❡♠ L ✱ s❡❣✉❡ q✉❡

  ′ 1/p 1/p

  Z Z

  p p

  |I |∇u | |u − u|

  1 (n)| ≤ M 1 n n B r B r

  • ε +ε

  p−1

  ≤ M ku k |u − u|

  1 n n p,B r

  • ε

  λ p−1

  ≤ M M |u − u| r → 0,

  1 n p,B +ε

  q✉❛♥❞♦ n → ∞✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ Z

  p−1

  |I (n)| ≤ (λa(x) + 1)|u | |u − u| ψ

  2 n n N R

  Z

  p−

  1

  1 p−1 p p

  = (λa(x) + 1) |u | (λa(x) + 1) |u − u| ψ

  n n N R

p−

  1

  1 Z Z p p p p p ≤ (λa(x) + 1)|u | ψ (λa(x) + 1)|u − u| . n n B r B r

  • ε +ε

  ∈ R

  r+ε

  ❈♦♠♦ a é ❝♦♥tí♥✉❛ ♥♦ ❝♦♠♣❛❝t♦ B ✱ ❡①✐st❡ a t❛❧ q✉❡ 0 ≤ |λa(x) + 1| ≤ a ❡♠ B

  r+ε

  ✳ ❆ss✐♠✱ ❧❡♠❜r❛♥❞♦ q✉❡ 0 ≤ ψ ≤ 1✱ t❡♠♦s

  1

  1 p p p−1 p−1

  |I ku k |u − u| ≤ a |u − u| → 0,

  2 (n)| ≤ a n n p,B r M n p,B r

  • ε +ε

  λ

  • ≤ |u |

  n

  q✉❛♥❞♦ n → ∞✳ P❛r❛ t♦❞♦ n ∈ N t❡♠♦s u n ✳ ❯s❛♥❞♦ ❡ss❡ ❢❛t♦✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡

  q N

  ֒→ L (R ) ❍ö❧❞❡r ❡ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ✐♠❡rsã♦ E λ ✱ ♦❜t❡♠♦s ✉♠❛ ❝♦♥st❛♥t❡ C > 0 t❛❧ q✉❡

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✸✹ Z

  q−1

  |I (n)| ≤ |u | ψ|u − u|

  3 n n N R

q−

  1

  1 Z Z q q q q q

  ≤ |u | ψ |u − u|

  n n N R

  B r

  • ε

  q−1

  ≤ Cku k |u − u|

  

n n q,B r +ε

λ q−1

  ≤ CM |u − u| → 0

  

n q,B r

  • ε

  ∗

  I

  4 (n) = 0

  q✉❛♥❞♦ n → ∞✱ ✉♠❛ ✈❡③ q✉❡ p < q < p ✳ P❛r❛ ♠♦str❛r q✉❡ lim ✱ ❝♦♥s✐❞❡r❡

  n→∞

  f : E → R

  λ

  ❞❛❞❛ ♣♦r Z

  p−2 f (z) = ψ|∇u| ∇u · ∇z.

  N R λ

  P❛r❛ ❝❛❞❛ z ∈ E ✱ t❡♠✲s❡

  p−

  1

  Z

  p p p−1 p p |f (z)| ≤ |∇u| |∇z|ψ ≤ |∇u| |∇z| ≤ C kzk . u λ

  1 Z Z

  N N N R R R p−

  1 Z p p

  |∇u|

  u =

  ♦♥❞❡ C ✳ ■ss♦ ♠♦str❛ q✉❡ f ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡✱ ♣♦r s❡r ❧✐♥❡❛r✱ ♠♦str❛

  N R

  ⇀ u (n) = f (u) − f (u ) → 0

  n λ 4 n

  t❛♠❜é♠ q✉❡ f é ❝♦♥tí♥✉❛✳ ❈♦♠♦ u ❡♠ E s❡❣✉❡ q✉❡ I q✉❛♥❞♦ n → ∞✳ ❈♦♥❝❧✉í♠♦s ❞❡ (2.13) q✉❡✱ ♣❛r❛ t♦❞♦ r > 0✱ Z

  → 0, P n q✉❛♥❞♦ n → ∞.

  ✭✷✳✶✹✮

  B r

  ❆✜r♠❛♠♦s q✉❡

  

p N

  ∇u → ∇u

  n (B r )) ,

  ❡♠ (L ♣❛r❛ t♦❞♦ r > 0. ✭✷✳✶✺✮ ❉❡ ❢❛t♦✱ ❛♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✈❡❥❛ ❬✷✺❪✮

  p−2 p−2 p

  (|a| a − |b| b)(a − b) ≥ M |a − b|

  p N

  > 0 q✉❡ ✈❛❧❡ ♣❛r❛ t♦❞♦ a, b ∈ R ❡ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ M p ✱ ❝♦♠ a = ∇u n ✱ b = ∇u ❡ ❧❡♠❜r❛♥❞♦ (2.14)✱ s❡❣✉❡ q✉❡

  Z Z

  1

  

p

  0 ≤ lim |∇u − ∇u| ≤ lim P = 0,

  n n n→∞ n→∞

  M p

  

B r B r

n (x) − ∇u(x)| → 0 r

  ♦ q✉❡ ♠♦str❛ (2.15) ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ |∇u q✳t✳♣✳ ❡♠ B ✱ s❡❥❛ q✉❛❧

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✸✺ ❞✐❛❣♦♥❛❧ ♣❛r❛ ❝♦♥❝❧✉✐r q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱

  N ∇u (x) → ∇u(x) . n

  q✳t✳♣✳ ❡♠ R ❘❡st❛✲♥♦s ♠♦str❛r q✉❡

  ′

  ∂u ∂u

  n p−2 p−2 p N |∇u | ⇀ |∇u| (R ), 1 ≤ i ≤ N. n ❢r❛❝❛♠❡♥t❡ ❡♠ L

  ∂x ∂x

  i i ′

  ∂u n

  p N p−2

  ≡ |∇u | (R )

  P❛r❛ ✐ss♦✱ ❝♦♥s✐❞❡r❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s ❡♠ L ❞❛❞❛ ♣♦r f n n ♣❛r❛ ∂x

  i

  ❝❛❞❛ n ∈ N✳ ❊♥tã♦✱

  

p

  Z Z

  

p−

  

1

  ∂u

  p n p p−2 p p ′

  |f | = |∇u | ≤ |∇u | ≤ ku k ≤ M

  n n n n p λ

  N N R ∂x R i

  ′ ′ p N p N

  (R ) (R ) ❉❛í✱ ❝♦♠♦ L é r❡✢❡①✐✈♦✱ ❡①✐st❡ f ∈ L t❛❧ q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱

  ′

  ∂u(x)

  p N p−2

  f ⇀ f (R ) (x) → |∇u(x)|

  n n

  ❢r❛❝❛♠❡♥t❡ ❡♠ L ✳ P❡❧♦ q✉❡ ❥á ❢♦✐ ♣r♦✈❛❞♦✱ f q✳t✳♣✳ ∂x

  i

  ∂u

  N p−2

  ❡♠ R ✳ ❙❡❣✉❡ ❞❛í ❡ ❞♦ ▲❡♠❛ ✶✳✷✸ q✉❡ f ≡ |∇u| ✳ ■ss♦ t❡r♠✐♥❛ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ∂x

  i

  ❧❡♠❛✳ ) ⊂ E

  n λ c λ,q

  ▲❡♠❛ ✷✳✾ ❙❡❥❛♠ λ > 0 ❡ (u ✉♠❛ s❡q✉ê♥❝✐❛ (P S) ♣❛r❛ I ✳ ❊♥tã♦✱ ❛ ♠❡♥♦s ❞❡

  • ⇀ u

  )

  n λ

  ✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱ u ❢r❛❝❛♠❡♥t❡ ❡♠ E ❝♦♠ u s❡♥❞♦ ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ (S ✳

  λ,q ′ ′

  = u − u = c − I (u)

  

n n c λ,q λ,q

  ❆❧é♠ ❞✐ss♦✱ v é ✉♠❛ s❡q✉ê♥❝✐❛ (P S) ♣❛r❛ I ✱ ♦♥❞❡ c ✳ )

  n λ

  ❉❡♠♦♥str❛çã♦✿ P❡❧♦ ▲❡♠❛ 2.4 (i)✱ (u é ❧✐♠✐t❛❞❛ ❡♠ E ✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ u ⇀ u ,

  n ❢r❛❝❛♠❡♥t❡ ❡♠ E λ s N ∗

  → u u n (R ) , ❡♠ L ♣❛r❛ t♦❞♦ p ≤ s < p

  loc N

  u (x) → u(x) .

  n

  q✳t✳♣✳ ❡♠ R

  λ n ) c λ,q

  ❙❡❥❛ v ∈ E ✳ ❈♦♠♦ (u é ✉♠❛ s❡q✉ê♥❝✐❛ (P S) ♣❛r❛ I ✱ Z Z Z

  • ′ p−2 p−2 q−1

  hI (u ), vi = |∇u | ∇u · ∇v + (λa(x) + 1)|u | u v − (u ) v

  n n n n n λ,q n

  N N N R R R

  = o(1), ✭✷✳✶✻✮

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✸✻ q✉❛♥❞♦ n → ∞✳ ❱❛♠♦s ❝❛❧❝✉❧❛r ♦ ❧✐♠✐t❡ ❞❡ ❝❛❞❛ ♣❛r❝❡❧❛ ❡♠ ✭✷✳✶✻✮✳ P❡❧♦ ▲❡♠❛ 2.8 ❡

  

p N N

  (R ))

  λ

  ✉s❛♥❞♦ q✉❡ v ∈ E ❡✱ ♣♦rt❛♥t♦✱ ∇v ∈ (L ✱

  N

  Z Z

  X ∂u ∂v

  n p−2 p−2

  |∇u | ∇u · ∇v = |∇u | ·

  n n n N N

  

R R ∂x ∂x

i i i=1 N

  Z

  X ∂u ∂v

  n p−2

  ✭✷✳✶✼✮ = |∇u | ·

  n N

  R ∂x ∂x i i i=1

  Z

  p−2

  → |∇u| ∇u · ∇v

  N R p p−2 p−

  1

  ≡ (λa(x) + 1) |u | u

  

n n n

  q✉❛♥❞♦ n → ∞✳ ❆❣♦r❛✱ s❡❥❛ g ✳ ❊♥tã♦✱ Z Z

  ′ p p p

  |g | = (λa(x) + 1)|u | ≤ ku k ,

  n n n λ N N R R

  ′

  p

  ′ p N

  = n ) λ n ) (R ) ♦♥❞❡ p ✳ ❈♦♠♦ (u é ❧✐♠✐t❛❞❛ ❡♠ E ✱ s❡❣✉❡ q✉❡ (g é ❧✐♠✐t❛❞❛ ❡♠ L ✳ p − 1

  ′ ′ p N p N

  (R ) (R )

  ❙❡♥❞♦ L ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ r❡✢❡①✐✈♦✱ ❡①✐st❡ g ∈ L t❛❧ q✉❡✱ ❛ ♠❡♥♦s

  ′ p N

  ⇀ g (R ) (x) →

  n n

  ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ g ❢r❛❝❛♠❡♥t❡ ❡♠ L ✳ ❏á t❡♠♦s ❛ ❝♦♥✈❡r❣ê♥❝✐❛ g

  p p p−2 N p−2 p−

  1 p−

  1

  (λa(x) + 1) |u| u |u| u q✳t✳♣✳ ❡♠ R ✳ P❡❧♦ ▲❡♠❛ ✶✳✷✸✱ g ≡ (λa(x) + 1) ✳ ❆ss✐♠✱

  ′ ′ p N

  hϕ, g i → hϕ, gi (R )) ,

  n

  ♣❛r❛ t♦❞❛ ϕ ∈ (L ✭✷✳✶✽✮

  ′ ′ p N ′ p N

  (R )) (R ) ♦♥❞❡ (L ❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ❞✉❛❧ ❞❡ L ✳ P❡❧♦ ❚❡♦r❡♠❛ ❞❛ ❘❡♣r❡s❡♥t❛çã♦ ❞❡

  ′ p N ′ p N

  (R )) (R ) ❘✐❡s③ ✭✈❡❥❛ ❚❡♦r❡♠❛ ✹✳✶✶ ❡♠ ❬✾❪✮✱ ♣❛r❛ ❝❛❞❛ ϕ ∈ (L ❡①✐st❡ ✉♠❛ ú♥✐❝❛ w ∈ L t❛❧ q✉❡

  Z

  ′ p N hϕ, f i = f w, (R ).

  ♣❛r❛ t♦❞♦ f ∈ L

  N R

  ▲♦❣♦ ✭✷✳✶✽✮ é ❡q✉✐✈❛❧❡♥t❡ ❛ Z Z

  p N g w → gw, (R ). n

  ♣❛r❛ t♦❞♦ w ∈ L

  N N R R

  1 p p N

  v ∈ L (R )

  λ

  ❈♦♠♦ w ≡ (λa(x) + 1) ✱ ✉♠❛ ✈❡③ q✉❡ v ∈ E ✱ s❡❣✉❡ q✉❡ Z

  Z

  p p

  1

  1 p−2 p−2 p− p p− p

  1

  1

  (λa(x) + 1) |u | u (λa(x) + 1) v → (λa(x) + 1) |u| u(λa(x) + 1) v,

  n n N

  N R R

  ✐st♦ é✱ Z Z

  p−2 p−2

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✸✼ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ ♣❛r❛ ♠♦str❛r q✉❡

  Z Z

  • q−1 + q−1

  (u ) v → (u ) v,

  n

  ✭✷✳✷✵✮

  N N R R q−1

  • q

  N

  ≡ (u ) (R )

  n λ

  ❡s❝♦❧❤❡♠♦s g n ❡ ♦❜s❡r✈❛♠♦s q✉❡✱ ❞❡✈✐❞♦ ❛ ✐♠❡rsã♦ ❝♦♥tí♥✉❛ ❞❡ E ❡♠ L ✱

  ′ ′ q N + q−1 q N

  (g ) (R ) ) (R )

  n

  é ❧✐♠✐t❛❞❛ ❡♠ L ❡✱ ♣♦rt❛♥t♦✱ ❝♦♥✈❡r❣❡ ♣❛r❛ (u ❢r❛❝❛♠❡♥t❡ ❡♠ L ✳ ❉❡ ✭✷✳✶✼✮✱ ✭✷✳✶✾✮ ❡ ✭✷✳✷✵✮ ❡♠ ✭✷✳✶✻✮✱ s❡❣✉❡ q✉❡

  ′ hI (u), vi = 0, .

  λ

λ,q ♣❛r❛ t♦❞♦ v ∈ E

  (u) = 0 )

  λ

  ■ss♦ ♠♦str❛ q✉❡ I λ,q ❡✱ ♣♦rt❛♥t♦✱ u ∈ E é s♦❧✉çã♦ ❢r❛❝❛ ❞❡ (S ✳ ❆❣♦r❛ ♦❜s❡r✈❛✲

  λ,q

  ♠♦s q✉❡ I (v ) = I (u ) − I (u) + R

  

λ,q n λ,q n λ,q n

  ♦♥❞❡ Z

  1

  p p p

  R = I (u − u) − I (u ) + I (u) = (|∇(u − u)| − |∇u | + |∇u| )

  n λ,q n λ,q n λ,q n n N

  p R Z

  Z

  1 1 q

  • q + p p p q +

  − u| −|u | −u) −(u −(u (λa(x)+1)(|u +|u| + )− (u ) ) .

  n n n n N

  N

  p R q R )

  n λ

  ❆ ❧✐♠✐t❛çã♦ ❞❡ (u ❡♠ E ✱ ❛s ❝♦♥✈❡r❣ê♥❝✐❛s ♣♦♥t✉❛✐s ❞❛ Pr♦♣♦s✐çã♦ ✷✳✷ ❡ ❞♦ ▲❡♠❛ ✷✳✽ ♥♦s ♣❡r♠✐t❡♠ ❛♣❧✐❝❛r ♦ ▲❡♠❛ ❞❡ ❇ré③✐s✲▲✐❡❜ ✭▲❡♠❛ ✶✳✷✵ ❡ ❛ ❖❜s❡r✈❛çã♦ ✶✳✷✶✮ ♣❛r❛ ❝♦♥❝❧✉✐r

  → 0

  n

  q✉❡ R q✉❛♥❞♦ n → ∞✳ P♦rt❛♥t♦ I (v ) = I (u ) − I (u) + o(1)

  λ,q n λ,q n λ,q ′

  I (v ) = c−I (u) (v ) →

  n→∞ λ,q n λ,q n

  q✉❛♥❞♦ n → ∞✳ ❆ss✐♠ lim ✳ ❘❡st❛ ❛♣❡♥❛s ♠♦str❛r q✉❡ I λ,q

  ′

  (u), ϕi = 0

  λ

  q✉❛♥❞♦ n → ∞✳ P❛r❛ ✐ss♦✱ s❡❥❛ ϕ ∈ E q✉❛❧q✉❡r✳ ❯♠❛ ✈❡③ q✉❡ hI λ,q ✱ s♦♠❛♥❞♦

  ′ ′

  (u ), ϕi (v , ϕ)i ❡ s✉❜tr❛✐♥❞♦ hI n ♥❛ ❡①♣r❡ssã♦ ❞❡ hI n ♦❜t❡♠♦s

  λ,q λ,q ′ ′ ′

  hI (v ), ϕi = hI (u ), ϕi − hI (u), ϕi + I (n) + I (n) − I (n)

  n n

  5

  6

  7 λ,q λ,q λ,q

  ✭✷✳✷✶✮ = o(1) + I

  5 (n) + I

6 (n) − I

7 (n),

  ♦♥❞❡ Z

  p−2 p−2 p−2

  I (n) = (|∇v | ∇v + |∇u| ∇u − |∇u | ∇u ) · ∇ϕ,

  5 n n n n N R

  Z

  p−2 p−2 p−2

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✸✽ ❡

  Z

  q−2 q−2 q−2 I (n) = (|v | v + |u| u − |u | u )ϕ. 7 n n n n

  N R p N N

  ) := (∇v ) ⊂ (L (R ))

  n λ n n

  ❙❡♥❞♦ (v ❧✐♠✐t❛❞❛ ❡♠ E ✱ t❡♠♦s q✉❡ η s❛t✐s❢❛③ ❛s ❤✐♣ót❡s❡s ❞♦ ▲❡♠❛ 1.22 ❝♦♠ K = N✳ ❆ss✐♠✱ ❛♣❧✐❝❛♥❞♦ ❡ss❡ ❧❡♠❛ ❝♦♠ ω = ∇u✱ p = s ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✱ ♦❜t❡♠♦s

  p−

  1 Z p p p−2 p−2 p−2 p−

  1

  |I | ∇v ∇u − |∇u | ∇u kϕk

  

5 (n)| ≤ |∇v n n + |∇u| n n λ

N R ≤ o(1)kϕk .

  λ

  (n)

  7

  ❉❛ ♠❡s♠❛ ♠❛♥❡✐r❛✱ é ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ ❡ss❛ ❡st✐♠❛t✐✈❛ t❛♠❜é♠ ✈❛❧❡ ♣❛r❛ I ✳ ◆♦ (n)

  ❝❛s♦ ❞❡ I

  6 ✱ ❡ss❛ ❡st✐♠❛t✐✈❛ é ♦❜t✐❞❛ ❛♣❧✐❝❛♥❞♦✲s❡ ♥♦✈❛♠❡♥t❡ ♦ ▲❡♠❛ 1.22 ♣❛r❛ K = 1✱

  1

  1 p p

  s = p n = (λa(x) + 1) v n u n ) λ ✱ η ❡ ω = (λa(x) + 1) ✱ ✉♠❛ ✈❡③ q✉❡ (v é ❧✐♠✐t❛❞❛ ❡♠ E ❡ p

  ′

  |η | ≤ kv k =

  n p n λ

  ✳ P♦r ✐ss♦ ❡ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r ❝♦♠ ❡①♣♦❡♥t❡s p ❡ p ✱ p − 1 Z

  1

  1 p−2 p−2 p−2 p′ p

  |I | | |ϕ|

  6 (n)| ≤ (λa(x) + 1) |v n v n + |u| u − |u n u (λa(x) + 1) N R p−

  1 Z p p p−2 p−2 p−2 p−

  1

  ≤ (λa(x) + 1) |v | v + |u| u − |u | u · kϕk

  n n n n N R ≤ o(1) · kϕk .

  λ

  ❊♥tã♦✱ ❞❡ (2.21) s❡❣✉❡ q✉❡

  ′

  |hI (v ), ϕ)i| ≤ o(1)kϕk , .

  n λ ♣❛r❛ t♦❞❛ ϕ ∈ E λ λ,q ′

  hI (v ), ϕi

  n λ,q ′

  ′

  (v )k = sup → 0

  n E

  ▲♦❣♦ kI λ,q q✉❛♥❞♦ n → ∞✳ ■ss♦ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞♦

  λ

  kϕk

  \{0} λ ϕ∈E λ

  ❧❡♠❛✳ ❆❣♦r❛ t❡♠♦s ❢❡rr❛♠❡♥t❛s s✉✜❝✐❡♥t❡s ♣❛r❛ ♣r♦✈❛r ❛ Pr♦♣♦s✐çã♦ 2.3✳

  > 0 ❉❡♠♦♥str❛çã♦ ❞❛ Pr♦♣♦s✐çã♦ ✷✳✸✿ ❙❡❥❛ C ❞❛❞♦ ♣❡❧♦ ▲❡♠❛ 2.5✳ ❙❡❥❛ t❛♠✲ pq

  > 0 , R > 0

  1 ε ε

  ❜é♠ ε > 0 t❛❧ q✉❡ 2ε < C ✳ ❉❛❞♦ C ✱ s❡❥❛♠ Λ ❞❛❞♦s ♣❡❧♦ ▲❡♠❛ 2.6✳ p − q = Λ ) ⊂ E

  ❚♦♠❡ Λ ε ✳ ❙❡❥❛ (u n λ ✉♠❛ s❡q✉ê♥❝✐❛ (P S) c ♣❛r❛ I λ,q ❝♦♠ λ ≥ Λ ε ❡ c ≤ C

  1 ✳

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✸✾

  n ) λ

  P❡❧♦ ▲❡♠❛ 2.4 (i)✱ (u é ❧✐♠✐t❛❞❛ ❡♠ E ✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ u ⇀ u ,

  n λ

  ❢r❛❝❛♠❡♥t❡ ❡♠ E

  s N ∗

  u → u (R ) ,

  n

  ❡♠ L loc ♣❛r❛ t♦❞♦ p ≤ s < p

  N u (x) → u(x) . n

  q✳t✳♣✳ ❡♠ R

  ′ ′

  = u − u = c − I (u)

  n n c λ,q λ,q

  P❡❧♦ ▲❡♠❛ 2.9✱ v é ✉♠❛ s❡q✉ê♥❝✐❛ (P S) ♣❛r❛ I ❝♦♠ c ✳

  ′ ′

  = 0 ≥ C > 0

  ❆✜r♠❛♠♦s q✉❡ c ✳ ❉❡ ❢❛t♦✱ ❝❛s♦ ❝♦♥trár✐♦✱ ♣❡❧♦ ▲❡♠❛ ✷✳✺✱ c ✳ P♦r t❡r♠♦s

  q N

  → 0 v n (R ) ❡♠ L ✱ s❡❣✉❡ ❞♦s ▲❡♠❛s 2.4 (ii) ❡ 2.6 q✉❡

  loc

  pq pq

  ′ q +

  ≤ c |v | C = lim

  n q n→∞

  q − p q − p !

  Z Z

  • q + q

  = lim sup |v | + |v |

  n n c n→∞ B B

  Rε Rε

  Z Z

  q q

  ≤ lim sup |v | + lim sup |v |

  n n c n→∞ n→∞ B B

  Rε Rε

  ≤ ε.

  ′

  = 0 ■ss♦ ♥♦s ❞á ✉♠❛ ❝♦♥tr❛❞✐çã♦ ❝♦♠ ❛ ❡s❝♦❧❤❛ ❞❡ ε✳ ❊♥tã♦ c ✳ P❡❧♦ ▲❡♠❛ 2.4 (ii)✱ lim kv k = 0 → u

  n λ n λ

  ✱ ✐st♦ é✱ u ❡♠ E ✱ ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦✳

  n→∞ ✷✳✸ ❙♦❧✉çõ❡s ♣♦s✐t✐✈❛s ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛

  ◆♦ss♦ ♦❜❥❡t✐✈♦ ♥❡st❛ s❡çã♦ é ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ❆ q✉❡ ♥♦s ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ )

  ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ♣❛r❛ (S λ,q q✉❛♥❞♦ λ é s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳ ◆♦s ❛♣♦✐❛♠♦s ❡♠ ❛❧❣✉♥s ❧❡♠❛s✳ ◆♦ ❝❛s♦ ❞♦s ▲❡♠❛s 2.10 ❡ ✷✳✶✶ ❛❜❛✐①♦✱ ❛❞❛♣t❛♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✹✳✶ ❡♠ ❬✸✸❪ ✭✈❡❥❛ t❛♠❜é♠ ♦ ▲❡♠❛ ✶✳✸ ❡♠ ❬✷✵❪✮✳ ❈♦♥s✐❞❡r❛♠♦s

  • S

  = {u ∈ E : kuk = 1} =

  E λ λ E λ λ λ

  ❛ ❡s❢❡r❛ ✉♥✐tár✐❛ ❡♠ E ✱ ✐st♦ é✱ S ❡ ❞❡✜♥✐♠♦s S E

  λ

  • {w ∈ S : w 6= 0}

  E

  ✳

  • 6= 0

  λ

  ▲❡♠❛ ✷✳✶✵ ❙❡❥❛ u ∈ E ❝♦♠ u

  ✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ t(u) > 0 t❛❧ q✉❡ t(u)u ∈ N

  (tu)

  λ,q ✳ ❖ ♠á①✐♠♦ ❞❡ I λ,q ♣❛r❛ t ≥ 0 é ❛t✐♥❣✐❞♦ ❡♠ t = t(u)✳

  ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✹✵ 6= 0

  • p
  • ′ ′ p−1 q−1 q

  λ λ,q (tu)

  ❉❡♠♦♥str❛çã♦✿ ❉❛❞♦ u ∈ E ❝♦♠ u ✱ ❝♦♥s✐❞❡r❡ g(t) = I ♣❛r❛ t ≥ 0✳ ❊♥tã♦

  g (t) = hI (tu), ui = t kuk − t |u | , λ,q λ q ♣❛r❛ t ≥ 0.

  • 1

  | 6= 0

  q

  P♦r ❤✐♣ót❡s❡✱ |u ✳ ❙❡♥❞♦ 1 ≤ p − 1 < q − 1✱ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ✐♠♣❧✐❝❛ q✉❡ ♦ ú♥✐❝♦

  p q−p

  kuk

  ′ λ q

  ♣♦♥t♦ t 6= 0 ♥♦ q✉❛❧ g s❡ ❛♥✉❧❛ é t(u) = ✱ q✉❡ é ♦ ♠á①✐♠♦ ❞❡ g ♣❛r❛ t ≥ 0✳ |u | q

  • ❉❛í✱

  1

  ′ ′ ′ 0 = g (t(u)) = hI (t(u)u), ui = hI (t(u)u), t(u)ui.

  

λ,q λ,q

  t(u)

  λ,q

  ❆ss✐♠✱ t(u)u ∈ N ❡ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞♦ ❧❡♠❛✳

  • → N

  λ,q E λ ✱ s❡❥❛ t(u) ♦❜t✐❞♦ ❞♦ ▲❡♠❛ ✷✳✶✵✳ ❆ ❢✉♥çã♦ ϕ : S E λ

  • ▲❡♠❛ ✷✳✶✶ P❛r❛ ❝❛❞❛ u ∈ S

  ❞❛❞❛ ♣♦r ϕ(u) = t(u)u é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✳ ❉❡♠♦♥str❛çã♦✿ P❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ ❛ ❢✉♥çã♦ ϕ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ P❛r❛ ♣r♦✈❛r ❛ ❝♦♥t✐✲

  ∈ S → u

  n ) n λ

  ♥✉✐❞❛❞❡ ❞❡ ϕ✱ s❡❥❛♠ u ❡ (u ❡♠ S t❛❧ q✉❡ u ❡♠ E ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱

  E λ E λ 1,p N 1,p N q N

  (R ) (R ) (R ) → u

  n

  ❡♠ W ✳ P❡❧❛ ✐♠❡rsã♦ ❝♦♥tí♥✉❛ ❞❡ W ❡♠ L ✱ u ❢♦rt❡♠❡♥t❡ ❡♠

  q N

  L (R ) ✳ P❡❧♦ ❚❡♦r❡♠❛ ✹✳✾ ❡♠ ❬✾❪✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱

  N

  u (x) → u (x)

  n

  q✳t✳♣✳ ❡♠ R ❡

  • + N

    u (x) ≤ |u n (x)| ≤ h(x) .

  n

  q✳t✳♣✳ ❡♠ R u (x) + |u (x)|

  n n

  • 2
  •   (x) = ❈♦♠♦ u n ✱ s❡❣✉❡ q✉❡

    • N + u (x) → u (x) .

      

    n q✳t✳♣✳ ❡♠ R

      P❡❧♦ ❚✳❈✳❉✳▲✳✱

      |u − u | → 0

      q

    n q✉❛♥❞♦ n → ∞.

      | → |u |

      q q n ) → t(u )

      ▲♦❣♦ |u ❡ ♣♦r ❝♦♥s❡❣✉✐♥t❡ t(u ✳ ❈♦♠ ✐ss♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ϕ é

      n

      u

    • → S

      λ,q

      ❝♦♥tí♥✉❛✳ ❈♦♥s✐❞❡r❛♠♦s ❛❣♦r❛ ❛ ❢✉♥çã♦ θ : N ❞❛❞❛ ♣♦r θ(u) = ✳ ❊♥tã♦✱ θ

      E λ

      kuk

      λ p q

    • +

      | 6= 0

      λ,q

      = kuk λ = 1 ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♣♦✐s s❡ u ∈ N ❡ t❡♠✲s❡ kθ(u)k ✳ ❆❧é♠ ✱ ❡♥tã♦ |u

      q λ

    • E λ

      ❞✐ss♦✱ θ é ❝♦♥tí♥✉❛✳ ❆✜r♠❛♠♦s q✉❡ θ é ❛ ✐♥✈❡rs❛ ❞❛ ϕ✳ ❈♦♠ ❡❢❡✐t♦✱ s❡❥❛ u ∈ S ✳ ❊♥tã♦

      kuk = 1

      λ

      ✳ P❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ t(u) > 0 ❡✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ θ✱ t❡♠♦s ϕ(u) t(u)u u

      ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✹✶

      λ,q

      P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ u ∈ N ✱ ❡♥tã♦ u ϕ(θ(u)) = t(θ(u))θ(u) = t(θ(u)) . kuk

      λ

      ❙❡❣✉❡ ❞❛í q✉❡✱ ♣❛r❛ ♠♦str❛r ♥♦ss❛ ❛✜r♠❛çã♦ ❡ ♣♦r ❝♦♥s❡❣✉✐♥t❡ ♦ ❧❡♠❛✱ r❡st❛✲♥♦s ♠♦str❛r

      p

    • + q

      | = |u q✉❡ t(θ(u)) = kuk λ ✳ ❯s❛♥❞♦ q✉❡ kuk ✱ ❛ ❡①♣r❡ssã♦ ❞❡ t(u) ♥♦ ❧❡♠❛ ❛♥t❡r✐♦r ♥♦s

      λ q

      ❢♦r♥❡❝❡

      1

      1 q q q−p q−p

      kuk kuk u

      λ λ

      t(θ(u)) = t = = = kuk ,

      q p λ

    • kuk |u | q kuk

      λ λ

      ❝♦♠♦ q✉❡rí❛♠♦s✳ ❉❡✜♥✐♠♦s ♦s ♥í✈❡✐s ♠✐♥✐♠❛①

      = inf sup I (tu) c = inf sup I (γ(t)),

      λ λ,q λ λ,q

      bc ❡

      u∈Eλ γ∈Γ t≥0 t∈[0,1] u 6=0

    • ♦♥❞❡ Γ = {γ ∈ C([0, 1], E λ ) : γ(0) = 0; I λ,q (γ(1)) < 0}.

      (0) = 0 I (tu) = −∞ ❖❜s❡r✈❛♠♦s q✉❡✱ ❝♦♠♦ I λ,q ❡ lim λ,q ♣❛r❛ t♦❞❛ u 6≡ 0 ❡♠ E λ ✱ t❡♠♦s

      t→∞

      ∈ R

      λ

      ❜❡♠ ❞❡✜♥✐❞♦✳ bc

      λ λ > 0

      ▲❡♠❛ ✷✳✶✷ c ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ❡ c ✳

      6= 0 |

      λ q λ

      ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ u ∈ E ❝♦♠ u ✳ ❈♦♠♦ |u ❡ kuk sã♦ ✈❛❧♦r❡s ♥ã♦ ♥✉❧♦s ❡ p < q ✱ s❡❣✉❡ ❞❛ ❡①♣r❡ssã♦

      p q

      t t

      p +

      I (tu) = kuk − |u |

      

    λ,q q

    λ

      p q > 0

      (t u) < 0

      λ,q

      q✉❡ ❡①✐st❡ t s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ q✉❡ ❞❡♣❡♥❞❡ ❞❡ u t❛❧ q✉❡ I ✳ ❉❡✜♥✐♠♦s γ : [0, 1] → E (t) = tt u ∈ C([0, 1], E ) (0) = 0 (γ (1)) =

      λ λ λ,q

      ❞❛❞❛ ♣♦r γ ✳ ❊♥tã♦ γ ✱ γ ❡ I ∈ Γ

      I (t u) < 0

      

    λ,q ✱ ✐st♦ é✱ γ ✳ ▲♦❣♦ Γ 6= ∅✳ ❉❡✈✐❞♦ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ✐♠❡rsã♦ ❞❡ E λ ❡♠

    q N

      L (R )

      λ

      ✱ ❡①✐st❡ C > 0 t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ u ∈ E ✱

      q

    • q q |u | ≤ |u| ≤ Ckuk .

      q q λ

      ❉❛í✱

      1

      1

      1 C

      p p q

    • q I (u) = kuk − |u | ≥ kuk − kuk .

      λ,q λ q λ λ

      p q p q

      ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✹✷

      1 q−p

      q = ρ = > 0

      λ

      P❛r❛ kuk ✱ 2pC

      p

    1 C ρ

      p q I (u) ≥ ρ − ρ = := α > 0. λ,q

      ✭✷✳✷✷✮ p q 2p ❆❧é♠ ❞✐ss♦✱

      1 q−p

    1 C

      q

      p q

      kuk − kuk ≥ 0 ≤ ρ I λ,q (u) ≥ λ 1 := . s❡♠♣r❡ q✉❡ kuk ✭✷✳✷✸✮

      λ λ

      p q pC

      λ,q (γ(1)) < 0 λ > ρ

      ❙❡❥❛ γ ∈ Γ✳ ❊♥tã♦ I ✳ P♦r ✭✷✳✷✸✮✱ kγ(1)k ✳ P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ γ✱ ❡①✐st❡ t ∈ [0, 1] = ρ

      λ

      t❛❧ q✉❡ kγ(t)k ✳ P♦rt❛♥t♦ sup I λ,q (γ(t)) ≥ I λ,q (γ(t)) ≥ inf I λ,q (u) ≥ α, ♣❛r❛ t♦❞♦ γ ∈ Γ.

      kuk

    λ =ρ

    t∈[0,1]

      ❉❛í c λ ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ❡ c = inf sup I (γ(t)) ≥ α > 0.

      λ λ,q γ∈Γ t∈[0,1]

      ■ss♦ ♠♦str❛ ♦ ❧❡♠❛✳ = c = inf I (u)

      λ λ λ,q λ,q

      ▲❡♠❛ ✷✳✶✸ P❛r❛ t♦❞♦ λ > 0✱ ✈❛❧❡ c ≤ bc ✳

      u∈N λ,q

      ∈ Γ ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ γ ❞❡✜♥✐❞❛ ♥♦ ✐♥í❝✐♦ ❞❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✷✳✶✷✳ P❡❧❛

      λ

      ❞❡✜♥✐çã♦ ❞❡ γ ❡ c ✱ c ≤ sup I (γ (t)) = sup I (tt u) = sup I (tu) ≤ sup I (tu).

      λ λ,q λ,q λ,q λ,q t≥0 t∈[0,1] t∈[0,1] t∈[0,t ]

    • t≥0

      6= 0

      λ sup

      I λ,q (tu) : u ∈ E λ ▲♦❣♦✱ c é ✉♠❛ ❝♦t❛ ✐♥❢❡r✐♦r ❞♦ ❝♦♥❥✉♥t♦ ❝♦♠ u ✳ P♦rt❛♥t♦

    • c λ λ λ = c λ,q λ,q

      ≤ bc ✳ P❛r❛ ♠♦str❛r q✉❡ bc ✱ s❡❥❛ z ∈ N ✳ P❡❧♦ ▲❡♠❛ 2.11✱ ❡①✐st❡ u ∈ S t❛❧

      E λ λ

      q✉❡ z = t(u)u✳ P❡❧♦ ▲❡♠❛ 2.10 ❡ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ bc ✱ I λ,q (z) = I λ,q (t(u)u) = sup I λ,q λ .

      (tu) ≥ bc

      t≥0

      ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✹✸ }

      λ λ,q (z) : z ∈ N λ,q λ,q λ

      ▲♦❣♦ bc é ✉♠❛ ❝♦t❛ ✐♥❢❡r✐♦r ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ {I ❡✱ ♣♦rt❛♥t♦✱ c ≥ bc ✳

    • 6= 0

      λ

      λ,q

      P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ t♦❞♦ u ∈ E ❝♦♠ u

      ✱ t❡♠✲s❡ t(u)u ∈ N ❡ ✈❛❧❡ sup I (tu) = I (t(u)u) ≥ c .

      λ,q λ,q λ,q t≥0

      ≤ c

      λ λ,q

      ❆ss✐♠✱ bc ✱ ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦

      ∗

      ) > 0

      q

      ▲❡♠❛ ✷✳✶✹ ❙✉♣♦♥❤❛ q✉❡ q ∈ (p, p ❡ λ > 0✳ ❊♥tã♦ ❡①✐st❡ r t❛❧ q✉❡ kuk ≥ r ,

      

    λ q

      ✭✷✳✷✹✮ > 0

      ♣❛r❛ t♦❞♦ u ∈ N λ,q ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ c λ,q ✳

      q N

      (R ) ❉❡♠♦♥str❛çã♦✿ P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ✐♠❡rsã♦ ❞❡ E λ ❡♠ L ✱ ❡①✐st❡ C > 0 t❛❧ q✉❡✱

      λ,q

      ♣❛r❛ t♦❞♦ u ∈ N ✱

      p q q + q

      kuk = |u | ≤ |u| ≤ Ckuk

      

    q q

    λ λ

      1 q−p −1 p−q

      ≥ C

      q = C > 0

      ❡ ❛ss✐♠✱ kuk ✳ ❉❡✜♥✐♥❞♦ r ❝♦♥❝❧✉í♠♦s ✭✷✳✷✹✮✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱

      λ

      1

      1

      1

      1

      p

    p

      (u) = − kuk ≥ − r

      λ,q λ,q

      ❝♦♠♦ I q ♣❛r❛ t♦❞♦ u ∈ N ✱ s❡❣✉❡ q✉❡

      λ

      p q p q

      1

      1

      p

      c = inf I (u) ≥ − r > 0.

      λ,q λ,q q u∈N λ,q

      p q

      ∗

      ) ▲❡♠❛ ✷✳✶✺ P❛r❛ q ∈ (p, p ❡ λ > 0 ✈❛❧❡ c ≤ m = inf J (u).

      λ,q q,Ω q,Ω u∈M q, Ω

      N N

      → R \ Ω

      q,Ω

      ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ u ∈ M ✳ ❉❡✜♥❛ ˜u : R ♣♦r ˜u = u ❡♠ Ω ❡ u = 0 ❡♠ R ✳

      1,p N N

      (R ) P❡❧♦ ❚❡♦r❡♠❛ ✾✳✶✽ ❡♠ ❬✾❪✱ ˜u ∈ W ✳ ❈♦♠♦ ˜u(x)a(x) = 0 ❡♠ R ✱ s❡❣✉❡ q✉❡

      Z Z Z Z

      p p + p + p q q (|∇˜ u| + (λa(x) + 1)|˜ u| ) = (|∇u| + |u| ) = (u ) = (˜ u ) .

      N N R

      R Ω Ω

      (˜ u) = J (u)

      λ,q λ,q q,Ω

      ▲♦❣♦ ❛ ❡①t❡♥sã♦ ˜u ∈ N ✳ ❯s❛♥❞♦ ♥♦✈❛♠❡♥t❡ q✉❡ a = 0 ❡♠ Ω ✈❡♠♦s q✉❡ I ✳ ❆ss✐♠✱

      {J } ⊂ {I }

      

    q,Ω (u) : u ∈ M q,Ω λ,q (v) : v ∈ N λ,q

      ≤ m ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ c λ,q q,Ω ✳

      ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✹✹ > 0 λ,q

      λ

      ▲❡♠❛ ✷✳✶✻ ❊①✐st❡ Λ t❛❧ q✉❡ I s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ♥♦ ♥í✈❡❧ c ♣❛r❛ t♦❞♦ λ ≥ Λ ✳

      ≤ m

      λ,q q,Ω

      ❉❡♠♦♥str❛çã♦✿ P❡❧♦s ▲❡♠❛s ✷✳✶✹ ❡ ✷✳✶✺✱ 0 < c ✱ ♣❛r❛ t♦❞♦ λ > 0✳ ❉❛í ❡ ❞♦ ▲❡♠❛ ✷✳✶✸✱ c ≤ m

      λ q,Ω ♣❛r❛ t♦❞♦ λ > 0.

      = m > 0

      1 q,Ω λ,q

      ❆♣❧✐❝❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ ✷✳✸ ❝♦♠ C ✱ ♦❜t❡♠♦s Λ t❛❧ q✉❡ I s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦

      λ

      ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ♥♦ ♥í✈❡❧ c ♣❛r❛ t♦❞♦ λ ≥ Λ ✳ = {0, 1}

      λ λ,q

      ❆♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✷✳✾ ❡♠ ❬✸✸❪ ❝♦♠ X = E ✱ ϕ = I ✱ M = [0, 1]✱ M ✱ Γ = {γ : {0, 1} → E : γ (0) = 0, I (γ (1)) < 0}

      λ λ,q

      ♦❜t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

      1

      (X, R) ❚❡♦r❡♠❛ ✷✳✶✼ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡ I ∈ C t❛❧ q✉❡ I(0) = 0 ❡

      Γ = {γ ∈ C([0, 1], X) : γ(0) = 0 ❡ I(γ(1)) < 0} 6= ∅✳ sup I(γ(t))

      ❙❡❥❛ c := inf ✳ ❙❡ c > 0 ❡ s❡ I s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ♥♦ ♥í✈❡❧ c

      γ∈Γ t∈[0,1]

      ❡♥tã♦ c é ✉♠ ✈❛❧♦r ❝rít✐❝♦ ❞❡ I✳ → R

      P❛r❛ λ > 0✱ ❝♦♥s✐❞❡r❛♠♦s ψ : E λ ❞❛❞❛ ♣♦r

      p

    • ′ q ψ(u) = hI (u), ui = kuk − |u | .

      λ,q λ q

      = {u ∈ E : ψ(u) = 0}

      λ,q λ λ,q λ,q N λ,q

      ❊♥tã♦ N ✳ ❉❡♥♦t❛r❡♠♦s ♦ ❢✉♥❝✐♦♥❛❧ I r❡str✐t♦ ❛ N ♣♦r I ✳

      1 λ,q

      ❖ ❧❡♠❛ ❛ s❡❣✉✐r ♠♦str❛ q✉❡ N é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❝❧❛ss❡ C ✭✈❡❥❛ ▲❡♠❛ ✶✳✺ ❡♠ ❬✷✵❪✮✳

      λ,q

      ▲❡♠❛ ✷✳✶✽ ❙✉♣♦♥❤❛ q✉❡ u ∈ N ✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ δ > 0 t❛❧ q✉❡

      ′

      hψ (u), ui < −δ.

      1

      ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ N λ,q é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❝❧❛ss❡ C ✳ > 0 ≥ r

      ❉❡♠♦♥str❛çã♦✿ ❉❛❞♦ u ∈ N λ,q ✱ ♣❡❧♦ ▲❡♠❛ ✷✳✶✹✱ ❡①✐st❡ r q t❛❧ q✉❡ kuk λ q ✳ ❙❡❣✉❡ ❞❛í ❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ψ q✉❡

      p p p p ′ + q

      hψ (u), ui = pkuk − q|u | = pkuk − qkuk = −(q − p)kuk ≤ −(q − p)r

      q

      ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✹✺

      q

      ❉❡✜♥✐♥❞♦ δ := (q − p)r ❝♦♥❝❧✉í♠♦s q✉❡

      ′ hψ (u), ui < −δ.

      ✭✷✳✷✺✮

      1

      ∈ C (E , R)

      λ,q λ

      ❉❛ ♠❡s♠❛ ♠❛♥❡✐r❛ q✉❡ s❡ ❞❡♠♦♥str❛ q✉❡ I é ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ ψ ∈

      1 ′

      C (E , R) (u) 6≡ 0

      λ

      ❡✱ ❛❧é♠ ❞✐ss♦✱ ♣♦r ✭✷✳✷✺✮✱ ψ ✳ ❉❛í✱ ❛r❣✉♠❡♥t❛♥❞♦ ❝♦♠♦ ♥♦ ▲❡♠❛ ✶✳✺ ❡♠

      1

      ❬✷✵❪✱ ❝♦♥❝❧✉í♠♦s q✉❡ N λ,q é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❝❧❛ss❡ C ✳

      λ,q λ,q λ,q

      ▲❡♠❛ ✷✳✶✾ ❙❡❥❛ λ > 0✳ ❙❡ u ∈ N é ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ I r❡str✐t♦ ❛ N ✱ ❡♥tã♦ u é

      λ,q

      ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ I ✳ ❉❡♠♦♥str❛çã♦✿ ❙❡ u ∈ N λ,q é ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ♥ã♦ tr✐✈✐❛❧ ❞❡ I λ,q rstr✐t♦ ❛ N λ,q ✱ ❡♥tã♦

      ′

      ∈ R I (u) = 0 λ,q λ,q

      N ✳ ❆♣❧✐❝❛♥❞♦ ♦ ▲❡♠❛ ✶✳✾ ❝♦♠ V = N ❡ ϕ = I ✱ ♦❜t❡♠♦s t t❛❧ q✉❡ λ,q

      ′ ′ hI (u), vi = t hψ (u), vi, .

      ♣❛r❛ t♦❞♦ v ∈ E λ ✭✷✳✷✻✮

      λ,q

      ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ v = u

      ′ ′ 0 = ψ(u) = hI (u), ui = t hψ (u), ui. λ,q ′

      (u), ui 6= 0 = 0 P❡❧♦ ▲❡♠❛ ✷✳✶✽✱ hψ ✳ ▲♦❣♦✱ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ❞✐③ q✉❡ t ❡✱ ♣♦r ✭✷✳✷✻✮✱

      ′

      I (u) = 0 λ,q λ ✳ ❆ss✐♠✱ u é ♣♦♥t♦ ❝rít✐❝♦ ❞❡ I s♦❜r❡ E ✳ ❖ ❧❡♠❛ ❡stá ❞❡♠♦♥str❛❞♦✳

      λ,q

      ◆❡ss❡ ♣♦♥t♦✱ é ♣♦ssí✈❡❧ ❞❡♠♦♥str❛r ♦ ❚❡♦r❡♠❛ ❆✳ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❆✿ P❡❧♦ ▲❡♠❛ ✷✳✶✻✱ I λ,q s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ P❛❧❛✐s✲❙♠❛❧❡

      λ λ,q

      ♥♦ ♥í✈❡❧ c ♣❛r❛ λ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳ ❉❛í ❡ ❞♦ ▲❡♠❛ ✷✳✶✷✱ s❡❣✉❡ q✉❡ I = I s❛t✐s❢❛③

      λ λ λ,q

      ❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛ ✷✳✶✼ ❝♦♠ X = E ✳ P♦rt❛♥t♦ c é ✈❛❧♦r ❝rít✐❝♦ ❞❡ I ✱ ✐st♦ é✱

      ′

      (u) = c (u) = 0

      λ λ,q λ

      ❡①✐st❡ u ∈ E t❛❧ q✉❡ I ❡ I λ,q ✱ ♣❛r❛ λ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳ ▲♦❣♦ u ∈ N (u) = c > 0 (0) = 0

      λ,q ✳ P❡❧♦ ▲❡♠❛ ✷✳✶✷✱ I λ,q λ ❡✱ ♣♦rt❛♥t♦✱ u 6= 0 ❥á q✉❡ I λ,q ✳ ❙❡❣✉❡ ❞♦

      ≤ I ≤ c

      

    λ,q λ,q (u) = c λ λ,q λ,q (u) = c λ,q λ,q

      ▲❡♠❛ ✷✳✶✸ q✉❡ c ✱ ♦✉ s❡❥❛✱ I ✳ ❈♦♠♦ N ❝♦♥té♠ t♦❞❛s

      ) )

      ❛s s♦❧✉çõ❡s ❞❡ (S ✱ s❡❣✉❡ q✉❡ u é s♦❧✉çã♦ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ❞❡ (S ✳ ❆❧é♠ ❞✐ss♦✱

      λ,q λ,q

      Z

      p ′ − p−2 − p−2 − −

      0 = hI (u), u i = |∇u| ∇u · ∇u + (λa(x) + 1)|u| u · u = ku k

      λ,q λ

      N R −

    • ≥ 0

      = 0 ❞♦♥❞❡ s❡❣✉❡ q✉❡ u ✳ P♦rt❛♥t♦ u = u

      ✳ P❡❧♦ Pr✐♥❝í♣✐♦ ❞♦ ▼á①✐♠♦ ❋♦rt❡ ✭✈❡❥❛

      N

      ❚❡♦r❡♠❛ ✸✳✺ ❡ ♦❜s❡r✈❛çã♦ ♣♦st❡r✐♦r ❡♠ ❬✶✾❪✮✱ u > 0 ❡♠ R ❡✱ ❞❡ss❡ ♠♦❞♦✱ u t❛♠❜é♠ é

      ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✹✻ N

      λ,q ) λ,q

      s♦❧✉çã♦ ♣❛r❛ (S ✳ ❙❡❣✉❡ ❞❡ss❡s ❢❛t♦s q✉❡ u ∈ e ❡ c = I (u) = e I .

      

    λ,q λ,q λ,q λ,q

      (u) ≥ ec ∈ e N (u ) < e I (u)

      λ,q λ,q 1 λ,q λ,q 1 λ,q

      ❙✉♣♦♥❞♦ ♣♦r ❝♦♥tr❛❞✐çã♦ q✉❡ eI (u) > ec ✱ ❡①✐st❡ u t❛❧ q✉❡ eI ✳ ❉❛í c = I (u) = e

      I (u) > e I (u ) = e I (|u |) = I (|u |, )

      λ,q λ,q λ,q λ,q 1 λ,q 1 λ,q

      1

      | ∈ N

      1 λ,q λ,q λ,q

      ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ✈✐st♦ q✉❡ |u ✳ ❊♥tã♦✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ eI (u) = ec ✳ ■ss♦ )

      ♠♦str❛ q✉❡ u é s♦❧✉çã♦ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ❞❡ (S λ,q ✱ ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛✳

      ✷✳✹ ❈♦♥❝❡♥tr❛çã♦ ❞❛s s♦❧✉çõ❡s

      ❆❣♦r❛ ♥♦s ❝♦♥❝❡♥tr❛♠♦s ❡♠ ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ❇✳ ❈♦♠❡ç❛♠♦s ❝♦♠ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ✭✈❡❥❛ ▲❡♠❛ ✸✳✶✵ ❡♠ ❬✶✽❪✮✳

      ≥ 1 ) ⊂ E n → ∞ k n ≤ M ▲❡♠❛ ✷✳✷✵ ❙❡❥❛ M > 0✱ λ n ❡ (u n λ t❛✐s q✉❡ λ n ❡ ku n λ ✳

      1,p

      (Ω)

      n ⇀ u

      ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ u ∈ W t❛❧ q✉❡✱ ❛ ♠❡♥♦s ❞❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱ u

      1,p N s N + +

      ⇀ u (R ) → u → u (R )

      1 n n

      ❢r❛❝❛♠❡♥t❡ ❡♠ E ✱ u ❢r❛❝❛♠❡♥t❡ ❡♠ W ✱ u ❡ u n ❡♠ L

      ∗

      ♣❛r❛ t♦❞♦ p ≤ s < p ✳ k ≤ ku k ≤ M

      n 1 n λ n

      ❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ ku ✱ ♣❛r❛ t♦❞♦ n ∈ N✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷

      1

      ❡①✐st❡ u ∈ E t❛❧ q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ u n ⇀ u

      1 ,

      ❢r❛❝❛♠❡♥t❡ ❡♠ E

      

    p

    N

      u → u (R ),

      n

      ❡♠ L

      

    loc

    N

      u (x) → u(x) ,

      n

      q✳t✳♣✳ ❡♠ R

      1,p N u ⇀ u (R ). n

      ❢r❛❝❛♠❡♥t❡ ❡♠ W

      1,p c

      (Ω) ❱❛♠♦s ♣r♦✈❛r q✉❡ u ∈ W ✳ P❛r❛ ✐ss♦✱ ✈❛♠♦s ♠♦str❛r q✉❡ u = 0 ❡♠ Ω ✳ ❙❡❥❛

      N

      C = {x ∈ R : a(x) > 1/j} ∩ B (0)

      

    j j ✱ ♣❛r❛ j ∈ N✳ ❊♥tã♦ ❝❛❞❛ C j é ❧✐♠✐t❛❞♦ ❡ ✈❛❧❡

      Z

      ∞ p c N

      Ω = ∪ C |u|

      j j=1 ✳ P❛r❛ ❝❛❞❛ ❝♦♥❥✉♥t♦ A ⊂ R ▲❡❜❡s❣✉❡ ♠❡♥s✉rá✈❡❧ ❞❡✜♥✐♠♦s µ(A) = ✳ A p N

      ֒→ L (R )

      1

      ❈♦♠♦ u ∈ E ❝♦♥t✐♥✉❛♠❡♥t❡✱ t❡♠♦s q✉❡ µ ❞❡✜♥❡ ✉♠❛ ♠❡❞✐❞❛ ❡ ❞❛í !

      ∞ ∞

      Z Z [

      X

      p p

      ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✹✼ Z

      p

      |u| = 0

      j

      ❆✜r♠❛♠♦s q✉❡✱ ♣❛r❛ ❝❛❞❛ j ∈ N✱ t❡♠♦s ✳ ❈♦♠ ❡❢❡✐t♦✱ ❝♦♠♦ a(x)j > 1 ❡♠ C ✱

      C j

      Z Z j j j

      p p p p

      0 ≤ |u | < λ a(x)u ≤ ku k ≤ M → 0

      n n n n λ n

      λ λ λ

      n n n C j C j

      Z Z

      p p p N

      |u | → |u| → u (R )

      n n

      q✉❛♥❞♦ n → ∞✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ✉♠❛ ✈❡③ q✉❡ u ❡♠ L loc ❡

      C j C j

      Z

      p

      C |u| = 0

      j

      é ❧✐♠✐t❛❞♦✳ ❉❛í ❡ ❞❛ ♦❜s❡r✈❛çã♦ ❛❝✐♠❛✱ s❡❣✉❡ q✉❡ ♣❛r❛ t♦❞♦ j ∈ N✳ ❊ss❡

      C j

      r❡s✉❧t❛❞♦ ❝♦♠❜✐♥❛❞♦ ❝♦♠ (2.27) ♣r♦✈❛ ♥♦ss❛ ❛✜r♠❛çã♦✳

      s N ∗ N

      → u (R ) = {x ∈ R : a(x) ≤

      n M

      P❛r❛ ✈❡r✐✜❝❛r q✉❡ u ❡♠ L ♣❛r❛ p ≤ s < p s❡❥❛ A

      N c

      M } > 0 ) \ A ⊂ Ω

    2 M

      ✱ ♦♥❞❡ M é ❞❛❞❛ ♣❡❧❛ ❤✐♣ót❡s❡ (A ✳ ❈♦♠♦ R ✱ t❡♠♦s u ≡ 0 ❡♠

      N

      R \ A

      M ✳ ❆ss✐♠ p

      Z Z Z

      p

      ku k 1 n M

      λ n p p p

      |u − u| = |u | ≤ λ a(x)|u | ≤ ≤ .

      n n n n N N N

      

    R R λ n M R λ n M λ n M

    \A \A \A

    M0 M0 M0

      → ∞ ∈ N

      n

      1

      ❈♦♠♦ λ ✱ ❞❛❞♦ ε > 0✱ ❡①✐st❡ n t❛❧ q✉❡ Z

      ε

      p |u − u| < , . n

      1

      s❡♠♣r❡ q✉❡ n ≥ n ✭✷✳✷✽✮

      N R

      3

      \A M0

      (0) ❉❛❞♦ R > 0✱ ♣❛r❛ s✐♠♣❧✐✜❝❛r ❛ ♥♦t❛çã♦✱ ❞❡♥♦t❛♠♦s B R ♣♦r B R ✳ ❙❡❥❛ r ∈ (1, N/(N−p))✳

      − uk ≤ ku k

      n 1 n λ n + kuk

      1

      ❆r❣✉♠❡♥t❛♥❞♦ ❝♦♠♦ ❡♠ ✭✷✳✶✶✮ ❡ ♦❜s❡r✈❛♥❞♦ q✉❡ ku é ❧✐♠✐t❛❞❛✱ > 0

      é ♣♦ssí✈❡❧ ♦❜t❡r ✉♠❛ ❝♦♥st❛♥t❡ C t❛❧ q✉❡ Z

      ′ p N 1/r

      |u − u| ≤ C L(A ∩ (R \ B n M R )) .

      N A ∩(R \B R )

      M0

      > 0 P❡❧♦ ▲❡♠❛ ✷✳✼✱ ❡①✐st❡ R t❛❧ q✉❡

      Z ε

      p |u − u| ≤ . n

      ✭✷✳✷✾✮

      N

      3 A ∩(R \B )

    M0 R0

      p N

      → u ∈ N

      n (R )

      2

      ❈♦♠♦ u ❡♠ L ✱ ❡①✐st❡ n t❛❧ q✉❡

      loc

      Z ε

      p

      |u − u| ≤ , .

      n s❡♠♣r❡ q✉❡ n ≥ n

      2

      3

      ∩B A

    M0 R0

      ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✹✽ }

      =

      1 , n

      2

      ❙❡❥❛ n ♠❛①{n ✳ ❯s❛♥❞♦ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱ ✭✷✳✷✾✮ ❡ ✭✷✳✷✽✮✱ ❝♦♥❝❧✉í♠♦s q✉❡✱ ♣❛r❛ n ≥ n ✱

      Z Z Z

      p p p

      |u − u| |u − u| |u − u|

      n = + n n N N R R

      \A A

    M0 M0

      Z Z Z

      p p p

      |u − u| |u − u| |u − u|

    • =
    • n n n

      N N R

      \A A ∩B A ∩(R \B )

    M0 M0 R0 M0 R0

      < ε,

      p N

      → u (R )

      n

      ✐st♦ é✱ u ❡♠ L ✳

      ∗ ∗

      ) P❛r❛ s ∈ (p, p ✱ t♦♠❛♠♦s θ ∈ (0, 1) t❛❧ q✉❡ s = (1−θ)p+θp ✳ ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡

      p∗ N

      ֒→ L (R ) − uk ≤ ku k + kuk

      1 n 1 n λ n

      1

      ❞❡ ❍ö❧❞❡r✱ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ✐♠❡rsã♦ E ❡ q✉❡ ku é , C > 0

      1

      2

      ❧✐♠✐t❛❞❛✱ ♦❜t❡♠♦s ❝♦♥st❛♥t❡s C t❛✐s q✉❡

      θ 1−θ

      Z Z Z Z

      ∗ ∗ s θp (1−θ)p p p

      |u − u| |u − u| |u − u| ≤ |u −u| |u −u|

      n = n n n n N N N N R R R R

      ∗ θp (1−θ)p

      ≤ C ku − uk |u − u| p

      1 n n

      1 (1−θ)p

      ≤ C |u − u| .

      2 n p p N s N

      → u (R ) → u (R )

      n n

      ❈♦♠♦ u ❡♠ L ✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ✐♠♣❧✐❝❛ q✉❡ u ❡♠ L ✳ ❊♥tã♦✱

      s N

      (R ) ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ ❡①✐st❡ h ∈ L t❛❧ q✉❡

      N

    • u (x) ≤ |u (x)| ≤ h(x) .

      n

    n q✳t✳♣✳ ❡♠ R

      u + |u |

      n n

    • N

      = → u (x) → u

      ❈♦♠♦ u n q✳t✳♣✳ ❡♠ R ✱ s❡❣✉❡ ❞♦ ❚✳❈✳❉✳▲✳ q✉❡ u n ❡♠

      2

      s N

      L (R ) ✳

      = u > 0 )

      n λ n ,q

      ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❇✿ P♦r ❤✐♣ót❡s❡✱ u n é s♦❧✉çã♦ ❞❡ (S ❡✱

      ) ∈ N

      n λ n ,q

      ♣♦rt❛♥t♦✱ ❞❡ (S n ✳ ❉❛í u ❡

      λ ,q

      1

      1

      p

      − ku k = I (u ) = e I (u ).

      n λ n ,q n λ n ,q n λ n

      p q

      ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✹✾ k → ∞

      λ n ,q (u n ) n λ n n

      ❙❡♥❞♦ eI ❧✐♠✐t❛❞❛✱ s❡❣✉❡ q✉❡ ku é ❧✐♠✐t❛❞❛✳ ❈♦♠♦ λ ✱ ♣♦❞❡♠♦s s✉♣♦r

      1,p

      λ > 1 (Ω)

      n

      ✳ ❆♣❧✐❝❛♥❞♦ ♦ ▲❡♠❛ ✷✳✷✵✱ ♦❜t❡♠♦s u ∈ W t❛❧ q✉❡ u ⇀ u ,

      n

      1

      ❡♠ E

      ∗

    • s N

      u → u → u (R ), p ≤ s < p

      n

      ✭✷✳✸✵✮ ❡ u n ❡♠ L

      N u n (x) → u(x) .

      q✳t✳♣✳ ❡♠ R

      ∞

      (Ω) ❆r❣✉♠❡♥t❛♥❞♦ ❝♦♠♦ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✷✳✽ ❛❣♦r❛ ❝♦♠ ψ ∈ C ❡ ✉s❛♥❞♦ q✉❡

      ′

      I (u n ) = 0 ❡ a(x) ≡ 0 ❡♠ Ω✱ ♦❜t❡♠♦s

      λ n ,q N p

      ∇u → ∇u (B )) ,

      

    n r

      ❡♠ (L

      N

      ⊂ Ω

      r

      ♣❛r❛ t♦❞♦ r > 0 t❛❧ q✉❡ B ✳ ❈♦♠♦ Ω ⊂ R é ❛❜❡rt♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ▲✐♥❞❡❧ö❢ ⊂

      n j

      ✭✈❡❥❛ ❚❡♦r❡♠❛ ✷✷ ❞♦ ❈❛♣ít✉❧♦ ✶ ❡♠ ❬✷✸❪✮✱ ❡①✐st❡ ✉♠❛ ❢❛♠í❧✐❛ ❡♥✉♠❡rá✈❡❧ ❞❡ ❜♦❧❛s B

      ∞

      Ω, j ∈ N B ✱ t❛✐s q✉❡ Ω = ∪ n j ✳ P♦r ✉♠ ♣r♦❝❡ss♦ ❞✐❛❣♦♥❛❧ s❡♠❡❧❤❛♥t❡ ❛♦ ❞❛ Pr♦♣♦s✐çã♦

      j=1

      ✷✳✷ (iii)✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ ∇u (x) → ∇u(x),

      n

      q✳t✳♣✳ ❡♠ Ω. ✭✷✳✸✶✮ ❆❧é♠ ❞✐ss♦✱

      ′

      ∂u n ∂u

      p−2 p−2 p

      |∇u | ⇀ |∇u| (Ω), 1 ≤ i ≤ N.

      n ❢r❛❝❛♠❡♥t❡ ❡♠ L ✭✷✳✸✷✮

      ∂x ∂x

      i i 1,p ′

      (Ω) (u ), φi = 0

      n

      P❛r❛ t♦❞❛ φ ∈ W ✱ t❡♠✲s❡ hI λ n ,q ❡ ❞❛í Z Z Z

      p−2 p−2 q−1

    • n Ω Ω Ω p N

      |∇u | ∇u · ∇φ + |u | n n n u n φ = (u ) φ.

      ✭✷✳✸✸✮

      (Ω)) ❈♦♠♦ ∇φ ∈ (L ✱ ❞❡ ✭✷✳✸✷✮ s❡❣✉❡ q✉❡

      Z Z

      

    p−2 p−2

      |∇u | ∇u · ∇φ −→ |∇u| ∇u · ∇φ,

      n n N N R R

      q✉❛♥❞♦ n → ∞✳ ❈♦♠ ❛r❣✉♠❡♥t♦s s❡♠❡❧❤❛♥t❡s ❛♦s ❞❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✷✳✾✱ é ♣♦ssí✈❡❧ ♠♦str❛r q✉❡

      Z Z

      p−2 p−2

      |u | |u|

      n u n φ −→ uφ Ω Ω

      ❝♦♠♦ t❛♠❜é♠ Z Z

    • + q−1 q−1 +

      (u ) φ −→ (u ) φ.

      n Ω Ω

      ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✺✵ ❋❛③❡♥❞♦ n → ∞ ❡♠ ✭✷✳✸✸✮ ❡ ✉s❛♥❞♦ ❛s ❝♦♥✈❡r❣ê♥❝✐❛s ❛❝✐♠❛✱ ♦❜t❡♠♦s

      1,p ′

      hJ (u), φi = 0, (Ω).

      q,Ω ♣❛r❛ t♦❞♦ φ ∈ W

    • q 1,p N

      ) → u

      q,Ω n

      ❆ss✐♠ u é s♦❧✉çã♦ ❞❡ (D ✳ ❊♠ ♣❛rt✐❝✉❧❛r u ∈ M ✳ ❱❛♠♦s ♠♦str❛r q✉❡ u ❡♠

      W (R ) ✳ P❛r❛ t❛♥t♦✱ ✉s❛♥❞♦ ✭✷✳✸✶✮ ❡ ♦ ▲❡♠❛ ❞❡ ❇ré③✐s✲▲✐❡❜ ✭✈❡❥❛ ❖❜s❡r✈❛çã♦ ✶✳✷✶✮

      ♦❜t❡♠♦s Z Z Z

      p p p

      |∇u − ∇u| |∇u | − |∇u| n = n + o(1).

      ✭✷✳✸✹✮

      N N N R R R c

      ❆❧é♠ ❞✐ss♦✱ ✉s❛♥❞♦ q✉❡ u ≡ 0 ❡♠ Ω ❡ a ≡ 0 ❡♠ Ω✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ Z Z

      p p a(x)|u − u| = a(x)|u | . n n

      N N R R

      ∈ N

      n λ n ,q q,Ω

      ❉❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ ❞❡ ✭✷✳✸✹✮✱ ❞❡ ✭✷✳✸✵✮ ❡ ❞♦ ❢❛t♦ ❞❡ q✉❡ u ❡ u ∈ M s❡❣✉❡ q✉❡ Z Z Z

      p p p p p

      ku − uk ≤ ku − uk = |∇u −∇u| −u| + λ a(x)|u −u| + |u

      n 1,p n n n n n W λ n N N N R R R

      Z Z Z

      p p p

      = |∇u | − |∇u| +o(1)+ λ a(x)|u | +o(1)

      n n n N N N R R R

      Z Z Z

    • q p p

      = (u ) − |u | − |∇u| + o(1)

      n n N N N R R R

      Z Z Z

    • q p p

      = (u ) − |u| − |∇u| + o(1)

      N N N R R R

      = 0 + o(1) ≥ 1 − uk ≤ ku − uk n q✉❛♥❞♦ n → ∞✳ P❛r❛ n s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ λ n ✳ ❆ss✐♠ ku n

      1 n λ

      → u k ≥ r

      n 1 n 1 q > 0

      ❡✱ ♣♦rt❛♥t♦✱ u ❡♠ E ✳ P❡❧♦ ▲❡♠❛ ✷✳✶✹✱ ku ✳ ❉❛í✱ ❢❛③❡♥❞♦ n → ∞✱

    • ≥ r > 0

      )

      1 q

      ♦❜t❡♠♦s kuk ✳ ▲♦❣♦ u 6= 0✳ ❆❧é♠ ❞✐ss♦✱ s❡♥❞♦ u s♦❧✉çã♦ ❞❡ (D q ✱

      ′ − − p

      i = ku k 0 = hJ (u), u ,

      q,Ω

      ❞♦♥❞❡ s❡❣✉❡ q✉❡ u ≥ 0✳ P❡❧♦ Pr✐♥❝í♣✐♦ ❞♦ ▼á①✐♠♦ ❋♦rt❡✱ u > 0 ❡♠ Ω✳ ▲♦❣♦ u é s♦❧✉çã♦ )

      ❞❡ (D q ✳ ■ss♦ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛✳ → ∞ ≥ Λ

      n n

      ❉❡♠♦♥str❛çã♦ ❞♦ ❈♦r♦❧ár✐♦ ❈✿ ❈♦♠♦ λ ✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ λ ♣❛r❛

      ❈❛♣ít✉❧♦ ✷✳ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ❆ ❡ ❇ ✺✶

      λ n ,q λ n ,q = c 0,λ n

      t❡♦r❡♠❛✱ ♣❛r❛ t♦❞♦ n ∈ N✱ ❡①✐st❡ ✉♠❛ s♦❧✉çã♦ ♥♦ ♥í✈❡❧ ec ✳ ▲♦❣♦ t❡♠✲s❡ ec ✱ n o e

      = inf I (v) : v )

      0,λ n λ n ,q λ n ,q

      ♦♥❞❡ c é s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ❞❡ (S ✳ Pr♦❝❡❞❡♥❞♦ s✐♠✐❧❛r♠❡♥t❡ m

      λ n ,q q,Ω n

      ❝♦♠♦ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✷✳✶✺✱ ♦❜t❡♠♦s ec ≤ e ✳ ❈♦♠♦ u é s♦❧✉çã♦ ❞❡ ❡♥❡r❣✐❛ ) ∈ e N

      λ n ,q n λ n ,q

      ♠í♥✐♠❛ ❞❡ (S ✱ t❡♠✲s❡ u ✳ ▲♦❣♦

      1

      1

      p

      − ku k = e I (u ) = c m .

      n λ n ,q n 0,λ n λ n ,q q,Ω n = ec ≤ e

      λ

      p q

      1,p q N N

      → u (R ) (R )

      n

      ❆ss✐♠✱ ❛♣❧✐❝❛♥❞♦ ♦ ▲❡♠❛ ✷✳✷✵ ♣♦❞❡♠♦s s✉♣♦r u ❡♠ L ❝♦♠ u ∈ W ✳

      1,p N

      → u (R )

      ▼❛✐s ❛✐♥❞❛✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❇✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ u n ❡♠ W ✱ s❡♥❞♦ u ✉♠❛ M

      q ) q,Ω

      s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❞❡ (D ✳ ▲♦❣♦ u ∈ f ✳ ❱❛♠♦s ♠♦str❛r q✉❡ u é ✉♠❛ s♦❧✉çã♦ ❞❡ ❡♥❡r❣✐❛

      ′

      ) ∈ e N (u ), u i = 0

      q n λ n ,q λ n ,q n n

      ♠í♥✐♠❛ ❞❡ (D ✳ P❛r❛ ✐ss♦✱ ❝♦♠♦ u ♣❛r❛ ❝❛❞❛ n✱ t❡♠✲s❡ heI ✳ ❊♥tã♦

      Z

      1

      1

      1

      q

      m n = e I n (u ) − he I n (u ), u i = − |u | .

      

    q,Ω λ ,q λ ,q n λ ,q n n n

      e ≥ ec

      N

      p p q R M

      q,Ω

      ❋❛③❡♥❞♦ n → ∞ ❡ ✉s❛♥❞♦ q✉❡ u ∈ f t❡♠♦s Z

      1

      1

      q m ≥ − |u| = e J m .

    q,Ω q,Ω q,Ω

      e (u) ≥ e p q

      Ω

      e M J q,Ω m q,Ω = inf J q,Ω (v) q,Ω

      ❆ss✐♠✱ e ✳ ❈♦♠♦ f ❝♦♥té♠ t♦❞❛s ❛s s♦❧✉çõ❡s ♥ã♦ tr✐✈✐❛✐s (u) = e

      M v∈ f q, Ω

      ) )

      q q

      ❞❡ (D ✱ t❡♠✲s❡ q✉❡ u é s♦❧✉çã♦ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ❞❡ (D ✳ ■ss♦ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❝♦r♦❧ár✐♦✳

      ❈❛♣ít✉❧♦ ✸ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s

      ◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❉✳ ❈♦♥t✐♥✉❛♠♦s ❛ ✉s❛r ❛s ♠❡s♠❛s ♥♦t❛çõ❡s ✐♥tr♦❞✉③✐❞❛s ♥❛ ❙❡çã♦ ✷✳✶✳ ▲❡♠❜r❛♠♦s q✉❡ ♥♦ ❚❡♦r❡♠❛ ❉✱ ♦ ❝♦♥❥✉♥t♦

      −1

      Ω = (0) ✐♥t a é ❧✐♠✐t❛❞♦✳

      ❉❛q✉✐ ❡♠ ❞✐❛♥t❡✱ ✜①❛♠♦s r > 0 t❛❧ q✉❡ ♦s ❝♦♥❥✉♥t♦s

      − N

    • Ω = {x ∈ R : = {x ∈ Ω :

      2r r

      ❞✐st(x, Ω) < 2r} ❡ Ω ❞✐st(x, ∂Ω) ≥ r}

      s❡❥❛♠ ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s ❛ Ω✳ ❚❛❧ r ❡①✐st❡ ♣❡❧♦ ▲❡♠❛ ✶✳✸✳ ❙❡♠ ♣❡r❞❛ ❞❡

      1,p N

      ⊂ Ω

      r q : W (Ω) \ {0} → R

      ❣❡♥❡r❛❧✐❞❛❞❡✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ B ✳ ❉❡✜♥✐♠♦s t❛♠❜é♠ β ❛ ❢✉♥çã♦ ❝❡♥tr♦ ❞❡ ♠❛ss❛ ♣♦r

      Z

      q

      |u| xdx

      

    β (u) = . q Z q

      |u| dx

      

    1,p

    q

      (Ω) ֒→ L (Ω)

      q

      P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ✐♠❡rsã♦ W ❡ ♣❡❧♦ ❢❛t♦ ❞❡ Ω s❡r ❧✐♠✐t❛❞♦✱ β ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳

      1,p N

      (D) ֒→ ❙❡❥❛ D ✉♠ ❞♦♠í♥✐♦ ❞♦ R ✳ ❉❡♥♦t❛♠♦s ♣♦r S ❛ ♠❡❧❤♦r ❝♦♥st❛♥t❡ ❞❛ ✐♠❡rsã♦ W

      ∗ p

      L (D) ✱ ✐st♦ é✱

      Z

      1,p

    p p

      S(D) = S = inf (|∇u| + |u| ) : u ∈ W (D), |u| = 1

      p∗,D D

      ❈❛♣ít✉❧♦ ✸✳ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✺✸ )

      (R

      p p

      |∇u| + |u|

      1,p Ω

      = inf : u ∈ W (D) \ {0} .

      p ∗

      |u|

      p ,D

      ▲❡♠❜r❛♠♦s q✉❡ S ♥ã♦ ❞❡♣❡♥❞❡ ❞♦ ❝♦♥❥✉♥t♦ D t♦♠❛❞♦ ❡✱ ❛❧é♠ ❞✐ss♦✱ ♥ã♦ é ❛t✐♥❣✐❞♦ ❛

      N

      = inf J (u)

      q,D q,D r q,B r

      ♠❡♥♦s q✉❡ D = R ✳ ❉❡✜♥✐♠♦s m ✳ ◗✉❛♥❞♦ D = B ❞❡♥♦t❛♠♦s m

      u∈M

    q,D

    q,r

      ♣♦r m ✱ ♣❛r❛ s✐♠♣❧✐✜❝❛r ❛ ♥♦t❛çã♦✳

      N p

      S

      N

      m =

      q,D

      ▲❡♠❛ ✸✳✶ ❙❡❥❛ D ⊂ R ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦✳ ❊♥tã♦ lim inf ✳

      ∗ q→p

      N

      ∗

      ❉❡♠♦♥str❛çã♦✿ P❛r❛ p < q < p ❞❡✜♥✐♠♦s Z Z

      p 1,p + p q

      α = inf (|∇u| + |u| ) : u ∈ W (D), (u ) = 1

      q,D D D

      ) (R

      p p

      (|∇u| + |u| )

    • D 1,p

      = inf : u ∈ W (D), u 6= 0

      p

    • |u |

      q,D

      Z

      1,p

    • q

      = u ∈ W (D) : (u ) = 1

      q,D

      ❙❡❥❛ A ✳ ▲❡♠❜r❛♠♦s q✉❡

      D

      Z Z

      1,p p + p q M = u ∈ W (D) \ {0} : (|∇u| + |u| ) = (u ) . q,D D D

      → M

      q,D q,D q,D q,D

      ❆✜r♠❛♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ❜✐❥❡çã♦ ❡♥tr❡ A ❡ M ✳ ❉❡ ❢❛t♦✱ s❡❥❛ θ : A

      p q−p D u q,D

      ❞❛❞❛ ♣♦r θ(u) = kuk ✳ ❖❜s❡r✈❛♠♦s q✉❡ θ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♣♦✐s s❡ u ∈ A ❡♥tã♦ Z p2 pq

      q−p q−p p p p

      kuk (|∇θ(u)| + |θ(u)| ) = kuk = kuk

      D D D D

      ❝♦♠♦ t❛♠❜é♠✱ Z Z p pq Z pq

      q−p q−p q−p

    q q q

    + + +

      (θ(u) ) = (kuk u ) = kuk (u ) = kuk ,

      D D D D D D

      v → A

      

    q,D q,D q,D

      ♦ q✉❡ ♠♦str❛ q✉❡ θ(u) ∈ M ✳ ❉❡✜♥✐♠♦s Φ : M ♣♦r Φ(v) = ✳ ❆ss✐♠✱ |v |

    • q,D

      ♣❛r❛ u ∈ A q,D ✱

      p q−p p q−p kuk u

      D p Φ(θ(u)) = Φ kuk u = = u.

      D

    • q−p

      kuk |u |

      q,D D

      ❈❛♣ít✉❧♦ ✸✳ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✺✹

      q,D

      ❆❣♦r❛✱ s❡ v ∈ M ✱

      1 p

      !

      q−p p q−p

      v v v kvk

      D θ(Φ(v)) = θ = = v = v. q

      |v

    • | |v | |v |

      |v |

      q,D q,D q,D q,D q,D q,D

      ■ss♦ ♣r♦✈❛ q✉❡ ❡①✐st❡ ✉♠❛ ❜✐❥❡çã♦ ❡♥tr❡ A ❡ M ✳ ▲♦❣♦

      p p {kuk : u ∈ M } = {kθ(v)k : v ∈ A }. q,D q,D D D

      1

      1

      p

      − kuk

      q,D q,D (u) =

      ▲❡♠❜r❛♥❞♦ q✉❡ ♣❛r❛ u ∈ M ✈❛❧❡ J s❡❣✉❡ q✉❡

      D

      p q

      −1

      1

      1

      p p

      − m = inf kuk = inf kθ(v)k

      q,D D D u∈M v∈A q,D q,D

      p q

      p2 pq q−p q−p p

      kvk kvk kvk = inf = inf

      D D D v∈A q,D v∈A q,D q q−p q p q−p

      = inf kvk = (α ) ,

      q,D D v∈A q,D

      ♦✉ s❡❥❛✱

      q

      1

      1

      ✭✸✳✶✮ p q

      q−p m = − (α ) . q,D q,D

      ∗

      1

      1 1 p N

      − ∗ = ∗ = α = S

      ❯♠❛ ✈❡③ q✉❡ ❡ é s✉✜❝✐❡♥t❡ ♠♦str❛r♠♦s q✉❡ lim inf q,D

      p p N p −p p ∗ q→p

      ♣❛r❛ ❝♦♥❝❧✉✐r ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛✳ P❛r❛ ✐ss♦✱ ❛♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r

      ∗ ∗ p p

      > 1 ∗ ❝♦♠ ❡①♣♦❡♥t❡s ❡ ♦❜t❡♠♦s

      −q q p ∗ p − q p∗q ∗

      |u| ≤ L(D) |u| .

      

    q,D p ,D

    1,p

      6= 0 (D)

    • Z Z Z  

      ❆ss✐♠✱ ♣❛r❛ u ∈ W ❝♦♠ u t❡♠✲s❡

      p p p p p p

      |∇u| + |u| |∇u| + |u| |∇u| + |u|

      ∗ − p − q (p )

       

      D D D qp∗

      ≥ ≥ L(D)  

      

    p p p

       

      |u | |u| |u|

      

    q,D q,D p ,D

    ∗ − p (p − q ) qp∗

      ≥ L(D) S ❡✱ ♣♦rt❛♥t♦✱

      ∗ p − q (p )

      − qp∗

      ≥ L(D) α q,D S

      ❈❛♣ít✉❧♦ ✸✳ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✺✺ α q,D

      ❆ ❡①♣r❡ssã♦ ❛❝✐♠❛ ✐♠♣❧✐❝❛ q✉❡ ba = lim inf é ✜♥✐t♦ ❡ a ≥ S✳ ❱❛♠♦s ♠♦str❛r q✉❡

      ∗ q→p

      a = S ✳ ❆r❣✉♠❡♥t❛♥❞♦ ♣♦r ❝♦♥tr❛❞✐çã♦✱ s✉♣♦♠♦s q✉❡ ba > S✳ ❙❡❥❛ ε ∈ (0, ba − S)✳ P❡❧❛

      1,p

      ∈ W (D) \ {0}

      1

      ❞❡✜♥✐çã♦ ❞❡ S ❡①✐st❡ u t❛❧ q✉❡ Z

      

    p p

      |∇u | + |u |

      1

      1

      ε

      D < S + . p

      ✭✸✳✷✮

      ∗

      |u |

      2

      1 p ,D ∗

      R R

      1,p q p

      = |u | ∈ W (D) \ {0} |u | → |u |

      1

      ❙❡❥❛ u ✳ ❊♥tã♦ u ✳ ❆✜r♠❛♠♦s q✉❡ q✉❛♥❞♦

      D D ∗ ∗ ∗

      ∈ (p, p → p q → p n ) n ) n ♣❡❧❛ ❡sq✉❡r❞❛✳ ❉❡ ❢❛t♦✱ s❡❥❛ (q ✉♠❛ s❡q✉ê♥❝✐❛ t❛❧ q✉❡ q ❡ q ✳

      q n

      (x) = |u (x)|

      n

      ❉❡✜♥✐♠♦s✱ ♣❛r❛ ❝❛❞❛ n ∈ N ❡ x ∈ D✱ f ✳ ❊♥tã♦

      q n q n q n

      f (x) = |u (x)| = |u (x)| χ({x ∈ D : |u (x)| ≤ 1})+|u (x)| χ({x ∈ D : |u (x)| > 1})

      n ∗ p

      1

      ≤ 1 + |u (x)| χ({x ∈ D : |u (x)| > 1}) ∈ L (D),

      ∗

      R R

      q n p

      |u | → |u | ✉♠❛ ✈❡③ q✉❡ D é ❧✐♠✐t❛❞♦✳ ❉❛í✱ s❡❣✉❡ ❞♦ ❚✳❈✳❉✳▲✳ q✉❡ ✳ ❈♦♠♦ ❛

      D D n )

      s❡q✉ê♥❝✐❛ (q ❢♦✐ t♦♠❛❞❛ ❛r❜✐tr❛r✐❛♠❡♥t❡✱ ♥♦ss❛ ❛✜r♠❛çã♦ ❡stá ♣r♦✈❛❞❛✳ ❆❣♦r❛✱ ❢❛③❡♥❞♦

      ∗ ∗

      q → p ∈ (p, p )

      1

      ✱ ♦❜t❡♠♦s ❞❛ ❛✜r♠❛çã♦ ❛❝✐♠❛ ✉♠ q t❛❧ q✉❡ Z Z

      p p p p

      |∇u | + |u | |∇u | + |u | ε

      ∗ D D

      − < , 1 , p ).

      p p s❡♠♣r❡ q✉❡ q ∈ (q ✭✸✳✸✮ ∗

      ku k ku k

      2

      q,D p ,D

    • |

      q,D

      = u = |u

      1

      ❉❛ ❞❡✜♥✐çã♦ ❞❡ α ✱ ❞♦ ❢❛t♦ ❞❡ u ❡ ❞❛s ❞❡s✐❣✉❛❧❞❛❞❡s ✭✸✳✷✮ ❡ ✭✸✳✸✮ s❡❣✉❡ q✉❡ Z Z

      p p p p

      |∇u | + |u | |∇u | + |u | ε

      D D

      α ≤

    • <

      q,D p p

    • |u | ∗

      |u |

      2

      q,D p ,D

      Z

      p p

      |∇u | |

    • |u

      1

      1

      ε

      D

    • =

      p ∗

      |u |

      2

      1 p ,D

      ε ε + < S + = S + ε.

      2

      2 ❉❛í✱

      α

      

    q,D

      ba = lim inf ≤ S + ε < ba,

      ∗ q→p

      q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ▲♦❣♦ ba = S ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞♦ ❧❡♠❛✳

      ❈❛♣ít✉❧♦ ✸✳ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✺✻

      q n N

      ≥ 1

      n ) n n ) ⊂ L (R )

      ▲❡♠❛ ✸✳✷ ❙❡❥❛♠ (q ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s t❛❧ q✉❡ q ❡ (u

      c q n

      ≡ 0 | dx ⇀ ν

      n n

      ❝♦♠ u ❡♠ Ω ♣❛r❛ t♦❞♦ n ∈ N✳ ❙❡ |u ❢r❛❝❛♠❡♥t❡ ♥♦ s❡♥t✐❞♦ ❞❛s ♠❡❞✐❞❛s ♦♥❞❡ ν é ✉♠❛ ♠❡❞✐❞❛ ❞❡ ❘❛❞♦♥✱ ❡♥tã♦

      Z Z

      n q

      |u |

      n f (x)dx → f (x)dν

      q✉❛♥❞♦ n → ∞

      N R

    N

      , R) ♣❛r❛ q✉❛❧q✉❡r f ∈ C(R ✳

      Z Z Z

    • q n q n q n −

      |u | f (x)dx = |u | f (x)dx − |u | f (x)dx

      n n n

      ❉❡♠♦♥str❛çã♦✿ ❊s❝r❡✈❡♥❞♦

      N R Ω Ω

      ✈❡♠♦s q✉❡✱ ♣❛r❛ ❞❡♠♦♥str❛r ♦ ❧❡♠❛✱ é s✉✜❝✐❡♥t❡ ♠♦str❛r♠♦s q✉❡ Z Z

      n q

      |u |

      n g(x)dx → g(x)dν N R

      Ω N q n

      | , [0, ∞)) n dx ⇀ ν

      ♣❛r❛ t♦❞❛ g ∈ C(R ✳ P❛r❛ ✐ss♦✱ ❧❡♠❜r❛♠♦s q✉❡ |u ❢r❛❝❛♠❡♥t❡ ♥♦ s❡♥t✐❞♦ ❞❛s ♠❡❞✐❞❛s s✐❣♥✐✜❝❛ Z Z

      q n

      |u | Φ(x)dx → Φ(x)dν,

      n

      ✭✸✳✹✮

      N N R R N N

      (R , R) ∈ C (R , R) ≤ 1 ♣❛r❛ t♦❞❛ Φ ∈ C ✳ P❛r❛ ε > 0 q✉❛❧q✉❡r✱ s❡❥❛ ϕ ε t❛❧ q✉❡ 0 ≤ ϕ ε

      N +

      ≡ 1 ≡ 0 \ Ω

      ε ε ε

      ❡ ϕ ❡♠ Ω ❡ ϕ ❡♠ R ✳ ❚❛❧ ϕ ❡①✐st❡ ♣❡❧♦ ▲❡♠❛ ❞❡ ❯r②s♦❤♥ ✭✈❡❥❛ ▲❡♠❛

      ε N

      ∈ C (R , R)

      

    ε

      ✹✳✸✷ ❡♠ ❬✶✺❪✮✳ ❉❛í✱ ❞❡ ✭✸✳✹✮ ❡ ❞♦ ❢❛t♦ ❞❡ gϕ s❡r ♥ã♦ ♥❡❣❛t✐✈❛ s❡❣✉❡ q✉❡ Z Z Z

      

    q n q n q n

      lim |u | g(x)dx = lim ϕ (x)|u | g(x)dx ≤ lim |u | ϕ (x)g(x)dx

      

    n ε n n ε

    n→∞ n→∞ n→∞ N R Ω Ω

      Z Z = ϕ (x)g(x)dν = ϕ (x)g(x)dν

      

    ε ε

    • N

      R Ω ε

      ✭✸✳✺✮ Z Z

      ≤ g(x)dν = g(x)χ (x)dν

    • Ω ε
    • N

      R Ω ε

    • Ω ε ε Ω ε (x) → bg(x) ♣❛r❛ t♦❞♦ N

      ♦♥❞❡ χ é ❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ Ω ✳ ❖❜s❡r✈❡ q✉❡ g(x)χ

      c

      x ∈ R q✉❛♥❞♦ ε → 0 ♦♥❞❡ bg ≡ g ❡♠ Ω ❡ bg ≡ 0 ❡♠ Ω ✳ P♦❞❡♠♦s s✉♣♦r q✉❡ ♣❛r❛

    • ε ⊂ B ≤ g
    • Ω ε c

      R R

      s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ Ω ε ♣❛r❛ ❛❧❣✉♠ R > 0✳ ❈♦♠♦ gχ ❡♠ B ❡

      gχ ≡ 0

      B R Ω ε ❡♠ B R t❡♠♦s q✉❡ gχ Ω ε ❡stá ❧✐♠✐t❛❞❛ s✉♣❡r✐♦r♠❡♥t❡ ♣❡❧❛ ❢✉♥çã♦ h = gχ q✉❡

      Z

    1 N

      ❈❛♣ít✉❧♦ ✸✳ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✺✼ ❘❛❞♦♥✳ ❋❛③❡♥❞♦ ε → 0 ❡♠ ✭✸✳✺✮ ❡ ❛♣❧✐❝❛♥❞♦ ♦ ❚✳❈✳❉✳▲✳✱ ♦❜t❡♠♦s

      Z Z Z

      n q lim |u | g(x)dx ≤ gdν. n

      bgdν = ✭✸✳✻✮

      n→∞ N R

    Ω Ω

    N − c

      ∈ C ≤ 1 ≡ 1 ≡ 0

      ε (R , R) ε ε ε

      ❈♦♥s✐❞❡r❡ ψ t❛❧ q✉❡ 0 ≤ ψ ✱ ψ ❡♠ Ω ❡ ψ ❡♠ Ω ✳ ❆ss✐♠✱

      ε

      ❝♦♠♦ ✜③❡♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ Z Z Z

      

    n n n

    q q q

      |u | | |u | lim n g(x)dx ≥ lim ψ ε (x)|u n g(x)dx = lim n ψ ε (x)g(x)dx

      n→∞ n→∞ n→∞ N R Ω Ω

      Z Z = ψ (x)g(x)dν = ψ (x)g(x)dν

      ε ε N R

      Ω

      Z Z

      −

      ≥ g(x)dν = g(x)χ (x)dν

      Ω ε −

      N R Ω ε

      ❡✱ ❢❛③❡♥❞♦ ε → 0✱ ♣❡❧♦ ❚✳❈✳❉✳▲✳✱ Z Z

      q n lim |u | g(x)dx ≥ g(x)dν. n n→∞

      

    Ω Ω

      ❉❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ❡ ❞❡ ✭✸✳✻✮✱ ❝♦♥❝❧✉í♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛✳

      N N

      → R )

      ▲❡♠❛ ✸✳✸ ❙❡❥❛♠ f : R ❝♦♥tí♥✉❛ ❡ ν ∈ M(R ✉♠❛ ♠❡❞✐❞❛ q✉❡ s❡ ❝♦♥❝❡♥tr❛ ❡♠ Z f (x)dν = f (y)|ν|

      ✉♠ ú♥✐❝♦ ♣♦♥t♦ y ∈ Ω✳ ❊♥tã♦ ✳

      Ω

      ❉❡♠♦♥str❛çã♦✿ P❛r❛ t♦❞♦ t > 0✱ t❡♠✲s❡ Z Z Z f (x)dν = f (x)dν = f (x)χ B t (y) (x)dν

      ✭✸✳✼✮

      N R Ω B t (y) N t (x) → f (y)χ {y} (x)

      ❚❡♠♦s q✉❡ f(x)χ B (y) q✉❛♥❞♦ t → 0 ❡♠ R ✳ ❈♦♠♦ f é ❝♦♥tí♥✉❛✱ ≤ f (x)χ ≤ M χ ∈

      B t (y) B (y) B (y)

      ♣❛r❛ t♦❞♦ t ≤ 1 t❡♠♦s q✉❡ ❡①✐st❡ M > 0 t❛❧ q✉❡ f(x)χ

      1

      1

    1 N

      L (R , ν) ✳ ❋❛③❡♥❞♦ t → 0 ❡♠ ✭✸✳✼✮✱ ♣❡❧♦ ❚✳❈✳❉✳▲✳ ♦❜t❡♠♦s

      Z Z f (x)dν = f (y)χ (x)dν = f (y)ν({y}).

      {y} N R

      Ω

      ❘❡st❛ ♠♦str❛r q✉❡ ν({y}) = |ν|✳ ▲❡♠❜r❛♠♦s q✉❡✱ ♣♦r ❞❡✜♥✐çã♦✱ Z

      N |ν| = sup φ(x)dν (R ), |φ| ≤ 1 .

      ∞

      ✭✸✳✽✮ : φ ∈ C

      N R

      ❈❛♣ít✉❧♦ ✸✳ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✺✽

      N

      (R )

      1 (y)

      ❙❡❥❛ φ ∈ C ❝♦♠ 0 ≤ φ ≤ 1 ❡ φ ≡ 1 ❡♠ B ✳ P♦r ✭✸✳✽✮✱ Z Z |ν| ≥ φ(x)dν ≥ 1dν = ν({y}).

      N R B (y)

      1 N

      ≤ 1 (R ) ∞

      P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ t♦❞❛ ϕ ∈ C ❝♦♠ |ϕ| ✱ Z Z Z ϕdν |ϕ|dν ≤ 1dν = ν({y}).

      N ≤ N N R R R

      P♦rt❛♥t♦ |ν| ≤ ν({y}) ❡✱ ❛ss✐♠✱ ν({y}) = |ν|✱ ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦✳

      

    ∗ ∗

    • ∈ (p, p ) , p ) (u) ∈ Ω

      2 2 q

      ▲❡♠❛ ✸✳✹ ❊①✐st❡ q t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ q ∈ (q ✱ t❡♠✲s❡ β r s❡♠♣r❡ (u) ≤ m q✉❡ u ∈ M q,Ω ❝♦♠ J q,Ω q,r ✳

      ❉❡♠♦♥str❛çã♦✿ ❆r❣✉♠❡♥t❛♥❞♦ ♣♦r ❝♦♥tr❛❞✐çã♦ s✉♣♦♥❤❛ q✉❡ ♦ ❧❡♠❛ é ❢❛❧s♦✳ ❊♥tã♦ ♣❛r❛

      ∗ p −p ∗ ∗ ∗

      ∈ (p − , p ) ∈ (q , p ) ) ⊂ M (u ) ≤ m

      2,n n 2,n n q n ,Ω q n ,Ω n q n ,r

      t♦❞♦ q ❡①✐st❡ q ❡ (u ❝♦♠ J

      n ∗

    • (u ) / ∈ Ω → p

      )

      q n n n n

      ❡ β r ✳ ❖❜s❡r✈❡ q✉❡ q ✳ ❆ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ ♣♦❞❡♠♦s s✉♣♦r (q

      ❝r❡s❝❡♥t❡✳ ❆ss✐♠

      1

      1

      p m ≤ J (u ) = − ku k ≤ m .

    q n ,Ω q n ,Ω n n q n ,r

      p q

      n

      P❡❧♦ ▲❡♠❛ 3.1✱

      N N

      1

      1

      1

      1

      p p p S ≤ lim inf − ku k ≤ S . n Ω n→∞

      N p q N

      n

      1

      1

      1 − =

      ❯♠❛ ✈❡③ q✉❡ ❡①✐st❡ lim s❡❣✉❡ q✉❡

      n→∞

      p q N

      n N p q n p lim inf |u | = lim inf ku k = S .

    n n

    q n Ω n→∞ n→∞

      P❛ss❛♥❞♦ ❛ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱ ♣♦❞❡♠♦s s✉♣♦r

      N

    n p

    q p lim |u | = lim ku k = S . n n q n

      ✭✸✳✾✮

      Ω n→∞ n→∞ ∗ ∗ p p

      > 1 ∗ P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r ❝♦♠ ❡①♣♦❡♥t❡s ❡ ✱ ♦❜t❡♠♦s

      q n p −q n

      Z

      ∗ p − qn

    q n q n

    q n p∗

      |u | |u | |u | ∗

    n = n dx ≤ L(Ω) n .

      q n ,Ω p ,Ω Ω

      ▲♦❣♦

      ∗

      1

      ❈❛♣ít✉❧♦ ✸✳ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✺✾ P♦r ✭✸✳✾✮✱

      N

      1 p p∗ ∗ (S ) ≤ lim inf |u | . n p ,Ω

      ✭✸✳✶✵✮

      

    n→∞

      P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ S✱

      1 − p ∗

      |u | ≤ S ku k .

      

    n p ,Ω n Ω

      ❆ss✐♠✱ s❡❣✉❡ ❞❡ ✭✸✳✾✮ q✉❡

      1 N

      1 N

      1

    p p p p p∗ ∗

      |u | ≤ S lim sup n p ,Ω (S ) = (S ) .

      n→∞

      ❉❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ❡ ❞❡ ✭✸✳✶✵✮ ♦❜t❡♠♦s

      N

      1 p p∗ ∗

      lim |u | = (S ) ,

      n p ,Ω n→∞

      ♦✉ ❛✐♥❞❛✱

      N

    p

    p

      lim |u | = S .

      n p ,Ω ✭✸✳✶✶✮ n→∞

    1,p

      ) (Ω) P♦r ✭✸✳✾✮ t❡♠♦s q✉❡ (u n é ❧✐♠✐t❛❞❛ ❡♠ W ✳ P❛ss❛♥❞♦ ❛ ♦✉tr❛ s✉❜s❡q✉ê♥❝✐❛ s❡ ♥❡✲

      1,p 1,p

      (Ω) n ⇀ u (Ω) ❝❡ssár✐♦✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ ❡①✐st❡ u ∈ W t❛❧ q✉❡ u ❢r❛❝❛♠❡♥t❡ ❡♠ W

      1,p p p

      (Ω) ֒→ L (Ω) → u (Ω)

      n

      ❡✱ ❝♦♠♦ W ❝♦♠♣❛❝t❛♠❡♥t❡✱ u ❢♦rt❡♠❡♥t❡ ❡♠ L ❡✱ ❝♦♥s❡q✉❡♥t❡✲ u

      n ∗

      (x) → u(x) = | = 1

      n n n p ,Ω

      ♠❡♥t❡✱ u q✳t✳♣✳ ❡♠ Ω✳ ❆❣♦r❛✱ s❡❥❛ v ✳ ❊♥tã♦ |v ✳ ❉❡ ✭✸✳✾✮

      ∗

      |u |

      n p ,Ω

      ❡ ✭✸✳✶✶✮ s❡❣✉❡ q✉❡

      N p p

      ku k

      n S p Ω

      kv k lim n = lim = N = S.

      p Ω n→∞ n→∞

      ∗ p∗

      |u |

      n p ,Ω

      S

      p

      k )

      n

      ❆ss✐♠✱ ❛ s❡q✉ê♥❝✐❛ (kv Ω é ✉♠❛ s❡q✉ê♥❝✐❛ ♠✐♥✐♠✐③❛♥t❡ ♣❛r❛ S✳ P❡❧♦ ▲❡♠❛ ✶✳✶✾✱ ❡①✐st❡

      1,p

      v ∈ W (Ω) t❛❧ q✉❡

      1,p

      v n ⇀ v (Ω), ❢r❛❝❛♠❡♥t❡ ❡♠ W v (x) → v(x)

      n

      q✳t✳♣✳ ❡♠ Ω, ∇v (x) → ∇v(x)

      n q✳t✳♣✳ ❡♠ Ω.

      ∗

      (x) = |u | v (x)

      n n p ,Ω n

      ❉❛í✱ t♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ q✉❛♥❞♦ n → ∞ ♥❛ ✐❣✉❛❧❞❛❞❡ u ✱ ♦❜t❡♠♦s✱ ♣♦r ✭✸✳✶✶✮✱

      N

      1 · p p∗

      u(x) = S v(x), q✳t✳♣✳ ❡♠ Ω. ✭✸✳✶✷✮ (x) → ∇v(x)

      n

      ❙❡❣✉❡ ❞❡ ✭✸✳✶✶✮✱ ✭✸✳✶✷✮ ❡ ❞♦ ❢❛t♦ ❞❡ ∇v q✳t✳♣✳ ❡♠ Ω q✉❡

      

    N

      1

    p p∗

    p∗

      ∇u (x) = |u | ∇v (x) → S · ∇v(x) = ∇u(x)

      n n n L (Ω) q✳t✳♣✳ ❡♠ Ω.

      ❈❛♣ít✉❧♦ ✸✳ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✻✵

      N c n n = u = 0

      ❊st❡♥❞❡♠♦s u ❡ u ♣❛r❛ R ♣♦♥❞♦ u ❡♠ Ω ✳ P♦r ✭✸✳✾✮✱ ❡①✐st❡ C > 0 t❛❧ q✉❡ Z Z

      n q p |u − u| ≤ C |∇u − ∇u| ≤ C. n n

      ❡

      N N R R

      P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✻✱ ❡①✐st❡♠ ♠❡❞✐❞❛s ♣♦s✐t✐✈❛s ❞❡ ❘❛❞♦♥ ν ❡ ω t❛✐s q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱

      n q p

      |u − u| − ∇u|

      

    n ⇀ ν n ⇀ ω

      ❡ |∇u ❢r❛❝❛♠❡♥t❡ ♥♦ s❡♥t✐❞♦ ❞❛s ♠❡❞✐❞❛s✳ ❉❛s ❝♦♥s✐❞❡r❛çõ❡s ❛♥t❡r✐♦r❡s✱ ✈❡♠♦s q✉❡ ❛s ❤✐♣ót❡s❡s ❞♦s ▲❡♠❛ ✶✳✶✼ sã♦ s❛t✐s❢❡✐t❛s✳ ❖❜s❡r✈❛♠♦s q✉❡ ❝♦♠♦ Ω é ❧✐♠✐t❛❞♦✱ t❡♠✲s❡

      Z Z

      p q n ω ∞ := lim lim sup |∇u | = 0 ∞ := lim lim sup |u | = 0. n n

      ❡ ν

      R→∞ R→∞ n→∞ n→∞ |x|>R |x|>R

      P❡❧♦ ▲❡♠❛ ✶✳✶✼ ♦❜t❡♠♦s

      ∗ p/p −1

      ≤ S ⑤ν⑤ ⑤ω⑤, ✭✸✳✶✸✮

      p p

      lim sup |∇u | = |∇u| +

      n p p ⑤ω⑤, ✭✸✳✶✹✮ n→∞

      ❡

      ∗ q n p

      ∗

      lim sup |u | = |u| +

      n q n p ⑤ν⑤. ✭✸✳✶✺✮ n→∞ p

      → u (Ω)

      n

      ❈♦♠♦ u ❢♦rt❡♠❡♥t❡ ❡♠ L ✱ t❡♠♦s

      p p p p p ku k = |u | + |∇u | = |u| + |∇u | + o(1). n n n n p p p p

      ❉❛í✱ ❞❡ ✭✸✳✾✮ ❡ ❞❡ ✭✸✳✶✹✮ s❡❣✉❡ q✉❡

      N p p p p S = |u| + |∇u| + |ω| = kuk + |ω|. p p

      ✭✸✳✶✻✮ ❉❡ ✭✸✳✾✮ ❡ ✭✸✳✶✺✮ t❡♠♦s

      N ∗

    p

    p ∗

      S = |u| + |ν|.

      p ✭✸✳✶✼✮

      P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ S✱

      p −1 p ∗

      |u| ≤ S kuk .

      p

      ✭✸✳✶✽✮

      ∗ N ∗ N p p p p ∗ ∗

      6= S = S

      ❆✜r♠❛♠♦s q✉❡ ♦✉ u ≡ 0 ♦✉ |u| ✳ ❈❛s♦ ❝♦♥trár✐♦✱ u 6≡ 0 ❡ |u| ✳ ❆ss✐♠

      p p ∗ p t t t

      ∗

      |u| 6= 0 < a + b

      p ❡✱ ♣♦r ✭✸✳✶✼✮✱ |ν| 6= 0✳ ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ (a + b) ♣❛r❛ a, b > 0 ❡

      ❈❛♣ít✉❧♦ ✸✳ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✻✶ 0 < t < 1 ✱ s❡❣✉❡ ❞❡ ✭✸✳✶✼✮✱ ✭✸✳✶✽✮✱ ✭✸✳✶✸✮ ❡ ✭✸✳✶✻✮ q✉❡

      ∗ ∗ N −p p/p p/p

    ∗ ∗

    p p

    p/p p

    ∗ ∗

      |u| |u| S = + |ν| < + |ν|

      

    p p

    N −p

      −1 p p

      ≤ S (kuk + |ω|) = S ,

      ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳

      1,p N

      ⇀ u (R ) ❈♦♠♦ u n ❡♠ W ✱ t❡♠♦s q✉❡

      N p p p kuk ≤ lim inf ku k = S . n n→∞

      ∗ N p p ∗

      = S ❉❛í✱ s❡ |u| p ✱ ❡♥tã♦

      N p p

      kuk S S ≤ ≤ = S

      N −p p

      ∗

      |u| p

      p

      S

      1,p

      (Ω) ✐st♦ é✱ ❛ ❝♦♥st❛♥t❡ S é ❛t✐♥❣✐❞❛ ♣♦r ❛❧❣✉♠❛ ❢✉♥çã♦ u ∈ W ✳ ■ss♦ ♥♦s ❞á ✉♠❛

      N

      ❝♦♥tr❛❞✐çã♦✱ ✉♠❛ ✈❡③ q✉❡ Ω 6= R ✳ ❆ss✐♠ u ≡ 0 ❡✱ ♣♦rt❛♥t♦✱ ❞❡ ✭✸✳✾✮ ❡ ✭✸✳✶✼✮✱ Z

      N q n p

      |u | dx → S = |ν| 6= 0,

      n

      ✭✸✳✶✾✮

      Ω

      ❡✱ ❛❧é♠ ❞✐ss♦✱

      

    q n N

    |u | → ν ). n

      ❢r❛❝❛♠❡♥t❡ ❡♠ M(R (x) = x

      i i

      ❙❡❣✉❡ ❞♦ ▲❡♠❛ ✸✳✷ ❛♣❧✐❝❛❞♦ ♣❛r❛ ❝❛❞❛ ❢✉♥çã♦ ♣r♦❥❡çã♦ f ✱ i ∈ {1, . . . , N}✱ q✉❡ Z Z

      q n |u | xdx → xdν. n

      ✭✸✳✷✵✮

      N R Ω

      P♦r s❡r u ≡ 0✱ ❞❡ ✭✸✳✶✼✮ ❡ ✭✸✳✶✻✮ s❡❣✉❡ q✉❡

      N ∗

    p/p −1 −1

    p |ν| |ω|.

      = S S = S ▲♦❣♦✱ ♣❡❧♦ ▲❡♠❛ ✶✳✶✼✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ❛ ♠❡❞✐❞❛ ν ❡stá ❝♦♥❝❡♥tr❛❞❛ ❡♠ ✉♠ ú♥✐❝♦

      N

      (x) = x ♣♦♥t♦ y ∈ R ✳ P❡❧♦ ▲❡♠❛ ✸✳✸ ❛♣❧✐❝❛❞♦ às ❢✉♥çõ❡s ♣r♦❥❡çõ❡s f i i ✱ s❡❣✉❡ q✉❡

      Z xdν = y|ν|.

      ✭✸✳✷✶✮

      Ω

      ❈❛♣ít✉❧♦ ✸✳ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✻✷ ❉❡ ✭✸✳✶✾✮✱ ✭✸✳✷✵✮ ❡ ✭✸✳✷✶✮ ✈❡♠ q✉❡

      R R

      q n

    N |u | x dx xdν

    n y|ν| R

      Ω

      → β q n (u n ) = R = = y.

      ✭✸✳✷✷✮

      n q

      N |u | dx |ν| |ν| n R c N

      ⊂ Ω

      2R (R )

      ❆✜r♠❛♠♦s q✉❡ y ∈ Ω✳ ❈❛s♦ ❝♦♥trár✐♦✱ ♣♦❞❡rí❛♠♦s t♦♠❛r B

      1 ❡ ϕ ∈ C t❛❧ N

      \ B

      R

      2R

      q✉❡ ϕ ≥ 0✱ ϕ ≡ 1 ❡♠ B

      1 ❡ ϕ ≡ 0 ❡♠ R 1 ❡ ✈❛❧❡r✐❛

      Z Z Z

      q n

      0 = ϕ|u | dx → ϕdν ≥ 1dν = ν({y}) = |ν| > 0,

      n N N R R

      B (y) R1

      ✉♠ ❛❜s✉r❞♦✳ ■ss♦ ♠♦str❛ q✉❡ y ∈ Ω ❡ ♥♦s ♣❡r♠✐t❡ ❝♦♥❝❧✉✐r ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛✱ ♣♦✐s

    • q n r

      n (u ) / ∈ Ω

      ✭✸✳✷✷✮ ♥♦s ❞á ✉♠❛ ❝♦♥tr❛❞✐çã♦ ❝♦♠ ♦ ❢❛t♦ ❞❡ β ✳

      ❙❡❣✉✐♥❞♦ ❛ ✐❞❡✐❛ ❞❡ ❇❛rts❝❤ ❡ ❲❛♥❣ ❡♠ ❬✺❪✱ s❡♥❞♦ Ω ❧✐♠✐t❛❞♦✱ ❡s❝♦❧❤❡♠♦s R > 0

      R

      t❛❧ q✉❡ Ω ⊂ B ❡ ❞❡✜♥✐♠♦s (

      1, s❡ 0 ≤ t ≤ R, ξ(t) =

      R/t, s❡ t ≥ R.

      N

      : N → R ❚❛♠❜é♠ ❞❡✜♥✐♠♦s ❛ ❢✉♥çã♦ ❝❡♥tr♦ ❞❡ ♠❛ss❛ tr✉♥❝❛❞❛ β λ,q ♣♦r

      q

      R

      q N |u| ξ(|x|)x dx

      R β (u) = .

      R

      q

      ✭✸✳✷✸✮

      q N |u|

      dx

      

    R

      ❖❜s❡r✈❛♠♦s q✉❡ β q ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡✱ ❛❧é♠ ❞✐ss♦✱ é ❝♦♥tí♥✉❛✳ ❉❡ ❢❛t♦✱ t♦♠❛♥❞♦ ✉♠❛

      q N

      → u

      n λ,q λ (R )

      s❡q✉ê♥❝✐❛ u ❡♠ N ✱ ♣❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ✐♠❡rsã♦ ❞❡ E ❡♠ L ✱ t❡♠♦s

      q N q

      → u (R ) | ≤ h

      n n

      q✉❡ u ❡♠ L ✳ P♦rt❛♥t♦✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ |u ♣❛r❛ ❛❧❣✉♠❛

      q N

      h ∈ L (R ) (u ) → β (u)

      n

      ✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ξ ❡ ♦ ❚✳❈✳❉✳▲✳✱ ♦❜t❡♠♦s β q q ✳ ▲♦❣♦ β q é ❝♦♥tí♥✉❛✳

      ❖ ❧❡♠❛ s❡❣✉✐♥t❡ ❝♦rr❡s♣♦♥❞❡ ❛♦ ▲❡♠❛ ✸✳✶✹ ❡♠ ❬✶✽❪✳

      ∗ ∗

      ∈ (p, p ) , p )

      2

      2

      ▲❡♠❛ ✸✳✺ ❙❡❥❛ q ♦❜t✐❞♦ ♥♦ ▲❡♠❛ ✸✳✹✳ ❊♥tã♦ ♣❛r❛ ❝❛❞❛ q ∈ (q ✱ ❡①✐st❡ ✉♠

    • = Λ

      (q) (u) ∈ Ω

      2

      2

      

    2

    λ,q

      ♥ú♠❡r♦ Λ t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ λ ≥ Λ ✱ t❡♠♦s q✉❡ β q 2r s❡♠♣r❡ q✉❡ u ∈ N

      λ,q (u) ≤ m q,r

      ❡ I ✳ ❉❡♠♦♥str❛çã♦✿ ❆r❣✉♠❡♥t❛♥❞♦ ♣♦r ❝♦♥tr❛❞✐çã♦✱ s✉♣♦♠♦s q✉❡ ♦ ❧❡♠❛ é ❢❛❧s♦✳ ❊♥tã♦

      ❈❛♣ít✉❧♦ ✸✳ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✻✸

      ∗

      ∈ N → ∞

      2 , p ) n ) ⊂ R n λ n ,q n

      ❡①✐st❡ q ∈ (q ✱ ✉♠❛ s❡q✉ê♥❝✐❛ (λ ✱ u t❛✐s q✉❡ λ ✱

      1

      1

      p

      I (u ) = − ku k ≤ m

      

    λ n ,q n n q,r

    λ n ✭✸✳✷✹✮

      p q

    • (u ) / ∈ Ω

      )

      n n

      ❡ β

      q 2r ✳ ❊♥tã♦ ❛s ❤✐♣ót❡s❡s ❞♦ ▲❡♠❛ ✷✳✷✵ sã♦ s❛t✐s❢❡✐t❛s ♣❛r❛ ❛ s❡q✉ê♥❝✐❛ (u ✱ ❞❡ 1,p

      (Ω) ♠♦❞♦ q✉❡ ❡①✐st❡ u ∈ W t❛❧ q✉❡

      1,p N

      u ⇀ u (R ),

      n

      ❢r❛❝❛♠❡♥t❡ ❡♠ W

      s N ∗

      → u u (R ), p ≤ s < p ,

      n ❡♠ L

      ✭✸✳✷✺✮

    • + s ∗ + N

      → u u (R ), p ≤ s < p .

      n ❡♠ L

      ∈ N

      n λ n ,q

      ❈♦♠♦✱ ♣❛r❛ ❝❛❞❛ n ∈ N✱ u ✱ t❡♠♦s Z Z

      p

    • p p q (|∇u | + |u | ) ≤ ku k = (u ) .

      n n n λ n n N N R R

      ❉❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ❡ ❞❡ ✭✸✳✷✺✮ ♦❜t❡♠♦s Z Z Z

      p p p p q +

      (|∇u| + |u| ) ≤ lim inf (|∇u | + |u | ) ≤ (u ) ,

      n n

      ✭✸✳✷✻✮

      N n→∞ N N R R R c

      ♦✉ ❛✐♥❞❛✱ ❝♦♠♦ u ≡ 0 ❡♠ Ω ✱ Z Z

      p p q + (|∇u| + |u| ) ≤ (u ) .

      ✭✸✳✷✼✮

      Ω Ω

      R R

      p

    • q p q
    • N (u ) = ku k ≥ r > 0 (u ) ≥

      n

      P❡❧♦ ▲❡♠❛ ✷✳✶✹✱ t❡♠♦s R n λ n q ✳ ❉❡ ✭✸✳✷✺✮ s❡❣✉❡ q✉❡

      Ω

      1 p q−p

      kuk

      p

    • > 0 6≡ 0

      r ✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡ u ✳ ❆ss✐♠✱ t♦♠❛♥❞♦ t =

      q ✱ t❡♠♦s t ∈ (0, 1] ❡ q

    • q

      |u |

      tu ∈ M

      q,Ω

      ✳ ▲♦❣♦✱ ❞❡ ✭✸✳✷✹✮ ❡ ❞❡ ✭✸✳✷✺✮ t❡♠♦s Z

      1

      1

      p p

      J (tu) = − (|∇(tu)| + |tu| )

      q,Ω

      p q

      

      Z

      1

      1

      p p

      ≤ − (|∇u| + |u| )

      N

      p q R Z

      1

      1

      p p

      ≤ lim inf − (|∇u | + |u | )

      n n n→∞ N

      p q

      R

      ❈❛♣ít✉❧♦ ✸✳ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✻✹ Z

      1

      1

      p p

      ≤ lim inf − (|∇u | + (λ a(x) + 1)|u | )

      n n n n→∞ N

      p q R = lim inf I (u ) ≤ m .

      λ n ,q n q,r n→∞

    • (tu) ∈ Ω

      q

      ❉❛í ❡ ❞♦ ▲❡♠❛ ✸✳✹ s❡❣✉❡ q✉❡ β

      r ✳

      )

      n

      ❉❡ ✭✸✳✷✺✮✱ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ (u ❡stá ♣♦r ❜❛✐①♦ ❞❡

      q N N

      (R ) (x) → u(x) ✉♠❛ ❢✉♥çã♦ f ∈ L ❡ u n q✳t✳♣✳ ❡♠ R ✳ ❉✐ss♦ ❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ξ✱ ♦❜t❡♠♦s

      

      q

      |f (x)| R, R ,

       s❡ x ∈ B

      q

      |u (x)| |ξ(|x|)||x| ≤

      n R q

      |f (x)| |x|,  R s❡ x 6∈ B

      |x|

      q

      1 N

      = |f (x)| R ∈ L (R ), ❝♦♠♦ t❛♠❜é♠✱

      q q N

      |u

    n (x)| ξ(|x|)x → |u(x)| ξ(|x|)x .

    q✳t✳♣✳ ❡♠ R

      c R

      ❙❡♥❞♦ u ≡ 0 ❡♠ Ω ❡ Ω ⊂ B ✱ ♣❡❧♦ ❚✳❈✳❉✳▲✳✱ ♣♦r ✭✸✳✷✺✮ ❡ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ξ ✈❡♠ q✉❡ R R

      q q N |u| ξ(|x|)x dx |u| x dx

    • R Ω lim β (u ) = = = β (u) = β (tu) ∈ Ω .

      

    n R R q q

    q r q q n→∞

      N |u| dx |u| dx R Ω

    • q 2r ❡ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛✳ − Ω (Ω) = (Ω )

      (u n ) 6∈ Ω ■ss♦ ❝♦♥tr❛❞✐③ ♦ ❢❛t♦ ❞❡ β

    • Ω r

      ▲❡♠❛ ✸✳✻ ❝❛t ❝❛t ✳

      2r − +

      ⊂ Ω ❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ Ω é ❝♦♠♣❛❝t♦✱ ❛ Pr♦♣♦s✐çã♦ ✶✳✶✷ (iv) ♥♦s ❣❛r❛♥t❡ q✉❡

      r 2r − − − N

      cat (Ω ) < ∞ (Ω ) ⊂ A ∪ . . . ∪ A ⊂ R

      1 k i r r r

      ✳ ❙❡❥❛ k := cat ✳ P♦r ❞❡✜♥✐çã♦✱ Ω ❝♦♠ A

      Ω Ω 2r 2r

    • − −

      ∩ Ω

      i

      ❢❡❝❤❛❞♦ ❡ ❝♦♥trát✐❧ ❡♠ Ω 2r ✱ ♣❛r❛ i ∈ {1, . . . , k}✳ ❈♦♠♦ A r é ❝♦♥trát✐❧ ❡♠ Ω 2r ❡ Ω r

      − −

      = A ∪ . . . ∪ A

      1 k

      é ❢❡❝❤❛❞♦✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ Ω r ✳ ❙❡♥❞♦ Ω r ❡ Ω ❤♦♠♦t♦♣✐❝❛♠❡♥t❡

      − −

      → Ω ❡q✉✐✈❛❧❡♥t❡s✱ ❡①✐st❡♠ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s f : Ω ❡ g : Ω → Ω t❛✐s q✉❡ f ◦ g ≃ Id Ω ✱

      r r

      ✐st♦ é✱ ❡①✐st❡ H : [0, 1] × Ω → Ω ❝♦♥tí♥✉❛ t❛❧ q✉❡ H(0, x) = f(g(x)) ❡ H(1, x) = x ♣❛r❛ t♦❞♦ x ∈ Ω✳ ◆♦ss♦ ♦❜❥❡t✐✈♦ ❛❣♦r❛ é ❡st❡♥❞❡r f ❝♦♥t✐♥✉❛♠❡♥t❡ ❛ Ω 2r ✳ P❡❧♦ ❚❡♦r❡♠❛ ✶✳✷✺ ✭❉✉❣✉♥❞❥✐✮

    • ❜❛st❛ ♠♦str❛r q✉❡ Ω é ✉♠ ❡s♣❛ç♦ ❛✜♠ ❞♦ t✐♣♦ m ✭❝♦♥❢♦r♠❡ ❉❡✜♥✐çã♦ ✶✳✷✹✮✳ P❛r❛ ✐ss♦✱ s❡❥❛

      X ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ q✉❛❧q✉❡r✳ P❛r❛ t♦❞❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ F : X → Ω✱ s❡❥❛♠ x ∈ X ❡

      N

      W ⊃ F (x) ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ F (x) ❞❛❞♦s ❛r❜✐tr❛r✐❛♠❡♥t❡✳ ❈♦♠♦ W, Ω ⊂ R sã♦ ❛❜❡rt♦s✱

      −1

      > 0 (F (x)) ⊂ W (F (x)) ⊂ Ω (B (F (x)))

      R R R

      ❡①✐st❡ R t❛❧ q✉❡ B ❡ B ✳ ❙❡❥❛ U = F ✳

      ❈❛♣ít✉❧♦ ✸✳ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✻✺ W R (F (x))

      ✳ ❙❡♥❞♦ ❛ ❜♦❧❛ B ❝♦♥✈❡①❛✱ Ω é ✉♠ ❡s♣❛ç♦ ❛✜♠ ❞♦ t✐♣♦ m✳ ❉❛í✱ ♣❡❧♦ ❚❡♦r❡♠❛

    • r ✳ −1

      −

      f : Ω → Ω → Ω ✶✳✷✺✱ ❡①✐st❡ ˜ 2r ❝♦♥tí♥✉❛ q✉❡ ❡st❡♥❞❡ f : Ω

      = g (A ) ∪ . . . ∪ B

      i i 1 k

      P❛r❛ ❝❛❞❛ i ∈ {1, . . . , k}✱ ❞❡✜♥✐♠♦s B ✳ ❊♥tã♦ Ω = B ✳ ❈♦♠♦

    • g

      i i

      é ❝♦♥tí♥✉❛ ❡ A é ❢❡❝❤❛❞♦ ❡♠ Ω 2r ✱ t❡♠♦s q✉❡ B é ❢❡❝❤❛❞♦ ❡♠ Ω✳ ❱❛♠♦s ♠♦str❛r q✉❡

      B : [0, 1] × A → Ω

      i i i i

      é ❝♦♥trát✐❧ ❡♠ Ω✳ P❛r❛ ✐ss♦✱ s❡♥❞♦ A ❝♦♥trát✐❧ ❡♠ Ω ✱ ❡①✐st❡ ψ

      2r 2r

    • ∈ Ω

      i (0, x) = x i (1, x) = x i i i 2r

      ❝♦♥tí♥✉❛ t❛❧ q✉❡ ψ ❡ ψ ♣❛r❛ t♦❞♦ x ∈ A ❡ ♣❛r❛ ❛❧❣✉♠ x ✳

      : [0, 1] × B → Ω

      i i

      ❉❡✜♥❛ θ ♣♦r 

      1  

      H(1 − 2t, x), , s❡ 0 ≤ t ≤

      2 θ (t, x) =

      i

      1  ˜ ≤ t ≤ 1.  f [ψ (2t − 1, g(x))],

      i s❡

      2

      i

      P❡❧❛s ❝♦♥s✐❞❡r❛çõ❡s ❥á ❢❡✐t❛s✱ θ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡ ✈❛❧❡ θ (0, x) = H(1, x) = x

      

    i i

      ♣❛r❛ t♦❞♦ x ∈ B ❡ θ (1, x) = ˜ f [ψ (1, g(x))] = ˜ f (x ) ∈ Ω .

      

    i i i i

      ♣❛r❛ t♦❞♦ x ∈ B

      1

      i

      P❛r❛ t = ❡ x ∈ B t❡♠♦s

      2

      1 θ i , x = H(0, x) = f (g(x)) = ˜ f (g(x)),

      2

      −

      ♣♦✐s g(x) ∈ Ω ✳ P♦r ♦✉tr♦ ❧❛❞♦✱

      r

      1 θ i , x = ˜ f [ψ i (0, g(x))] = ˜ f (g(x)).

      2

      − i i Ω (Ω) ≤ k = cat (Ω )

    • Ω r

      ▲♦❣♦ θ é ❝♦♥tí♥✉❛ ♠♦str❛♥❞♦ q✉❡ B é ❝♦♥trát✐❧ ❡♠ Ω✳ ❆ss✐♠ cat ✳

      2r

      ⊂ Ω P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ Ω

      r 2r ❡ Ω 2r é ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ Ω✱ s❡❣✉❡ q✉❡ − +

      Ω r Ω 2r 2r 2r

    • cat (Ω ) ≤ cat (Ω ) = cat Ω (Ω).

      ■ss♦ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦ ❧❡♠❛✳

      ∗

      , p ) = Λ (q)

      2

      2

      2

      2

      ▲❡♠❛ ✸✳✼ ❙❡❥❛♠ q ♦❜t✐❞♦ ♥♦ ▲❡♠❛ ✸✳✺✱ q ∈ (q ❡ Λ s❛t✐s❢❛③❡♥❞♦ ❛ ♣r♦♣r✐✲ ❡❞❛❞❡ ❞♦ ❡♥✉♥❝✐❛❞♦ ❞♦ ▲❡♠❛ ✸✳✺✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ λ ≥ Λ

      2 ✱

      ❈❛♣ít✉❧♦ ✸✳ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✻✻

      m q,r

      } = {u ∈ N λ,q : I λ,q (u) ≤ m q,r

      ♦♥❞❡ Y = I ✳

      λ,q 1,p p q q,r = u ∈ W (B r ) \ {0} : kuk = |u|

      ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ U ∈ M ✉♠❛ ❢✉♥çã♦

      B r q,B r −

      (U ) = m (Y ) ≥ cat (Ω )

      

    q,B r q,r Y

    • r ✳ P❛r❛ Ω

      r❛❞✐❛❧ ♣♦s✐t✐✈❛ t❛❧ q✉❡ J ✳ ❱❛♠♦s ♠♦str❛r q✉❡ cat

      2r −

      : B (y) → R (x) = U (x − y) (Y ) = +∞

      y r y Y

      ✐ss♦✱ ❞❛❞♦ y ∈ Ω r ✱ ❞❡✜♥✐♠♦s U ♣♦r U ✳ ❙❡ cat ✱ (Y ) = k < +∞

      

    Y

      ❛ ❞❡s✐❣✉❛❧❞❛❞❡ é ✈á❧✐❞❛✳ ❙✉♣♦♥❤❛ q✉❡ cat ✳ ❊♥tã♦ ❡①✐st❡♠ ❝♦♥❥✉♥t♦s A , . . . , A ⊂ Y

      ∪ . . . ∪ A

      1 k ❢❡❝❤❛❞♦s ❡ ❝♦♥trát❡✐s ❡♠ Y t❛✐s q✉❡ Y = A 1 k ✳ P♦r ❞❡✜♥✐çã♦ ❞❡

      → Y

      i : [0, 1] × A i

      ❝♦♥❥✉♥t♦ ❝♦♥trát✐❧✱ ♣❛r❛ ❝❛❞❛ i ∈ {1, . . . , k}✱ ❡①✐st❡ ψ ❝♦♥tí♥✉❛ t❛❧ q✉❡

      −

      ψ (0, u) = u (1, u) = u ∈ Y → Y

      i i i i i

      ❡ ψ ♣❛r❛ t♦❞♦ u ∈ A ❡ ❛❧❣✉♠ u ✳ ❉❡✜♥✐♠♦s α : Ω r ♣♦r

      N

      α(y) = U \B (y)

      y y r

      ✳ ❚❡♠♦s q✉❡ α ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ ❉❡ ❢❛t♦✱ ❡st❡♥❞❡♥❞♦ U ❝♦♠♦ ✵ ❡♠ R

      −

      (y)

      r

      ❡ s❡♥❞♦ a ≡ 0 ❡♠ Ω ⊃ B ✱ ❥á q✉❡ y ∈ Ω r ✱ s❡❣✉❡ q✉❡

      p p p

      kU k = kU k = kU k ,

      y y r ✭✸✳✷✽✮

      λ B r (y) B

      ♦♥❞❡ ♥❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ✉s❛♠♦s ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ✉♠❛ ✈❡③ q✉❡ U ∈ M

      q,Ω

      ✱

      p q q q

      kU k = |U | = |U | = |U | .

      y y B r q,B r q ✭✸✳✷✾✮ q,B r (y)

      ∈ N

      y λ,q

      ❉❡ ✭✸✳✷✽✮ ❡ ✭✸✳✷✾✮✱ U ✳ ❆❧é♠ ❞✐ss♦✱ ✉s❛♥❞♦ ♥♦✈❛♠❡♥t❡ ♦ ❚❡♦r❡♠❛ ❞❛ ▼✉❞❛♥ç❛ ❞❡

      r

      ❱❛r✐á✈❡✐s ❡ ♦ ❢❛t♦ ❞❡ a ≡ 0 ❡♠ Ω ⊃ B ✱ ♦❜t❡♠♦s I (U ) = I (U ) = J (U ) = m .

      

    λ,q y λ,q q,B r q,r

      ∈ Y (y)

      ▲♦❣♦ U y ❡ α ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ P❛r❛ y ∈ Ω ✱ ✉♠❛ ✈❡③ q✉❡ ξ ≡ 1 ❡♠ Ω ⊃ B r ✱

      r

      s❡❣✉❡ ❞❡ ✭✸✳✷✸✮ q✉❡ R R

      q q

      |U (x − y)| 1x dx

      N |U

      (x)| ξ(|x|)x dx

      R y B r (y)

      β (α(y)) = β (U y ) = R = R

      q q q q N |U (x)| dx |U (x − y)| dx y R

      B r (y)

      R

      q

      |U (z)| (z + y)dz

      B r

      = R ✭✸✳✸✵✮

      q

      |U (z)| dz

      B r

      R

      q

      |U (z)| z dz

      B r

      = + y = y, R

      q

      |U (z)| dz

      B r −1

      = α (A )

      i i

      ♣♦✐s ❛ ✐♥t❡❣r❛❧ ❛ ❡sq✉❡r❞❛ é ♥✉❧❛✱ ✈✐st♦ q✉❡ U é r❛❞✐❛❧✳ ❆❣♦r❛✱ s❡❥❛ B ✳ ❊♥tã♦

      − −

      Ω = B ∪ . . . ∪ B

      1 k i

    r ✳ ❆❞♠✐t✐♥❞♦ q✉❡ α é ❝♦♥tí♥✉❛✱ t❡♠♦s q✉❡ B é ❢❡❝❤❛❞♦ ❡♠ Ω r ✱

      ❈❛♣ít✉❧♦ ✸✳ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✻✼

      N + +

      ∩ B B i i = Ω i i

      é ❢❡❝❤❛❞♦ ❡♠ R ✳ ❙❡♥❞♦ B ✱ t❡♠♦s q✉❡ B é ❢❡❝❤❛❞♦ ❡♠ Ω ✳ ❱❛♠♦s

      2r 2r

      : [0, 1] × B → Ω

      i i i

      ♠♦str❛r q✉❡ B é ❝♦♥trát✐❧ ❡♠ Ω 2r ✳ P❛r❛ t❛♥t♦✱ ❝♦♥s✐❞❡r❡ θ

      2r ❞❛❞❛ ♣♦r

      θ (t, y) = β (ψ (t, α(y)))

      i i q ✳ ❊♥tã♦✱ ♣❡❧❛s ❝♦♥s✐❞❡r❛çõ❡s ❢❡✐t❛s ❛❝✐♠❛ ❡ ♣❡❧♦ ▲❡♠❛ ✸✳✺✱ ♣❛r❛

      λ ≥ Λ

      2 i i i

      ✱ θ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ ❚❡♠♦s t❛♠❜é♠ q✉❡ θ é ❝♦♥tí♥✉❛✱ ✈✐st♦ q✉❡ β q ✱ ψ ❡ α ♦ sã♦✳ ❉❡ ✭✸✳✸✵✮ ❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ θ i s❡❣✉❡ q✉❡ θ (0, y) = β (ψ (0, α(y))) = β (α(y)) = y,

      i i i q q ♣❛r❛ t♦❞♦ y ∈ B

      ❡ θ (1, y) = β (ψ (1, α(y))) = β (u ), .

      

    i i i i

    q q ♣❛r❛ t♦❞♦ y ∈ B

      (Ω ) ≤ k = cat (Y )

      i Y

    • 2r r ✳ ❙❡❣✉❡ ❞♦ ▲❡♠❛ ✸✳✻ q✉❡ Ω

      ▲♦❣♦ B é ❝♦♥trát✐❧ ❡♠ Ω ✳ ❆ss✐♠ cat

      2r − cat (Y ) ≥ cat (Ω ) = cat (Ω).

    • +

      Y Ω

      Ω r

      

    2r

      P❛r❛ ❝♦♥❝❧✉✐r ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛✱ r❡st❛ ❛♣❡♥❛s ♠♦str❛r q✉❡ ❛ ❢✉♥çã♦ α é ❝♦♥tí♥✉❛✳

      

    − −

      ) ⊂ Ω → y ∈ Ω ) → P❛r❛ ✐ss♦✱ s❡❥❛ (y n ✉♠❛ s❡q✉ê♥❝✐❛ t❛❧ q✉❡ y n ✳ ◗✉❡r❡♠♦s q✉❡ α(y n

      r r c

      ≡ 0 α(y) λ y n , U y n ) − α(y)k λ =

      ❡♠ E ✳ ❈♦♠♦ a ≡ 0 ❡♠ Ω ❡ U ❡♠ Ω ✱ t❡♠♦s q✉❡ kα(y

      ∞ N 1,p N

      kα(y ) − α(y)k ) ⊂ C (R ) ⊂ W (R ) → U

      n k k

      ✳ ❈♦♥s✐❞❡r❡ (U ✉♠❛ s❡q✉ê♥❝✐❛ t❛❧ q✉❡ U

      1,p N

      (R ) ∈ N

      ❡♠ W q✉❛♥❞♦ k → ∞✳ ❉❛❞♦ ε > 0✱ ❡①✐st❡ k t❛❧ q✉❡ ε kU (· − y k <

      

    n ) − U k (· − y n )k = kU − U k

      ✭✸✳✸✶✮

      3 ❡

      ε kU (· − y) − U k <

      k (· − y)k = kU − U k .

      ✭✸✳✸✷✮

      3 ∈ N

      ❆✜r♠❛♠♦s q✉❡ ❡①✐st❡ n t❛❧ q✉❡ ε kU (· − y ) − U (· − y)k < , .

      k n k

      s❡♠♣r❡ q✉❡ n ≥ n ✭✸✳✸✸✮

      3 N (x) = U (x − y ) − U (x − y), x ∈ R

      n k n k k

      ❈♦♠ ❡❢❡✐t♦✱ ❝♦♥s✐❞❡r❡ f ✳ ❈♦♠♦ s✉♣♣ U é

      c

      → y (x − y ) = 0 (x − y) = 0 (y)

      n k n k

      ❝♦♠♣❛❝t♦ ❡ y ✱ ♣♦❞❡♠♦s s✉♣♦r U ❡ U ❡♠ B t ♣❛r❛

      k

      ❛❧❣✉♠ t > 0 ❡ n ≥ n s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳ ❙❡♥❞♦ U ❝♦♥tí♥✉❛✱ t❡♠♦s

      1

      |f (x)| ≤ C ∈ L (B (y)),

      n t N

      → 0

      n

      ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ C > 0✳ ❈♦♠♦ f q✳t✳♣✳ ❡♠ R ✱ s❡❣✉❡ ❞♦ ❚✳❈✳❉✳▲✳✱ Z Z

      p p

      ❈❛♣ít✉❧♦ ✸✳ ❖ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞♦ ❝♦♥❥✉♥t♦ Ω ♥♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ✻✽ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱

      Z

      p

      |∇U (x − y ) − ∇U (x − y)| → 0,

      k n k N R

      ∞ N ∞ N

      ∈ C (R ) ∈ C (R )

      k k

      ♦❜s❡r✈❛♥❞♦ q✉❡ U ❡ ❡♥tã♦ ∇U ✳ ■ss♦ ♠♦str❛ ✭✸✳✸✸✮✳ ❉❛í✱ ❞❡ ✭✸✳✸✶✮ ❡ ❞❡ ✭✸✳✸✷✮✱ s❡❣✉❡ q✉❡ kα(y

      

    n ) − α(y)k = kU (· − y n ) − U (· − y)k ≤ kU (· − y n ) − U k (· − y n )k

    • kU (· − y ) − U (· − y)k

      k n k

    • kU k (· − y) − U (· − y)k < ε, ♣❛r❛ n ≥ n ✳ ■ss♦ ♠♦str❛ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❢✉♥çã♦ α✳

      = q

      2

      2

      ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❉✿ ❙❡❥❛ q ❞❛❞♦ ♣❡❧♦ ▲❡♠❛ ✸✳✺ ❡ ❝♦♥s✐❞❡r❡ q ✳ ❉❛❞♦

      ∗

      q ∈ (q , p ) = Λ (q)

      2

      2

      ✱ s❡❥❛ Λ ❞❛❞♦ ♣❡❧♦ ▲❡♠❛ ✸✳✺✳ ❆♣❧✐❝❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ ✷✳✸ ❝♦♠ C = m = Λ (q)

      

    1 q,r ✱ ♦❜t❡♠♦s Λ t❛❧ q✉❡ I λ,q s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ♥♦ ♥í✈❡❧

      } c q,r , Λ

      2 λ,q

      ✱ ♣❛r❛ t♦❞♦ c ≤ m ❡ λ ≥ Λ ✳ ❉❡✜♥✐♠♦s Λ = Λ(q) = max{Λ ✳ ❚❡♠♦s q✉❡ I

      λ,q λ,q

      r❡str✐t♦ à N é ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡✳ P❡❧♦ ▲❡♠❛ ✷✳✶✽✱ N é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❝❧❛ss❡

    1 C

      λ λ,q λ,q

      ✳ P❛r❛ λ > Λ✱ ♣❡❧♦ ▲❡♠❛ ✸✳✼ ❡ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✶✸ ❝♦♠ X = E ✱ I = I ✱ M = N ✱

      d

      d = m = Y (Y ) ≥ cat (Ω)

      q,r ❡ I ✱ ✈❡♠♦s q✉❡ I λ,q r❡str✐t♦ à N λ,q t❡♠ ♣❡❧♦ ♠❡♥♦s cat Y Ω i λ,q (u i ) ≤ m q,r λ,q Ω (Ω)

      ♣♦♥t♦s ❝rít✐❝♦s u t❛✐s q✉❡ I ✳ P❡❧♦ ▲❡♠❛ ✷✳✶✾✱ I ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s cat ♣♦♥t♦s ❝rít✐❝♦s✳ ❈♦♠♦ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❆✱

      p ′ − −

      i = ku k 0 = hI (u ), u

      i λ,q i i λ

    • = u ≥ 0 > 0

      i i

      ❡✱ ♣♦rt❛♥t♦✱ u

      i ✳ P❡❧♦ Pr✐♥❝í♣✐♦ ❞♦ ▼á①✐♠♦ ❋♦rt❡✱ u ✳ ❆ss✐♠✱ ♦❜t❡♠♦s ♣❡❧♦

    • (Ω) )

      ) ♠❡♥♦s cat Ω s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ❞❡ (S ✱ q✉❡ t❛♠❜é♠ sã♦ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ❞❡ (S λ,q ✳

      λ,q

      ■ss♦ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛✳

      ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

      ❬✶❪ ❆❧✈❡s✱ ❈✳❖✳ ❊①✐st❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ❢♦r ❛ ♣r♦❜❧❡♠ ✇✐t❤ ❧❛❝❦ ♦❢ ❝♦♠♣❛❝t♥❡ss ✐♥✈♦❧✈✐♥❣ t❤❡ ♣✲▲❛♣❧❛❝✐❛♥✳ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✺✶✱ ✶✶✽✼✲✶✷✵✻✱ ✷✵✵✷✳

      ❬✷❪ ❆❧✈❡s✱ ❈✳❖❀ ❉✐♥❣✱ ❨✳❍✳ ▼✉❧t✐♣❧✐❝✐t② ♦❢ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s t♦ ❛ ♣✲❧❛♣❧❛❝✐❛♥ ❡q✉❛t✐♦♥ ✐♥✈♦❧✈✐♥❣ ❝r✐t✐❝❛❧ ♥♦♥❧✐♥❡❛r✐t②✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✷✼✾✱ ✺✵✽✲✺✷✶✱ ✷✵✵✸✳

      ❬✸❪ ❆♠❜r♦s❡tt✐✱ ❆❀ ▼❛❧❝❤✐♦❞✐✱ ❆✳ ◆♦♥❧✐♥❡❛r ❛♥❛❧②s✐s ❛♥❞ s❡♠✐❧✐♥❡❛r ❡❧❧✐♣t✐❝ ♣r♦✲ ❜❧❡♠s✳ ❈❛♠❜r✐❞❣❡✱ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✷✵✵✼✳

      ❬✹❪ ❇❛rt❧❡✱ ❘✳●✳ ❚❤❡ ❡❧❡♠❡♥ts ♦❢ ✐♥t❡❣r❛t✐♦♥ ❛♥❞ ❧❡❜❡s❣✉❡ ♠❡❛s✉r❡✳ ❏♦❤♥ ❲✐❧❡② ❛♥❞ ❙♦♥s✱ ◆❡✇ ❨♦r❦✱ ✶✾✾✺✳

      ❬✺❪ ❇❛rts❝❤✱ ❚❀ ❲❛♥❣✱ ❩✳◗✳ ▼✉❧t✐♣❧❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ❢♦r ❛ ♥♦♥❧✐♥❡❛r ❙❝❤rö❞✐♥❣❡r ❡q✉❛✲ t✐♦♥✱ ❩❆▼P ✺✶✱ ✸✻✻✲✸✽✹✱ ✷✵✵✵✳ ❬✻❪ ❇❡♥❝✐✱ ❱❀ ❈❡r❛♠✐✱ ●✳ ❚❤❡ ❡✛❡❝t ♦❢ t❤❡ ❞♦♠❛✐♥ t♦♣♦❧♦❣② ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ♦❢ ♥♦♥❧✐♥❡❛r ❡❧❧✐♣t✐❝ ♣r♦❜❧❡♠s✱ ❆r❝❤✳ ❘❛t✐♦♥❛❧ ▼❡❝❤✳ ❆♥❛❧✳ ✶✶✹✱ ✼✾✲✾✸✱

      ✶✾✾✶✳ ❬✼❪ ❇❡♥❝✐✱ ❱❀ ❈❡r❛♠✐✱ ●✳ ▼✉❧t✐♣❧❡ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ♦❢ s♦♠❡ ❡❧❧✐♣t✐❝ ♣r♦❜❧❡♠s ✈✐❛ t❤❡

      ▼♦rs❡ t❤❡♦r② ❛♥❞ t❤❡ ❞♦♠❛✐♥ t♦♣♦❧♦❣②✱ ❈❛❧✳ ❱❛r✳ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✷✱ ✷✾✲✹✽✱ ✶✾✾✹✳

      ❬✽❪ ❇❡♥❝✐✱ ❱❀ ❈❡r❛♠✐✱ ●❀ P❛ss❛s❡♦✱ ❉✳ ❖♥ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ♦❢ s♦♠❡ ♥♦♥❧✐♥❡❛r ❡❧❧✐♣t✐❝ ♣r♦❜❧❡♠s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✱ ❆ tr✐❜✉t❡ ✐♥ ❤♦♥♦✉r ♦❢ ●✳ Pr♦❞✐✱ ◗✉❛❞❡r♥♦ ❙❝✉♦❧❛ ◆♦r♠✳ ❙✉♣✳✱ P✐s❛✱ ✾✸✲✶✵✼✱ ✶✾✾✶✳

      ❬✾❪ ❇ré③✐s✱ ❍✳ ❋✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s✱ ❙♦❜♦❧❡✈ s♣❛❝❡s ❛♥❞ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛✲ t✐♦♥s✳ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✱ ✷✵✶✵✳ ❬✶✵❪ ❇ré③✐s✱ ❍❀ ▲✐❡❜✱ ❊✳ ❆ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ♣♦✐♥t✇✐s❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ❛♥❞ ❝♦♥✲

      ❘❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s ✼✵

      ❬✶✶❪ ❈❡r❛♠✐✱ ●❀ P❛ss❛s❡♦✱ ❉✳ ❊①✐st❡♥❝❡ ❛♥❞ ♠✉❧t✐♣❧✐❝✐t② ♦❢ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ❢♦r ♥♦♥❧✐♥❡❛r ❡❧❧✐♣t✐❝ ♣r♦❜❧❡♠s ✐♥ ❡①t❡r✐♦r ❞♦♠❛✐♥s ✇✐t❤ ✏r✐❝❤✑ t♦♣♦❧♦❣②✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✶✽✱ ✶✵✾✲ ✶✶✾✱ ✶✾✾✷✳

      ❬✶✷❪ ❞♦ Ó✱ ❏✳ ▼✳ ❇✳ ❚❡♦r✐❛ ❞❡ ♣♦♥t♦s ❝rít✐❝♦s ❞❡ ▲✉st❡r♥✐❦✲❙❝❤♥✐r❡❧♠❛♥♥ ❡ ❛♣❧✐✲ ❝❛çõ❡s ❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s✱ ▼✐♥✐❝✉rs♦ ❞♦ ■ ❊❇❊❉✱ ■♠❡❝❝ ❯♥✐❝❛♠♣✱ ✷✵✵✸✳

      ❬✶✸❪ ❞♦s Pr❛③❡r❡s✱ ❉✳ P✳ ▼✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ♣r♦❜❧❡♠❛s ❡❧í♣t✐❝♦s s❡♠✐✲ ❧✐♥❡❛r❡s ❡♥✈♦❧✈❡♥❞♦ ♦ ❡①♣♦❡♥t❡ ❝rít✐❝♦ ❞❡ ❙♦❜♦❧❡✈✱ ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦✱ ❯❋P❇✱ ✷✵✶✵✳

      ❬✶✹❪ ❉✉❣✉♥❞❥✐✱ ❏✳ ❚♦♣♦❧♦❣②✱ ❆❧❧②♥ ❛♥❞ ❇❛❝♦♥✱ ❇♦st♦♥✱ ✶✾✻✻✳ ❬✶✺❪ ❋♦❧❧❛♥❞✱ ●✳ ❇✳ ❘❡❛❧ ❛♥❛❧②s✐s ♠♦❞❡r♥ t❡❝❤♥✐q✉❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s✱ ❏♦❤♥

      ❲✐❧❡② ❛♥❞ ❙♦♥s✱ ✶✾✾✾✳ ❬✶✻❪ ❋✉rt❛❞♦✱ ▼✳ ❋✳ ❆ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❞♦♠❛✐♥ t♦♣♦❧♦❣② ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ♠✐♥✐♠❛❧

      ♥♦❞❛❧ s♦❧✉t✐♦♥s ❢♦r ❛ q✉❛s✐❧✐♥❡❛r ❡❧❧✐♣t✐❝ ♣r♦❜❧❡♠✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s ✻✷✱ ✻✶✺✲✻✷✽✱ ✷✵✵✺✳

      ❬✶✼❪ ❋✉rt❛❞♦✱ ▼✳ ❋✳ ▼✉❧t✐♣❧❡ ♠✐♥✐♠❛❧ ♥♦❞❛❧ s♦❧✉t✐♦♥s ❢♦r ❛ q✉❛s✐❧✐♥❡❛r ❡q✉❛t✐♦♥ ✇✐t❤ s②♠♠❡tr✐❝ ♣♦t❡♥t✐❛❧✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✵✹✱ ✶✼✵✲✶✽✽✱ ✷✵✵✺✳ ❬✶✽❪ ❋✉rt❛❞♦✱ ▼✳ ❋✳ ▼✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ♥♦❞❛✐s ♣❛r❛ ♣r♦❜❧❡♠❛s ❡❧í♣t✐❝♦s q✉❛s✐❧✐♥❡❛r❡s✱ ❚❡s❡ ❞❡ ❞♦✉t♦r❛❞♦✱ ❯♥✐❝❛♠♣✱ ✷✵✵✹✳ ❬✶✾❪ ●✐❧❜❛r❣✱ ❉❀ ❚r✉♥❞✐♥❣❡r✱ ◆✳ ❙✳ ❊❧❧✐♣t✐❝ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♦❢ s❡❝♦♥❞

      ♦r❞❡r✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✱ ✶✾✽✸✳ ❬✷✵❪ ●♦✉❧❛rt✱ ❈✳ ❙✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞❡ ❙❝❤rö❞✐♥❣❡r ♥ã♦ ▲✐♥❡❛r❡s ❝♦♠ ❆❝♦♣❧❛✲

      ♠❡♥t♦✱ ❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦✱ ❯♥❇✱ ✷✵✶✶✳ ❬✷✶❪ ❑❛✈✐❛♥✱ ❖✳ ■♥tr♦❞✉❝t✐♦♥ à ❧❛ t❤é♦r✐❡ ❞❡s ♣♦✐♥ts ❝r✐t✐q✉❡s ❡t ❛♣♣❧✐❝❛t✐♦♥s ❛✉①

      ♣r♦❜❧è♠❡s ❡❧❧✐♣t✐q✉❡s✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✱ ✶✾✾✶✳ ❬✷✷❪ ▲❛③③♦✱ ▼❀ ❙♦❧✉t✐♦♥s ♣♦s✐t✐✈❡s ♠✉❧t✐♣❧❡s ♣♦✉r ✉♥❡ éq✉❛t✐♦♥ ❡❧❧✐♣t✐q✉❡ ♥♦♥ ❧✐♥é❛✐r❡ ❛✈❡❝

      ❧é①♣♦s❛♥t ❝r✐t✐q✉❡ ❞❡ ❙♦❜♦❧❡✈✱ ❈✳ ❘✳ ❆❝❛❞✳ ❙❝✐✳✱ P❛r✐s✱ ✸✶✹✱ ✻✶✲✻✹✱ ✶✾✾✷✳ ❬✷✸❪ ▲✐♠❛✱ ❊✳ ▲✳ ❈✉rs♦ ❞❡ ❆♥á❧✐s❡✱ ✈♦❧✳ ✷✱ ✹

      ❛

      ❡❞✳✱ ■▼P❆✱ ✶✾✾✺✳

      ❘❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s ✼✶

      ❬✷✺❪ ▲✐♥❞q✈✐st✱ P✳ ❖♥ t❤❡ ❡q✉❛t✐♦♥ ❞✐✈(|∇u|

      

    p−2

      ∇u) + λ|u|

      p−2

      u = 0 ✳ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡

      ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ❱✳ ✶✵✾✱ ♥✳ ✶✱ ♣✳ ✶✺✼✲✶✻✹✱ ✶✾✾✵✳ ❬✷✻❪ ▲✐♦♥s✱ P✳▲✳ ❚❤❡ ❝♦♥❝❡♥tr❛t✐♦♥ ❝♦♠♣❛❝t♥❡ss ♣r✐♥❝✐♣❧❡ ✐♥ t❤❡ ❝❛❧❝✉❧✉s ♦❢ ✈❛r✐❛t✐♦♥s✳

      ❚❤❡ ❧♦❝❛❧❧② ❝♦♠♣❛❝t ❝❛s❡✱ ❆♥♥✳ ■♥s✳ ❍❡♥r✐ P♦✐♥❝❛ré✱ ❆♥❛❧②s❡ ◆♦♥ ▲✐♥é❛✐r❡ ✶✱ ✶✵✾✲✶✹✺ ❡ ✷✷✸✲✷✽✸✱ ✶✾✽✹✳

      ❬✷✼❪ ▲✐♦♥s✱ P✳▲✳ ❚❤❡ ❝♦♥❝❡♥tr❛t✐♦♥ ❝♦♠♣❛❝t♥❡ss ♣r✐♥❝✐♣❧❡ ✐♥ t❤❡ ❝❛❧❝✉❧✉s ♦❢ ✈❛r✐❛t✐♦♥s✳ ❚❤❡ ❧✐♠✐t ❝❛s❡✱ ❘❡✈✳ ▼❛t✳ ■❜❡r♦❛♠❡r✐❝❛♥❛ ✶ ✶✹✺✲✷✵✶✱ ✶✾✽✺ ❡ ✷ ✹✺✲✶✷✶✱ ✶✾✽✺✳

      ❬✷✽❪ ❘❡②✱ ❖❀ ❆ ♠✉❧t✐♣❧✐❝✐t② r❡s✉❧t ❢♦r ❛ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ✇✐t❤ ❧❛❝❦ ♦❢ ❝♦♠♣❛❝t♥❡ss✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✶✸✱ ✶✷✹✶✲✶✷✹✾✱ ✶✾✽✾✳

      ❬✷✾❪ ❙❝❤✇❛rt③✱ ❏✳ ❚❀ ◆♦♥❧✐♥❡❛r ❢✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s✱ ●♦r❞♦♥ ❛♥❞ ❇r❡❛❝❤ ❙❝✐❡♥❝❡✱ ◆❡✇ ❨♦r❦✱ ✶✾✻✾✳

      ❬✸✵❪ ❙✐❧✈❛✱ ❊✳❆✳❇❀ ❙♦❛r❡s✱ ❙✳❍✳▼✳ ◗✉❛s✐❧✐♥❡❛r ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠s ✐♥ R

      N

      ✇✐t❤ ❝r✐t✐❝❛❧ ❣r♦✇t❤✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✹✸✱ ✶✲✷✵✱ ✷✵✵✶✳

      ❬✸✶❪ ❙♠❡ts✱ ❉✳ ❆ ❝♦♥❝❡♥tr❛t✐♦♥✲❝♦♠♣❛❝t❡♥ss ❧❡♠❛ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s t♦ s✐♥❣✉❧❛r ❡✐❣❡♥✈❛✲ ❧✉❡s ♣r♦❜❧❡♠s✱ ❏✳ ❋✉♥❝t✳ ❆♥❛❧✳ ✶✻✼✱ ✹✻✸✲✹✽✵✱ ✶✾✾✾✳

      ❬✸✷❪ ❙tr✉✇❡✱ ▼✳ ❱❛r✐❛t✐♦♥❛❧ ▼❡t❤♦❞s✳ ❆♣♣❧✐❝❛t✐♦♥s t♦ ♥♦♥❧✐♥❡❛r ♣❛rt✐❛❧ ❞✐✛❡r❡♥✲ t✐❛❧ ❡q✉❛t✐♦♥s ❛♥❞ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠s✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✱ ✶✾✾✵✳ ❬✸✸❪ ❲✐❧❧❡♠✱ ▼✳ ▼✐♥✐♠❛① t❤❡♦r❡♠s✳ ❇✐r❦❤ä✉s❡r✱ ❇❛s❡❧✱ ✶✾✾✻✳

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Universidade de Brasília Instituto de Ciências Exatas Departamento de Matemática Programa de Mestrado Profissional em Matemática em Rede Nacional
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Universidade Federal de Uberlândia Faculdade de Matemática Licenciatura em Matemática
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