Dissertação Jeferson Camilo Silva

Livre

0
0
78
1 year ago
Preview
Full text

  ❏❊❋❊❘❙❖◆ ❈❆▼■▲❖ ❙■▲❱❆ ❊❙❚■▼❆❚■❱❆❙ P❆❘❆ ❆❯❚❖❱❆▲❖❘❊❙ ❉❊ ❙■❙❚❊▼❆❙

  ❊▲❮P❚■❈❖❙ ◗❯❆❙❊ ▲■◆❊❆❘❊❙ ❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❡①✐✲ ❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳

  ❱■➬❖❙❆ ▼■◆❆❙ ●❊❘❆■❙ ✲ ❇❘❆❙■▲

  ✷✵✶✻ Ficha catalográfica preparada pela Biblioteca Central da Universidade

Federal de Viçosa - Câmpus Viçosa

  T Silva, Jeferson Camilo, 1992- S586e Estimativas para autovalores de sistemas elípticos quase 2016 lineares / Jeferson Camilo Silva. – Viçosa, MG, 2016. viii, 68f. ; 29 cm. Orientador: Anderson Luis Albuquerque de Araujo. Dissertação (mestrado) - Universidade Federal de Viçosa. Inclui bibliografia.

  Departamento de Matemática. Programa de Pós-graduação em Matemática. II. Título.

  

CDD 22. ed. 519.544

  ❏❊❋❊❘❙❖◆ ❈❆▼■▲❖ ❙■▲❱❆ ❊❙❚■▼❆❚■❱❆❙ P❆❘❆ ❆❯❚❖❱❆▲❖❘❊❙ ❉❊ ❙■❙❚❊▼❆❙

  ❊▲❮P❚■❈❖❙ ◗❯❆❙❊ ▲■◆❊❆❘❊❙ ❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s ❡①✐✲ ❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳

  ❆P❘❖❱❆❉❆✿ ✷✺ ❞❡ ❥✉❧❤♦ ❞❡ ✷✵✶✻✳ ▼❛r❣❛r❡t❤ ❞❛ ❙✐❧✈❛ ❆❧✈❡s

  ❊❞❡r ▼❛r✐♥❤♦ ▼❛rt✐♥s ❆♥❞❡rs♦♥ ▲✉✐s ❆❧❜✉q✉❡rq✉❡ ❞❡ ❆r❛✉❥♦

  ✭❖r✐❡♥t❛❞♦r✮

  ❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❛♦s ♠❡✉s ♣❛✐s✱ ❆❞ã♦ ❡ ■♦❧❛♥❞❛✳

  ❆ ♠❡♥t❡ q✉❡ s❡ ❛❜r❡ ❛ ✉♠❛ ♥♦✈❛

✐❞❡✐❛ ❥❛♠❛✐s ✈♦❧t❛rá ❛♦ s❡✉

t❛♠❛♥❤♦ ♦r✐❣✐♥❛❧✳ ❆❧❜❡rt ❊✐♥st❡✐♥

  ❆❣r❛❞❡❝✐♠❡♥t♦s

  ❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛♦s ♠❡✉s ♣❛✐s✱ ❆❞ã♦ ❡ ■♦❧❛♥❞❛✱ ♣❡❧♦ ❛♠♦r✱ ✐♥❝❡♥t✐✈♦ ❡ ❛♣♦✐♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧✳

  ❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❆♥❞❡rs♦♥✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❛♣r❡♥❞✐③❛❞♦ ✈❛❧✐♦s♦✱ ♣❡❧❛s s✉❛s ❝♦rr❡çõ❡s ❡ ✐♥❝❡♥t✐✈♦✳ ❊♥✜♠ ♣❡❧❛ ❣r❛♥❞❡ ♣❡ss♦❛ q✉❡ é✳

  ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s ❞❡ ❝✉rs♦ ♣❡❧❛ ❛♠✐③❛❞❡✱ ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦ ❡ ❞❡ ❡st✉❞♦s✳ ❱♦❝ês ✜③❡r❛♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❡ ✈ã♦ ❝♦♥t✐♥✉❛r ♣r❡s❡♥t❡s ❡♠ ♠✐♥❤❛ ✈✐❞❛ ❝♦♠ ❝❡rt❡③❛✳

  ❆♦s ♣r♦❢❡ss♦r❡s ❡ ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉▼❆✲❯❋❱✱ ♣♦r ❝♦❧❛❜♦r❛r❡♠ ❝♦♠ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❡ ♣❡❧♦s ❡✜❝✐❡♥t❡s s❡r✈✐ç♦s ♣r❡st❛❞♦s✳

  ❆❣r❛❞❡ç♦ ❛ t♦❞♦s q✉❡ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❋✐♥❛❧♠❡♥t❡✱ ❛❣r❛❞❡ç♦ à ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ✐♥❞✐s♣❡♥sá✈❡❧ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

  ◆♦t❛çõ❡s n

  • R r❡♣r❡s❡♥t❛rá ♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ n✲❞✐♠❡♥s✐♦♥❛❧✳

  n

  é ✉♠ ❛❜❡rt♦ ❧✐♠✐t❛❞♦ ❞♦ R ❝♦♠ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ |Ω| ❡ ❢r♦♥t❡✐r❛ ∂Ω✳

  n

  ∂

  • ∇ =

  ❞❡♥♦t❛rá ♦ ♦♣❡r❛❞♦r ❣r❛❞✐❡♥t❡✳ ∂x

  i i=1 n

  X ∂u

  i

  • div u =

  ❞❡♥♦t❛rá ♦ ❞✐✈❡r❣❡♥t❡ ❞❡ ✉✳ ∂x

  i i=1 p−2

  • ∆ u = div (|∇u| ∇u)

  p

  ❞❡♥♦t❛rá ♦ ♦♣❡r❛❞♦r ♣✲▲❛♣❧❛❝✐❛♥♦ ❛♣❧✐❝❛❞♦ ❡♠ u✳

  • |u| := |u(x)|

  ✳

  • |∇u| := |∇u(x)|

  ✳

  • → :

  ❝♦♥✈êr❣❡♥❝✐❛ ❢♦rt❡✳

  • ⇀ :

  ❝♦♥✈❡❣ê♥❝✐❛ ❢r❛❝❛✳

  1

  1

  ! !

  n 2 n

  2

  2 X

  X ∂u

  2

  • |x| = x

  i ❡ |∇u| = ❞❡♥♦t❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛

  ∂x

  i i=1 i=1 n

  ♥♦r♠❛ ❡✉❝❧✐❞✐❛♥❛ ❞❡ x ∈ R ❡ ❛ ♥♦r♠❛ ❞♦ ✈❡t♦r ❣r❛❞✐❡♥t❡✳

  1/p

  Z

  p p 1,p

  • kuk |∇u|

  p = kuk = k∇uk L (Ω) =

  ✳

  W (Ω) Ω

  ❘❡s✉♠♦

  ❙■▲❱❆✱ ❏❡❢❡rs♦♥ ❈❛♠✐❧♦✱ ▼✳❙❝✳ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❥✉❧❤♦ ❞❡ ✷✵✶✻✳ ❊st✐♠❛t✐✈❛s ♣❛r❛ ❛✉t♦✈❛❧♦r❡s ❞❡ s✐st❡♠❛s ❡❧í♣t✐❝♦s q✉❛s❡ ❧✐♥❡❛r❡s✳ ❖r✐❡♥t❛❞♦r✿ ❆♥❞❡rs♦♥ ▲✉✐s ❆❧❜✉q✉❡rq✉❡ ❞❡ ❆r❛✉❥♦✳ ◆♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛✲s❡ ❡st✐♠❛t✐✈❛s ♣❛r❛ ❛✉t♦✈❛❧♦r❡s ❞❡ s✐st❡♠❛s ❡❧í♣t✐❝♦s✳ ❖ ♦❜❥❡t✐✈♦ é ❡st❛❜❡❧❡❝❡r ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ❛ ❡st✐♠❛t✐✈❛ ♦❜t✐❞❛ ♣❛r❛ ♦s ❛✉t♦✈❛❧♦r❡s ❞♦ s✐st❡♠❛ ❝♦♠ ❞♦♠í♥✐♦ ❡♠ R s❡❥❛ ❛ ♠❡s♠❛ ♣❛r❛ ✉♠❛ ❝❧❛ss❡ ❞❡

  n n

  ❝♦♥❥✉♥t♦s ❡♠ R q✉❛♥❞♦ ❝♦♥s✐❞❡r❛✲s❡ ♦ s✐st❡♠❛ ❝♦♠ ❞♦♠í♥✐♦ ❡♠ R ✳ P❛r❛ ✐ss♦✱ ✉t✐❧✐③❛✲s❡ ✈ár✐❛s ♠✉❞❛♥ç❛s ❞❡ ✈❛r✐á✈❡✐s s♦❜r❡ ♦ s✐st❡♠❛ ❝♦♥s✐❞❡r❛❞♦✳

  ❆❜str❛❝t

  ❙■▲❱❆✱ ❏❡❢❡rs♦♥ ❈❛♠✐❧♦✱ ▼✳❙❝✳ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❥✉❧②✱ ✷✵✶✻✳ ❊st✐♠❛t❡s ❢♦r ❡✐❣❡♥✈❛❧✉❡s ♦❢ ❛❧♠♦st ❧✐♥❡❛r ❡❧❧✐♣t✐❝ s②st❡♠s✳ ❆❞✈✐s❡r✿ ❆♥❞❡rs♦♥ ▲✉✐s ❆❧❜✉q✉❡rq✉❡ ❞❡ ❆r❛✉❥♦✳ ■♥ t❤✐s ♠❛❣✐st❡r✬s ❞✐ss❡rt❛t✐♦♥ ✇❡ ♣r❡s❡♥t ❡st✐♠❛t❡s ❢♦r ❡✐❣❡♥✈❛❧✉❡s ♦❢ ❡❧❧✐♣t✐❝ s②st❡♠s✳ ❚❤❡ ❣♦❛❧ ✐s t♦ ❡st❛❜❧✐s❤ ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ❡st✐♠❛t❡ ♦❜t❛✐♥❡❞ ❢♦r t❤❡

  n

  ❡✐❣❡♥✈❛❧✉❡s ♦❢ t❤❡ s②st❡♠ ✇✐t❤ ❞♦♠❛✐♥ ✐♥ R ✐s t❤❡ s❛♠❡ ❢♦r ❛ ❝❧❛ss ♦❢ s❡ts ✐♥ R ✱

  n

  ✇❤❡♥ ✇❡ ❝♦♥s✐❞❡r t❤❡ s②st❡♠ ✇✐t❤ ❞♦♠❛✐♥ ✐♥ R ✳ ❋♦r t❤✐s✱ ✇❡ ✉s❡ s❡✈❡r❛❧ ❝❤❛♥❣❡s ♦❢ ✈❛r✐❛❜❧❡s ♦♥ t❤❡ s②st❡♠ ❝♦♥s✐❞❡r❡❞✳

  ❙✉♠ár✐♦

   ✶

   ✹

  ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

   ✶✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

  

  ✹✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

  

  

   ✺✷

  

   ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻

   ✻✻

   ✻✼

  ■♥tr♦❞✉çã♦

  ❊st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❡♥❝♦♥tr❛r ❧✐♠✐t❡s ✐♥❢❡r✐♦r❡s ❡ s✉♣❡r✐♦r❡s ♣❛r❛ ♦s ❛✉t♦✈❛❧♦r❡s ❞❡ s✐st❡♠❛s ❡❧í♣t✐❝♦s ♥ã♦✲❧✐♥❡❛r❡s✳ ❊①✐st❡♠ ✈ár✐♦s ❧✐♠✐t❡s ♣❛r❛ ✈❛❧♦r❡s ♣ró♣r✐♦s ❞❡ ✉♠❛ ú♥✐❝❛ ❡q✉❛çã♦ ❡❧í♣t✐❝❛✱ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❧✐♥❡❛r✱ ❝♦♠ ❜❛s❡ ❡♠ ❞✐❢❡r❡♥t❡s té❝♥✐❝❛s✳ ◆♦ ❡♥t❛♥t♦ ❛ s✐t✉❛çã♦ é ❞✐❢❡r❡♥t❡ ♣❛r❛ s✐st❡♠❛s ❡❧í♣t✐❝♦s✱ ❡ ❤á ♣♦✉❝♦s r❡s✉❧t❛❞♦s ♥❡st❡ ❝❛s♦✳ P♦❞❡rí❛♠♦s ❝✐t❛r ♦ tr❛❜❛❧❤♦ ❞❡ Pr♦tt❡r q✉❡

  , · · · , λ ) ∈

  1 m

  ✐♥tr♦❞✉③✐✉ ❛ ♥♦çã♦ ❞❡ ❡s♣❡❝tr♦ ❣❡♥❡r❛❧✐③❛❞♦✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s λ = (λ

  m

  C t❛❧ q✉❡ ♦ s❡❣✉✐♥t❡ s✐st❡♠❛

  m m

  X X L u + λ r u = 0 , 1 ≤ j ≤ m

  ij j i,j i

  ✭✶✮

  i=1 i=1

  , · · · , u )

  i m

  t❡♠ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ u = (u s✉❥❡✐t❛ ❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝♦♥❞✐çõ❡s ❞❡

  ij

  ❝♦♥t♦r♥♦ ❤♦♠♦❣ê♥❡❛s✱ ♦♥❞❡ L sã♦ ♦♣❡r❛❞♦r❡s ❡❧í♣t✐❝♦s ❧✐♥❡❛r❡s✳ ❊♠ s❡❣✉✐❞❛ ♦ ❡s♣❡❝tr♦ ❣❡♥❡r❛❧✐③❛❞♦ ❢♦✐ ❡st❡♥❞✐❞♦ ❛ s✐st❡♠❛s ❡❧í♣t✐❝♦s ❣❡r❛✐s✳ ❈♦♠ ❛ ♠❡s♠❛ ❛❜♦r❞❛❣❡♠ Pr♦tt❡r ♦❜t❡✈❡ ♦s s❡❣✉✐♥t❡s r❡s✉❧t❛❞♦s✿

  X

  2

  ∈ R < λ

  Ω Ω

  ✭✐✮ ❊①✐st❡ ✉♠ ❛✉t♦✈❛❧♦r ♣♦s✐t✐✈♦ r t❛❧ q✉❡ r i ♣❛r❛ t♦❞♦ λ ♥♦

  i

  ❡s♣❡❝tr♦ ❣❡♥❡r❛❧✐③❛❞♦✳

  m

  ✭✐✐✮ P❛r❛ ❝❛❞❛ λ ∈ C ✱ s❡ Ω ❡stá ❝♦♥t✐❞♦ ❡♠ ✉♠❛ ❡s❢❡r❛ ❞❡ r❛✐♦ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ❞❡♣❡♥❞❡♥❞♦ ❞♦ s❡✉ t❛♠❛♥❤♦ ❛♣❡♥❛s ♥♦s ❝♦❡✜❝✐❡♥t❡s✱ ❡♥tã♦ ♥ã♦ ❡①✐st❡♠ s♦❧✉çõ❡s ♥ã♦ tr✐✈✐❛✐s ❞♦ s✐st❡♠❛✳

  ❆❧é♠ ❞✐ss♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❛✉t♦✈❛❧♦r❡s ♥ã♦✲❧✐♥❡❛r✱

  m

  X L + λf (x, u) = 0 , 1 ≤ j ≤ m,

  ij

  ✭✷✮

  i=1

  ❡♠ q✉❡ f(x, u) é ♥ã♦ ❧✐♥❡❛r ❡ s❛t✐s❢❛③ ❝❡rt❛s ❝♦♥❞✐çõ❡s ❞❡ ❝r❡s❝✐♠❡♥t♦✱ Pr♦tt❡r ♦❜t❡✈❡ r❡❧❛çõ❡s ❡♥tr❡ ❛ ♥♦r♠❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ❡ λ✱ ✉s❛♥❞♦ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❞❡ ❋❛❜❡r✲❑r❛❤♥ ❡ ❙♦❜♦❧❡✈✳ P❛r❛ ✉♠❛ ú♥✐❝❛ ❡q✉❛çã♦ ♣✲▲❛♣❧❛❝✐❛♥♦✱ ❛❧❣✉♥s ❧✐♠✐t❡s ✐♥❢❡r✐♦r❡s ❢♦r❛♠ ♦❜t✐❞♦s✳ ❆❧é♠ ❞✐ss♦✱ ❢♦r❛♠ ❛♣❧✐❝❛❞❛s té❝♥✐❝❛s ❞❡ s✐♠❡tr✐③❛çã♦ ♣❛r❛ ♦❜t❡r ❧✐♠✐t❡s ❞❡ ✈❛❧♦r❡s ♣ró♣r✐♦s✳ ◆♦ ❡♥t❛♥t♦✱ ❛té ❛♥t❡s ❞♦s r❡s✉❧t❛❞♦s ❞❡ ◆á♣♦❧✐ ❡ P✐♥❛s❝♦ ♥ã♦ t✐♥❤❛♠♦s ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ tr❛❜❛❧❤♦s s✐♠✐❧❛r❡s ♣❛r❛ s✐st❡♠❛s ❞♦ t✐♣♦ ♣✲▲❛♣❧❛❝✐❛♥♦✳

  ✷ ❈♦♥s✐❞❡r❛♠♦s ✉♠ s✐st❡♠❛ ❡❧í♣t✐❝♦ q✉❛s❡ ❧✐❡❛r ❞♦ t✐♣♦ r❡ss♦♥❛♥t❡

  

  α−2 β

  −∆ u = λαr(x)|u| u|v| , x ∈ Ω

  p

   ✭✸✮

  α β−2

   −∆ v = µβr(x)|u| v|v| , x ∈ Ω

  q

  ♦♥❞❡ ❛s ❢✉♥çõ❡s u ❡ v s❛t✐s❢❛③❡♠ ❛ ❝♦♥❞✐çã♦ ❤♦♠♦❣ê♥❡❛ ❞❡ ❢r♦♥t❡✐r❛ ❞❡ ❉✐r✐❝❤❧❡t u(x) = v(x) = 0 para x ∈ ∂Ω.

  n s−2

  u = div(|∇u| ∇u)

  s

  ❆q✉✐✱ Ω ⊂ R é ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠ ❜♦r❞♦ s✉❛✈❡ ∂Ω✱ ❡ ∆ ✳ ❖s ❡①♣♦❡♥t❡s s❛t✐s❢❛③❡♠ 1 < p, q < +∞✱ ♦s ♣❛râ♠❡tr♦s ♣♦s✐t✐✈♦s α, β s❛t✐s❢❛③❡♠

  α β = 1,

  • p q

  ✭✹✮

  ∞

  (Ω) ❡ r ∈ L ✳

  ❆ ❝♦♥❞✐çã♦ ❢♦r♥❡❝❡ ✉♠ t✐♣♦ ❞❡ ❤♦♠♦❣❡♥❡✐❞❛❞❡✱ ✈✐st♦ q✉❡✱ ❛s s♦❧✉çõ❡s ❞❡ sã♦ ✐♥✈❛r✐❛♥t❡s s♦❜r❡ ♦ r❡❡s❝❛❧♦♥❛♠❡♥t♦

  1

  1 p q

  (u, v) 7−→ (θ u, θ v), θ > 0. P♦r s✐♠♣❧✐❝✐❞❛❞❡ ♥♦s ❧✐♠✐t❛r❡♠♦s ❛♣❡♥❛s ❛ ❞✉❛s ❡q✉❛çõ❡s✱ ♠❛s ♦s r❡s✉❧t❛❞♦s s❡❣✉❡♠ ❛ ♠❡♥♦s ❞❡ ♠✉❞❛♥ç❛s ♣❛r❛ m ❡q✉❛çõ❡s✳

  ❖s ✈❛❧♦r❡s ♣ró♣r✐♦s ❞❡ s✐st❡♠❛s ❡❧í♣t✐❝♦s q✉❛s❡ ❧✐♥❡❛r❡s t❡♠ ♠❡r❡❝✐❞♦ ✉♠❛ ❣r❛♥❞❡ ❛t❡♥çã♦ ♥♦s ú❧t✐♠♦s ❛♥♦s✳ ❉❡✜♥✐♠♦s ♦ ❡s♣❡❝tr♦ S ❞❡ ✉♠ s✐st❡♠❛ ♥ã♦ ❧✐♥❡❛r ❡❧í♣t✐❝♦ ❣❡♥❡r❛❧✐③❛❞♦ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛r❡s (λ, µ) ∈ R × R ❞❡ t❛❧ ♠♦❞♦ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❛✉t♦✈❛❧♦r ❛❞♠✐t❡ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧✳ ❆ ❞✐❛❣♦♥❛❧ λ = µ ❞♦ ❡s♣❡❝tr♦ ❣❡♥❡r❛❧✐③❛❞♦ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦s ✈❛❧♦r❡s ♣ró♣r✐♦s ❞♦ s✐st❡♠❛ ❡❧í♣t✐❝♦ ❝♦♥s✐❞❡r❛❞♦ ❡♠ t❡♠ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ❛✉t♦✈❛❧♦r❡s ❞❛❞♦s ♣♦r✿

  Z Z

  1

  1

  p q

  • |∇u| |∇v| p q

  Ω Ω

  λ = inf sup

  k Z

  ✭✺✮

  C∈C k α β (u,v)∈C

  r(x)|u| |v|

  Ω k

  ♦♥❞❡ C é ❛ ❝❧❛ss❡ ❞♦s ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦s s✐♠étr✐❝♦s (C = −C) ❞❡ s✉❜❝♦♥❥✉♥t♦s

  1,p 1,q

  (Ω) × W (Ω) ❞♦ ❡s♣❛ç♦ W ❞❡ ❣ê♥❡r♦ ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ k✱ q✉❡ s❡rá ❞❡✈✐❞❛♠❡♥t❡ ❞❡✜♥✐❞♦ ♥♦ ❝❛♣ít✉❧♦ ♣r❡❧✐♠✐♥❛r✳

  ❈♦♥s✐❞❡r❛♥❞♦ ❡♠ Ω = (a, b), a, b ∈ R✱ ❢♦✐ ♣r♦✈❛❞♦ ♣♦r ◆á♣♦❧✐ ❡ P✐♥❛s❝♦ ❡♠ s❛t✐s❢❛③❡♥❞♦

  q −1+q/p

  Λ m p

  k q/p

  λ (t) ≤ Λ , +

  k

  ✭✻✮

  k

  p qt q

  k

  ♦♥❞❡ Λ é ♦ ❦✲és✐♠♦ ❛✉t♦✈❛❧♦r ✈❛r✐❛❝✐♦♥❛❧ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❉✐r✐❝❤❧❡t

  ′ ′ p−2 ′ p−2

  −|u (x)| u (x) = λr(x)|u| u, x ∈ (a, b).

  ✸

  n

  ❆✐♥❞❛ ❡♠ q✉❛♥❞♦ Ω é ✉♠ ❞♦♠í♥✐♦ ❛r❜✐trár✐♦ ❡♠ R ✱ ♦s ❛✉t♦r❡s ♣r♦✈❛r❛♠ q✉❡

  q −1+q/p

  Λ

  

1 m p

q/p

1 (t) ≤ Λ ,

  • λ

  1

  p qt q

  1

  ♦♥❞❡ Λ é ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ✈❛r✐❛❝✐♦♥❛❧ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❉✐r✐❝❤❧❡t

  p−2

  −∆ p u = λr(x)|u| u, x ∈ Ω. ❚❛♠❜é♠ ❡♠ é ✈á❧✐❞❛ ♣❛r❛

  n

  (t), k ≥ 1

  k

  t♦❞♦s ♦s ❛✉t♦✈❛❧♦r❡s ✈❛r✐❛❝✐♦♥❛✐s λ ♣❛r❛ Ω ❡♠ R ✳ ◆♦ ❝❛♣ít✉❧♦ q✉❛tr♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✱ ❞❛r❡♠♦s ✉♠❛ r❡s♣♦st❛ ♣♦s✐t✐✈❛ ♣❛r❛ ❛

  ❝♦♥❥❡❝t✉r❛ ❞❡ ◆á♣♦❧✐ ❡ P✐♥❛s❝♦ ❢❡✐t❛ ❡♠ q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s p = q ❡ ❡st❛♠♦s

  n

  / R < |x| < ¯ R} s♦❜r❡ ❛ r❡t❛ µ = tλ ♦♥❞❡ Ω = {x ∈ R ✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦ Ω é ✉♠

R

  ❛♥❡❧ ♣❛r❛ ❝❡rt♦s ✈❛❧♦r❡s ❞❡ R ❡ ¯ ✳ P❛r❛ t♦r♥❛r ♦ t❡①t♦ ♠❛✐s ❞✐♥â♠✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❢♦✐ ❡str✉t✉r❛❞♦ ❞❛ s❡❣✉✐♥t❡

  ♠❛♥❡✐r❛✿ ◆♦ ❈❛♣ít✉❧♦ ✶ ❞❡s❝r❡✈❡✲s❡ ❛s ♥♦t❛çõ❡s✱ ♦s ❡s♣❛ç♦s ❢✉♥❝✐♦♥❛✐s✱ ❛❧❣✉♠❛s

  ❞❡s✐❣✉❛❧❞❛❞❡s ✐♠♣♦rt❛♥t❡s✱ ❛❧❣✉♥s r❡s✉❧t❛❞♦s s♦❜r❡ ❣ê♥❡r♦✱ ❡ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ✈❛r✐❛❝✐♦♥❛❧ q✉❡ s❡rã♦ ✉s❛❞♦s ♥♦ tr❛❜❛❧❤♦✳

  ◆♦ ❈❛♣ít✉❧♦ ✷ ❛♣r❡s❡♥t❛✲s❡ ❛ ❞❡♠♦♥tr❛çã♦ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ❛✉t♦✈❛❧♦r❡s ❣❡♥❡r❛❧✐③❛❞♦s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ s♦❜r❡ ❛ r❡t❛ µ = λt✳ ❙❡❣✉✐♥❞♦ ❛s ♠❡s♠❛s ✐❞❡✐❛s ✉s❛❞❛s ♣♦r

  ◆♦ ❈❛♣ít✉❧♦ ✸ ❛♣r❡s❡♥t❛✲s❡ ❡st✐♠❛t✐✈❛s ♣❛r❛ ❛✉t♦✈❛❧♦r❡s ❞❡ s✐st❡♠❛s ❡❧í♣t✐❝♦s q✉❛s❡ ❧✐♥❡❛r❡s ❡♠ R✱ s♦❜r❡ ❛ r❡t❛ µ = λt✱ ♦❜t❡♥❞♦ ❧✐♠✐t❡s ♠á①✐♠♦s ♣❛r❛ ❛✉t♦✈❛❧♦r❡s ❣❡♥❡r❛❧✐③❛❞♦s✳ ❚❛♠❜é♠ ♠♦str❛r❡♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ▲②❛♣✉♥♦✈ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❡ ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❣❛r❛♥t✐r❡♠♦s ✉♠❛ ❝✉r✈❛ ❞♦ t✐♣♦ ❤✐♣ér❜♦❧❡ q✉❡ ❧✐♠✐t❛ ✐♥❢❡r✐♦r♠❡♥t❡ ♦s ❛✉t♦✈❛❧♦r❡s ❞♦ ♣r♦❜❧❡♠❛ ❡♠ R✳

  ◆♦ ❈❛♣ít✉❧♦ ✹ ❛♣r❡s❡♥t❛✲s❡ ❡st✐♠❛t✐✈❛s ♣❛r❛ s✐st❡♠❛s ❡❧í♣t✐❝♦s q✉❛s❡ ❧✐♥❡❛r❡s

  n

  ♥♦ R ✳ Pr✐♠❡✐r❛♠❡♥t❡ ❢❛r❡♠♦s ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ ♥♦ ♣r♦❜❧❡♠❛ s♦❜r❡

  n

  ❛ r❡t❛ µ = tλ ❡ ❝♦♥s✐❞❡r❛r❡♠♦s Ω ✉♠ ❛♥❡❧ ❡♠ R ✳ ❙♦❜r❡ ❡st❛s ❝♦♥❞✐çõ❡s ❝♦♥s❡❣✉✐♠♦s ❛ ♠❡s♠❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ ♦s ❛✉t♦✈❛❧♦r❡s ❞❡ ❛♣r❡s❡♥t❛❞❛s ♥♦

  n

  ❈❛♣ít✉❧♦ ✸✱ ♠❛s ❛❣♦r❛ ❡♠ R ✱ ❡ ❛✐♥❞❛ ❝♦♥s❡❣✉✐r❡♠♦s ✉♠❛ ❝✉r✈❛ q✉❡ ❧✐♠✐t❛ ✐♥❢❡r✐♦r♠❡♥t❡ ❡st❡s ❛✉t♦✈❛❧♦r❡s✳

  ❈❛♣ít✉❧♦ ✶ Pr❡❧✐♠✐♥❛r❡s

  ◆❡st❡ ❝❛♣ít✉❧♦ ✈❛♠♦s r❡✈❡r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ❡st✉❞♦ ❞♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✳

  ✶✳✶ ❊s♣❛ç♦s ❋✉♥❝✐♦♥❛✐s ❡ ❘❡s✉❧t❛❞♦s

  ◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛s ❞❡✜♥✐çõ❡s ❞♦s ❡s♣❛ç♦s ❢✉♥❝✐♦♥❛✐s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ❛♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡s ❝♦♥s✉❧t❛r ❇ré③✐s

  n

  ◆❛s ❞❡✜♥✐çõ❡s s❡❣✉✐♥t❡s ❝♦♥s✐❞❡r❛♠♦s Ω ⊂ R ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦✳ ❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛ u : Ω → R ❝♦♥tí♥✉❛✳ ❖ s✉♣♦rt❡ ❞❡ u✱ q✉❡ s❡rá ❞❡♥♦t❛❞♦ ♣♦r supp(u)

  ✱ é ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ❢❡❝❤♦ ❡♠ Ω ❞♦ ❝♦♥❥✉♥t♦ {x ∈ Ω; u(x) 6= 0}. ❙❡ supp(u) ❢♦r ✉♠ ❝♦♠♣❛❝t♦ ❞♦ Ω ❡♥tã♦ ❞✐③❡♠♦s q✉❡ u ♣♦ss✉✐ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✳ ❉❡♥♦t❛♠♦s

  (Ω) ♣♦r C ❛♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❡♠ Ω ❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✳

  m

  (Ω) ❉❡✜♥✐çã♦ ✶✳✷✳ C é ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❝♦♠ t♦❞❛s ❛s ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s ❞❡ ♦r❞❡♠ ≤ m ❝♦♥tí♥✉❛s ❡♠ Ω ✭m ✐♥t❡✐r♦ ♥ã♦✲♥❡❣❛t✐✈♦ ♦✉ m = ∞✮✳ ❉❡♥♦t❛r❡♠♦s (Ω) = C(Ω).

  ♣♦r C ❉❡✜♥✐çã♦ ✶✳✸✳ ❖ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ϕ : Ω → R q✉❡ ♣♦ss✉❡♠ t♦❞❛s ❛s ❞❡r✐✈❛❞❛s ❛té ❛ ♦r❞❡♠ m ❝♦♥tí♥✉❛s ❡♠ Ω ❡ q✉❡ tê♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✱ s❡♥❞♦

  m ∞

  (Ω) q✉❡ ❡ss❡ s✉♣♦rt❡ ❞❡♣❡♥❞❡ ❞❡ ϕ✱ é ❞❡♥♦t❛❞♦ ♣♦r C ✭♦✉ C s❡ m = ∞✮✳

  ∞

  ) (Ω)

  ν ν∈N

  ❉❡✜♥✐çã♦ ✶✳✹✳ ❯♠❛ s✉❝❡ssã♦ (ϕ ❞❡ ❢✉♥çõ❡s ❞❡ C ❝♦♥✈❡r❣❡ ♣❛r❛ ③❡r♦ q✉❛♥❞♦ ❡①✐st❡ K ⊂ Ω ❝♦♠♣❛❝t♦ t❛❧ q✉❡✿ ∗ supp ϕ ⊂ K, ∀ ν ∈ N;

  ν n

  ∗ P❛r❛ ❝❛❞❛ α ∈ N

  α

  D ϕ → 0

  ν ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ K,

  ✺ ✶✳✶✳ ❊❙P❆➬❖❙ ❋❯◆❈■❖◆❆■❙ ❊ ❘❊❙❯▲❚❆❉❖❙

  α

  ♦♥❞❡ D ❞❡♥♦t❛ ♦ ♦♣❡r❛❞♦r ❞❡r✐✈❛çã♦ ❞❡ ♦r❞❡♠ α ❞❡✜♥✐❞♦ ♣♦r

  

|α|

  ∂ ,

  α α

  1

  

2

α n

  ∂x ∂x ...∂x

  1 2 n n

  , α , ..., α ) ∈ N + α + ... + α

  1 2 n

  

1

2 n

  ❝♦♠ α = (α ❡ |α| = α ✳

  ∞

  (Ω) ❉❡✜♥✐çã♦ ✶✳✺✳ ❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ C ❝♦♠ ❛ ♥♦çã♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡✜♥✐❞❛ ❛❝✐♠❛ é r❡♣r❡s❡♥t❛❞♦ ♣♦r D(Ω) ❡ ❞❡♥♦♠✐♥❛❞♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s t❡st❡s ❡♠ Ω.

  )

  n n∈N

  ❉❡✜♥✐çã♦ ✶✳✻✳ ❙❡❥❛ (f ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ n ∈ N

  : S → X t❡♠♦s f n ❡♠ q✉❡ S ❡ X sã♦ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳ ❉✐③❡♠♦s q✉❡ ❛ )

  n n∈N

  s❡q✉ê♥❝✐❛ (f ❝♦♥✈❡r❣❡ ♣♦♥t✉❛❧♠❡♥t❡ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ f : S → X q✉❛♥❞♦✱

  n (x)) n∈N

  ♣❛r❛ t♦❞♦ x ∈ S✱ ❛ s❡q✉ê♥❝✐❛ (f ❝♦♥✈❡r❣❡ ♣❛r❛ f(x)✳

  n

  ❉❡✜♥✐çã♦ ✶✳✼✳ ❉✐③✲s❡ q✉❡ ✉♠ ❝♦♥❥✉♥t♦ S ⊂ R t❡♠ ♠❡❞✐❞❛ ♥✉❧❛ s❡✱ ❞❛❞♦ ε > 0✱

  ∞

  }

  k

  ❡①✐st❡ ✉♠❛ ❝♦❧❡çã♦ ❡♥✉♠❡rá✈❡❧ ❞❡ ❝♦♥❥✉♥t♦s ❛❜❡rt♦s {I t❛✐s q✉❡

  k=1 ∞

  [

  I

  k

  ❛✮ S ⊂

  k=1 ∞

  X vol(I ) < ε

  k

  ❜✮

  k=1 ∞

  }

  k

  ❉✐③✲s❡ q✉❡ ❛ ❝♦❧❡çã♦ {I k=1 é ✉♠❛ ❝♦❜❡rt✉r❛ ❞❡ S✳ P❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ ❛ ❝♦❜❡rt✉r❛ é ✜♥✐t❛✱ ❞✐③✲s❡ q✉❡ S t❡♠ ❝♦♥t❡ú❞♦ ♥✉❧♦✳

  )

  n n∈N

  ❉❡✜♥✐çã♦ ✶✳✽✳ ❙❡❥❛ (f ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s✱ t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ n t❡♠✲

  n : S → X

  s❡ f ✱ ❡♠ q✉❡ S é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛ ❡ X é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ )

  n n∈N

  ❉✐③❡♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ (f ❝♦♥✈❡r❣❡ q✳t✳♣ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ f : S → X s❡ )

  n n∈N

  ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛ ♥✉❧❛ Z t❛❧ q✉❡ (f ❝♦♥✈❡r❣❡ ♣♦♥t✉❛❧♠❡♥t❡ ♣❛r❛ f ❡♠ S \ Z✳

  p

  (Ω) ❉❡✜♥✐çã♦ ✶✳✾✳ ❙❡❥❛ 1 ≤ p ≤ +∞. ❉❡♥♦t❛♠♦s ♣♦r L ♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❞❛s

  p

  ✭❝❧❛ss❡s ❞❡✮ ❢✉♥çõ❡s ❞❡✜♥✐❞❛s ❡♠ Ω ❝♦♠ ✈❛❧♦r❡s ❡♠ R, t❛✐s q✉❡ |u| é ✐♥t❡❣rá✈❡❧ ♥♦ s❡♥t✐❞♦ ❞❡ ▲❡❜❡s❣✉❡ ❡♠ Ω, ❝♦♠ ♥♦r♠❛

  1 Z p p p

  ||u|| |u(x)| = dx

  L ♣❛r❛ 1 ≤ p < +∞ Ω ∞

  (Ω) ❡✱ ♣❛r❛ p = ∞✱ ❞❡♥♦t❛♠♦s L ♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❞❛s ✭❝❧❛ss❡s ❞❡✮ ❢✉♥çõ❡s ♠❡♥s✉rá✈❡s ❞❡ u ❞❡✜♥✐❞❛s s♦❜r❡ Ω✱ q✉❡ sã♦ ❡ss❡❝✐❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛s✱ ❝♦♠ ❛ ♥♦r♠❛ ❞❛❞❛ ♣♦r

  ∞

  ||u|| = sup ess |u(x)| = inf {C ∈ R; |u(x)| ≤ C

  L q✳t✳♣✳ ❡♠ Ω} . x∈Ω n

  ❉❡✜♥✐çã♦ ✶✳✶✵✳ ❙❡❥❛ Ω ⊂ R ✉♠ ❛❜❡rt♦ ❧✐♠✐t❛❞♦✱ ❡ p ∈ R ❝♦♠ 1 ≤ p ≤ ∞✳ ❖

  1,p

  (Ω) ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ W é ❞❡✜♥✐❞♦ ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♥❥✉♥t♦

  1,p p p W (Ω) = {u ∈ L (Ω); ∇u ∈ L (Ω)} .

  ✻ ✶✳✶✳ ❊❙P❆➬❖❙ ❋❯◆❈■❖◆❆■❙ ❊ ❘❊❙❯▲❚❆❉❖❙

  1,p

  (Ω) ❖ ❡s♣❛ç♦ W é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❝♦♠ ❛ ♥♦r♠❛

  1 p p p

1,p p p

  kuk = (kuk + k∇uk ) .

  W L L

  1

  ❚❡♦r❡♠❛ ✶✳✶✶✳ ❙❡❥❛ Ω ✉♠ ❛❜❡rt♦ ❧✐♠✐t❛❞♦ ❞❡ ❝❧❛ss❡ C ✳ ❊♥tã♦ ❛s s❡❣✉✐♥t❡s ✐♠❡rsõ❡s sã♦ ❝♦♥tí♥✉❛s✿

  

1,p q ∗

  W (Ω) ֒→ L (Ω) ], ♣❛r❛ p < n, ∀ q ∈ [1, p

  1,p q

  W (Ω) ֒→ L (Ω) ♣❛r❛ p = n, ∀ q ∈ [p, +∞),

  1,p ∞

  W (Ω) ֒→ L (Ω) ♣❛r❛ p > n, np

  ∗

  = ❝♦♠ p ✳ n − p ❉❡♠♦♥str❛çã♦✳ ❱❡r ▲✳❆✳ ❞❛ ❏✳▼❡❞❡✐r♦s ♣✳ ✼✺✳

  1

  ❚❡♦r❡♠❛ ✶✳✶✷✳ ❙❡❥❛ Ω ✉♠ ❛❜❡rt♦ ❧✐♠✐t❛❞♦ ❞❡ ❝❧❛ss❡ C ✳ ❊♥tã♦ ❛s s❡❣✉✐♥t❡s ✐♠❡rsõ❡s sã♦ ❝♦♠♣❛❝t❛s✿

  

1,p q ∗

  W (Ω) ֒→ L (Ω) ), ♣❛r❛ p < n, ∀ q ∈ [1, p

  1,p q

  W (Ω) ֒→ L (Ω) ♣❛r❛ p = n, ∀ q ∈ [p, +∞),

  1,p ∞

  W (Ω) ֒→ L (Ω) ♣❛r❛ p > n, np

  ∗

  = ❝♦♠ p ✳ n − p ❉❡♠♦♥str❛çã♦✳ ❱❡r ▲✳❆✳ ❞❛ ❏✳▼❡❞❡✐r♦s ♣✳ ✼✺✳

  1,p

  1

  (Ω) (Ω) ❉❡✜♥✐çã♦ ✶✳✶✸✳ ❉❛❞♦ 1 ≤ p < ∞✱ ❞❡♥♦t❛♠♦s ♣♦r W ♦ ❢❡❝❤♦ ❞❡ C ❡♠

  1,p 1,p

  W (Ω) (Ω) ✱ ❡q✉✐♣❛❞♦ ❝♦♠ ❛ ♥♦r♠❛ ❞❡ W ✳

  1,p 1,p

  (Ω) (Ω) ❚❡♦r❡♠❛ ✶✳✶✹✳ ❙❡❥❛ u ∈ W ✳ ❊♥tã♦ u ∈ W s❡✱ s♦♠❡♥t❡ s❡✱ u = 0 ❡♠ ∂Ω ♥♦ s❡♥t✐❞♦ ❞♦ tr❛ç♦✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ♣✳ ✷✶✼✳

  ∗

  ❙❡❥❛ E ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡ s❡❥❛ f ∈ E ✱ ♦♥❞❡

  ∗

  E = {f : E → R/ f é ❧✐♥❡❛r ❡ ❧✐♠✐t❛❞❛},

  : E → R

  f

  é ♦ ❡s♣❛ç♦ ❞✉❛❧ ❞❡ E✳ ❉❡♥♦t❛♠♦s ♣♦r ϕ ♦ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r

  ∗ ∗

  ϕ (x) = hf, xi )

  

f ✳ ❈♦♠♦ f ♣❡r❝♦rr❡ E ♦❜t❡♠♦s ✉♠❛ ❝♦❧❡çã♦ (ϕ f f ∈E ❞❡ ❢✉♥çõ❡s

  ❞❡ E ❡♠ R✳ ■❣♥♦r❛♠♦s ❛ t♦♣♦❧♦❣✐❛ ✉s✉❛❧ ❞❡ E ✭❛ss♦❝✐❛❞❛ ❛ k k✮ ❡ ❞❡✜♥✐♠♦s ✉♠❛ ♥♦✈❛ t♦♣♦❧♦❣✐❛ ♥♦ ❝♦♥❥✉♥t♦ E ❝♦♠♦ s❡ s❡❣✉❡✿

  ∗

  ) ❉❡✜♥✐çã♦ ✶✳✶✺✳ ❆ t♦♣♦❧♦❣✐❛ ❢r❛❝❛ σ(E, E ❡♠ ❊ é ❛ t♦♣♦❧♦❣✐❛ ❛ss♦❝✐❛❞❛ ❛

  ∗ f ) f ∈E

  ❝♦❧❡çã♦ (ϕ ✳

  f

  ◆♦t❡ q✉❡ t♦❞❛ ❢✉♥çã♦ ϕ é ❝♦♥tí♥✉❛ ♥❛ t♦♣♦❧♦❣✐❛ ✉s✉❛❧ ♣♦rt❛♥t♦ ❛ t♦♣♦❧♦❣✐❛ ❢r❛❝❛ é ♠❛✐s ❢r❛❝❛ q✉❡ ❛ t♦♣♦❧♦❣✐❛ ✉s✉❛❧✳

  ✼ ✶✳✶✳ ❊❙P❆➬❖❙ ❋❯◆❈■❖◆❆■❙ ❊ ❘❊❙❯▲❚❆❉❖❙ )

  n

  ◆♦t❛çã♦✳ ❙❡ ❛ s❡q✉ê♥❝✐❛ (x ❡♠ E ❝♦♥✈❡r❣❡ ♣❛r❛ x ♥❛ t♦♣♦❧♦❣✐❛ ❢r❛❝❛

  ∗

  σ(E, E ) ❡s❝r❡✈❡♠♦s x ⇀ x

  n ∗

  )

  n

  ❡ ❞✐③❡♠♦s q✉❡ ✧x ❝♦♥✈❡r❣❡ ❢r❛❝❛♠❡♥t❡ ♣❛r❛ x ❡♠ σ(E, E ✧✳

  n )

  Pr♦♣♦s✐çã♦ ✶✳✶✻✳ ❙❡❥❛ (x ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ ❊✳ ❊♥tã♦

  ∗ ∗

  ⇀ x ) ⇔ hf, x i → hf, xi ∀ x ∈ E

  n n

  ✭✐✮ x ❢r❛❝❛♠❡♥t❡ ❡♠ σ(E, E ✳

  ∗

  → x ⇀ x )

  n n

  ✭✐✐✮ ❙❡ x ❢♦rt❡♠❡♥t❡✱ ❡♥tã♦ x ❢r❛❝❛♠❡♥t❡ ❡♠ σ(E, E ✳

  ∗

  ⇀ x ) k) ✭✐✐✐✮ ❙❡ x n ❢r❛❝❛♠❡♥t❡ ❡♠ σ(E, E ✱ ❡♥tã♦ (kx n é ❧✐♠✐t❛❞❛ ❡ kxk ≤ lim inf kx k

  n

  ✳

  ∗ ∗

  → f

  n ⇀ x ) n

  ✭✐✈✮ ❙❡ x ❢r❛❝❛♠❡♥t❡ ❡♠ σ(E, E ❡ s❡ f ❢♦rt❡ ❡♠ E ✭✐st♦ é✱

  ∗

  kf − f k → 0 , x i → hf, xi

  n E n n

  ✮✱ ❡♥tã♦ hf ✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ♣✳ ✺✽✳ ❉❡✜♥✐çã♦ ✶✳✶✼✳ ❯♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ é ❞✐t♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦ s❡

  ∀ǫ > 0, ∃ δ > 0 t❛❧ q✉❡ x − y x, y ∈ E, kxk ≤, kyk ≤ 1 < 1 − δ. ❡ kx − yk > ǫ ⇒

  2

  p

  ❚❡♦r❡♠❛ ✶✳✶✽✳ ❖s ❡s♣❛ç♦s L sã♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦s ♣❛r❛ 1 < p < ∞✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ♣✳ ✾✻✳

  p

  1

  ∈ L (Ω), f ∈

  1

  2

  ❚❡♦r❡♠❛ ✶✳✶✾ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r ●❡♥❡r❛❧✐③❛❞❛✮✳ ❙❡❥❛♠ f

  1

  1

  n p 2 p

  L (Ω), ..., f ∈ L (Ω), n ∈ N, com p , ..., p > 1 + ... + = 1

  n 1 n

  ❡ ✳ ❊♥tã♦ p p

  1 n

  1

  f · ... · f ∈ L (Ω)

  1 n

  ❡ Z

  1 n

  1 L n L Ω

  pn |f · ... · f | dx ≤ ||f || p1 · ... · ||f || .

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇ré③✐s ♣✳ ✾✷✳

  ′

  1

  1

  = 1 ❈♦♥s✐❞❡r❡ p > 1✱ ❡♥tã♦ p s❡rá ♦ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ q✉❡ s❛t✐s❢❛③ ✳

  p p 1,p ′

  (Ω) ❘❡♣r❡s❡♥t❛r❡♠♦s ♣♦r V ♦ ❡s♣❛ç♦ ♥♦r♠❛❞♦ W ❝♦♠ ❛ ♥♦♠❛ k·k ❡ V ♦ ❡s♣❛ç♦

  ′ −1,p

  ′ ′

  W (Ω)

  V ,V

  ❝♦♠ ♦ ♣❛r ❞❡ ❞✉❛❧✐❞❛❞❡ ❞❛❞♦ ♣♦r V ❡ V ♣♦r h·, ·i ✳ ❉❡✜♥❛♠♦s ♣❛r❛

  1,p

  (Ω) ❝❛❞❛ u ∈ W ✱ ❛ ❛♣❧✐❝❛çã♦ Au : V → R ❞❛❞❛ ♣♦r

  Z

  p−2 ′ hAu, vi = |∇u| ∇u∇v dx = h−∆ u, vi . p V ,V Ω p p−2 p p −2

  ∈ L (Ω) (Ω) ❈♦♠♦ |∇u| ❡ ∇u, ∇v ∈ L ✱ ❡♥tã♦ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ H¨older

  p−2

  1

  ∇u∇v ∈ L (Ω) ❣❡♥❡r❛❧✐③❛❞❛ t❡♠♦s q✉❡ |∇u| ✱ ♣♦rt❛♥t♦ ❛ ❛♣❧✐❝❛çã♦ Au ❡stá

  ✽ ✶✳✶✳ ❊❙P❆➬❖❙ ❋❯◆❈■❖◆❆■❙ ❊ ❘❊❙❯▲❚❆❉❖❙ ❜❡♠ ❞❡✜♥✐❞❛✳ ❆❧é♠ ❞✐ss♦✱

  Z Z

  p−2 p−1 ′

  |hAu, vi | = |∇u| ∇u∇v dx ≤ |∇u| |v|dx

  V ,V

Ω Ω

  ✭✶✳✶✮

  p−1 p−1 p 1,p

  ≤ k∇uk p k∇vk ≤ kuk kvk

  L (Ω) 1,p , L (Ω) W (Ω) W (Ω) ′

  ♦ q✉❡ ✐♠♣❧✐❝❛✱ Au ∈ V ❝♦♠

  p−1 ′ kAuk ≤ Ckuk 1,p .

  V

  ✭✶✳✷✮

  W (Ω) ′

  P♦rt❛♥t♦✱ A : V → V ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ❡

  1,p ′ hAu, vi = h−∆ u, vi, ∀ u, v ∈ V = W (Ω).

  V ,V p

  ❚r❛❜❛❧❤❛r❡♠♦s ♥♦ s❡❣✉✐♥t❡ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤

  1,p 1,q

  W = W (Ω) × W (Ω) ❡q✉✐♣❛❞♦ ❝♦♠ ❛ ♥♦r♠❛✿ q

  

2

  2

  k(u, v)k = kuk + kvk

  W 1,p 1,q W (Ω) W (Ω)

  ′ ′ ∗ ∗ −1,p −1,q ∗ ∗

  , v ) ∈ W (Ω) L W (Ω) , v ) ∈ ◆♦t❡ q✉❡ s❡ ❝♦♥s✐❞❡r❛r♠♦s (u ✈❡♠♦s q✉❡ (u

  ∗

  W ❡ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ s❡❣✉✐♥t❡ ❛♣❧✐❝❛çã♦

  

∗ ∗ ∗ ∗

  h(u , v ), (u, v)i = hu , ui + hv , vi.

  ′ ′ ∗ −1,p −1,q

  ∼ (Ω) L W (Ω)

  = W ❊♥tã♦ t❡♠♦s W ✭✐s♦♠♦r✜s♠♦ ✐s♦♠étr✐❝♦✮ ♦♥❞❡ ❛

  ∗

  ♥♦r♠❛ ❡♠ W é ❞❛❞❛ ♣♦r✿ q

  ∗ ∗

  

2

  2 ∗ ∗ ∗ k(u , v )k = ku k + kv k .

  W −1,p′ −1,q′

  W (Ω) W (Ω)

  ❙❛❜❡♠♦s ❞❛ t❡♦r✐❛ ❞❡ ❢✉♥❝✐♦♥❛✐s ❧✐♥❡❛r❡s q✉❡

  ∗ ∗ ∗ ∗ ∗ |h(u , v ), (u, v)i| ≤ k(u , v )k k(u, v)k , ∀(u, v) ∈ W.

  W W

  ✭✶✳✸✮ ❉❡✜♥✐çã♦ ✶✳✷✵✳ ❙❡❥❛♠ 1 ≤ p < ∞✳ ❉✐r❡♠♦s q✉❡ f : Ω → R é ❧♦❝❛❧♠❡♥t❡

  p p

  (Ω) (Ω) ✐♥t❡❣rá✈❡❧ ❡♠ L ✱ ❡ ❞❡♥♦t❛r❡♠♦s ♣♦r f ∈ L ✱ s❡ f ❢♦r ✉♠❛ ❢✉♥çã♦

  loc

  ♠❡♥s✉rá✈❡❧ ❡ ♣❛r❛ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ K ⊂ Ω t✐✈❡r♠♦s Z

  p |f (x)| dx < ∞.

  1

  (I) ❚❡♦r❡♠❛ ✶✳✷✶✳ ❙❡❥❛♠ I = (a, b)✱ −∞ ≤ a < b ≤ ∞ ❡ s❡❥❛ u ∈ L t❛❧ q✉❡

  loc

  Z

  1 uϕ dx = 0 ∀ϕ ∈ C (I). x

  I

  ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C t❛❧ q✉❡ u = C ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡ I✳

  ✾ ✶✳✶✳ ❊❙P❆➬❖❙ ❋❯◆❈■❖◆❆■❙ ❊ ❘❊❙❯▲❚❆❉❖❙ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ♣✳ ✷✵✺✳ ❉❡✜♥✐çã♦ ✶✳✷✷✳ ❙❡❥❛♠ I = (a, b)✱ −∞ ≤ a < b ≤ ∞✱ ❡ p ∈ R ❝♦♠ 1 ≤ p ≤ ∞✳

  1,p

  (I) ❖ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ W é ❞❡✜♥✐❞♦ ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♥❥✉♥t♦

  Z Z

  b b 1,p p p

  1 W (I) = u ∈ L (I); ∃u ∈ L (I) ; uϕ dx = − u ϕ dx ∀ϕ ∈ C (I) x x x a a 1,p

  (I) ❖ ❡s♣❛ç♦ W é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❝♦♠ ❛ ♥♦r♠❛

  1 p p p

1,p p p

  kuk k = (kuk + ku x ) .

  W L L 1 1,2

  1

  (I) = W (I) (I) ◗✉❛♥❞♦ p = 2✱ ❞❡♥♦t❛♠♦s H ✳ ❖ ❡s♣❛ç♦ H é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❡q✉✐♣❛❞♦ ❝♦♠ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦

  Z b

  1

  2

  H L x x L x x a 1,p

  2 hu, vi = hu, vi + hu , v i = (uv + u v ) dx.

  (I) Pr♦♣♦s✐çã♦ ✶✳✷✸✳ ❖ ❡s♣❛ç♦ W é r❡✢❡①✐✈♦ ♣❛r❛ 1 < p < ∞ ❡ s❡♣❛rá✈❡❧ ♣❛r❛ 1 ≤ p < ∞

  ✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ♣✳ ✷✵✸✳ ❉❡✜♥✐çã♦ ✶✳✷✹✳ ❉❛❞♦ ✉♠ ✐♥t❡✐r♦ m ≥ 2 ❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ 1 ≤ p ≤ ∞ ❞❡✜♥✐♠♦s✱ ♣♦r ✐♥❞✉çã♦✱ ♦ ❡s♣❛ç♦

  m,p m−1,p 1 m−1,p

  W (I) = u ∈ W (I); D u ∈ W (I) ,

  1

  u = u

  x

  ❝♦♠ ❛ ♥♦t❛çã♦ D ✱ ❡q✉✐♣❛❞♦ ❝♦♠ ❛ ♥♦r♠❛

  m

  X

  i

  • m,p p p

    kuk = kuk kD uk .

  

W L L

i=1

  ❊ t❛♠❜é♠ ❞❡✜♥✐♠♦s

  m m,2

  H (I) = W (I), ❡q✉✐♣❛❞♦ ❝♦♠ ♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r

  

m m

  Z Z

  b b

  X X

  i i i i

  2

  2

  

2

  hu, vi hD = hu, vi + u, D vi = uv dx + D uD v dx.

  H L L a a

i=1 i=1

  ❚❡♦r❡♠❛ ✶✳✷✺✳ ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ C ✭q✉❡ ❞❡♣❡♥❞❡ s♦♠❡♥t❡ ❞❡ |I| ≤ ∞

  ✮ t❛❧ q✉❡

  1,p ∞ 1,p

  kuk ≤ Ckuk ∀u ∈ W

L , (I), ∀1 ≤ p ≤ ∞.

  W 1,p ∞

  (I) ֒→ L (I) ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ W ❝♦♥t✐♥✉❛♠❡♥t❡ ♣❛r❛ t♦❞♦ 1 ≤ p ≤ ∞✳

  ❆❧é♠ ❞✐ss♦✱ ❙❡ I é ✉♠ ✐♥t❡r✈❛❧♦ ❧✐♠✐t❛❞♦ ❡♥tã♦

  ✶✵ ✶✳✶✳ ❊❙P❆➬❖❙ ❋❯◆❈■❖◆❆■❙ ❊ ❘❊❙❯▲❚❆❉❖❙

  1,p

  (I) ֒→ C(I) ❆ ✐♠❡rsã♦ W é ❝♦♠♣❛❝t❛ ♣❛r❛ t♦❞♦ 1 < p ≤ ∞✳

  1,1 q

  (I) ֒→ L (I) ❆ ✐♠❡rsã♦ W é ❝♦♠♣❛❝t❛ ♣❛r❛ t♦❞♦ 1 ≤ q < ∞✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ♣✳ ✷✶✷✳

  1,p

  (I) ❈♦r♦❧ár✐♦ ✶✳✷✻✳ ❙❡❥❛♠ u✱ v ∈ W ❝♦♠ 1 ≤ p ≤ ∞✳ ❊♥tã♦

  

1,p

  uv ∈ W (I) ❡ (uv) x = u x v + u v x .

  ❆❞❡♠❛✐s✱ ✈❛❧❡ ❛ ❢♦r♠✉❧❛ ❞❡ ✐♥t❡❣r❛çã♦ ♣♦r ♣❛rt❡s Z Z

  z z ∀x, y ∈ I.

  u v dx = u(z)v(z) − u(y)v(y) − uv dx,

  x x y y

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ♣✳ ✷✶✺✳

  1 1,p

  (R) (I) ❈♦r♦❧ár✐♦ ✶✳✷✼✳ ❙❡❥❛ G ∈ C t❛❧ q✉❡ G(0) = 0✱ ❡ s❡❥❛ u ∈ W ❝♦♠ 1 ≤ p ≤ ∞

  ✳ ❊♥tã♦

  1,p G ◦ u ∈ W (I) (G ◦ u) = (G ◦ u)u . x x x

  ❡ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ♣✳ ✷✶✺✳

  )

  n

  ❚❡♦r❡♠❛ ✶✳✷✽✳ ✭❚❡♦r❡♠❛ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦♠✐♥❛❞❛✮✳ ❙❡❥❛ (u ✉♠❛ s✉❝❡ssã♦ ❞❡ ❢✉♥çõ❡s ✐♥t❡❣rá✈❡✐s ❡♠ Ω✱ ❝♦♥✈❡r❣❡♥t❡ q✉❛s❡ s❡♠♣r❡ ♣❛r❛ ❛ ❢✉♥çã♦ u

  | ≤ u

  n

  ✳ ❙❡ ❡①✐st✐r ✉♠❛ ❢✉♥çã♦ ✐♥t❡❣rá✈❡❧ u t❛❧ q✉❡ |u q✉❛s❡ s❡♠♣r❡ ♣❛r❛ t♦❞♦ n ∈ N ✱ ❡♥tã♦ u é ✐♥t❡❣rá✈❡❧ ❡ t❡♠✲s❡

  Z Z u = lim u .

  n

n−→∞

Ω Ω

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s

  n

  ❚❡♦r❡♠❛ ✶✳✷✾✳ ❙❡ K ⊂ R é ❝♦♠♣❛❝t♦✱ ❡♥tã♦ t♦❞❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ ✐♥❥❡t✐✈❛

  n

  f : K −→ R é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ s♦❜r❡ s✉❛ ✐♠❛❣❡♠✳

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ❊❧♦♥ ♣✳ ✷✹✳ ❚❡♦r❡♠❛ ✶✳✸✵✳ ✭❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦✮✳ ❉❛❞❛ f : U → R ❞✐❢❡r❡♥❝✐á✈❡❧

  n

  ♥♦ ❛❜❡rt♦ U ⊂ R ✱ s❡ ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❬❛✱❛✰✈❪ ❡st✐✈❡r ❝♦♥t✐❞♦ ❡♠ ❯ ❡♥tã♦ ❡①✐st❡ θ ∈ (0, 1) t❛❧ q✉❡

  ∂f f (a + v) − f (a) = (a + θv) = hgrad f (a + θv), vi ∂v

  n

  X ∂f

  = (a + θv).α

  i

  ∂x i

  ✶✶ ✶✳✷✳ ❉❊❙■●❯❆▲❉❆❉❊❙ ■▼P❖❘❚❆◆❚❊❙ , · · · , α )

  1 n

  ♦♥❞❡ v = (α ✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❊❧♦♥ ♣✳ ✻✶✳

  ❈♦♥s✐❞❡r❡ X ❡ Y ❞♦✐s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ♥♦r♠❛❞♦s✱ U ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ✉♠ ♣♦♥t♦ x ❡♠ X ❡ F ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ U ❡♠ Y ✳ ❉❡✜♥✐çã♦ ✶✳✸✶✳ ❉✐③❡♠♦s q✉❡ ❋ t❡♠ ✉♠❛ ❞❡r✐✈❛❞❛ ❞✐r❡❝✐♦♥❛❧ ♥♦ ♣♦♥t♦ x ♥❛ ❞✐r❡çã♦ ❤✱ s❡ ♦ ❧✐♠✐t❡

  F (x + ǫh) − F (x ) lim

  − ǫ→0

  ǫ ).h

  ❡①✐st❡✳ ❉❡♥♦t❛r❡♠♦s ♣♦r DF (x ✳ ).h

  ❉❡✜♥✐çã♦ ✶✳✸✷✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ♣❛r❛ t♦❞♦ h ∈ X ❛ ❞❡r✐✈❛❞❛ DF (x ♥❛ F (x , ·) : X → Y F (x , h) =

  ❞✐r❡çã♦ ❞❡ ❤ ❡①✐st❛✳ ❆ ❛♣❧✐❝❛çã♦ δ ❞❡✜♥✐❞❛ ♣♦r δ DF (x ).h

  é ❞❡♥♦♠✐♥❛❞❛ ❛ ♣r✐♠❡✐r❛ ✈❛r✐❛çã♦ ❞❛ ❛♣❧✐❝❛çã♦ ❋ ♥♦ ♣♦♥t♦ x ❉❡✜♥✐çã♦ ✶✳✸✸✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❋ ♣♦ss✉✐ ✉♠❛ ♣r✐♠❡✐r❛ ✈❛r✐❛çã♦ ♥♦ ♣♦♥t♦ x

  F (x , h) = Λh ❡ q✉❡ ❡①✐st❡ ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❝♦♥tí♥✉♦ Λ ∈ L(X, Y ) t❛❧ q✉❡ δ ✳

  • ❊♥tã♦ ♦ ♦♣❡r❛❞♦r Λ é ❞❡♥♦♠✐♥❛❞♦ ❞❡r✐✈❛❞❛ ❞❡ ●ât❡❛✉① ❞❛ ❛♣❧✐❝❛çã♦ ❋ ♥♦

  ).(·) ♣♦♥t♦ x ❡ ❞❡♥♦♠✐♥❛♠♦s ♣♦r DF (x ✳

  ).(·) ❆ss✐♠ DF (x é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ L(X, Y ) t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ h ∈ X t❡♠♦s

  ❛ r❡❧❛çã♦ F (x + ǫh) = F (x ) + DF (x ).h + o(ǫ), q✉❛♥❞♦ ǫ → 0 ✳

  • ✶✳✷ ❉❡s✐❣✉❛❧❞❛❞❡s ✐♠♣♦rt❛♥t❡s

  Pr♦♣♦s✐çã♦ ✶✳✸✹✳ ❙❡❥❛♠ ♥ú♠❡r♦s r❡❛✐s a, b ≥ 0 ❡ p ≥ 1✳ ❊♥tã♦

  p p p p (a + b) ≤ 2 (a + b ).

  ❉❡♠♦♥str❛çã♦✳ ❯s❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♠á①✐♠♦ ♦❜t❡♠♦s

  p p

  (a + b) ≤ (2 max{a, b})

  p p p

  } = 2 max{a , b

  p p p ≤ 2 (a + b ).

  Pr♦♣♦s✐çã♦ ✶✳✸✺ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣✮✳ ❙❡ a, b ≥ 0 ❡ p, q > 1 sã♦ t❛✐s q✉❡

  1

  1 = 1

  • p q

  ❡♥tã♦

  p q

  a b + ab ≤ . p q

  ✶✷ ✶✳✷✳ ❉❊❙■●❯❆▲❉❆❉❊❙ ■▼P❖❘❚❆◆❚❊❙ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❘✳●✳ ❇❛rt❧❡ ♣✳ ✺✻✳

  ❯♠❛ ✈❛r✐❛çã♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣ q✉❡ s❡rá ♠✉✐t♦ ✉t✐❧✐③❛❞♦ ♥❡st❡ tr❛❜❛❧❤♦ é ❞❛❞❛ ♣❡❧♦ s❡❣✉✐♥t❡ ❝♦r♦❧ár✐♦✳

  1

  1 = 1

  ❈♦r♦❧ár✐♦ ✶✳✸✻✳ ❙❡❥❛♠ a, b ≥ 0 ❡ p, q > 1 t❛✐s q✉❡ ✳ P❛r❛ t♦❞♦ ε > 0

  • p q t❡♠✲s❡

  p q

  ab ≤ c(ε)a + εb , ♦♥❞❡ c(ε) é ✉♠❛ ❝♦♥st❛♥t❡ q✉❡ ❞❡♣❡♥❞❡ ❞❡ ε, p ❡ q✳ ❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s

  1

  1

  q

  ab = (qε) ab

  1 q

  (qε) !

  1

  a

  

1

q

  q = (qε) b .

  (qε) ❆♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣ s❡❣✉❡

  ! p

  q

  1

  1 a

  1

  ab ≤ (qε) b

  • q

  1 q

  p q (qε)

  1

  p q p = a + εb ∀ ε > 0. q

  p(qε)

  1

  p

  ❚♦♠❛♥❞♦ c(ε) = t❡♠♦s✱

  q

  p(qε)

  

p q

ab ≤ c(ε)a + εb ∀ ε > 0.

  ❚❡♦r❡♠❛ ✶✳✸✼ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✮✳ ❙❡❥❛ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧

  V

  ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ h·, ·i ✳ ❊♥tã♦ ♣❛r❛ t♦❞♦s u✱ v ∈ V t❡♠♦s |hu, vi | ≤ kuk kvk ;

  V V

  V

  ❛ ✐❣✉❛❧❞❛❞❡ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ {u, v} é ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❖❧✐✈❡✐r❛ ♣✳ ✶✷✶✳

  1,p

  1

  (I) (I) ❉❡✜♥✐çã♦ ✶✳✸✽✳ ❉❛❞♦ 1 ≤ p < ∞✱ ❞❡♥♦t❛♠♦s ♣♦r W ♦ ❢❡❝❤♦ ❞❡ C ❡♠

  1,p 1,p

  W (I) (I) ✱ ❡q✉✐♣❛❞♦ ❝♦♠ ❛ ♥♦r♠❛ ❞❡ W ✳

  1,2

  1

  1

  (I) = W (I) (I)

  ❖ ❡s♣❛ç♦ H é ❡q✉✐♣❛❞♦ ❝♦♠ ♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❞❡ H ✳

  1,p 1,p

  (I) (I) ❚❡♦r❡♠❛ ✶✳✸✾✳ ❙❡❥❛ u ∈ W ✳ ❊♥tã♦ u ∈ W s❡✱ s♦♠❡♥t❡ s❡✱ u = 0 ❡♠ ∂I

  ✳

  ✶✸ ✶✳✷✳ ❉❊❙■●❯❆▲❉❆❉❊❙ ■▼P❖❘❚❆◆❚❊❙ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ♣✳ ✷✶✼✳ ❚❡♦r❡♠❛ ✶✳✹✵ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ P♦✐♥❝❛ré✮✳ ❙✉♣♦♥❤❛♠♦s I ✉♠ ✐♥t❡r✈❛❧♦

  ≥ 0 ❧✐♠✐t❛❞♦✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C p ✱ q✉❡ ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ✐♥t❡r✈❛❧♦ I✱ t❛❧ q✉❡

  1,p

p

1,p

  kuk ≤ C ku k ∀u ∈ W (I).

  W p x L 1,p

p

  (I) k

  x L

  ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❡♠ W ✱ ku é ✉♠❛ ♥♦r♠❛ ❡q✉✐✈❛❧❡♥t❡ à ♥♦r♠❛ ❞❡

  1,p

  W (I) ✳

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳ ❇r❡③✐s ♣✳ ✷✶✽✳ > 0

  p

  ❚❡♦r❡♠❛ ✶✳✹✶✳ ❙❡❥❛ p > 1✳ ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ c t❛❧ q✉❡ ♣❛r❛ t♦❞♦

  n

  s , s ∈ R

  1 2 ✱

  

  p

  c |s − s | ,

  p

  2

  1

  s❡ p ≥ 2   

  p−2 p−2

  |s | − |s | − s ≥

  2 s

  2 1 s 1 , s

  1

  2

  2

  c |s − s |

  p

  2

  1

   

  2−p

   (|s | + |s |)

  , s❡ p ≤ 2

  2

  1 n

  ♦♥❞❡ h·, ·i é ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ✉s✉❛❧ ❡♠ R ✳ ❉❡♠♦♥str❛çã♦✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ❚❡♦r❡♠❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠

  1,p

  , u ∈ W (Ω)

  1

   R

  p

  c ¯ |∇(u − u )| dx; p ≤ 2

  p

  2

  1

   Ω 

  Z 

  p−2 p−2

  2

  (|∇u | ∇u − |∇u | ∇u )∇(u − u )dx ≥

  2

  2

  1

  1

  2

  

1

R p p p c ¯ ( |∇(u 2 −u 1 )| dx)

  Ω Ω

    2−p ; p ≥ 2.  R

  p p ( (|∇u |+|∇u |) dx)

  2

  1 Ω

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ♣✳ ✼✸✳ ❖❜s❡r✈❛çã♦ ✶✳✹✸✳ ❆ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ♣♦❞❡ s❡ ❡s❝r✐t❛ ❡♠ t❡r♠♦s ❞❛ ♥♦r♠❛

  1,p

  (Ω) ❡♠ W ✳ ❉❡ ❢❛t♦✱ ❝♦♠♦

  1 Z p p p

  (|∇u | + |∇u |) dx = k|∇u | + |∇u |k

  1

  2

  1

  2 L (Ω) Ω 1,p 1,p

  ≤ ku k + ku k ,

  1

  2 W (Ω) W (Ω)

  ❡♥tã♦

  1

  1 ≥ .

  1 Z 1,p 1,p p

  ku k + ku k

  1

  2 W (Ω) W (Ω) p

  (|∇u | + |∇u |) dx

  1

  2 Ω

  ❆ss✐♠✱ s❡ p ≤ 2 ✈❛❧❡ Z

  p p−2 p−2

  |∇u | ∇u − |∇u | ∇u ∇(u − u ku − u k )dx ≥ ¯ c 1,p ,

  2

  2

  1

  1

  2 1 p

  2

  1 W (Ω)

  ✶✹ ✶✳✸✳ ●✃◆❊❘❖

  ❡ s❡ p ≥ 2 ✈❛❧❡

  2 Z

  ku − u k 1,p c ¯ p

  2

  1 W (Ω) p−2 p−2

  |∇u | ∇u − |∇u | ∇u ∇(u − u )dx ≥ .

  2

  2

  1

  1

  

2

  1 2−p Ω

  1,p

  ku k + ku k

  2

  1 W (Ω) p−2

  ◦ ϕ

  p : R −→ R p (s) = |s| s p q =

  ❉❡✜♥❛ ❛ ❢✉♥çã♦ ϕ ♣♦r ϕ ✳ ❖❜s❡r✈❡ q✉❡ ϕ

  1

  1

  id ◦ ϕ = id = 1

  • p q p > 0

  q p

  ❡ ϕ ✱ ♦♥❞❡ ✳

  ❚❡♦r❡♠❛ ✶✳✹✹✳ ❙❡❥❛ p > 1✳ ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ c t❛❧ q✉❡ ♣❛r❛ t♦❞♦

  n

  t , t ∈ R

  1

  2

  ✱ 

  p−1

  c ¯ |t − t | ,

  

p

  2

  1

  s❡ p ≤ 2 

  |ϕ (t ) − ϕ (t )| ≤

  p 2 p

  1 p−2

   ¯ c |t − t | (|t | + |t |) ,

  p

  2

  

1

  2

  1 s❡ p ≥ 2.

  n o

  1

  1 p

  = min , −1

  p

  ♦♥❞❡ ¯c ✳

  c p c p

  ❉❡♠♦♥str❛çã♦✳ ❱❡r ♣✳ ✼✺✳

  ✶✳✸ ●ê♥❡r♦

  ❆s ❞❡♠♦♥str❛çõ❡s ❞♦s r❡s✉❧t❛❞♦s ❞❡st❛ s❡çã♦ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ◆❡st❛ s❡çã♦ ❝♦♥s✐❞❡r❛r❡♠♦s E ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳ ❉❡✜♥✐çã♦ ✶✳✹✺✳ ❉❡♥♦t❛r❡♠♦s ♣♦r Υ ❛ ❢❛♠í❧✐❛ ❞❡ ❝♦♥❥✉♥t♦s A ⊂ E \ {0} t❛❧ q✉❡ A

  é ❢❡❝❤❛❞♦ ❡♠ E ❡ é s✐♠étr✐❝♦✱ ♦✉ s❡❥❛ A = −A✳ P❛r❛ A ∈ Υ✱ ❞❡✜♥✐♠♦s ♦ ❣ê♥❡r♦ ❞❡ A ❝♦♠♦ ♦ ♠❡♥♦r ♥❛t✉r❛❧ ♥ t❛❧ q✉❡ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ í♠♣❛r

  n

  ϕ ∈ C(A, R \ {0}) ✳ ◆❡ss❡ ❝❛s♦✱ ❞❡♥♦t❛r❡♠♦s ♣♦r gen(A) = n✳ ◗✉❛♥❞♦ ♥ã♦

  ❡①✐st❡ ♥✱ ❡s❝r❡✈❡♠♦s gen(A) = ∞ ❡ gen(∅) = 0 ❊①❡♠♣❧♦ ✶✳✹✻✳ ❙❡❥❛ A = B ∪ (−B)✱ ♦♥❞❡ B ⊂ E é ❢❡❝❤❛❞♦ ❡ B ∩ (−B) = ∅✳ ❊♥tã♦✱ gen(A) = 1✱ ♣♦✐s ❛ ❢✉♥çã♦

  R \ {0}

  ϕ : A −→ 

  1 s❡ x ∈ B  x 7−→ 

  −1 s❡ x 6∈ B

  1

  (A, R \ {0}) ♣❡rt❡♥❝❡ ❛ C ✳

  → x

  n ) ⊂ A n

  ❉❡ ❢❛t♦✱ s❡❥❛♠ x ∈ B ❡ (x ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ A t❛❧ q✉❡ x ✳ )

  n

  ❈♦♠♦ −B t❛♠❜é♠ é ❢❡❝❤❛❞♦ ❡ t♦❞❛s ❛s s✉❜s❡q✉ê♥❝✐❛s ❞❡ (x ❝♦♥✈❡r❣❡♠ ♣❛r❛ x ✱ ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ss❛ s❡q✉ê♥❝✐❛ ❡♠ −B é ✜♥✐t♦✱ ♣♦✐s ❝❛s♦ ❝♦♥trár✐♦ t❡rí❛♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ −B q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ x✱ q✉❡ ❡stá ❡♠ B✳ ❆ss✐♠

  ✶✺ ✶✳✹✳ ❯▼❆ ❈❆❘❆❈❚❊❘■❩❆➬➹❖ ❱❆❘■❆❈■❖◆❆▲ ϕ(x ) = 1 = ϕ(x)

  ) → ϕ(x)

  n n

  ❛ ♣❛rt✐r ❞❡ ✉♠ ❝❡rt♦ í♥❞✐❝❡ ❡✱ ❡♥tã♦ ϕ(x ✳ ▲♦❣♦ ϕ é ❝♦♥tí♥✉❛ ❡♠ x✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣❛r❛ x ∈ −B✳ P♦rt❛♥t♦✱ ϕ é ❝♦♥tí♥✉❛✳ ❖❜s❡r✈❛çã♦ ✶✳✹✼✳ ❙❡ A ∈ Υ ❡ gen(A) > 1 ❡♥tã♦ ❆ ❝♦♥té♠ ✐♥✜♥✐t♦s ♣♦♥t♦s✳

  ❈♦♠ ❡❢❡✐t♦✱ s❡ ❢♦ss❡ ✜♥✐t♦✱ ❝♦♠♦ ❆ é s✐♠étr✐❝♦ ❡ A ⊂ E \ {0}✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r A = B ∪ (−B)✱ ♦♥❞❡ ❇ é ❢❡❝❤❛❞♦ ✭♣♦✐s é ✜♥✐t♦✮ ❡ B ∩ (−B) = ∅✳ ▼❛s ♣❡❧♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r t❡rí❛♠♦s gen(A) = 1✳ Pr♦♣♦s✐çã♦ ✶✳✹✽✳ ❙❡❥❛♠ A, B ∈ Υ ❡♥tã♦✿

  ✶✳ ❙❡ x 6= 0, gen({x} ∪ {−x}) = 1✳ ✷✳ ❙❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ í♠♣❛r f ∈ C(A, B) ❡♥tã♦ gen(A) ≤ gen(B)✳ ✸✳ ❙❡ A ⊂ B, gen(A) ≤ gen(B)✳ ✹✳ gen(A ∪ B) ≤ gen(A) + gen(B)✳

  ❖❜s❡r✈❛çã♦ ✶✳✹✾✳ ❙❡ gen(B) < ∞, gen(A \ B) ≥ gen(A) − gen(B)✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ A ⊂ A \ B ∪ B✱ t❡♠♦s gen(A) ≤ gen(A \ B ∪ B),

  ♣♦r (3) ≤ gen(A \ B) + gen(B),

  ♣♦r (4), ♦✉ s❡❥❛ gen(A \ B) ≥ gen(A) − gen(B).

  k

  Pr♦♣♦s✐çã♦ ✶✳✺✵✳ ❙❡ A ⊂ E, Ω é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❧✐♠✐t❛❞❛ ❞❡ 0 ∈ R ❡ ❡①✐st❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ í♠♣❛r h ∈ C(A, ∂Ω) ❡♥tã♦ gen(A) = k✳

  k

  ❉❡✜♥✐çã♦ ✶✳✺✶✳ ❉❡✜♥✐♠♦s ♣♦r C ✱ ❛ ❝❧❛ss❡ ❞♦s ❝♦♥❥✉♥t♦s s✐♠étr✐❝♦s q✉❡ t❡♠ ❣ê♥❡r♦ ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ k✳ ▲❡♠❜r❛♥❞♦ q✉❡ ✉♠ ❝♦♥❥✉♥t♦ C é s✐♠étr✐❝♦ s❡ C = −C✳

  ✶✳✹ ❯♠❛ ❈❛r❛❝t❡r✐③❛çã♦ ❱❛r✐❛❝✐♦♥❛❧

  ❈♦♥s✐❞❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ✉♥✐❞✐♠❡♥❝✐♦♥❛❧ ❡♠ Ω = (0, L)✱

  

′ s−2 ′ ′ s−2

  −(|u | u ) = Λ|u| u ✭✶✳✹✮

  ❝♦♠ ❝♦♥❞✐çã♦ ❞❡ ❢r♦♥t❡✐r❛ ❞❡ ❉✐r✐❝❤❧❡t✱ u(0) = u(L) = 0.

  ❙❛❜❡♠♦s q✉❡ ❛✉t♦✈❛❧♦r❡s ❞❡st❡ ♣r♦❜❧❡♠❛ sã♦ ❞❛❞♦s ♣♦r❀ Z L

  ′ s

  |u | dx Λ (s) = inf sup ,

  k

  Z L

  C∈C k

u∈C

s

  |u| dx

  ✶✻ ✶✳✹✳ ❯▼❆ ❈❆❘❆❈❚❊❘■❩❆➬➹❖ ❱❆❘■❆❈■❖◆❆▲

  1,s

  (0, L)

  k

  ♦♥❞❡ u ∈ W ❡ C s❡ r❡❢❡r❡ ❛ ❉❡✜♥✐çã♦ ❆ss✐♠ t♦❞♦s ♦s s❡✉s ❛✉t♦✈❛❧♦r❡s ❡ ❛s ❛✉t♦❢✉♥çõ❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡①♣❧✐❝✐t❛♠❡♥t❡✱ ❡ ❡st❡ r❡s✉❧t❛❞♦ ♣♦❞❡ s❡r ✈✐st♦ ❡♠ ✈❡❥❛ q✉❡ ❛ ♥♦t❛çã♦ é ❞✐❢❡r❡♥t❡ ❡♠ ❛♠❜❛s ❛s

  (x)

  s

  r❡❢❡rê♥❝✐❛s✮✳ ❈❤❛♠❛r❡♠♦s ❞❡ sin ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧

  ′ s−2 ′ ′ s−2

  −(|u | u ) = (s − 1)|u| u

  ′

  u(0) = 0 , u (0) = 1 ❡ ♣♦r ✐♥t❡❣r❛çã♦ ♣♦❞❡♠♦s ✈❡r q✉❡ ❡st❛ s♦❧✉çã♦ ❡stá ❞❡✜♥✐❞❛ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ❝♦♠♦

  Z sin s (x) dt x = .

  1 s s

  (1 − t ) (x)

  s s

  ❉❡♥♦t❡ ♣♦r π ♦ ♣r✐♠❡✐r♦ ③❡r♦ ❞❡ sin ✱ ❞❛❞♦ ♣♦r Z

  

1

  dt π = 2 .

  s

  1 s s

  (1 − t )

  ′

  (x) (x)

  s

  ❖❜s❡r✈❡ q✉❡ sin ❡ sin s s❛t✐s❢❛③❡♠

  ′

  |sin (x)| ≤ 1 , |sin (x)| ≤ 1

  s s

  ❡ t❛♠❜é♠ t❡♠♦s ♣❡❧❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ P✐tá❣♦r❛s q✉❡

  s ′ s

  |sin s (x)| + |sin (x)| = 1.

  s

  ❖ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ é ♦ s❡❣✉✐♥t❡ ❚❡♦r❡♠❛❀ (s)

  ❚❡♦r❡♠❛ ✶✳✺✷✳ ❖s ❛✉t♦✈❛❧♦r❡s Λ k ❡ ❛s ❛✉t♦❢✉♥çõ❡s u s,k ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❉✐r✐❝❤❧❡t ♥♦ ✐♥t❡r✈❛❧♦ [0, L] sã♦ ❞❛❞♦s ♣♦r

  s s

  π k

  s

  Λ k (s) = (s − 1)

  s

  L kx π s u s,k (s) = sin s

  L ❉❡♠♦♥str❛çã♦✳ ❱❡r ❉❡❧ P✐♥♦✱ ❉r❛❜❡❦ ❡ ▼❛♥ás❡✈✐❝❤✱ ❖❜s❡r✈❛çã♦ ✶✳✺✸✳ ◆♦t❡♠♦s q✉❡ ♦ ❦✲és✐♠♦ ❛✉t♦✈❛❧♦r é s✐♠♣❧❡s✱ ❡ ❛ss♦❝✐❛❞♦ ❛

  s,k s,k

  ❛✉t♦❢✉♥çã♦ u t❡♠ ❦ ❞♦♠í♥✐♦s ♥♦❞❛✐s✳ ❊q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ u t❡♠ ❦✰✶ ③❡r♦s s✐♠♣❧❡s ❡♠ [0, L]✳ ▼❛✐s ❛✐♥❞❛✱ ♦❜s❡r✈❛♠♦s q✉❡ Z π s

  s

  K s = sin (t)dt

  s

  é ✉♠❛ ❝♦♥st❛♥t❡ ✜①❛❞❛ ❞❡♣❡♥❞❡♥❞♦ ❛♣❡♥❛s ❞❡ s✳ ❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ❞❡ ❛✉t♦✈❛❧♦r ♣❛r❛ ♦ ♣✲▲❛♣❧❛❝✐❛♥♦ ❡♠ (a, b)✿

  

′ p−2 ′ ′ p−2

  −(|u | u ) = λw(x)|u| u,

  ✶✼ ✶✳✹✳ ❯▼❆ ❈❆❘❆❈❚❊❘■❩❆➬➹❖ ❱❆❘■❆❈■❖◆❆▲ ❝♦♠ ❝♦♥❞✐çã♦ ❞❡ ❢r♦♥t❡✐r❛ ❞❡ ❉✐r✐❝❤❧❡t u(a) = u(b) = 0,

  ✭✶✳✻✮

  1

  (a, b) ♦♥❞❡ ♦ ♣❡s♦ w(x) ∈ L é ✉♠❛ ❢✉♥çã♦ ♣♦s✐t✐✈❛✱ ❡ ♦ ♣❛r❛♠❡tr♦ λ ∈ R é ♦ ❛✉t♦✈❛❧♦r✳

  P❛r❛ ❡st❡ ♣r♦❜❧❡♠❛ t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿ }

  k k

  ❚❡♦r❡♠❛ ✶✳✺✹✳ ❙❡❥❛ {λ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❛✉t♦✈❛❧♦r❡s ❞♦ ♣r♦❜❧❡♠❛ ♦♠ ❛ ❝♦♥❞✐çã♦ ❊♥tã♦✿

  ✶✳ ❚♦❞❛ ❛✉t♦❢✉♥çã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ❦✲❛✉t♦✈❛❧♦r λ k t❡♠ ❡①❛t❛♠❡♥t❡ ❦✰✶ ③❡r♦s ❡♠ [a, b]✳

  k

  ✷✳ P❛r❛ ❝❛❞❛ ❦✱ λ é s✐♠♣❧❡s✳ < λ < λ

  k k+1

  ✸✳ ❙❡ λ ✱ ❛ ú♥✐❝❛ s♦❧✉ç❛♦ ❞♦ ♣r♦❜❧❡♠❛ é u ≡ 0✳ ❉❡♠♦♥str❛çã♦✳ ❆ ❞❡♠♦♥str❛çã♦ ♣♦r s❡r ❡♥❝♦♥tr❛❞❛ ❡♠

  ❈❛♣ít✉❧♦ ✷ ❊①✐stê♥❝✐❛ ❞❡ ❆✉t♦✈❛❧♦r❡s ●❡♥❡r❛❧✐③❛❞♦s

  ❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ s✐st❡♠❛ ❡❧í♣t✐❝♦ q✉❛s❡ ❧✐♥❡❛r ❞♦ t✐♣♦ r❡ss♦♥❛♥t❡✱ 

  α−2 β

  −∆ u = λαr(x)|u| u|v| ;

  p ❡♠ Ω

   ✭✷✳✶✮

  α β−2

   −∆ v = µβr(x)|u| v|v| ;

  q ❡♠ Ω

  ♦♥❞❡ ❛s ❢✉♥çõ❡s u ❡ v s❛t✐s❢❛③❡♠ ❛ ❝♦♥❞✐çã♦ ❞❡ ❢r♦♥t❡✐r❛ ❞❡ ❉✐r✐❝❤❧❡t u(x) = v(x) = 0 para x ∈ ∂Ω, 1 < p, q < ∞ s❛t✐s❢❛③❡♠

  α β = 1,

  • p q

  ✭✷✳✷✮

  ∞

  (Ω) ❡ r ∈ L ✳

  ◆❡st❡ ❝❛♣ít✉❧♦ ♠♦str❛r❡♠♦s ❛ ❡①✐stê♥❝✐❛ ❞♦s ❛✉t♦✈❛❧♦r❡s ❣❡♥❡r❛❧✐③❛❞♦s ❞♦ s✐st❡♠❛ s♦❜r❡ ❛ r❡t❛ µ = tλ✱ ❡ t❛♠❜é♠ ❡①✐❜✐r❡♠♦s ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ♣❛r❛ ❡st❡s ❛✉t♦✈❛❧♦r❡s✳

  ✷✳✶ ❊①✐stê♥❝✐❛ ❞❡ ❆✉t♦✈❛❧♦r❡s ●❡♥❡r❛❧✐③❛❞♦s k

  ❘❡❧❡♠❜r❛♥❞♦ q✉❡ ♣❛r❛ C ∈ C ♦ ❣ê♥❡r♦ ❑r❛s♥♦s❡❧s❦✐✐ gen(C) é ❞❡✜♥✐❞♦ ❝♦♠♦

  n

  \ {0} ♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ n ♠í♥✐♠♦ ❞❡ t❛❧ ♠♦❞♦ q✉❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ϕ : C −→ R ❝♦♥tí♥✉❛ í♠♣❛r✳ ◆♦ ❡♥t❛♥t♦ q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s µ = tλ ♥ã♦ s❛❜❡♠♦s s❡ ❡st❡ ❝♦♥❥✉♥t♦ ❞❡ ❛✉t♦✈❛❧♦r❡s ❡s❣♦t❛ ♦ ❡s♣❡❝tr♦✳

  ❆♣r❡s❡♥t❛r❡♠♦s ❛❣♦r❛ ✉♠ ❚❡♦r❡♠❛ q✉❡ s❡rá ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♠♦str❛r ❛ ❡①✐stê♥❝✐❛ ❞♦s ❛✉t♦✈❛❧♦r❡s ❛ss✐♠ ❝♦♠♦ ❛s ❛✉t♦❢✉♥çõ❡s✳ ❚❡♦r❡♠❛ ✷✳✶✳ ❙✉♣♦♥❤❛ q✉❡ ❛s s❡❣✉✐♥t❡s ❤✐♣ót❡s❡s sã♦ s❛t✐s❢❡✐t❛s✿ ✭✐✮ ❳ é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ r❡❛❧ ❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛✱ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦✳

  ✶✾ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙

  ∗

  ✭✐✐✮A : X −→ X é ✉♠ ♦♣❡r❛❞♦r ♣♦t❡♥❝✐❛❧ í♠♣❛r✭✐st♦ é ❆ é✱ ❛ ❞❡r✐✈❛❞❛ ❞❡ ●❛t❡❛✉① ❞❡ A : X −→ R) q✉❡ é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥tí♥✉❛ ❡♠ ❝♦♥❥✉♥t♦s ❧✐♠✐t❛❞♦s ❡ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ (S) : −→ u ) −→ v −→ u

  1 j j j

  ❙❡ u ✭ ❢r❛❝❛♠❡♥t❡ ❡♠ ❳✮ ❡ A(u ✱ ❡♥tã♦ u ✭❢♦rt❡♠❡♥t❡ ❡♠ ❳✮✳ ✭✐✐✐✮P❛r❛ ✉♠❛ ❞❛❞❛ ❝♦♥st❛♥t❡ m > 0 ♦ ❝♦♥❥✉♥t♦ ❞❡ ♥í✈❡❧

  M m = {u ∈ X / A(u) = m}

  m

  é ❧✐♠✐t❛❞♦ ❡ ❝❛❞❛ r❛✐♦ q✉❡ ♣❛ss❛ ♣❡❧❛ ♦r✐❣❡♠ ❝r✉③❛ M ✳ ❆❧é♠ ❞✐ss♦ ♣❛r❛ ❝❛❞❛ u 6= 0, hA(u), ui > 0 > 0

  m

  ❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ρ t❛❧ q✉❡ hA(u), ui ≥ ρ em M .

  m m ∗

  ✭✐✈✮❆ ❢✉♥çã♦ B : X −→ X é ✉♠ ♦♣❡r❛❞♦r ♣♦t❡♥❝✐❛❧ í♠♣❛r ❢♦rt❡♠❡♥t❡ s❡q✉❡♥❝✐❛❧ ❝♦♥tí♥✉❛✭❝♦♠ ♣♦t❡♥❝✐❛❧ B✱ ❞❡ ♠♦❞♦ q✉❡ B(u) 6= 0 ✐♠♣❧✐❝❛ q✉❡ B(u) 6= 0✳ ❙❡❥❛

  β = sup inf B(u)

  k

C u∈C

∈Ck

  C ⊂Mm

  ♦♥❞❡ C k é ❛ ❝❧❛ss❡ ❞♦s ❝♦♥❥✉♥t♦s ❝♦♠♣❛❝t♦s s✐♠étr✐❝♦s (C = −C) ❞❡ s✉❜❝♦♥❥✉♥t♦s

  1,p 1,q

  (Ω) × W (Ω) ❞♦ ❡s♣❛ç♦ W ❞❡ ❣ê♥❡r♦ ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ ❦✳

  ∈ M

  k > 0 k m

  ❙❡ β ✱ ❡①✐st❡ ✉♠❛ ❛✉t♦❢✉♥çã♦ u ❝♦♠ m B(u ) = β = .

  k k

  λ

  k

  }

  k k∈N

  ❉✐③❡♠♦s q✉❡ {λ é ♦ ❝♦♥❥✉♥t♦ ❞❡ ❛✉t♦✈❛❧♦r❡s ✈❛r✐❛❝✐♦♥❛✐s✳ ▼❛✐s ❛✐♥❞❛✱ s❡

  γ({u ∈ M / B(u) 6= 0}) = ∞

  m

  ✭✷✳✸✮ ♦♥❞❡ ♦ ❣ê♥❡r♦ s♦❜r❡ ❝♦♥❥✉♥t♦s ❝♦♠♣❛❝t♦s é ❞❡✜♥✐❞♦ ♣♦r

  γ(S) = sup{gen(C) : C ⊂ S; C ∈ C; C compacto}, ❡♠ q✉❡ C é ❛ ❝❧❛ss❡ ❞♦s ❝♦♥❥✉♥t♦s s✐♠étr✐❝♦s✱ ♦✉ s❡❥❛ C = −C✳ ❊♥tã♦ ❡①✐st❡♠ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ❛✉t♦❢✉♥çõ❡s✳ ❉❡♠♦♥str❛çã♦✳ ❱❡r ❍✳❆♠❛♥♥

  ❘❡❧❡♠❜r❛♥❞♦✱ ❡st❛♠♦s tr❛❜❛❧❤❛♥❞♦ ♥♦ s❡❣✉✐♥t❡ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤

  1,p 1,q

  W = W (Ω) × W (Ω) ❡q✉✐♣❛❞♦ ❝♦♠ ❛ ♥♦r♠❛✿ q

  

2

  2

  k(u, v)k = kuk 1,p + kvk 1,q

  W W (Ω) W (Ω) ′ ′

  ∗ ∗ −1,p −1,q

  , v ) ∈ W (Ω) L W (Ω) ❉❛❞♦ (u ♣♦❞❡♠♦s ♣❡♥sá✲❧♦ ❝♦♠♦ ✉♠ ❡❧❡♠❡♥t♦ ❞❡

  ✷✵ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙

  ∗

  W ✿

  

∗ ∗ ∗ ∗

h(u , v ), (u, v)i = hu , ui + hv , vi.

  ′ ′ ∗ −1,p −1,q

  ∼ (Ω) L W (Ω)

  = W ❊♥tã♦ t❡♠♦s W ✭✐s♦♠♦r✜s♠♦ ✐s♦♠étr✐❝♦✮ ♦♥❞❡ ❛

  ∗

  ♥♦r♠❛ ❡♠ W é ❞❛❞❛ ♣♦r✿ q

  ∗ ∗

  

2

  2 ∗ ∗ ∗ k(u , v )k = ku k + kv k .

  W −1,p′ −1,q′

  W (Ω) W (Ω)

  ❈♦♠ ❛ ♥♦t❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❝♦♥s✐❞❡r❡ ♦s s❡❣✉✐♥t❡s ❢✉♥❝✐♦♥❛✐s Z Z

  1

  1

  p q

  A |∇u| |∇v|

  • t (u, v) = , p tq

  

Ω Ω

  1 A (u, v) = (−∆ u, − ∆ v),

  t p q

  t ✭✷✳✹✮

  Z

  

α β

  B(u, v) = |v| r(x)|u| ,

  Ω α−2 β β β−2

  B(u, v) = (r(x)α|u| u|v| , r(x)|u| β|v| v). ❖ ♣r✐♥❝✐♣❛❧ t❡♦r❡♠❛ ❞❡st❡ ❝❛♣ít✉❧♦ é ♦ s❡❣✉✐♥t❡✱ ❚❡♦r❡♠❛ ✷✳✷✳ ❈♦♥s✐❞❡r❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❛✉t♦✈❛❧♦r s♦❜r❡ ❛ r❡t❛ µ = tλ ❝♦♠

  ∞

  r ∈ L (Ω) ✳ ❙❡❥❛

  Z

  α β

  |v| β k = sup inf r(x)|u| ,

  C (u,v)∈C

  ∈Ck Ω C ⊂Mm k m

  ❡♠ q✉❡ ❛ ❝❧❛ss❡ ❞♦s ❝♦♥❥✉♥t♦s C ❢♦✐ ♠❡♥❝✐♦♥❛❞♦ ❉❡✜♥✐çã♦ ❡ M é ❞❛❞♦ ♣♦r

  Z Z

  1

  1

  p q M (u, v) ∈ W ; + = |∇u| |∇v| = m . m

  p tq

  Ω Ω

  ❖ ♣r♦❜❧❡♠❛ t❡♠ ✐♥✜♥✐t♦s ❛✉t♦✈❛❧♦r❡s ✈❛r✐❛❝✐♦♥❛✐s ❝❛r❛❝t❡r✐③❛❞♦s ♣♦r m λ = .

  k

  β

  k k

  ❊ ♠❛✐s✱ q✉❛♥❞♦ Ω = (a, b) ❝❛❞❛ ❛✉t♦✈❛❧♦r λ t❡♠ ✐♥✜♥✐t❛s ❛✉t♦❢✉♥çõ❡s ❛ss♦❝✐❛❞❛s✳ ❉❡♠♦♥str❛çã♦✳ ❉✐✈✐❞✐r❡♠♦s ❛ ♣r♦✈❛ ❡♠ ✈ár✐❛s ❡t❛♣❛s✳ ❖❜s❡r✈❡ q✉❡ W é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛ ❡ ♣❡❧♦ ❚❡♦r❡♠❛ ❣❛r❛♥t✐♠♦s q✉❡ W

  é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥✈❡①♦✳ ❱❡r✐✜❝❛r❡♠♦s ❛❣♦r❛ q✉❡ ❡st❛s ❢✉♥çõ❡s ❡stã♦ ♥❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛

  ❊t❛♣❛ ✶ Pr✐♠❡✐r❛♠❡♥t❡✱ ♠♦str❛r❡♠♦s q✉❡ ❛ ❞❡r✐✈❛❞❛ ❞❡ ●❛t❡❛✉① ❞❡

  Z Z

  1

  1

  p q

  • A (u, v) = |∇u| |∇v| ,

  t

  p tq

  Ω Ω

  ✷✶ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ é ♦ ♦♣❡r❛❞♦r ❞❛❞♦ ♣♦r

  1 A (u, v) = (−∆ u, − ∆ v).

  

t p q

  t P❛r❛ ✐ss♦ ❢❛r❡♠♦s ❛❧❣✉♥s ❝á❧❝✉❧♦s ❡ ♦❜s❡r✈❛çõ❡s✳

  p

  ❖❜s❡r✈❛çã♦ ✷✳✸✳ ❆ ❞❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ |u| ❡ ❞❛❞❛ ♣♦r d

  p p−2 |u| = p|u| u.

  du

  p p

  = u ❈♦♠ ❡❢❡✐t♦✱ s❡ u ≥ 0 t❡♠♦s q✉❡ |u| ❞❛í d

  p p−1 p−2 p−2

  |u| = pu = pu u = p|u| u. du

  p p

  = (−u) ❆❣♦r❛ s❡ u ≤ 0 t❡♠♦s q✉❡ |u| ✱ ❧♦❣♦ d

  p p−1 p−2 p−2 p−2 |u| = −p(−u) = −p(−u) (−u) = p(−u) u = p|u| u.

  du P♦rt❛♥t♦ d

  p p−2 |u| = p|u| u.

  du ∪ {0} → R

  ❖❜s❡r✈❛çã♦ ✷✳✹✳ ❉❡✜♥❛ ❛ ❛♣❧✐❝❛çã♦ φ : R ❞❛❞❛ ♣♦r✿

  • p φ(s) = |∇u + s∇z| .

  P♦❞❡♠♦s r❡❡s❝r❡✈❡r ❡st❛ ❛♣❧✐❝❛çã♦ ❞❡ ❢♦r♠❛ ❝♦♥✈❡♥✐❡♥t❡ ♣❛r❛ ♦❜t❡r♠♦s ❛ s✉❛ ❞❡r✐✈❛❞❛✱ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛

  

p

  φ(s) = |∇u + s∇z|

  p

  

2

  2

  = (|∇u + s∇z| )

  p

  2

  2

  2

  2 = (|∇u| + 2sh∇u, ∇zi + s |∇z| ) .

  ❊♥tã♦ ❝❛❧❝✉❧❛♥❞♦ s✉❛ ❞❡r✐✈❛❞❛ t❡r❡♠♦s

  

p

  p

  −1 ′

  2

  

2

  2

  φ (s) = (|∇u + s∇z| ) . (2h∇u, ∇zi + 2s|∇z| )

  2 p

  p−2

  = |∇u + s∇z| .2h∇u + s∇z, ∇zi

  2

  p−2 = p|∇u + s∇z| .h∇u + s∇z, ∇zi.

  (u, v)

  t

  ❙❛❜❡♠♦s q✉❡ ❛ ❞❡r✐✈❛❞❛ ❞❡ ●❛t❡❛✉① ❞❡ A ♥❛ ❞✐r❡çã♦ ❞❡ (z, w) ∈

  1,p 1,q

  W (Ω) × W (Ω) ♣❡❧❛ ❞❡✜♥✐çã♦ é ♦ s❡❣✉✐♥t❡ ❧✐♠✐t❡

  A

  t ((u, v) + s(z, w)) − A t (u, v) DA (u, v).(z, w) = lim . t

  ✷✷ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙

  t

  ❆❣♦r❛ ✉s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ A t❡♠♦s q✉❡ DA (u, v).(z, w) =

  t

  Z Z

  1

  p p q q

  1

  . (|∇(u + sz)| − |∇u| ) + . (|∇(v + sw)| − |∇v| )

  p

  tq

  Ω Ω

  = lim

  s→0

  s Z p p Z q q 1 |∇(u + sz)| − |∇u|

  1 |∇(v + sw)| − |∇v| = lim

  . +

  s→0

  p s tq s

  Ω Ω

  ◆♦t❡ q✉❡ ♣♦❞❡♠♦s ♣❛ss❛r ♦ ❧✐♠✐t❡ ♣❛r❛ ❞❡♥tr♦ ❞❛s ✐♥t❡❣r❛✐s✳ ❉❡ ❢❛t♦✱ ♦s ✐♥t❡❣r❛♥❞♦s ❡stã♦ ♥❛s ❝♦♥❞✐çõ❡s ❞♦ ❚❡♦r❡♠❛ ♣❛r❛ ❛ ❢✉♥çã♦ φ ❞❡✜♥✐❞❛ ♥❛ ❖❜s❡r✈❛çã♦ ❝♦♥❝❧✉í♠♦s q✉❡ ❡①✐st❡ θ ∈ [0, 1] ❞❡ ♠♦❞♦ q✉❡ ♣❛r❛ s > 0 ✈❛❧❡

  p p

  |∇(u + sz)| − |∇u| φ(s) − φ(0) = s s

  ′

  φ (0 + θs).s = s

  ′ = φ (θs).

  ▲♦❣♦✱

  p p

  |∇(u + sz)| − |∇u|

  p−2 = p|∇(u + sθz)| h∇(u + sθz), ∇zi.

  s ❆ss✐♠✱ s❡ s −→ 0 t❡♠♦s q✉❡

  • p p

  |∇(u + sz)| − |∇u|

  p−2

  −→ p|∇u| ∇u.∇z, q.t.p ❡♠ Ω. s

  ❖s ♠❡s♠♦s ❝á❧❝✉❧♦s ♣♦❞❡♠ s❡r ❢❡✐t♦s ♣❛r❛ ♦ q✉♦❝✐❡♥t❡ ❞❛ s❡❣✉♥❞❛ ✐♥t❡❣r❛❧ ♣❡r♠✐t✐♥❞♦ ❝♦♥❝❧✉✐r q✉❡

  q q

  |∇(v + sw)| − |∇v|

  q−2

  −→ q|∇v| ∇v.∇w, q.t.p ❡♠ Ω, s q✉❛♥❞♦ s −→ 0 ✳

  • ❆❣♦r❛ ♦❜s❡r✈❡ q✉❡

  p p

  |∇(u + sz)| − |∇u|

  p−2

  ≤ p|∇(u + sθz)| |∇(u + sθz)||∇z| s

  p−1

  = p|∇u + θs∇z| |∇z|

  p−1

  ≤ p (|∇u| + θs|∇z|) |∇z|, ❝♦♠♦ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s q✉❛♥❞♦ s −→ 0 ✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ s é

  ✷✸ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ > 0 s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ ❞❡ ♠♦❞♦ q✉❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ c t❛❧ q✉❡

  |s| ≤ c . ❆❧é♠ ❞✐ss♦✱ ♣♦❞❡♠♦s ✉s❛r ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡ ❝♦♥❝❧✉✐r q✉❡

  p p

  |∇(u + sz)| − |∇u|

  p−1

  ≤ p (|∇u| + θs|∇z|) |∇z| s

  p−1 p−1 p−1 p−1 p−1

  ≤ 2 p (|∇u| + θ s |∇z| ) |∇z|

  p−1 p−1 p−1 p−1

  ≤ 2 p |∇u| + c |∇z| |∇z|

  p−1 p−1 p−1 p

  = 2 p |∇u| |∇z| + c |∇z| .

  p p−1 p p p

  1 −1

  ∈ L (Ω) (Ω) ∈ L (Ω) ❈♦♠♦ |∇u| ✱ |∇z| ∈ L ✱ |∇z| ❡ ❛✐♥❞❛ p − 1

  1 = 1, + p p

  ❝♦♥❝❧✉✐♠♦s✱ ♣❡❧♦ ❚❡♦r❡♠❛ q✉❡

  p−1 p−1 p−1 p 1 2 p |∇u| |∇z| + c |∇z| ∈ L (Ω).

  ▲♦❣♦ ♦❜t❡♠♦s q✉❡ ♦ q✉♦❝✐❡♥t❡

  p p

  |∇(u + sz)| − |∇u| s é ❧✐♠✐t❛❞♦ ♣♦r ✉♠❛ ❢✉♥çã♦ ✐♥t❡❣rá✈❡❧✱ ❛♦ q✉❛❧ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ s✳ ❖s ♠❡s♠♦s ❝á❧❝✉❧♦s ♣♦❞❡♠ s❡r ❢❡✐t♦s ♣❛r❛ ♦ s❡❣✉♥❞♦ q✉♦❝✐❡♥t❡ q✉❡ ❛♣❛r❡❝❡ ♥♦ ❧✐♠✐t❡✱ ❡ ❝♦♥❝❧✉✐r❡♠♦s q✉❡ ❡st❡ t❛♠❜é♠ é ❧✐♠✐t❛❞♦ ♣♦r ✉♠❛ ❢✉♥çã♦ ✐♥t❡❣rá✈❡❧ q✉❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ s✳ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ♣❛ss❛r ♦ ❧✐♠✐t❡ ♣❛r❛ ❞❡♥tr♦ ❞❛s ✐♥t❡❣r❛✐s ✉s❛♥❞♦

  u, −

  (|∇w|

  Ω

  u), vi| = Z

  p

  w − (−∆

  p

  |h−∆

  v , é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥tí♥✉♦ ❡♠ ❝♦♥❥✉♥t♦s ❧✐♠✐t❛❞♦s✳ ❈♦♠ ❡❢❡✐t♦✱ ♦❜s❡r✈❡ q✉❡

  q

  1 t ∆

  p

  ∇w − |∇u|

  (u, v) = −∆

  t

  ❆❣♦r❛ ♠♦str❛r❡♠♦s q✉❡ ♦ ♦♣❡r❛❞♦r A

  q v , (·, ·) .

  1 t ∆

  u, −

  p

  (u, v).(·, ·) = −∆

  t

  ❆ss✐♠ ♦❜t❡♠♦s DA

  p−2

  p−2

  1 t ∆

  u), vi| ≤ Z

  |∇w − ∇u|

  

  Z

  p

  ∇u |∇v|dx ≤ ˜c

  p−2

  ∇w − |∇u|

  p−2

  |∇w|

  Ω

  p

  ∇u)∇vdx ≤

  w − (−∆

  p

  t❛❧ q✉❡ |h−∆

  p

  ∇u |∇v|dx, ❡ s❡ 1 < p ≤ 2✱ ♦ ❚❡♦r❡♠❛ ♥♦s ❣❛r❛♥t❡ q✉❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ˜c

  p−2

  ∇w − |∇u|

  p−2

  |∇w|

  Ω

  Z

  q v , (z, w) .

  u, −

  ✷✹ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ ♦ ❚❡♦r❡♠❛ ❞❛í

  p

  q

  − |∇v|

  q

  |∇(v + sw)|

  s→0

  lim

  Ω

  Z

  1 tq

  s

  − |∇u|

  1 p

  p

  |∇(u + sz)|

  s→0

  lim

  Ω

  Z

  1 p

  (u, v).(z, w) =

  t

  DA

  s =

  Z

  p

  ∇u∇z +

  v, w = −∆

  q

  1 t ∆

  u, zi + −

  p

  ∇v∇w = h−∆

  q−2

  |∇v|

  Ω

  1 t Z

  p−2

  Ω

  |∇u|

  Ω

  Z

  ∇v∇w =

  q−2

  q|∇v|

  Ω

  1 tq Z

  ∇u∇z +

  

p−2

  p|∇u|

  p−1 |∇v|dx.

  ✷✺ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙

  p p−1 p p

  −1

  ∈ L ❆♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ♣❛r❛ |∇w − ∇u| ❡ ∇v ∈ L ♦❜t❡r❡♠♦s✿

  Z

  p−1

  |h−∆ w − (−∆ u), vi| ≤ ˜ c |∇w − ∇u| |∇v|dx

  p p p Ω p −1

  1 Z Z p p p p−1 p

  −1 p

  ≤ ˜c |∇w − ∇u| dx . |∇v| dx

  p Ω Ω

p−1

  1,p

  kw − uk = ˜ c p 1,p .kvk .

  W (Ω) W (Ω)

  ❉❛í ❡♥tã♦ ♦❜t❡♠♦s ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡

  p−1

−1,q

k − ∆ w − (−∆ u)k ≤ ˜c kw − uk 1,p . p p W (Ω) p W (Ω)

  1,p −1,q

  : W (Ω) → W (Ω)

  p

  P♦rt❛♥t♦✱ ❝❤❡❣❛♠♦s q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ −∆ é H¨older ❝♦♥tí♥✉♦ ❡✱ ❧♦❣♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥tí♥✉♦✳

  ❙❡ p > 2✱ t❛♠❜é♠ ♣❡❧♦ ❚❡♦r❡♠❛ ❣❛r❛♥t✐♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ❝♦♥st❛♥t❡ c ˜ p t❛❧ q✉❡

  Z

  p−2 p−2

  |h−∆ w − (−∆ u), vi| ≤ |∇w| ∇w − |∇u| ∇u |∇v|dx

  p p Ω

  Z

  p−2 ≤ ˜c |∇w − ∇u| (|∇w| + |∇u|) |∇v|dx. p Ω p p−2 p p p

  −2

  (Ω), (|∇w| + |∇u|) ∈ L (Ω), |∇v| ∈ L (Ω) ❈♦♠♦ |∇w − ∇u| ∈ L

  ❡ 1 p − 2

  1 = 1 + + p p p

  ♣♦❞❡♠♦s ✉s❛r ♦ ❚❡♦r❡♠❛ ❡ ❝♦♥❝❧✉✐r Z

  p−2

  |h−∆ w − (−∆ u), vi| ≤ ˜ c |∇w − ∇u| (|∇w| + |∇u|) |∇v|dx

  p p p Ω p 1 −2

  1 Z Z Z p p p (p−2)p p p p −2

  ≤ ˜c |∇w − ∇u| dx (|∇w| + |∇u|) dx |∇v| dx

  p

Ω Ω Ω

p −2

  Z

  p p

  1,p 1,p

  kw − uk kvk = ˜ c p (|∇w| + |∇u|) dx

  W (Ω W (Ω) Ω p −2

  Z

  p p p p−2

  1,p 1,p

  ≤ ˜c .2 kw − uk (|∇w| + |∇u| )dx kvk

  p W (Ω W (Ω) Ω p −2 p

p p

p−2

  1,p 1,p = ˜ c .2 kw − uk kwk 1,p + kuk 1,p kvk . p W (Ω W (Ω)

  W (Ω) W (Ω)

  ✷✻ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙

  p−2

  .2

  p

  ❉❡♥♦t❛♥❞♦ c = ˜c ✱ ♦❜t❡♠♦s

  p −2 p p p

  −1,q 1,p

  k − ∆ w − (−∆ u)k ≤ ckw − uk kwk 1,p + kuk 1,p ,

  p p W (Ω) W (Ω) W (Ω) W (Ω) 1,p

  (Ω) q✉❡ r❡str✐t♦ ❛ ✉♠ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦✱ A ⊂ W ✱ ♦✉ s❡❥❛✱ s❡ ❡①✐st❡ M > 0 t❛❧ q✉❡

  1,p 1,p

  kwk ≤ M ≤ M, ∀ w, u ∈ A ❡ kuk

  W (Ω) W (Ω)

  t❡♠♦s❀

  p −2 p p −1,q 1,p

  k − ∆ w − (−∆ u)k ≤ c(2M ) kw − uk ,

  p p W (Ω) W (Ω) 1,p

  : A ⊂ W (Ω) → ♦✉ s❡❥❛✱ r❡str✐t♦ ❛♦ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦ A✱ ♦ ♦♣❡r❛❞♦r −∆ p

  −1,q

  W (Ω) é ❧✐♣s❝❤✐t③✱ ♣♦rt❛♥t♦✱ é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥tí♥✉♦✳ ▼♦str❛♠♦s ❡♥tã♦ q✉❡

  ♦ ♦♣❡r❛❞♦r ♣✲▲❛♣❧❛❝✐❛♥♦ é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥tí♥✉♦ ❡♠ ❝♦♥❥✉♥t♦s ❧✐♠✐t❛❞♦s✳ ❊t❛♣❛ ✷✿

  t (u, v) 1 )

  ❆❣♦r❛ ♠♦str❛r❡♠♦s q✉❡ ♦ ♦♣❡r❛❞♦r A s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ (S ✱ ✐st♦ é✱ s❛t✐s❢❛③ , v ) ⇀ (u, v) (u , v ) −→ (z, w) , v ) −→ •

  j j t j j j j

  ❙❡ (u ❡♠ W ❡ A ❡♠ W ❡♥tã♦ (u

  1,p 1,q (u, v) (Ω) × W (Ω).

  ❡♠ W ✱ ♦♥❞❡ W = W , v ) ⇀ (u, v)

  j j

  ❉❡ ❢❛t♦✱ s❡ (u ❡♠ W ✱ ❡♥tã♦ ♣❡❧❛ Pr♦♣♦s✐çã♦ ❛ s❡q✉ê♥❝✐❛ {(u

  j , v j )} j∈N t (u j , v j ) −→ (z, w)

  é ❧✐♠✐t❛❞❛ ❡♠ W ✱ ❡ s❡ A ❡♠ W ✱ t❡♠♦s q✉❡ A (u , v )

  t j j

  é ❞❡ ❈❛✉❝❤② ❡♠ W ✳ ◆♦t❡ q✉❡✱

  ∗

  hA

  t (u j , v j ) − A t (u k , v k ), (u j , v j ) − (u k , v k )i W ,W =

  1 = h−∆ u − (−∆ u ), u − u i + h−∆ v − (−∆ v ), v − v i

  p j p k j k q j q k j k

  t Z

  p−2 p−2

  = |∇u | ∇u − |∇u | ∇u .∇(u − u )dx

  j j k k j k Ω

  Z

  1

  q−2 q−2

  |∇v | ∇v − |∇v | ∇v − v + j j k k .∇(v j k )dx. t

  Ω

  ❆❣♦r❛ t❡r❡♠♦s q✉❡ ❝♦♥s✐❞❡r❛r ❞♦✐s ❝❛s♦s✿ 1 < p ≤ 2 ❡ p ≥ 2.

  • 1 t
  • c
  • c
  • ku
  • kv

  • c
  • 1 t c q kv

  )i

  W ∗

  ,W

  ≤ kA

  t

  (u

  j

  , v

  j

  ) − A

  t

  (u

  k

  

k

  , v

  )k

  W ∗

  k(u

  j

  − u

  k

  , v

  j

  − v

  k

  )k

  W .

  ❈♦♠♦ (A

  k

  j

  − v

  t

  k

  k

  p W 1,p

  (Ω)

  

q

  kv

  j

  − v

  k

  k

  q W 1,q

  (Ω)

  t ≤ hA

  (u

  , v

  j

  , v

  j

  ) − A

  t

  (u

  k

  , v

  

k

  ), (u

  j

  − u

  k

  t

  (u

  n

  W 1,q

  − u

  

k

  k

  W 1,p

  (Ω)

  = 0, ❡ lim

  j,k→∞

  kv

  j

  − v

  

k

  k

  (Ω) = 0.

  ku

  ❖✉ s❡❥❛✱ ❛s s❡q✉ê♥❝✐❛s (u n )

  n∈N ❡ (v n

  )

  n∈N sã♦ ❞❡ ❝❛✉❝❤② ❡♠ W 1,p

  (Ω) ❡ W

  1,q

  (Ω) r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ ❝♦♠♦ ❡st❡s ❡s♣❛ç♦s sã♦ ❝♦♠♣❧❡t♦s t❡♠✲s❡ q✉❡ ❛ s❡q✉ê♥❝✐❛ (u n , v n ) n∈N

  ❝♦♥✈❡r❣❡ ❢♦rt❡ ❡♠ W ✱ ♠❛s ❥á t❡♠♦s q✉❡ ❡st❛ s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡ ❢r❛❝♦ ❡♠ W ❝♦♠ ❧✐♠✐t❡ ❢r❛❝♦ (u, v) ∈ W ✳ P❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ❧✐♠✐t❡ ❝♦♥❝❧✉í♠♦s q✉❡

  (u

  n

  , v

  n

  ) −→ (u, v) ❡♠ W

  j

  j,k→∞

  , v

  − u

  n

  ))

  n∈N

  é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✱ ❡ (u

  n

  , v

  n

  )

  n∈N

  é ❧✐♠✐t❛❞❛ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ lim

  j,k→∞

  c p ku

  j

  k

  sã♦ ♣♦s✐t✐✈❛s ❡ ❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ t > 0✱ ❝♦♥❝❧✉í♠♦s q✉❡ lim

  k

  p W 1,p

  (Ω)

  j

  − v

  k

  k

  q W 1,q

  (Ω) = 0.

  ❈♦♠♦ t❡♠♦s q✉❡ ❛s ❝♦♥st❛♥t❡s c

  p

  ❡ c

  q

  − u

  j

  ku

  − u

  |

  p−2

  ∇u

  j

  − |∇u

  k

  |

  p−2

  ∇u

  k

  .∇(u

  j

  k

  |∇u

  )dx

  Z

  Ω

  |∇v

  j

  |

  q−2

  ∇v

  j

  − |∇v

  k

  |

  q−2

  j

  Ω

  

k

  k

  ✷✼ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ P❡❧❛ ❖❜s❡r✈❛çã♦ ❣❛r❛♥t✐♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ❝♦♥st❛♥t❡s c

  p

  ❡ c

  q

  t❛✐s q✉❡ hA

  t

  (u

  j

  , v

  j

  ) − A

  t

  (u

  , v

  Z

  k

  ), (u

  j

  , v

  j

  ) − (u

  k

  , v

  k

  )i

  W ∗

  ,W

  = =

  ∇v

  .∇(v

  p

  k

  (Ω)

  k

  k

  W 1,p

  (Ω)

  )

  2−p

  q

  kv

  j

  − v

  k

  2 W 1,q

  k

  (Ω)

  t(kv

  j

  k

  W 1,q

  (Ω)

  k

  k

  W 1,q

  (Ω)

  )

  2−q , p ≤ 2.

  ▲♦❣♦✱ ❝♦♥s✐❞❡r❛♥❞♦ p ≥ 2 ❡ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ t❡r❡♠♦s c

  W 1,p

  j

  j

  t kv

  − v

  k

  )dx ≥

             c

  p

  ku

  j

  − u

  k

  k

  p W 1,p

  (Ω)

  q

  j

  (ku

  − v

  

k

  k

  q W 1,q

  (Ω)

  , p ≥ 2 c

  p

  ku

  j

  − u

  k

  k

  1,p (Ω)

  ❝♦♠♦ q✉❡rí❛♠♦s✳

  ✷✽ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ ❙❡ 1 < p ≤ 2 t❡r❡♠♦s✿

  ∗

  hA (u , v ) − A (u , v ), (u , v ) − (u , v )i =

  t j j t k k j j k k W ,W

  Z

  p−2 p−2

  = |∇u | ∇u − |∇u | ∇u .∇(u − u )dx

  j j k k j k Ω

  Z

  1

  q−2 q−2

  |∇v | ∇v | + − |∇v ∇v .∇(v − v )dx

  j j k k j k

  t

  Ω

  2

  2

  ku − u k 1,p kv − v k 1,q

  j k j k W (Ω)

  1 W (Ω) ≥ c c +

  p q 2−p 2−q 1,p 1,p 1,q 1,q

  (ku k + ku k ) t (kv k + kv k )

  

j k j k

W (Ω) W (Ω) W (Ω) W (Ω)

  , v )

  n n n∈N

  ❝♦♠♦ ❛ s❡q✉ê♥❝✐❛ (u é ❧✐♠✐t❛❞❛ ❡♠ W ✱ s❡❣✉❡ q✉❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ M > 0 t❛❧ q✉❡

  1,p

  ku k ≤ M

  j

  ❡

  W (Ω) 1,q

  kv k ≤ M

  j W (Ω)

  ♣❛r❛ t♦❞♦ j ∈ N✳ ❉❛í

  2−p 2−p 1,p 1,p

  (ku k + ku k ) ≤ (2M )

  j k W (Ω) W (Ω)

  ❡

  2−q 2−p 1,q 1,q (kv k + kv k ) ≤ (2M ) . j k W (Ω) W (Ω)

  P♦rt❛♥t♦

  ∗

  hA (u , v ) − A (u , v ), (u , v ) − (u , v )i =

  t j j t k k j j k k W ,W

  Z

  p−2 p−2

  = |∇u | ∇u − |∇u | ∇u .∇(u − u )dx

  j j k k j k Ω

  Z

  1

  q−2 q−2

  |∇v | ∇v − |∇v | ∇v − v

  • j j k k .∇(v j k )dx t

  Ω

  2

  2

  ku − u k 1,p kv − v k 1,q

  j k j k W (Ω)

  1 W (Ω) ≥ c + c ,

  p q 2−p 2−p

  (2M ) t (2M ) ❛ss✐♠ r❡❡s❝r❡✈❡♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ❡ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ t❡r❡♠♦s

  2

  2

  c ku − u k 1,p c kv − v k 1,q

  p j k q j k W (Ω) W (Ω)

  • 2−p 2−p

  (2M ) t(2M )

  ∗

  ≤ hA (u , v ) − A (u , v ), (u , v ) − (u , v )i

  t j j t k k j j k k W ,W ∗

  ≤ kA k(u − u − v

  t (u j , v j ) − A t (u k , v k )k W j k , v j k )k W ,

  (u , v )) , v )

  t n n n∈N n n n∈N

  ❡✱ ♥♦✈❛♠❡♥t❡ ♣♦r (A s❡r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✱ ❡ (u s❡r

  ✷✾ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ ❧✐♠✐t❛❞❛ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡

  2

  2

  ku − u k kv − v k

  j k 1,p j k 1,q W (Ω)

  1 W (Ω) lim c c + = 0.

  p q 2−p 2−p j,k−→∞

  (2M ) t (2M ) ❉❛í ♦❜t❡♠♦s q✉❡

  1,p

  lim ku − u k = 0

  j k W (Ω) j,k−→∞

  ❡

  1,q

  lim kv − v k = 0,

  j k W (Ω) j,k−→∞

  , v )

  n n n∈N

  ♦✉ s❡❥❛ ❛ s❡q✉ê♥❝✐❛ (u é ❞❡ ❈❛✉❝❤② ❡♠ W ✱ q✉❡ é ✉♠ ❡s♣❛ç♦ ❝♦♠♣❧❡t♦✱ ❧♦❣♦ ❝♦♥✈❡r❣❡ ❡♠ W ✳ P♦r ❡st❛ s❡q✉ê♥❝✐❛ ❥á ♣♦ss✉✐r ❧✐♠✐t❡ ❢r❛❝♦ (u, v) ∈ W ❝♦♥❝❧✉í♠♦s ♣❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ❧✐♠✐t❡ q✉❡

  (u , v ) −→ (u, v)

  j j

  ❡♠ W ❝♦♠♦ q✉❡rí❛♠♦s✳

  ❊t❛♣❛ ✸✿ ▼♦str❛r❡♠♦s ❛❣♦r❛ ❛ t❡r❝❡✐r❛ ❝♦♥❞✐çã♦✱ q✉❡ é ❛ s❡❣✉✐♥t❡❀

  • M = {(u, v) ∈ W/A (u, v) = m}

  ❉❛❞♦ ✉♠❛ ❝♦♥st❛♥t❡ m > 0 ♦ ❝♦♥❥✉♥t♦

  m t m

  é ❧✐♠✐t❛❞♦✱ ❡ ❝❛❞❛ r❛✐♦ q✉❡ ♣❛ss❛ ♣❡❧❛ ♦r✐❣❡♠ ✐♥t❡rs❡❝t❛ M ✳ ▼❛✐s ❛✐♥❞❛✱ (u, v), (u, v)i > 0

  

t

  ♣❛r❛ t♦❞♦ (u, v) 6= 0 t❡♠♦s q✉❡ hA ❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ρ > 0 (u, v), (u, v)i ≥ ρ

  m t m

  t❛❧ q✉❡ hA ✳ ▲❡♠❜r❡♠♦s q✉❡

  1

  2

  2

  2 1,p 1,q

  k(u, v)k = kuk + kvk ,

  W W (Ω) W (Ω)

  ❡ s❡ (u, v) ∈ M m ✱ ❡♥tã♦ Z Z

  1

  1

  p q

  • |∇u| |∇v| = m p tq

  Ω Ω

  ♦✉ s❡❥❛✱

  1

  1

  p q

  • kuk 1,p kvk 1,q = m,

  

W (Ω) W (Ω)

  p tq ❧♦❣♦

  1

  p q

  kuk kvk

  

1,p = p m − 1,q

W (Ω) W (Ω)

  tq ≤ pm. P♦rt❛♥t♦

  2

  2 p 1,q

  kuk ≤ (pm) ,

  ✸✵ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙

  2 1,q

  ❝♦♠ ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ ❝♦♥s❡❣✉✐♠♦s ❧✐♠✐t❛r kuk ✱ ♣♦✐s

  W (Ω)

  1

  q p

  kvk 1,q = tq m − kuk 1,p

  

W (Ω) W (Ω)

  p ≤ tqm.

  ❉❛í✱

  2

  2 q

  kvk 1,q ≤ (tqm) .

  W (Ω)

  P♦rt❛♥t♦

  1

  2

  2

  2

  k(u, v)k = kuk 1,p + kvk 1,q

  W W (Ω) W (Ω)

  1

  

2

  2

  2 p q

  ≤ (pm) + (tqm) = C(t).

  m

  ❈♦♠♦ ❡st❛♠♦s ✜①❛♥❞♦ t > 0 s❡❣✉❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ M é ❧✐♠✐t❛❞♦✳

  m

  ▼♦str❡♠♦s ❛❣♦r❛ q✉❡ t♦❞♦ r❛✐♦ c(u, v), c ∈ R ❝♦♠ (u, v) 6= 0 ✐♥t❡rs❡❝t❛ M ✳ ❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❡ (u, v) ∈ W \ {(0, 0)} ❡ ❛ s❡❣✉✐♥t❡ ❢✉♥çã♦ f : R ∪ {0} −→ R

  • c 7−→ A (c(u, v)),

  t

  ♦❜s❡r✈❡ q✉❡ f(0) = 0 ❡

  

p Z q Z

  c c

  p q + f (c) = |∇u| |∇v| .

  p tq

  Ω Ω

  ❆ss✐♠✱ s❡ t é ✜①❛❞♦ ❡ ❢❛③❡♠♦s c → +∞ ✈❡♠♦s q✉❡ f(c) → +∞✳ P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ f s❡❣✉❡ q✉❡ ❡①✐st❡ d ∈ R t❛❧ q✉❡ f(d) = m✱ ♣♦rt❛♥t♦ ♦ r❛✐♦ c(u, v) ✐♥t❡rs❡❝t❛ M

  m

  ✳ ❆❣♦r❛ ❝♦♥s✐❞❡r❡ (u, v) ∈ W \ {(0, 0)}✳ ❉❛í t❡♠♦s

  1 hA

  t (u, v), (u, v)i = h(−∆ p u, − ∆ q v), (u, v)i

  t

  1 = h−∆ u, ui + h−∆ , vi

  p q

  t Z Z

  1

  p−2 q−2

  = |∇u| ∇u.∇u + |∇v| ∇v.∇v t

  Ω Ω

  Z Z

  1

  p q

  = |∇u| |∇v| + t

  

Ω Ω

  1

  p q + = kuk kvk > 0.

  1,p 1,q W (Ω) W (Ω)

  t

  ✸✶ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙

  m

  ◆♦t❡ q✉❡ ❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ p, q > 1 ❡ s❡ (u, v) ∈ M ♦❜t❡♠♦s q✉❡

  1

  1

  p q

  hA kuk kvk

  t (u, v), (u, v)i = p 1,p + q 1,q W (Ω) W (Ω)

  p qt

  1

  1

  p q

  ≥ min{p, q} kuk + 1,p kvk 1,q

  W (Ω) W (Ω)

  p qt = min{p, q}A (u, v)

  t

  = min{p, q}m P♦rt❛♥t♦✱ hA

  t (u, v), (u, v)i ≥ ρ m m = min{p, q}m > 0

  t❛❧ q✉❡ ρ ✳ ▲♦❣♦ ❝♦♥❝❧✉✐♠♦s ❛ t❡r❝❡✐r❛ ❝♦♥❞✐çã♦ ❞♦ ❚❡♦r❡♠❛✳ ❊t❛♣❛ ✹✿ P♦r ✜♠✱ ♠♦str❡♠♦s ❛ ú❧t✐♠❛ ❝♦♥❞✐çã♦✱ q✉❡ é ❛ s❡❣✉✐♥t❡❀

  ∗

  ❆ ❢✉♥çã♦ B : W −→ W é ✉♠ ♦♣❡r❛❞♦r ♣♦t❡♥❝✐❛❧ í♠♣❛r ✭❝♦♠ ♣♦t❡♥❝✐❛❧

  • B ✮ ❝♦♥tí♥✉♦ ♥♦ s❡♥t✐❞♦ ❞❛ ♥♦r♠❛ ✭❢♦rt❡♠❡♥t❡✮✱ t❛❧ q✉❡ B(u, v) 6= 0 ✐♠♣❧✐❝❛

  B(u, v) 6= 0 ✳

  ❉❡ ❢❛t♦✱ ♣♦r ❞❡✜♥✐çã♦✱ ❛ ❞❡r✐✈❛❞❛ ❞❡ ●❛t❡❛✉① ❞❡ B ♥❛ ❞✐r❡çã♦ ❞❡ (z, w) ∈ W é ♦ ❧✐♠✐t❡✿

  B((u, v) + s(z, w)) − B(u, v) DB(u, v).(z, w) = lim

  • s−→0

  s Z Z

  α β α β

  |v + sw| − |v| r(x)|u + sz| r(x)|u|

  Ω Ω

  = lim s

  • s−→0

  Z

  α β α β

  |v + sw| − |u| |v| |u + sz| = lim r(x) . s

  • s−→0 Ω

  ◆♦t❡ q✉❡ ♣♦❞❡♠♦s tr♦❝❛r ♦ ❧✐♠✐t❡ ❞❡ ❧✉❣❛r ❝♦♠ ❛ ✐♥t❡❣r❛❧✳ ❈♦♠ ❡❢❡✐t♦✱ ❞❡✜♥❛ ❛ ❢✉♥çã♦✿

  ϕ : R ∪ {0} −→ R

  • α β

  7−→ |u + sz| |v + sw| s . ❖❜s❡r✈❡ q✉❡ ϕ é ❞❡r✐✈á✈❡❧✱ ❞❛í ✉s❛♥❞♦ ❡ ❛ r❡❣r❛ ❞❛ ❝❛❞❡✐❛ ♦❜t❡♠♦s q✉❡❀

  ′ α−2 β α β−2

  ϕ (s) = α|u + sz| (u + sz)z|v + sw| + |u + sz| β|v + sw| (v + sw)w, ❛ss✐♠✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❡①✐st❡ θ ∈ (0, 1) t❛❧ q✉❡❀

  ′

  ϕ(s) − ϕ(0) = ϕ (θs)s

  ✸✷ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ ♦✉ s❡❥❛✱ ♣❛r❛ s > 0 t❡♠♦s✿

  α β α β

  |u + sz| |v + sw| − |u| |v|

  α−2 β

  = α|u + θsz| + (u + θsz)z|v + θsw| s ✭✷✳✺✮

  α β−2 + |u + θsz| β|v + θsw| (v + θsw)w.

  ▲♦❣♦ ♦❜t❡♠♦s q✉❡

  α β α β

  |u + sz| |v + sw| − |u| |v|

  α−2 β α β−2 lim = α|u| u|v| z + |u| β|v| vw.

  ✭✷✳✻✮ s

  • s−→0

  ❱♦❧t❛♥❞♦ ❡♠ ♦❜s❡r✈❛♠♦s q✉❡

  α β α β

  |u + sz| |v + sw| − |u| |v|

  α−1 β

  ≤ α|u + θsz| |v + θsw| |z|+ s

  α β−1

  |w|,

  • |u + θsz| β|v + θsw| ❡ ✉s❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ t❡♠♦s✿

  α β α β

  |u + sz| |v + sw| − |u| |v| s

  α+β−1 α−1 α−1 α−1 β β β

  ≤ α2 (|u| + (sθ) |z| ) |v| + (θs) |w| |z|+

  α+β−1 α α α β−1 β−1 β−1 |z| |w| |w|.

  • β2 (|u| + (θs) ) |v| + (θs) ❈♦♠♦ θ ∈ (0, 1) ❡ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s q✉❛♥❞♦ s −→ 0 ♣♦❞❡♠♦s t♦♠❛r ✉♠❛
  • ❝♦♥st❛♥t❡ c > 0 t❛❧ q✉❡ |s| ≤ c ❡✱ ♣♦rt❛♥t♦ ♦❜t❡♠♦s ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡ θs ≤ c.

  ❆ss✐♠✱

  

α β α β

  |v + sw| − |u| |v| |u + sz| r(x) s

  α+β−1 α−1 α−1 α−1 β β β

  ≤ r(x) |z| |w| |z|+ α2 (|u| + c ) |v| + c

  α+β−1 α α α β−1 β−1 β−1

  |z| |w| |w| + β2 (|u| + c ) |v| + c .

  p q

α−1 α−1 α−1 β β β

  α β −1

  |z| |v| |w| ∈ L

  • c ) ∈ L (Ω), + c (Ω), |z| ∈ ❖❜s❡r✈❡ q✉❡ (|u|

  q p p α α α β−1 β−1 β−1 q

  β α −1

  |z| |w| ∈ L L (Ω), (|u| + c ) ∈ L (Ω), |v| + c (Ω), |w| ∈ L (Ω)

  ✱ ❡ ❞❡ ♦❜t❡♠♦s

  α − 1 β

  1

  • = 1 p q p

  ❡ α β − 1

  1 = 1. + + p q q

  ✸✸ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ ❆ss✐♠✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❝♦♥❝❧✉✐♠♦s q✉❡ ❛ s♦♠❛ ❡♥tr❡ ❝♦❧❝❤❡t❡s é ✉♠❛ ❢✉♥çã♦

  ∞

  (Ω) |r(x)| ✐♥t❡❣rá✈❡❧✳ ❈♦♠♦ r ∈ L ✱ s❡ M = sup ess x∈Ω t❡♠♦s q✉❡ r(x) ≤ M, q✳t✳♣ ❡♠ Ω,

  ❧♦❣♦

  α β α β

  |u + sz| |v + sw| − |u| |v| r(x) s

  α+β−1 α−1 α−1 α−1 β β β

  ≤ M α2 (|u| + c |z| ) |v| + c |w| |z|+

  α+β−1 α α α β−1 β−1 β−1

  • β2 (|u| + c |z| ) |v| + c |w| |w| , q✳t✳♣ ❡♠Ω,

  α β α β

  |v + sw| − |u| |v| |u + sz|

  ♣♦rt❛♥t♦ r(x) é ❧✐♠✐t❛❞♦ ♣♦r ✉♠❛ ❢✉♥çã♦ s ✐♥t❡❣rá✈❡❧ ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ❞❡ Ω✱ ❛ss✐♠ ♣❡❧♦ ❚❡♦r❡♠❛ ♣♦❞❡♠♦s tr♦❝❛r ♦ ❧✐♠✐t❡ ❞❡ ❧✉❣❛r ❝♦♠ ❛ ✐♥t❡❣r❛❧✱ ♦✉ s❡❥❛

  Z

  α β α β

  |v + sw| − |u| |v| |u + sz|

  DB(u, v).(z, w) = lim r(x)

  • s−→0

  s

  Ω

  Z

  α β α β

  |v + sw| − |u| |v| |u + sz|

  = lim r(x)

  • s−→0

  s

  Ω

  ❡ ♣♦r ❝♦♥❝❧✉✐♠♦s q✉❡ Z

  α−2 β α β−2

  DB(u, v).(z, w) = r(x) α|u| u|v| z + |u| β|v| vw

  Ω

  Z Z

  α−2 β α β−2 = r(x)α|u| u|v| z + r(x)|u| β|v| vw. Ω Ω

  ❆ss✐♠✱ Z Z

  α−2 β α β−2

  DB(u, v).(z, w) = r(x)α|u| u|v| z + r(x)|u| β|v| vw

  Ω Ω α−2 β α β−2

  = hr(x)α|u| u|v| , zi + hr(x)|u| β|v| v, wi

  α−2 β α β−2

  = h(r(x)α|u| u|v| , r(x)|u| β|v| v), (z, w)i, ♦✉ s❡❥❛

  α−2 β α β−2

DB(u, v).(·, ·) = h(r(x)α|u| u|v| , r(x)|u| β|v| v), (·, ·)i

  ❝♦♠ q✉❡rí❛♠♦s✳ ❆❣♦r❛ ♦❜s❡r✈❡ q✉❡ ♦ ♦♣❡r❛❞♦r B é ❝♦♥tí♥✉♦ ♣♦✐s s✉❛s ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s sã♦

  ✸✹ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ ❝♦♥tí♥✉❛s✱ ❡ ❛✐♥❞❛ s❡ t❡♠♦s B(u, v) = (0, 0) ❡♥tã♦✱

  

α−2 β α β−2

  (r(x)α|u| u|v| , r(x)|u| β|v| v) = (0, 0), ❞❛í

  

  α−1 β

  |u| |v| = 0 

  α β−1

   |u| |v|

  = 0 ❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ ♣♦r |u| ❡ ❛ s❡❣✉♥❞❛ ♣♦r |v| ♦❜t❡r❡♠♦s

  α β

  |u| |v| = 0 ∀ x ∈ Ω ♣♦rt❛♥t♦ s❡❣✉❡ q✉❡

  Z

  α β B(u, v) = r(x)|u| |v| dx = 0. Ω

  > 0

  k

  ❆ss✐♠ ❝♦♥❝❧✉✐♠♦s ❛ ❛✜r♠❛çã♦✱ ❡ ♣❡❧♦ ❚❡♦r❡♠❛ s❡ β ✱ ❡①✐st❡ ✉♠❛ , v ) ∈ M

  ❛✉t♦❢✉♥çã♦ (u k k m ❝♦♠ B(u , v ) = β .

  k k k

  ❊t❛♣❛ ✺ ❙✉♣♦♥❞♦ Ω = (a, b) ❞❡♠♦♥str❛r❡♠♦s q✉❡ • γ({(u, v) ∈ M /B(u, v) 6= 0}) = ∞.

  m k

  ❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❡ ❡♠ R ❛ ♥♦r♠❛

  1

  !

  p

k

  X

  p

  |x| |x | = ,

  p i

i=1

k−1 k

  = {x ∈ R / |x| p = 1} k ❡ ❞❡✜♥❛ ♦ ❝♦♥❥✉♥t♦ S ✳ ❆❣♦r❛ ❝♦♥s✐❞❡r❡ φ ❛ ❛✉t♦❢✉♥çã♦

  k

  ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ❛✉t♦✈❛❧♦r λ ❞♦ ♣r♦❜❧❡♠❛ 

  ′ p−2 ′ ′ p−2

  (|φ | φ ) = λ|φ| φ, ❡♠ (a, b)

    φ(a) = φ(b) = 0.

  k

  P❡❧♦ ❚❡♦r❡♠❛ ❛ ❛✉t♦❢✉♥çã♦ φ t❡♠ k + 1 ③❡r♦s ❡♠

  k

  {s } , a = s < s < · · · < s = b

  i 1 k i=0 ✳ ❉❡✜♥❛ ❛ s❡❣✉✐♥t❡ ❢✉♥çã♦

   φ (s) , s )

  k i−1 i

  s❡ s ∈ (s  w (s) =

  i

   ❝❛s♦ ❝♦♥trár✐♦

  ❡♠ q✉❡ i ∈ {1, · · · k}✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ ♣♦❞❡♠♦s s✉♣♦r q✉❡

  1,p 1,q

  kw k = 1 k = 1

  ✱ i = 1, · · · , k − 1 ❡ kw ✱ ♦♥❞❡ ✉s❛♠♦s q✉❡

  • 1 qt
  • 1 qt

  ds + |x

  i=1

  Z s i −1

  s i

  |x

  i

  |

  p

  |w

  ′ i

  (s)|

  p

  

k

  k−1

  |

  p

  kw

  k

  (·)k

  q W 1,q

  (a,b)

  # = m

  "

  k−1

  X

  i=1

  X

  "

  i

  x i w

  x

  k

  w

  k

  (·)

  q W 1,q

  (a,b)

  = m Z

  b a k−1

  X

  i=1

  ′ i

  ds = m

  (s)

  p

  ds + m Z

  b a

  |x

  

k

  |

  p

  |w

  ′ k

  (s)|

  q

  |x

  |

  |

  "

  k−1

  X

  i=1

  |x

  i

  |

  p

  k

  |

  p

  # = m

  k

  # = m

  X

  i=1

  |x

  

i

  |

  p

  # = m|x|

  p = m.

  P♦rt❛♥t♦✱ h(S

  k−1

  ) ⊂ M

  m

  "

  p

  p

  "

  Z b

  a

  |w

  ′ i

  (s)|

  p

  ds + |x

  k

  |

  

p

  # = m

  k−1

  |

  X

  i=1

  |x

  i

  |

  p

  kw

  i

  (·)k

  p W 1,p

  (a,b)

  k

  1 q −1

  k

  ✱ ❡ ❝♦♥❝❧✉í♠♦s ❛ ❛✜r♠❛çã♦✳ ❆❣♦r❛ ♦❜s❡r✈❡ q✉❡ h é ❝♦♥tí♥✉❛ ❡ ✐♥❥❡t♦r❛✳ ❈♦♠ ❡❢❡✐t♦✱ h é ❝♦♥tí♥✉❛✱ ♣♦✐s s✉❛s

  k

  w

  i

  (s) , (qsα)

  1 q

  |x

  k

  |

  p q −1

  x

  k

  w

  (s) ! .

  x

  ❆✜r♠❛çã♦ ✷✳✺✳ h(S

  k−1

  ) ⊂ M

  m

  ❘❡❧❡♠❜r❛♥❞♦✱ M

  m

  = {(u, v) ∈ W/A

  

t

  (u, v) = m} ♦♥❞❡

  A

  t

  (u, v) =

  i

  i=1

  a

  = [w

  ✸✺ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ W

  1,p

  (a, b) ⊂ W

  1,q

  (a, b) ✳ ❈♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦

  B = (

  k−1

  X

  i=1

  c i w i , c k w k !

  ∈ W )

  1

  X

  , · · · , w

  k−1

  , w

  k ].

  ❉❡✜♥❛ ❛ ❢✉♥çã♦ h : S

  k−1

  → B (x

  1

  , · · · x

  k

  ) 7→ (pα)

  1 p k−1

  1 p Z b

  |u

  |x

  =

  (s), (qtm)

  1 q

  |x

  k

  |

  1 q −1

  x

  k

  w

  k

  (s) !

  1 p (pm)

  w

  1 p k−1

  X

  i=1

  x

  i

  w

  i

  (·)

  p W 1,p

  (a,b)

  (qtm)

  1 q

  i

  i

  ′

  ) ∈ S

  |

  p

  Z b

  a

  |v

  ′

  |

  q

  , ❡♠ q✉❡ ❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❝❛s♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ♦♥❞❡ Ω = (a, b)✳ P❛r❛ (x

  1

  , · · · , x

  k

  k−1

  x

  ❛r❜✐trár✐♦ t❡♠♦s❀ A

  t

  (h(x

  1

  , · · · , x

  k−1

  ) = A

  t

  (pm)

  

1

p

k−1

  X

  i=1

  • |x
  • |x

  ✸✻ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s sã♦ ❝♦♥tí♥✉❛s ❡ s❡ h(x , · · · , x ) = h(¯ x , · · · , ¯ x ),

  1 k 1 k i−1 , s i ), i = 1, · · · , k

  s❛❜❡♠♦s q✉❡ ♣❛r❛ s ∈ (s t❡♠✲s❡ w (s) 6= 0 (s) = 0, ∀ j 6= i.

  i j

  ❡ w ▲♦❣♦✱ s❡ h(x , · · · , x )(s) = h(¯ x , · · · , ¯ x )(s),

  1 k 1 k

  t❡♠♦s q✉❡

  1

  1 p p

  (pm) x w (s) = (pm) x ¯ w (s), i = 1, · · · , k − 1

  i i i i

  ❡

  1

  1

  1

  1 −1 −1 q q q q (qtm) |x | x w (s) = (qtm) |¯ x | ¯ x w (s). k k k k k k

  P♦rt❛♥t♦✱ x i = ¯ x i , i = 1, · · · , k − 1 ❡

  

p p

−1 −1

q q

  |x | x = |¯ x | x ¯ ,

  

k k k k

  ♦✉✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱

  p p

  |x | = |¯ x |

  

k k

  ❡ ❛ss✐♠ t❡♠♦s x k = ¯ x k .

  1 , · · · , x k ) = (¯ x 1 , · · · , ¯ x k )

  ❈♦♥❝❧✉í♠♦s ❡♥tã♦ q✉❡ (x ✱ ♣♦rt❛♥t♦ h é ✐♥❥❡t♦r❛✳

  k−1

  ❖❜s❡r✈❡ q✉❡ h é ✉♠❛ ❢✉♥çã♦ í♠♣❛r ❡ S é ❝♦♠♣❛❝t♦✳ P❡❧♦ ❚❡♦r❡♠❛

  k−1 k−1

  → h(S ) ⊂ M

  

m

  ♦❜t❡♠♦s q✉❡ h : S é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ í♠♣❛r✱ ♦✉ s❡❥❛ s❡

  −1 k−1 k−1

  g = h ) → S ✱ ❡♥tã♦ g : h(S é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ í♠♣❛r✱ ❡ ♣❡❧❛ Pr♦♣♦s✐çã♦

  ❝♦♥❝❧✉✐♠♦s q✉❡

  

k−1

gen(h(S )) = k. k−1

  ) ⊂ M ❈♦♠♦ h(S m s❡❣✉❡ q✉❡ gen(M ) ≥ k.

  

m

  ❱✐st♦ q✉❡ k é ✉♠ ♥❛t✉r❛❧ ❛r❜✐trár✐♦ ❝❤❡❣❛♠♦s q✉❡ γ({(u, v) ∈ M /B(u, v) 6= 0}) = ∞,

  m k−1

  ) ⊂ S = {(u, v) ∈ M /B(u, v) 6= 0} = M \ {(0, 0)}

  m m

  ♣♦✐s h(S ❡ t❡♠♦s q✉❡

  k−1 k−1

  h(S ) é ❝♦♠♣❛❝t♦✱ ✈✐st♦ q✉❡ S é ❝♦♠♣❛❝t♦ ❡ h é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳ ❆❧é♠

  k−1

  ❞✐ss♦✱ ❝♦♠♦ ❛ ❢✉♥çã♦ h é í♠♣❛r ❡ S é ✉♠ ❝♦♥❥✉♥t♦ s✐♠étr✐❝♦ s❡❣✉❡ q✉❡

  k−1 k−1 k−1

  − h(S ) = h(−S ) = h(S ).

  k−1

  ) ❆ss✐♠ ❝♦♥❝❧✉✐♠♦s q✉❡ h(S é ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ ❡ s✐♠étr✐❝♦✳ ❈♦♠♦

  k−1

  gen(h(S )) = k s❡❣✉❡ q✉❡ γ(S) = sup{gen(C)/ C ⊂ S, C ∈ C, C

  ❝♦♠♣❛❝t♦} ≥ k, ∀ k ∈ N,

  ✸✼ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ ♦✉ s❡❥❛✱ γ(S) = +∞. ▼♦str❛♠♦s ❡♥tã♦ q✉❡ ♦s ❢✉♥❝✐♦♥❛✐s ❛♣r❡s❡♥t❛❞♦s ♣❛r❛ ♥♦ss♦ ♣r♦❜❧❡♠❛ ❡stã♦ ♥❛s ❝♦♥❞✐çõ❡s ❞♦ ❚❡♦r❡♠❛ ❧♦❣♦ ❣❛r❛♥t✐♠♦s ❡①✐stê♥❝✐❛ ❡ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞♦s ❛✉t♦✈❛❧♦r❡s ❞♦ ♣r♦❜❧❡♠❛ s♦❜r❡ ❛ r❡t❛ µ = tλ✳ ▼❛✐s ❛✐♥❞❛✱ ❣❛r❛♥t✐♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ❛✉t♦❢✉♥çõ❡s ❛ss♦❝✐❛❞❛s ❛ ❝❛❞❛ ❛✉t♦✈❛❧♦r q✉❛♥❞♦ Ω = (a, b)

  ✳ ❉♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r ❣❛r❛♥t✐♠♦s ♣❛r❛ ❝❛❞❛ t ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛

  β (t)

  k

  ❞❛❞❛ ♣♦r✿ B(u, v)

  β k (t) = sup inf

  C (u,v)∈C

  ∈Ck C ∈Mm(t)

  ♦♥❞❡ M m (t) = {(u, v) ∈ W / A t (u, v) = m}

  ❡ ♦ k✲és✐♠♦ ❛✉t♦✈❛❧♦r ✈❛r✐❛❝✐♦♥❛❧ ❞♦ ♣r♦❜❧❡♠❛ s♦❜r❡ ❛ r❛t❛ µ = tλ é ❞❛❞♦ ♣♦r m λ k (t) = . β (t)

  k

  ◆♦ ❡♥t❛♥t♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞✐❢❡r❡♥t❡ ❞♦s ❛✉t♦✈❛❧♦r❡s✱ ❝♦♠ ♦s ❝♦♥❥✉♥t♦s ❞❡ ♥í✈❡❧ ❞♦ ❢✉♥❝✐♦♥❛❧ B ❡♠ ✈❡③ ❞♦ ❢✉♥❝✐♦♥❛❧ A t ✳ ❆ss✐♠✱ ❞❡✜♥✐r❡♠♦s

  (t)

  k

  ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❛✉t♦✈❛❧♦r❡s ˆλ ♣❡❧♦ q✉♦❝✐❡♥t❡ ❞❡ ❘❛②❧❡✐❣❤✱ Z Z

  1

  1

  p q

  |∇u| |∇v|

  • p tq

  Ω Ω

  ˆ λ (t) = inf sup .

  k Z C∈C k

  (u,v)∈C α β

  r(x)|u| |v|

  Ω (t) = λ (t). k k

  ▲❡♠❛ ✷✳✻✳ ˆλ ❉❡♠♦♥str❛çã♦✳ ▼str❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡

  β B(u, v)

  k ≤ sup inf .

  C (u,v)∈C

  m A (u, v)

  ∈Ck t C ⊂W \{0}

  C m ❙❡ C ⊂ W \ {(0, 0)}✱ ❝♦♥str✉✐♠♦s ✉♠ ❝♦♥❥✉♥t♦ ˜ ❡♠ M t♦♠❛♥❞♦ ❛ ✐♠❛❣❡♠ ❞❡ C

  ♣❡❧❛ r❡tr❛çã♦✿ ˜ h : C −→ C

  !

  1/p 1/q

  um vm (u, v) 7−→ , .

  1/p 1/q

  A A (u, v) (u, v)

  t t

  C) P♦r h s❡r ✉♠❛ ❢✉♥çã♦ í♠♣❛r ♦❜t❡♠♦s ♣❡❧❛ Pr♦♣♦s✐çã♦ q✉❡ gen(C) ≤ gen( ˜ ✳

  • m qt.A
  • 1 tq

  m Z

  r(x) um

  Ω

  1 m Z

  dx =

  α p

  (u, v))

  t

  (A

  β

  |v|

  α

  r(x)|u|

  Ω

  α p

  (A

  (u, v) = m

  t

  dx A

  

β

  |v|

  α

  r(x)|u|

  Ω

  Z

  (u, v) =

  t

  A

  ✳ ❙❡❥❛ (u, v) ∈ C✱ ♣❡❧❛ ❝♦♥❞✐çã♦ t❡r❡♠♦s B(u, v)

  m

  1/p

  

t

       = m.

  1/p

  (u,v)∈ ˜ C B(u, v).

  1 m inf

  (u, v) ≥

  t

  B(u, v) A

  (u,v)∈C

  ∈ ˜ C, ∀ (u, v) ∈ C. ❆ss✐♠✱ inf

  1/q

  (u, v))

  t

  (A

  

1/q

  , vm

  (u, v))

  (u, v))

  t

  (A

  1/p

  B(u, v), ∀ (u, v) ∈ C, ♣♦✐s um

  (u,v)∈ ˜ C

  1 m inf

  dx ≥

  1/q β

  (u, v))

  t

  (A

  1/q

  vm

  1/p α

  P♦rt❛♥t♦ t❡♠♦s q✉❡ ˜ C ⊂ M

  A t (u,v)

  ✭✷✳✼✮

  um

  1/p

  ∇ u.m

  

  1 p Z

  =

  (u, v) !

  1/q t

  A

  1/q

  (u, v) , vm

  1/p t

  A

  1/p

  t

  1/p t

  A

  C) ✳ ❆❣♦r❛ ♦❜s❡r✈❡ q✉❡

  C) ≤ gen(C) ✳ ❆ss✐♠ ❝♦♥❝❧✉í♠♦s q✉❡ gen(C) = gen( ˜

  7−→ (u, v), ✈❡♠♦s q✉❡ ˜h t❛♠❜é♠ é í♠♣❛r✱ ❧♦❣♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ♦❜t❡♠♦s q✉❡ gen( ˜

  (u, v) !

  1/q t

  A

  

1/q

  (u, v) , vm

  1/p t

  A

  1/p

  ˜h : ˜ C −→ C um

  ✸✽ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ ❙❡ ❞❡✜♥✐r♠♦s ❛ s❡❣✉✐♥t❡ ❢✉♥çã♦

  A

  (u, v) ! p

  | {z }

  Ω

  q

  |∇v|

  Ω

  Z

  p

  |∇u|

  Ω

  1 p Z

  (u, v)     

  t

  = m A

  q

  |∇v|

  (u, v) Z

  1 tq Z

  t

  p

  |∇(u)|

  Ω

  (u, v) Z

  

t

  = m p.A

  q

  (u, v) !

  1/q t

  A

  1/q

  ∇ v.m

  

  • β q
  • β q

  ✸✾ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ C (˜ u, ˜ v) = m C

  t

  P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ (˜u, ˜v) ∈ ˜ ✱ s❛❜❡♠♦s q✉❡ A ✳ ❈♦♠♦ ˜ é ❛ ✐♠❛❣❡♠ ❞✐r❡t❛ ❞❡ C ♣❡❧❛ r❡tr❛çã♦ h✱ s❡❣✉❡ ❞❛ s♦❜r❡❥❡t✐✈✐❞❛❞❡ ❞❡ h q✉❡ ❡①✐st❡ (u, v) ∈ C t❛❧ q✉❡

  1/p 1/q

  um vm (˜ u, ˜ v) = , .

  1/p 1/q

  (A t (u, v)) (A t (u, v)) P♦rt❛♥t♦

  α β

  Z

  1/p 1/q

  B(˜ u, ˜ v) 1 um vm = r(x) dx

  1/p 1/q

  m m (A t (u, v)) (A t (u, v))

  Ω

  B(u, v) = , ♣❛r❛ ❛❧❣✉♠ (u, v) ∈ C.

  A (u, v)

  t

  ▲♦❣♦ B(u, v) B(˜ u, ˜ v) inf ≤ , ∀ (˜ u, ˜ v) ∈ ˜

  C,

  (u,v)∈C

  A (u, v) m

  t

  ❞❛í s❡❣✉❡ q✉❡ B(u, v)

  1 inf ≤ inf B(u, v).

  ✭✷✳✽✮

  (u,v)∈C

  A (u, v) m (u,v)∈ ˜ C

  t

  ❆ss✐♠✱ ❞❡ t❡♠♦s ❛ ✐❣✉❛❧❞❛❞❡ B(u, v)

  1 B(u, v). inf = inf

  (u,v)∈C

  A (u, v) m (u,v)∈ ˜ C

  t

  ❆❣♦r❛ ♥♦t❡ q✉❡ B(u, v)

  1 B(u, v) sup inf = . sup inf

  C ˜ (u,v)∈C C

  A (u, v) m (u,v)∈ ˜ C

  ∈Ck t ∈Ck C ⊂W \{0}

  ˜ C ⊂W \{0}

  1 β k ≥ B(u, v) = . sup inf .

  C ˜

  m (u,v)∈ ˜ C m

  ∈Ck ˜ C ⊂Mm

  P♦r ♦✉tr♦ ❧❛❞♦ B(u, v) 1 β k

  ≤ B(u, v) = inf . sup inf , ∀ C ∈ W \ {(0, 0)},

  (u,v)∈C C ˜

  A (u, v) m (u,v)∈ ˜ C m

  t ∈Ck ˜

  C ⊂Mm

  ♦✉ s❡❥❛✱ B(u, v)

  β k ≥ sup inf .

  C (u,v)∈C

  m A (u, v)

  ∈Ck t C ⊂W \{0}

  ❈♦♥s❡q✉❡♥t❡♠❡♥t❡ B(u, v)

  β k (t) = sup inf .

  C (u,v)∈C

  m A (u, v)

  ∈Ck t C ⊂W \{0}

  ❊ s❡❣✉❡ q✉❡

  ✹✵ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ m A (u, v)

  t = inf sup .

  C ∈Ck

  β (t) B(u, v)

  k (u,v)∈C C ⊂W \{0}

  ❊♥✜♠ ❝♦♥❝❧✉í♠♦s q✉❡ λ (t) = ˆ λ (t)

  

k k

  ❚❡♠♦s ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛

  k (t), µ k (t))

  ❚❡♦r❡♠❛ ✷✳✼✳ ❊①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❝✉r✈❛s ❝♦♥tí♥✉❛s (λ q✉❡ , λ )

  k k k

  ❡♠❛♥❛♠ ❞❡ (λ ✱ ♦♥❞❡ λ é ♦ ❦✲❛✉t♦✈❛❧♦r ✈❛r✐❛❝✐♦♥❛❧ ❞♦ ♣r♦❜❧❡♠❛ q✉❛♥❞♦ µ = tλ✱ ❡ é ❞❛❞♦ ♣♦r Z Z

  1

  1

  p q

  • |∇u| |∇v| p q

  Ω Ω λ k = inf sup Z . C∈C k (u,v)∈C α β

  |v| r(x)|u|

  Ω

  ❈❧❛r❛♠❡♥t❡ ❥á ♠♦str❛♠♦s ❛ ❡①✐stê♥❝✐❛ ♣♦✐s ♦s ❢✉♥❝✐♦♥❛✐s q✉❡ ❛♣❛r❡❝❡♠ ♥♦ s✐st❡♠❛ ❝♦♥s✐❞❡r❛❞♦ s❛t✐s❢❛③❡♠ ❛s ❝♦♥❞✐çõ❡s ❞♦ ❚❡♦r❡♠❛ ▼♦str❡♠♦s ❛❣♦r❛ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦ ❛✉t♦✈❛❧♦r ❝♦♠ r❡s♣❡✐t♦ ❛ ❝✉r✈❛ ❡♠ t✱ ❛✜♠ ❞❡ ❝♦♥❝❧✉✐r ❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛

  (t), µ (t)) (t)

  

k k k

  ▲❡♠❛ ✷✳✽✳ ❆ ❝✉r✈❛ (λ é ❝♦♥tí♥✉❛✳ ▼❛✐s ❛✐♥❞❛ λ ✭ r❡s♣❡❝t✐✈❛♠❡♥t❡ µ (t)

  k ✮ é ♥ã♦ ❝r❡s❝❡♥t❡ ✭ r❡s♣❡❝t✐✈❛♠❡♥t❡ ♥ã♦ ❞❡❝r❡s❝❡♥t❡✮ ❡♠ t✳

  ❉❡♠♦♥str❛çã♦✳ ❉❡s❞❡ q✉❡ Z Z

  1

  1

  p q

  |∇u| + |∇v| p tq

  Ω Ω

  λ (t) = inf sup

  k Z C∈C k α β (u,v)∈C

  r(x)|u| |v|

  Ω k (t)

  ❡ 1/qt é ❞❡❝r❡s❝❡♥t❡✱ t❡♠♦s q✉❡ λ é ♥ã♦ ❝r❡s❝❡♥t❡✳ ❆❣♦r❛ ♠♦str❡♠♦s ❛ (t)

  k

  ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ λ ✳ ❋✐①❡♠♦s t ✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ í♥✜♠♦ t❡♠♦s q✉❡ ♣❛r❛ ❝❛❞❛ ε > 0

  ε

  ✱ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ s✐♠étr✐❝♦ q✉❡ ❞❡♣❡♥❞❡ ❞❡st❡ ε✱ ❞✐❣❛♠♦s C t❛❧ q✉❡✿ A (u, v)

  t

  sup ≤ λ (t ) + ε

  k

  ✭✷✳✾✮ B(u, v)

  ε (u,v)∈C

  • A
  • A
  • sup
  • λ

  • 1 t q

  ) ,

  ♦♥❞❡ (u

  ε

  , v

  ε

  ) ∈ C

  é ♦ ♣♦♥t♦ ♦❝♦rr❡ ♦ ♠á①✐♠♦ ❞❛ ❢✉♥çã♦ (u, v) 7−→

  ε

  1 t q Z

  Ω

  |∇v|

  q

  B(u, v) q✉❡ é ❛t✐♥❣✐❞♦ ❞❡✈✐❞♦ ❡st❛ ❢✉♥çã♦ s❡r ❝♦♥tí♥✉❛ ❡ ❡st❛♠♦s s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ C ε ✳ ❙♦♠❛♥❞♦✲s❡ ✐st♦ ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ t❡♠♦s q✉❡

  1 t q Z

  ε

  

ε

  , v

  q

  q

  B(u, v) ≥ sup

  (u,v)∈C ε

  1 t q Z

  Ω

  |∇v|

  B(u, v) ≥

  B(u

  1 t q Z

  Ω

  |∇v

  ε

  |

  q

  Ω

  ε

  |∇v

  |∇v

  = |t

  1

  − t | t

  1

  t q Z

  Ω

  ε

  t

  |

  q

  B(u

  ε

  , v

  ε

  (u, v) B(u, v)

  − A

  |

  k (t ) + ε.

  

q

  B(u

  ε

  , v

  ε

  ) ≤ λ

  ✭✷✳✶✶✮ ❯s❛♥❞♦ ❛s ❞❡✜♥✐çõ❡s ❞❡ A

  (u, v) B(u, v)

  t

  ❡ A

  t 1 t❡r❡♠♦s

  A

  t

  1

  |∇v|

  Ω

  Z

  1

  t (u, v)

  − A

  (u, v) B(u, v)

  1

  t

  A

  (u,v)∈C ε

  ≤ sup

  (u, v) B(u, v)

  t

  (u, v) B(u, v)

  t

  − A

  (u, v) B(u, v)

  t

  t (u, v)

  ) ≤ sup

  ✹✶ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ ❈♦♥s✐❞❡r❡♠♦s ❛❣♦r❛ t

  1

  ✱ ❧♦❣♦ λ

  k

  (t

  1

  (u,v)∈C ε

  A

  A

  t

  1

  (u, v) B(u, v)

  = sup

  (u,v)∈C ε

  B(u, v)

  B(u, v) ≤ sup

  p

  A

  t

  (u, v) B(u, v)

  k

  (t ) + ε, ♣♦r

  ✭✷✳✶✵✮ ❊ ❧❡♠❜r❛♥❞♦ q✉❡ sup

  (u,v)∈C ε

  t

  (u, v) B(u, v)

  (u, v) B(u, v)

  = sup

  (u,v)∈C ε

  1 p Z

  Ω

  |∇u|

  − A

  1

  (u,v)∈C ε

  (u, v) B(u, v)

  A

  t

  1

  (u, v) B(u, v)

  − A

  t

  (u,v)∈C ε

  t

  A

  t

  (u, v) B(u, v)

  ≤ sup

  (u,v)∈C ε

  A

  ) .

  ✹✷ ✷✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ ❉✐st♦ ❡ ❞❛s ❞❡s✐❣✉❛❧❞❛❞❡s ♦❜t❡♠♦s

  |t − t |

  1 λ (t ) ≤ (λ (t ) + ε) + λ (t ) + ε. k 1 k k ✭✷✳✶✷✮

  t

  1

  ▲♦❣♦✱ |t − t |

  1

  λ k (t 1 ) − λ k (t ) ≤ (λ k (t ) + ε) + ε. t

  

1

  1

  ❚r♦❝❛♥❞♦ t ♣♦r t ✱ ❡ ✈✐❝❡✲✈❡rs❛ t❡r❡♠♦s |t − t |

  1

  λ (t ) − λ (t ) ≤ (λ (t ) + ε) + ε

  k k

1 k

  1

  t ❆ss✐♠

  |t − t | |t − t |

  1

  1 |λ (t ) − λ (t )| ≤ max (λ (t ) + ε) + ε, (λ (t ) + ε) + ε . k k

1 k k

  1

  t t

  1

  ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ t❡♠♦s✿ |λ (t ) − λ (t )|

  k k

  1

  − t | |t − t | |t − t | |t

  1

  1

  1 ≤ max (λ (t ) + ε) + ε, (λ (t ) + ε) 1 + ε + . k k

  t

  1 t t

  1

  −→ t

  1

  ❋❛③❡♥❞♦ t ❡ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ❝♦♥❝❧✉✐♠♦s q✉❡✿ lim |λ (t ) − λ (t )| ≤ ε, ∀ε > 0.

  k k

  1 t −→t

  1

  ❖✉ s❡❥❛ lim |λ (t ) − λ (t )| = 0.

  

k k

  1 t −→t

  1 k (t)

  P♦rt❛♥t♦ t❡♠♦s q✉❡ λ é ❝♦♥tí♥✉❛✳

  ❈❛♣ít✉❧♦ ✸ ❊st✐♠❛t✐✈❛s ♣❛r❛ ❛✉t♦✈❛❧♦r❡s ❞❡ s✐st❡♠❛s ❡❧í♣t✐❝♦s q✉❛s❡ ❧✐♥❡❛r❡s ♥❛ r❡t❛

  ◆❡st❡ ❝❛♣ít✉❧♦ ♦❜t❡r❡♠♦s ❡st✐♠❛t✐✈❛s ♣❛r❛ ❛✉t♦✈❛❧♦r❡s ❞♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ 

  ′ p−2 ′ ′ α−2 β

  −(|u (x)| u (x)) = λαr(x)|u| u|v| , x ∈ (a, b),

   ✭✸✳✶✮

  ′ q−2 ′ ′ α β−2

   −(|v

  (x)| v (x)) = µβr(x)|u| v|v| , x ∈ (a, b), ♦♥❞❡ ❛s ❢✉♥çõ❡s u ❡ v s❛t✐s❢❛③❡♠ ❛ ❝♦♥❞✐çã♦ ❤♦♠❡❣ê♥❡❛ ❞❡ ❢r♦♥t❡✐r❛ ❞❡ ❉✐r✐❝❤❧❡t u(a) = v(b) = 0, ♦s ❡①♣♦❡♥t❡s s❛t✐s❢❛③❡♠ 1 < p, q < ∞✱ ❝♦♠

  α β = 1,

  • p q

  ✭✸✳✷✮

  ∞

  (Ω) ❡ r ∈ L ✳

  ❖s r❡s✉❧t❛❞♦s ❞❡st❡ ❝❛♣ít✉❧♦ s❡❣✉❡♠ ❞♦ ❛rt✐❣♦ ♣✉❜❧✐❝❛❞♦ ❡♠ ✷✵✵✻✳ ❆♦ ✜①❛r ✉♠❛ r❡t❛ µ = tλ✱ ❝♦♠♦ ♣r♦✈❛❞♦ ♥❛ ❙❡çã♦ ✷✳✶✱ ❡♥❝♦♥tr❛♠♦s ✉♠

  (t), µ (t)}

  k k k

  ❝♦♥❥✉♥t♦ ❞❡ ❛✉t♦✈❛❧♦r❡s ✈❛r✐❛❝✐♦♥❛✐s {λ ❞♦ ❡s♣❡❝tr♦ ❣❡♥❡r❛❧✐③❛❞♦✱ ❡ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ✈❛r✐❛♥❞♦ ♦ ♣❛râ♠❡tr♦ t ❢♦✐ ♣r♦✈❛❞♦ ♥♦ ❚❡♦r❡♠❛

  

✸✳✶ ▲✐♠✐t❡s ♠á①✐♠♦s ♣❛r❛ ❛✉t♦✈❛❧♦r❡s

❣❡♥❡r❛❧✐③❛❞♦s

  ◆❡st❛ s❡çã♦✱ ♦❜t❡r❡♠♦s ❧✐♠✐t❛♥t❡s s✉♣❡r✐♦r❡s ♣❛r❛ t♦❞♦s ♦s ❛✉t♦✈❛❧♦r❡s ✈❛r✐❛❝✐♦♥❛✐s ❞♦ ❡s♣❡❝tr♦ S ♥♦ ❝❛s♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧✱ ✐st♦ é✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ s✐st❡♠❛ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ (a, b) ❝♦♠ ❝♦♥❞✐çã♦ ♥❛ ❢r♦♥t❡✐r❛ ❞❡ ❉✐r✐❝❤❧❡t

  ✹✹ ✸✳✶✳ ▲■▼■❚❊❙ ▼➪❳■▼❖❙ P❆❘❆ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ ♦❜t❡r❡♠♦s ❧✐♠✐t❛♥t❡s s✉♣❡r✐♦r❡s ♣❛r❛ t♦❞♦s ♦s s❡✉s ❛✉t♦✈❛❧♦r❡s ✈❛r✐❛❝✐♦♥❛✐s✳ ❊st❛ ❣❡♥❡r❛❧✐③❛çã♦ é ♣♦ss✐✈❡❧ ❞❡✈✐❞♦ à ❡str✉t✉r❛ ❞❡ ❞♦♠í♥✐♦ ♥♦❞❛❧ ❞❡ ✉♠ k✲és✐♠♦ ❛✉t♦✈❛❧♦r ❞❡ ✉♠❛ ❡q✉❛çã♦✳

  

  (a, b) ❚❡♦r❡♠❛ ✸✳✶✳ ❙❡❥❛♠ p > q ❡ r ∈ L t❛❧ q✉❡ r(x) ≥ m > 0✳ ❊♥tã♦ ♦ ❦✲és✐♠♦ ❛✉t♦✈❛❧♦r ✈❛r✐❛❝✐♦♥❛❧ ❞♦ ♣r♦❜❧❡♠❛ s♦❜r❡ ❛ r❡t❛ µ = tλ s❛t✐s❢❛③

  q −1+q/p

  Λ m p

  k q/p

  λ (t) ≤ + Λ ,

  k k

  p qt q

  k

  ♦♥❞❡ Λ é ♦ ❦✲és✐♠♦ ❛✉t♦✈❛❧♦r ✈❛r✐❛❝✐♦♥❛❧ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❉✐r✐❝❤❧❡t

  ′ p−2 ′ ′ p−2

  −(|u (x)| u (x)) = λr(x)|u| u ✭✸✳✸✮

  ❡♠ ❛❧❣✉♠ ✐♥t❡r✈❛❧♦ (a, b)✳ ❉❡♠♦♥str❛çã♦✳ ❆♣r❡s❡♥t❛r❡♠♦s ✉♠ ❧✐♠✐t❛♥t❡ s✉♣❡r✐♦r ♣❛r❛ ♦ k✲❛✉t♦✈❛❧♦r

  (t), tλ (t))

  k k

  ✈❛r✐❛❝✐♦♥❛❧ (λ s♦❜r❡ ✉♠❛ r❡t❛ ✜①❛ µ = tλ✳

  k

  ❉❡ ❢❛t♦✱ s❡❥❛♠ Λ ♦ k ❛✉t♦✈❛❧♦r ✈❛r✐❛❝✐♦♥❛❧ ❞❡

  ′ ′ p−2 ′ p−2

  − |u (x)| u (x) = λr(x)|u| u,

  k

  ❡ ϕ ❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛✉t♦❢✉♥çã♦✳ ❊♥tã♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❙t✉r♠✲▲✐♦✉✈✐❧❧❡ ♣❛r❛

  k

  } , a = x < x < · · · < x = b ♦ ♣✲▲❛♣❧❛❝✐❛♥♦ ϕ k t❡♠ k + 1 ③❡r♦s ❡♠ {x j

  1 k ✳ j=0

  (x) = ϕ (x) , x ) (x) = 0

  i k i−1 i i

  ❙❡❥❛ w s❡ x ∈ (x ❡ w ❝❛s♦ ❝♦♥trár✐♦✳ ❆❣♦r❛ ❝♦♥s✐❞❡r❡

  1,p

  S (Ω)

  m ❛ ❡s❢❡r❛ ❡♠ W ❞❡ r❛✐♦ m✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ k p/q−1

  C = (u, |u| u) /u ∈ span{w , · · · , w } ∩ S

  1 k m

  t❡♠ ❣ê♥❡r♦ k✱ ✭❛ ❞❡♠♦♥str❛çã♦ ❞❡st❛ ❛✜r♠❛çã♦ s❡❣✉❡ ❛s ♠❡s♠❛s ❧✐♥❤❛s ❞❛ ❞❡♠♦♥str❛çã♦ ❞❡ ❡t❛♣❛ ✺ ❞❛ ❙❡çã♦ ✷✳✶✮ ❡ é ❛❞♠✐ssí✈❡❧ ♥❛ ❝❛r❛❝t❡r✐③❛çã♦ ✈❛r✐❛❝✐♦♥❛❧

  (t), tλ (t))

  k k

  ❞❡ (λ ✳ P♦r ❡st❛ ❝❛r❛❝t❡r✐③❛çã♦ ✈❛r✐❛❝✐♦♥❛❧ t❡♠♦s✿ Z b Z b

  1

  1

  ′ p ′ q

  • |u | |v | p tq

  a a

  λ (t) = inf sup

  k

  Z b

  C∈C k (u,v)∈C α β

  r(x)|u| |v|

  a

  ✭✸✳✹✮ Z Z

  b b

  1

  1

  ′ p ′ q

  |u | |v |

  • p tq

  a a ≤ sup .

  Z b

  k (u,v)∈C α β

  r(x)|u| |v|

  a k

  ❆❣♦r❛ r❡st❛ ❡❝♦❧❤❡r♠♦s ❛❞❡q✉❛❞❛♠❡♥t❡ ♦ ♣❛r ❞❡ ❢✉♥çõ❡s (u, v) ∈ C ♣❛r❛ ♦❜t❡r♠♦s ❛ ❝♦t❛ ❞❡s❡❥❛❞❛✳ ❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❡♠♦s ❛s ❢✉♥çõ❡s

  p/q−1

  v = |u| u ❡ u = t w + t w + · · · + t w .

  1

  1

  2 2 k k

  ✹✺ ✸✳✶✳ ▲■▼■❚❊❙ ▼➪❳■▼❖❙ P❆❘❆ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ ❆❣♦r❛ ❢❛r❡♠♦s ❛❧❣✉♥s ❝á❧❝✉❧♦s✳ ❈♦♠ ❡st❛s ❡s❝♦❧❤❛s t❡♠♦s q✉❡

  X

  a

  Z b

  dx =

  p

  |

  i

  |w

  p

  |

  i

  r(x) |t

  x i −1

  Z x i

  i=1

  k

  k

  r(x) |t

  dx =

  k

  X

  i=1

  Z

  x i x i −1

  i

  dx =

  |

  α+ pβ q

  |w

  

i

  |

  α+ pβ q

  r(x)

  X

  |

  p/q−1

  β

  dx = Z b

  a

  r(x)|u|

  p dx.

  ✭✸✳✺✮ ❆❣♦r❛ ♦❜s❡r✈❡ q✉❡ s❡ v = |u|

  u ❡♥tã♦ v

  α

  ′

  = p q

  |u|

  p/q−1

  u

  ′

  |v|

  r(x)|u|

  i=1

  |

  |t

  i

  |

  p

  |w

  i

  p

  a

  dx =

  Z b

  a

  r(x)|u|

  p dx.

  ❊♥tã♦ ❝♦♠ ❡st❛s ❡s❝♦❧❤❛s ❝♦♥❝❧✉í♠♦s q✉❡ Z b

  pβ q

  j

  Z b

  X

  β

  dx =

  Z b

  a

  r(x)

  k

  i=1

  p/q−1

  t i w i

  α

  |u|

  pβ q

  dx =

  Z b

  u

  |u|

  r(x)

  a

  a

  r(x)|u|

  α

  |v|

  β

  dx = Z b

  r(x)

  i α

  k

  X

  i=1

  t

  i

  w

  a

  k

  |w

  |

  |t

  i

  |

  α

  |w

  i

  α k

  X

  X

  j=1

  |t

  j

  |

  pβ q

  i=1

  k

  X

  j=1

  i=1

  t

  i

  w

  i α k

  X

  t

  r(x)

  

j

  w

  j pβ q

  dx =

  Z b

  a

  ✳ ❙♦♠❛♥❞♦✲s❡ ✐st♦ ❛

  ✹✻ ✸✳✶✳ ▲■▼■❚❊❙ ▼➪❳■▼❖❙ P❆❘❆ ❆❯❚❖❱❆▲❖❘❊❙ ●❊◆❊❘❆▲■❩❆❉❖❙ ❡q✉❛çã♦ ♦❜t❡♠♦s✱

  q

  Z b Z b Z b Z b

  1

  1

  1 1 p

  ′ p ′ q ′ p p−q ′ q

  |u | |u| |u | +

  • |u | |v | p tq p tq q

  a a a a

  = Z b Z b

  α β α β

  r(x)|u| |v| dx r(x)|u| |v| dx

  a a

  ✭✸✳✻✮

  q 1−

  Z p Z b b

  1

  ′ p p q

  |u | |u|

  q Z p b

  1 p p

  a a ′ p

  |u | + ≤ . Z b Z b tq q

  a p p

  r(x)|u| dx r(x)|u| dx

  a a

  e ϕ

  k k

  ❆❣♦r❛✱ s❡ Λ sã♦ ♦ k ❛✉t♦✈❛❧♦r ❡ ❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛✉t♦❢✉♥çã♦ ❞♦ ♣r♦❜❧❡♠❛

  ′ ′ p−2 ′ p−2

  − |w(x) | w (x) = Λr(x)|w(x)| w(x) ❝♦♠ x ∈ (a, b)

  , x ) = w

  i−1 i k i

  ❡ s❛❜❡♠♦s q✉❡ ♣❛r❛ x ∈ (x ✱ ϕ t❡♠♦s q✉❡✿

  ′ ′ p−2 ′ p−2

  − |w (x) | w (x) = Λ r(x)|w (x)| w (x); x ∈ (x , x ).

  i k i i i−1 i i

  ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❡q✉❛çã♦ ❛❝✐♠❛ ♣♦r w i ❡ ✐♥t❡❣r❛♥❞♦ ♣♦r ♣❛rt❡s ❞❡ x i−1 ❛té x i ♦❜t❡♠♦s✿

  Z x i Z x i

  ′ p p |w | dx = Λ r(x)|w | dx; ∀i ∈ {1, · · · , k}. k i i x i −1 x i −1 p

  |

  i

  ▼✉❧t✐♣❧✐❝❛♥❞♦ ❡st❛ ú❧t✐♠❛ ❡q✉❛çã♦ ♣♦r |t ♦❜t❡♠♦s✿ Z x i Z x i

  ′ p p

  |t | | i w dx = Λ k r(x)|t i w i dx; ∀i ∈ {1, · · · , k}.

  i x i x i −1 −1

  ❉❛í t❡♠♦s

  k k

  Z x i Z x i

  X X

  ′ p p |t w | dx = Λ r(x)|t w | dx. i k i i i x i x i

  −1 −1 i=1 i=1 i

  P❡❧❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ❢✉♥çã♦ w ✱ ❝♦♥❝❧✉í♠♦s ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛ q✉❡✿

  p p k k

  Z b Z b

  X X

  ′ t w dx = Λ r(x) t w dx. i k i i i a a i=1 i=1

  ❉❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ❢✉♥çã♦ u ❡s❝♦❧❤✐❞❛✱ ❡ ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛ ❝❤❡❣❛♠♦s ♥❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦✿

  Z b Z b

  p p ′

  |u | dx = Λ r(x) |u| dx.

  

k

  ✭✸✳✼✮

  a a

  ❈♦♠♦ ♣♦r ❤✐♣ót❡s❡ t❡♠♦s q✉❡ r(x) ≥ m > 0✱ ✉s❛♥❞♦ ❛ ❡q✉❛çã♦ ♦❜t❡♠♦s✿ Z Z

  b b p p ′

  |u | |u| dx ≥ Λ m dx,

  

k

a a

  • 1 tq
  • 1 tq p q
  • 1 tq p q

  =

  p

  dx Z b

  a

  |u

  ′

  |

  p q p

  1 p Λ

  b a

  k

  q

  m

  −1+q/p

  Λ

  

−1+q/p

k

     

  r(x)|u|

  Z

  b a

  (Λ

  ′

  |

  p q p

  ≤

  1 p Λ

  k

  q

  k

  p 1− q p

  m)

  −1

  Z b

  

a

  |u

  ′

  |

  Z

  |u

  b a

  .

  k

  p

  

k

  tp =

  Λ

  k

  p 1 + 1 t

  

✸✳✷ ❆ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ▲②❛♣✉♥♦✈ ❡ ❧✐♠✐t❡s

✐♥❢❡r✐♦r❡s

  k

  ❯♠❛ ✈❡③ q✉❡ ♦s ✈❛❧♦r❡s ♣ró♣r✐♦s ❞❡ ❢♦r❛♠ ❝❛❧❝✉❧❛❞♦s ♣♦r ❉r❛❜❡❦ ❡ ❡①♣❧✐❝✐t❛❞♦s ♣♦r ▼❛♥ás❡✈✐❝❤ q✉❛♥❞♦ r(x) = 1✱

  Λ

  k

  = kπ

  p

  b − a

  p

  (t) ≤ Λ

  λ

  ′

  =

  |

  p

  Z b

  a

  r(x)|u|

  p

  dx    

  1 p Λ

  ❖❜s❡r✈❛çã♦ ✸✳✷✳ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡ ♦ r❡s✉❧t❛❞♦ é ✈❡r❞❛❞❡✐r♦ q✉❛♥❞♦ p = q ❡ ♥❡st❡ ❝❛s♦✱ ✈❛❧❡ ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡✱

  k

  q

  m

  −1+q/p

  Λ

  

q/p

k

  .

  |u

  dx Z

  ,

  q

  |

  p

  Z b

  a

  |v

  ′

  |

  Z b

  |u

  a

  r(x)|u|

  α

  |v|

  β

  dx =

  1 p Z

  ′

  a

  |u

  m)

  ✹✼ ✸✳✷✳ ❆ ❉❊❙■●❯❆▲❉❆❉❊ ❉❊ ▲❨❆P❯◆❖❱ ❊ ▲■▼■❚❊❙ ■◆❋❊❘■❖❘❊❙ ♦✉ s❡❥❛✱

  Z b

  a

  |u|

  p

  dx ≤ (Λ

  

k

  −1

  1 p Z b

  Z b

  a

  |u

  ′

  |

  p dx.

  ✭✸✳✽✮ ❱♦❧t❛♥❞♦ ♥❛ ❡q✉❛çã♦ ♦❜t❡♠♦s✿

  b a

  ′

  p

  dx

  ′

  |

  p

  Z b

  a

  r(x)|u|

  p

  q

  a

  Z

  b a

  |u|

  p 1− q p

  Z b

  

a

  r(x)|u|

  |u

  1 p Z b

  |

  ′

  p

  q

  Z

  b a

  |u|

  p−q

  |u

  |

  dx ≤

  q

  Z b

  a

  r(x)|u|

  α

  |v|

  β

  • 1 tq p q
  • 1 tq p q
  • 1 tq p q
  • Λ

  ✹✽ ✸✳✷✳ ❆ ❉❊❙■●❯❆▲❉❆❉❊ ❉❊ ▲❨❆P❯◆❖❱ ❊ ▲■▼■❚❊❙ ■◆❋❊❘■❖❘❊❙ ♦❜t❡♠♦s ✉♠ ❧✐♠✐t❡ s✉♣❡r✐♦r ❡①♣❧í❝✐t♦ ♥♦ ❝❛s♦ ❞♦ s✐st❡♠❛ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ q✉❛♥❞♦ r(x) ≥ m > 0.

  ❋✐♥❛❧♠❡♥t❡ ❡♠ ❡ ♠❡❧❤♦r❛r❛♥❞♦ ♦s ❧✐♠✐t❡s ✐♥❢❡r✐♦r❡s ❞♦s ❛✉t♦✈❛❧♦r❡s✳ ❊♠ ✈❡③ ❞❡ ✉♠❛ ❜♦❧❛✱ ❢♦✐ ❡♥❝♦♥tr❛❞❛ ✉♠❛ ❝✉r✈❛ ❡♥✈♦❧✈❡♥t❡ t✐♣♦ ❤✐♣ér❜♦❧❡ ❡ q✉❡ ❝♦♥té♠ ♦s ❛✉t♦✈❛❧♦r❡s ♥❛ r❡❣✐ã♦✳

  ❆❣♦r❛ ♠♦str❛r❡♠♦s ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ▲②❛♣✉♥♦✈✱ q✉❡ s❡rá út✐❧ ♣❛r❛ ❡♥❝♦♥tr❛r ❛ ❝✉r✈❛ q✉❡ ❧✐♠✐t❛ ♦s ❛✉t♦✈❛❧♦r❡s✳ ❚❡♦r❡♠❛ ✸✳✸✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ♣❛r❛ ♦ s✐st❡♠❛

  

  ′ p−2 ′ ′ α−2 β

  −(|u (x)| u (x)) = f (x)|u| u|v| , x ∈ (a, b) 

  ′ q−2 ′ ′ α β−2

   −(|v (x)| v (x)) = g(x)|u| v|v| , x ∈ (a, b)

  ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ (a, b)✱ ❝♦♠ ❝♦♥❞✐çã♦ ❞❡ ❢r♦♥t❡✐r❛ ❤♦♠♦❣ê♥❡❛ ❞❡ ❉✐r✐❝❤❧❡t✳ ❊♥tã♦ t❡♠♦s q✉❡

  α/p β/q

  Z Z

  b b ′ ′

  α+β α/p +β/q

  ≤ (b − a) 2 f (x)dx . g(x)dx .

  ✭✸✳✾✮

  a a

  ❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡♠♦s ♦ s✐st❡♠❛✿ 

  ′ p−2 ′ ′ α−2 β

  −(|u (x)| u (x)) = f (x)|u| u|v| , x ∈ (a, b)

  

  ′ q−2 ′ ′ α β−2

   −(|v (x)| v (x)) = g(x)|u| v|v| , x ∈ (a, b)

  ❝♦♠ ❝♦♥❞✐çã♦ ❤♦♠♦❣ê♥❡❛ ❞❡ ❢r♦♥t❡✐r❛ ❞❡ ❉✐r✐❝❤❧❡t u(a) = u(b) = v(a) = v(b) = 0 ❡

  α β = 1.

  • p q

  ✭✸✳✶✵✮

  ❖❜s❡r✈❡ q✉❡ ♣❛r❛ q✉❛❧q✉❡r c ∈ [a, b]✱ t❡♠♦s✿ Z Z

  c b ′ ′

  2|u(c)| = u (x)dx + u (x)dx

  a c

  Z c Z b

  ′ ′

  ≤ |u (x)|dx + |u (x)|dx

  a c

  Z b

  ′ = |u (x)|dx. a

  ❆❣♦r❛✱ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ H¨oldder ♦❜t❡♠♦s✿

  1/p

  Z

  b

  1/p ′ p

  2|u(c)| ≤ (b − a) |u (x)| dx ,

  • 1 p

  e

  1

  e

  2

  ❆❣♦r❛ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❡st❛s ❡q✉❛çõ❡s ♦❜t❡r❡♠♦s✿

  e 2 /q .

  g(x)dx

  2 Z b a

  (β/q−1)e

  |v(d)|

  2

  (α/q)e

  |u(c)|

  ′

  2 /q

  ≤ (b − a)

  ≤ (b − a)

  |v(d)|

  e

  1 /p

  ′

  |u(c)|

  (α/p−1)e

  

1

  (β/p)e

  2

  1 Z b a

  f (x)dx

  e 1 /p

  ❡

  2

  e

  2

  e1 p′

  1

  g(x)dx

  2 = 0.

  − 1 e

  1

  = 0 β p e

  2

  

1

  α p − 1 e

  sã♦ s♦❧✉çõ❡s ❞♦ s✐st❡♠❛ ❧✐♥❡❛r ❤♦♠♦❣ê♥❡♦✿           

  2

  ❡ e

  1

  ❡ |v(d)| s❡ ❛♥✉❧❡♠✱ ✐st♦ é✱ e

  . P❛r❛ ♦❜t❡r ♦ r❡s✉❧t❛❞♦ ♥♦s r❡st❛ ❛❝❤❛r♠♦s ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ♦s ❡①♣♦❡♥t❡s ❞❡ |u(c)|

  e2 p

  e1 p Z b a

  |u(c)|

  ( β p

  

(

α

p

  −1)e

  1

  e

  2

  |v(d)|

  )e

  f (x)dx

  1

  β q −1)e

  2

  × ×

  Z

  b a

  ≤ (b − a)

  e

  ❯s❛♥❞♦ ❛ ❝♦♥❞✐çã♦ ✈❡♠♦s q✉❡ ❡st❡ s✐st❡♠❛ ♣♦ss✉✐ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧✱ ❛♦

  ❊s❝♦❧❤❡♥❞♦ ✉♠ ♣♦♥t♦ c ∈ [a, b] ♦♥❞❡ |u(x)| é ♠á①✐♠♦✱ ❡ d ∈ [a, b] ❝♦♠♦ ✉♠ ♣♦♥t♦ ♦♥❞❡ |v(x)| é ♠á①✐♠♦✱ t❡r❡♠♦s✿

  

α/p

  |u(c)|

  1/p ′

  ≤ (b − a)

  1/p

  dx

  β

  |v(x)|

  α

  f (x)|u(x)|

  b a

  Z

  1/p ′

  2|u(c)| ≤ (b − a)

  β dx.

  β/p

  |u

  ✹✾ ✸✳✷✳ ❆ ❉❊❙■●❯❆▲❉❆❉❊ ❉❊ ▲❨❆P❯◆❖❱ ❊ ▲■▼■❚❊❙ ■◆❋❊❘■❖❘❊❙ ♦♥❞❡

  1 p

  ′

  = 1 ✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ ❞♦ s✐st❡♠❛ ♣♦r u ❡ ✐♥t❡❣r❛♥❞♦

  ♣♦r ♣❛rt❡s ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❡ a ❛té b ♦❜t❡♠♦s q✉❡✿ Z b

  a

  ′

  |v(x)|

  (x)|

  p

  dx = Z b

  a

  f (x)|u(x)|

  α

  |v(d)|

  Z

  2

  1

  β/q

  Z

  b a

  g(x)dx

  1/q .

  ✭✸✳✶✷✮ ❙❡❥❛♠ e

  ❡ e

  

α/q

  2

  ❝♦♥st❛♥t❡s ❛ s❡r❡♠ ❡♥❝♦♥tr❛❞❛s✳ ❊❧❡✈❛♥❞♦ ❛s ❡q✉❛çõ❡s ❡ ♣♦r ❡st❛s ❝♦♥st❛♥t❡s e

  1

  ❡ e

  2

  r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦❜t❡r❡♠♦s✿

  |v(d)|

  |u(c)|

  b a

  b a

  f (x)dx

  1/p .

  ✭✸✳✶✶✮ ❋❛③❡♥❞♦ ♦s ♠❡s♠♦s ❝á❧❝✉❧♦s ♠❛s ❝♦♥s✐❞❡r❛♥❞♦ ❛ ❢✉♥çã♦ v ♣r✐♠❡✐r♦ ♦❜t❡r❡♠♦s✿

  2|v(d)| ≤ (b − a)

  1/q ′

  Z

  f (x)|u(x)|

  1/q ′

  α

  |v(x)|

  β

  dx

  1/p

  ≤ (b − a)

  • e2 q′
  • α q
  • (
  • e
  • α q e
  • β q

  ✺✵ ✸✳✷✳ ❆ ❉❊❙■●❯❆▲❉❆❉❊ ❉❊ ▲❨❆P❯◆❖❱ ❊ ▲■▼■❚❊❙ ■◆❋❊❘■❖❘❊❙ q✉❛❧ s❡ r❡s✉♠❡♠ ❛ ❡q✉❛çã♦✿ e β = e α.

  1

  2

  = α = β

  1

  β α

  Z p Z p

  b b α β

  • α+β

  p′ q′

  2 ≤ (b − a) f (x)dx g(x)dx ,

  a a

  ♦ q✉❡ ♣r♦✈❛ ♦ ❚❡♦r❡♠❛✳ ❆❣♦r❛ ✉s❛r❡♠♦s ❡st❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣❛r❛ ♣r♦✈❛r ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✿

  ❚❡♦r❡♠❛ ✸✳✹✳ ❊①✐st❡ ✉♠❛ ❢✉♥çã♦ h(λ) t❛❧ q✉❡ µ ≥ h(λ) ♣❛r❛ ❝❛❞❛ ❛✉t♦✈❛❧♦r ❣❡♥❡r❛❧✐③❛❞♦ (λ, µ) ❞♦ ♣r♦❜❧❡♠❛ ♦♥❞❡ h(λ) é ❞❛❞❛ ♣♦r

  ! q/β

  1 C h(λ) = , R

  b α/p

  β λ r(x)dx

  a

  ♦♥❞❡ ❛ ❝♦♥st❛♥t❡ C é ❞❛❞❛ ♣♦r

  α+β

  2 C = .

  ′ ′ α/p α/p +β/q

  α (b − a) ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ (λ, µ) ♦ ♣❛r ❞❡ ❛✉t♦✈❛❧♦r❡s ❣❡♥❡r❛❧✐③❛❞♦s ❡ (u, v) ♦ ♣❛r ❝♦rr❡s♣♦♥❞❡♥t❡ q✉❡ é s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ❞♦ s✐st❡♠❛✿

  

  ′ p−2 ′ ′ α−2 β

  −(|u (x)| u (x)) = λαr(x)|u| u|v| , x ∈ (a, b) 

  ′ q−2 ′ ′ α β−2

   −(|v (x)| v (x)) = µβr(x)|u| v|v| , x ∈ (a, b). ❈♦♥s✐❞❡r❡♠♦s ❛❣♦r❛ ❛ ❝♦♥st❛♥t❡✿

  α+β

  2 M = .

  α β

  • p′ q′

  (b − a) ❆ss✐♠✱ s✉❜st✐t✉✐♥❞♦ ❛s ❢✉♥çõ❡s ❛ s❡❣✉✐r✱ ♥❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ▲②❛♣✉♥♦✈ f (x) = λαr(x), g(x) = µβr(x) t❡r❡♠♦s✿

  β

α

  Z b p Z b q M ≤ λαr(x)dx . µβr(x)dx .

  a a

  ❘❡❛rr❛♥❥❛♥❞♦ ♦s t❡r♠♦s ❡ ✉s❛♥❞♦ ❛ ❝♦♥❞✐çã♦ ♦❜t❡♠♦s✿ Z b

  

α/p β/q

  M ≤ (λα) (µβ) r(x)dx,

  a

  ✺✶ ✸✳✷✳ ❆ ❉❊❙■●❯❆▲❉❆❉❊ ❉❊ ▲❨❆P❯◆❖❱ ❊ ▲■▼■❚❊❙ ■◆❋❊❘■❖❘❊❙ q✉❡ ♥♦s ❞á✿

  C λ

  α/p

  M α

  , ♦♥❞❡ C =

  q/β

  r(x)dx    

  b

a

  Z

  α/p

     

     

  1 β

  P♦rt❛♥t♦✱ µ ≥

  q/β ≤ µβ.

  r(x)dx    

  a

  Z b

  α/p

  M (λα)

  ✱ ❡ ✐st♦ t❡r♠✐♥❛ ❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛✳

  ❈❛♣ít✉❧♦ ✹ ❊st✐♠❛t✐✈❛s ♣❛r❛ ❛✉t♦✈❛❧♦r❡s ❞❡ s✐st❡♠❛s ❡❧í♣t✐❝♦s q✉❛s❡ ❧✐♥❡❛r❡s ♥♦ n

  R

  ❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ s✐st❡♠❛ ❡❧í♣t✐❝♦ q✉❛s❡ ❧✐♥❡❛r 

  α−2 β

  −∆ u = λαr(x)|u| u|v| , x ∈ Ω

  p

   ✭✹✳✶✮

  α β−2

   −∆

  q v = µβr(x)|u| v|v| , x ∈ Ω,

  ♦♥❞❡ ❛s ❢✉♥çõ❡s u ❡ v s❛t✐s❢❛③❡♠ ❛ ❝♦♥❞✐çã♦ ❞❡ ❢r♦♥t❡✐r❛ ❞❡ ❉✐r✐❝❤❧❡t u(x) = v(x) = 0 ♣❛r❛ x ∈ ∂Ω,

  ♦s ❡①♣♦❡♥t❡s s❛t✐s❢❛③❡♠ 1 < p, q < ∞✱ α β

  • p q

  = 1, ✭✹✳✷✮

  ∞

  (Ω) ❡ r ∈ L ✳

  ◆❡st❡ ❝❛♣ít✉❧♦ ❞❛r❡♠♦s ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ♣❛r❛ ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r (t), µ (t)}

  1

  1

  ✈❛r✐❛❝✐♦♥❛❧ {λ ❞♦ s✐st❡♠❛ ◗✉❛♥❞♦ Ω é ❛r❜✐trár✐♦✱ ❡st❡ r❡s✉❧t❛❞♦ ❢♦✐ ♣r♦✈❛❞♦ ❡♠

  ▼❛s ✉♠ ❞♦s ♦❜❥❡t✐✈♦s ♣r✐♥❝✐♣❛✐s ❞❡st❡ ❝❛♣ít✉❧♦ é ♠♦str❛r q✉❡✱ ❡♠ ❞♦♠í♥✐♦s

  n

  ❡s♣❡❝✐❛✐s✱ ♥❡st❡ ❝❛s♦ ❛♥é✐s ❡♠ R ✱ é ♣♦ssí✈❡❧ ❡st❡♥❞❡r ♦ r❡s✉❧t❛❞♦ ❞❡ ♣❛r❛ t♦❞♦s ♦s ❛✉t♦✈❛❧♦r❡s ❣❡♥❡r❛❧✐③❛❞♦s✱ s♦❜r❡ ❛ r❡t❛ µ = tλ✱ ♠❛s ❝♦♥s✐❞❡r❛♥❞♦ p = q✱ ❡ ❞❡st❛ ❢♦r♠❛ ♣♦❞❡♠♦s ♦❜t❡r r❡s✉❧t❛❞♦s s❡♠❡❧❤❛♥t❡s ❛♦s ♦❜t✐❞♦s ♥♦ ❈❛♣ít✉❧♦ ✸✱ ❢❛t♦ ❡st❡ q✉❡ ❢♦✐ ❝♦♥❥❡❝t✉r❛❞♦ ♣❡❧♦s ❛✉t♦r❡s ❡♠

  ❉❡st❛ ❢♦r♠❛✱ ♦s r❡s✉❧t❛❞♦s ❞❛ ❙❡çã♦ ❞❡st❡ ❝❛♣ít✉❧♦ sã♦ ♥♦✈♦s ❡ ♦r✐❣✐♥❛✐s✱ ♣♦✐s ♥ã♦ ♦s ❡♥❝♦♥tr❛♠♦s ❛✐♥❞❛ ♥❛ ❧✐t❡r❛t✉r❛ ✈✐❣❡♥t❡✳

  N

  ✺✸ ✹✳✶✳ ❊❙❚■▼❆❚■❱❆❙ P❆❘❆ ❖ P❘■▼❊■❘❖ ❆❯❚❖❱❆▲❖❘ ❊▼ R n

  ✹✳✶ ❊st✐♠❛t✐✈❛s ♣❛r❛ ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ❡♠ R

  ◆❡st❛ s❡çã♦ ❞❛r❡♠♦s ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ♣❛r❛ ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ✈❛r✐❛❝✐♦♥❛❧ {λ

  1 (t), µ 1 (t)}

  ❞♦ s✐st❡♠❛ q✉❛♥❞♦ Ω é ❛r❜✐trár✐♦✱ ❡st❡ r❡s✉❧t❛❞♦ ❢♦✐ ♣r♦✈❛❞♦ ♣♦r ❚❡♦r❡♠❛ ✹✳✶✳ [◆á♣♦❧✐ ❡ P✐♥❛s❝♦ ]. ❙❡❥❛ p > q, r(x) ≥ m > 0✳ ❊♥tã♦✱ ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ✈❛r✐❛❝✐♦♥❛❧ ❞♦ ♣r♦❜❧❡♠❛ s♦❜r❡ ❛ r❡t❛ µ = tλ s❛t✐s❢❛③

  q −1+q/p

  Λ m p

  1 q/p

  • λ ≤ Λ

  1

  1

  p qt q ♦♥❞❡ Λ

  1 é ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ✈❛r✐❛❝✐♦♥❛❧ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❉✐r✐❝❤❧❡t p−2

  −∆ u = λr(x)|u| u

  p

  ❡♠ ❛❧❣✉♠ ❞♦♠í♥✐♦ Ω✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦ A = {(u, v)} ∪ {−(u, v)} ♣❛r❛ t♦❞♦ (u, v) 6= (0, 0)

  ✳ P❡❧❛ Pr♦♣♦s✐çã♦ t❡♠♦s q✉❡ gen(A) = 1✱ ❛❧é♠ ❞✐ss♦ A é ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ s✐♠étr✐❝♦✱ ♦✉ s❡❥❛ A é ✉♠ ❝♦♥❥✉♥t♦ ❛❞♠✐ssí✈❡❧ ♥❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞♦s ❛✉t♦✈❛❧♦r❡s✱ ❞❛í t❡♠♦s✱

  Z Z

  1

  1

  p q

  |∇u| + |∇v| p qt

  Ω Ω

  λ (t) = inf sup

  1 Z C∈C

  1 α β (u,v)∈C

  r(x)|u| |v|

  Ω

  Z Z

  1

  1

  p q

  |∇u| |∇v|

  • p qt

  Ω Ω

  ≤ sup Z

  

(u,v)∈A α β

  r(x)|u| |v|

  

  Z Z

  1

  1

  p q

  |∇u| |∇v|

  • p qt

  Ω Ω = .

  Z

  α β

  r(x)|u| |v|

  Ω

  ❱✐st♦ q✉❡ (u, v) 6= (0, 0) é ❛r❜✐trár✐♦✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ✈❛❧❡ ♣❛r❛ t♦❞♦

  p/q

  (u, v) ∈ W \ {(0, 0)} ✳ ❆❣♦r❛ ❝♦♥s✐❞❡r❡ u = ϕ

  1 ❡ v = ϕ ✱ ♦♥❞❡ ϕ 1 ❞❡♥♦t❛ ❛

  1

  ♣r✐♠❡✐r❛ ❛✉t♦❢✉♥çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❉✐r✐❝❤❧❡t

  p−2

  −∆ w = Λr(x)|w| w

  p

  ❡♠ Ω✳ ❖❜s❡r✈❡ q✉❡ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r

  Z

  p

  r(x)|ϕ | = 1,

  1 Ω

  N

  ✺✹ ✹✳✶✳ ❊❙❚■▼❆❚■❱❆❙ P❆❘❆ ❖ P❘■▼❊■❘❖ ❆❯❚❖❱❆▲❖❘ ❊▼ R Z

  p

  r(x)|ϕ | 6= 1

  1

  ♣♦✐s ❝❛s♦ ❝♦♥s✐❞❡r❛♠♦s ❛ ❢✉♥çã♦

  Ω

  ϕ

  1 ϕ ˜ = .

  1 Z 1/p p

  r(x)|ϕ |

  1 Ω

  ❆ss✐♠ ♦❜t❡r❡♠♦s✿ Z

  p

  | r(x)| ˜ ϕ = 1.

  1 Ω p−2

  w = Λr(x)|w| w

  

p

  ❈♦♠ ✐ss♦✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❡q✉❛çã♦ −∆ ♣♦r w ❡ ✐♥t❡❣r❛♥❞♦ ♣♦r ♣❛rt❡s t❡r❡♠♦s✿

  Z Z

  ′ p−2 ′ 2 p |w | (w ) = Λ r(x)|w| . Ω Ω

  ❆ss✐♠✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ♥♦r♠❛❧✐③❛çã♦✱ ♦❜t❡r❡♠♦s q✉❡ ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❉✐r✐❝❤✐❧❡t é ❞❛❞♦ ♣♦r✿

  Z

  ′ p

  |ϕ | Λ = .

  1

  ✭✹✳✸✮

  1

  ❆❣♦r❛ ♦❜s❡r✈❡ q✉❡✿ p

  p/q−1

′ ′ p/q ′

  v = (ϕ ) = ϕ ϕ ,

  1

  1

  1

  q ❞❛í

  Z Z Z

  βp p/q α+

  α β α β q r(x)|u| .|v| = r(x)|ϕ | .|ϕ | = r(x)|ϕ | .

  

1

  1

  1 Ω Ω Ω

  α β βp = 1 = p

  • p q q

  ❈♦♠♦ t❡♠♦s q✉❡ α + ✱ ❞❛í✿

  Z Z

  α β p r(x)|u| .|v| = r(x)|ϕ | = 1.

  1 Ω Ω

  P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ H¨older t❡♠♦s✿

  q

  Z Z p

  ′ q p/q−1 ′ q

  |v | = |ϕ .ϕ |

  1

  q

  Ω Ω p

  Z p

  p−q ′ q

  = |ϕ | .|ϕ |

  

1

  1

  ✭✹✳✹✮ q

  Ω ′ p 1/s 1/s

  Z Z p

  ′ p p

  ≤ |ϕ | |ϕ |

  1 ,

  1

  q

  Ω Ω

  Z p p

  ′ p

  = r(x)|ϕ | = 1

  1

  ♦♥❞❡ s = ❡ s ✳ ❈♦♠♦ ❡ ♣♦r ❤✐♣ót❡s❡ t❡♠♦s q✉❡ q p − q

  Ω

  ✺✺ ✹✳✷✳ ❚❘❆◆❙❋❖❘▼❆➬➹❖ ❘❆❉■❆▲ ❉❖ P✲▲❆P▲❆❈■❆◆❖ r(x) ≥ m > 0 s❡❣✉❡ q✉❡✿

  Z Z Z

  p p p

  | ≥ | |ϕ | 1 = r(x)|ϕ

  1 m|ϕ 1 = m 1 .

Ω Ω Ω

  ❆ss✐♠ Z

  1

  

p

|ϕ | ≤ .

  1

  ✭✹✳✺✮ m

  Ω

  ▲♦❣♦ ❞❛s ❡q✉❛çõ❡s ❝♦♥❝❧✉✐♠♦s q✉❡✿

  q −1+q/p

  Λ m p

  1 q/p λ + (t) ≤ Λ .

  1

  1

  p qt q

  ✹✳✷ ❚r❛♥s❢♦r♠❛çã♦ r❛❞✐❛❧ ❞♦ ♣✲▲❛♣❧❛❝✐❛♥♦

  ❙❡❥❛ r(x) = r(|x|)✱ ♦✉ s❡❥❛✱ r é ✉♠❛ ❢✉♥çã♦ r❛❞✐❛❧✳ ▼♦str❡♠♦s q✉❡ ❛ ♠✉❞❛♥ç❛ p

  2

  2

  x + · · · + x ❞❡ ✈❛r✐á✈❡❧ ρ = |x| = tr❛♥s❢♦r♠❛ ❛ ❡q✉❛çã♦

  1 n p−2 n

  −∆ u = λr(x)|u| u; x ∈ Ω = {x ∈ R ; R < |x| < ¯ R}

  p

  ♥❛ ❡q✉❛çã♦

  ′ n−1 ′ p−1 ′ n−1 p−2

  ρ |v (ρ)| v (ρ) = −λr(ρ)ρ |v(ρ)| v(ρ) R)

  ♦♥❞❡ v(ρ) = u(|x|) = u(ρ); ρ ∈ (R, ¯ ✳ ❊st❛ é ❝❤❛♠❛❞❛ tr❛♥s❢♦r♠❛çã♦ r❛❞✐❛❧ ❞♦ p ✲▲❛♣❧❛❝✐❛♥♦✳ ❉❡ ❢❛t♦✱ ❧❡♠❜r❡♠♦s q✉❡

  p−2

  −∆ ∇u)

  p u = −div(|∇u|

  ❡✱ s❡ v(ρ) = u(x)✱ t❡r❡♠♦s✿ ∂u(x) ∂v(ρ) d ∂ρ ∂ρ

  ′ = = v(ρ). = v (ρ).

  ∂x ∂x dρ ∂x ∂x

  i i i i

  p

  2

  2

  x + · · · + x ❡ ❝♦♠♦ ρ =

  1 n t❡♠♦s✿

  ∂ρ 1 x i x i x i = .2x = = = . p i p

  2

  2

  2

  2

  ∂x |x| ρ

  i

  2 x + · · · + x x + · · · + x

  1 n 1 n

  ∂u(x) x

  i ′

  = v (ρ) P♦rt❛♥t♦ ✱ ❞❛í t❡♠♦s✿

  ∂x ρ

  i

  x

  ′ ∇u(x) = v (ρ). .

  |x| ❆ss✐♠✱

  p−2 ′ p−2

  |∇u(x)| = |v (ρ)| ,

  ✺✻ ✹✳✸✳ ❊❙❚■▼❆❚■❱❆❙ P❆❘❆ ❆❯❚❖❱❆▲❖❘❊❙ ❊▼ ❆◆➱■❙ ❡ ❡♥tã♦ x

  p−2 ′ p−2 ′ −∆ u = −div(|∇u| ∇u) = −div |v (ρ)| .v (ρ). . p

  ρ ❈♦♠♦

  n

  X x ∂ x

  i

′ p−2 ′ ′ p−2 ′

  −div |v (ρ)| .v (ρ). = − |v (ρ)| .v (ρ).

  ρ ∂x ρ

  i i=1

  ❡ ∂ x

  i ′ p−2 ′

  |v (ρ)| .v (ρ). = ∂x ρ

  i

  ∂ x ∂

  i x i ′ p−2 ′ ′ p−2 ′

  = (|v (ρ)| .v (ρ)) . + |v (ρ)| .v (ρ).

  ∂x i ρ ∂x i ρ   x

  i

  ρ − x

  i ′ ∂ρ x i

  ρ

  ′ p−2 ′ ′ p−2 ′

    = (|v (ρ)| .v (ρ)) . . + |v (ρ)| .v (ρ)

  

  2 

  ∂x ρ ρ

  i

  2

  2 ′ x

  1 x

  

′ p−2 ′ i ′ p−2 ′ i

  − = (|v (ρ)| .v (ρ)) . + |v (ρ)| .v (ρ) ,

  2

  3

  ρ ρ ρ t❡♠♦s

  n

  X

  2

  x

  i n

  2 X ′

  1 x

  i=1 p−2 ′ p−2 ′ ′ p−2 ′ i

  div |∇u| .∇u = |v (ρ)| .v (ρ) . + |v (ρ)| .v (ρ) −

  2

  3

  ρ ρ ρ

  i=1 n

  X n − 1

  2

  2

  x = ρ ❡✱ ❝♦♠♦ ✱ ❛ s❡❣✉♥❞❛ s♦♠❛ ♥♦s r❡st❛ ✱ ❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❡ ❞✐✈✐❞✐♥❞♦

  i

  ρ

  i=1 n−1

  ♣♦r ρ ♦❜t❡r❡♠♦s✿

  n−1

  n − 1 ρ

  ′ p−2 ′ p−2 ′ ′ p−2 ′

  div (|∇u| .∇u) = (|v (ρ)| .v (ρ)) + |v (ρ)| .v (ρ) .

  n−1

  ρ ρ

  ′

n−1 ′ p−2 ′ 1−n

  = (ρ .|v (ρ)| .v (ρ)) .ρ . P♦rt❛♥t♦✱ ♥♦ss❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ ♥♦s ❞á ❛ ❡q✉❛çã♦✿

  ′ n−1 ′ p−2 ′ n−1 p−2

  ρ .|v (ρ)| .v (ρ) = −λr(ρ)ρ |v(ρ)| v(ρ).

  ✹✳✸ ❊st✐♠❛t✐✈❛s ♣❛r❛ ❆✉t♦✈❛❧♦r❡s ❡♠ ❆♥é✐s n

  ; R < |x| < ¯ R} ◆❡st❛ s❡çã♦✱ ❝♦♥s✐❞❡r❛r❡♠♦s Ω = {x ∈ R ❡ p = q✳

  ✺✼ ✹✳✸✳ ❊❙❚■▼❆❚■❱❆❙ P❆❘❆ ❆❯❚❖❱❆▲❖❘❊❙ ❊▼ ❆◆➱■❙ ❈♦♠ ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ ❞❛ s❡çã♦ ❛♥t❡r✐♦r ♣♦❞❡♠♦s ♠♦str❛r q✉❡ ♦ s✐st❡♠❛

  

  α−2 β

  −∆ u = λαr(x)|u| u|v| , x ∈ Ω

  p

      

  α β−2

  −∆ v = µβr(x)|u| v|v| , x ∈ Ω

  p

  ✭✹✳✻✮      u(x) = v(x) = 0; x ∈ ∂Ω,

  ♣♦❞❡ s❡r tr❛♥s❢♦r♠❛❞♦ ❡♠✿  ′

  n−1 ′ p−2 ′ n−1 α−2 β

  (ρ .|u (ρ)| .u (ρ)) = −λαρ r(ρ)|u(ρ)| u(ρ)|v(ρ)| , ρ ∈ (R, ¯ R)     

  ′ n−1 ′ p−2 ′ n−1 α β−2

  (ρ .|v (ρ)| .v (ρ)) = −µβρ r(ρ)|u(ρ)| |v(ρ)| v(ρ), ρ ∈ (R, ¯ R) ✭✹✳✼✮

       u(R) = u( ¯ R) = v(R) = v( ¯ R) = 0.

  ❖❜s❡r✈❡ q✉❡ ❝♦♥s✐❞❡r❛♠♦s ❛s ♠❡s♠❛s ❢✉♥çõ❡s u ❡ v✱ ♠❛s ♥ã♦ ❤á ♣❡r✐❣♦ ❞❡ ❝♦♥❢✉sã♦✱ ♣♦✐s ❥á ♠♦str❛♠♦s ❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ q✉❡ tr❛♥s❢♦r♠❛ ♦ s✐st❡♠❛ ✐♥✐❝✐❛❧ ♥❡st❡ ú❧t✐♠♦✳ ❙♦❜r❡ ❡st❡ ú❧t✐♠♦ s✐st❡♠❛ ✈❛♠♦s ❢❛③❡r ✉♠❛ ♥♦✈❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧✳ ◆♦t❡ q✉❡ n ❡ q sã♦ ♥ú♠❡r♦s ❡ ♣♦❞❡♠ s❡r ❝♦♠♣❛r❛❞♦s✳ ❊st❛ ❝♦♠♣❛r❛çã♦ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ ❡s❝♦❧❤✐❞❛✳ ❆ ♥♦ss❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ ❢✉♥❝✐♦♥❛ ❡♠ ✉♠ ú♥✐❝♦ ❝❛s♦ q✉❡ é ♦ s❡❣✉✐♥t❡✿ n > p.

  ✭✹✳✽✮ ❆♥t❡s ❞❡ ❡♥✉♥❝✐❛r ♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ❢❛r❡♠♦s ❛❧❣✉♠❛s ♠✉❞❛♥ç❛s ♥♦ s✐st❡♠❛ q✉❡ s❡rã♦ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦✳

  ◆❛s ❡q✉❛çõ❡s ❞♦ s✐st❡♠❛ ❢❛ç❛ ❛ ♠✉❞❛♥ç❛✿

  p −1 n −p

  A

  p

  ρ = ρ = , ) )

  p p p

  t❛❧ q✉❡ ¯u(s) = u(ρ ❡ ¯v(s) = v(ρ B − s

  p

  ♦♥❞❡

  n n −p −p p p −1 −1

  ¯ R ¯ R R A = n n = n n .

  

p −p −p p −p −p

  ❡ B ✭✹✳✾✮

  p p p p −1 −1 −1 −1

  ¯ ¯ R − R R − R

  p

  ❖❜s❡r✈❡ q✉❡ ♣♦❞❡♠♦s ♦❜t❡r s ❡♠ ❢✉♥çã♦ ❞❡ ρ ♣♦r −A

  

p

  s = n + B

  −p p

p

−1

  ρ p ❈♦♠ ❡st❛s ♠✉❞❛♥ç❛s ♠♦str❛r❡♠♦s q✉❡ ♣♦❞❡♠♦s ❝❤❡❣❛r ♥♦ s❡❣✉✐♥t❡ s✐st❡♠❛✿

   ′

  ′ p−2 ′ α−2 β

  − (|¯ u (s)| .¯ u (s)) = λαr(ρ )q (s)|¯ u(s)| u(s)|¯ ¯ v(s)| , s ∈ (0, 1)

  p

  

1

      

  ′ ′ p−2 ′ α β−2

  − (|¯ v (s)| .¯ v (s)) = µβr(ρ )q (s)|¯ u(s)| |¯ v(s)| ¯ v(s), s ∈ (0, 1)

  p 1 ✭✹✳✶✵✮

      

  ¯ u(0) = ¯ u(1) = ¯ v(0) = ¯ v(1) = 0,

  • B
  • ¯
  • ¯

  

  p

  B

  p

  A

  = d ds

  ♠❛s ❞❡ t❡♠♦s dρ ds

  ds ,

  p

  ) dρ

  p

  (ρ

  (s) = u

  p −1 n −p

  ′

  t❡♠♦s q✉❡✱ ¯ u

  p )

  ❆❣♦r❛✱ ❝♦♥s✐❞❡r❡ ❛ ♠✉❞❛♥ç❛ ❛♣r❡s❡♥t❛❞❛✳ ❈♦♠♦ ¯u(s) = u(ρ

  = 1 ❝♦♥❝❧✉í♠♦s q✉❡ s ∈ [0, 1]✳

  n −p p −1

  − R

  n −p p −1

  ¯ R

  n −p p −1

  R

  n −p p −1

  − s

  ! = A

  n −p p −1

  p − 1 n − p . (B

  ✱ n > p ❡ B p > 1

  ❖❜s❡r✈❡ q✉❡ s ∈ [0, 1]✱ p > 1✱ ¯ R > R > 0

  1−n n −p .

  − s)

  p

  p − 1 n − p (B

  p −1 n −p p .

  = A

  −p+1−(n−p) n −p

  − s)

  p

  p −1 n −p p .

  p −1 n −p p .

  = A

  −1

  −(p−1) n −p

  − s)

  p

  p − 1 n − p (B

  p −1 n −p p

  = A

  −(p−1) n −p

  − s)

  

p

  d ds (B

  R

  − ¯ R

  ✺✽ ✹✳✸✳ ❊❙❚■▼❆❚■❱❆❙ P❆❘❆ ❆❯❚❖❱❆▲❖❘❊❙ ❊▼ ❆◆➱■❙ ♦♥❞❡ q

  p

  ) = n − p p − 1

  p

  (ρ

  ′

  ❞❡r✐✈á✈❡❧ ❡ s

  p

  ✭✹✳✶✷✮ ◆♦t❡♠♦s q✉❡ s é ✉♠❛ ❢✉♥çã♦ ❞❡ ρ

  p −1 n −p .

  − s

  

p

  B

  (s) = A

  p

  p

  ❡ ρ

  , ✭✹✳✶✶✮

  (n−1)p n −p

  − s)

  p

  (B

  (p−1)p n −p p

  A

  

p

.

  (s) = p − 1 n − p

  1

  A

  ρ

  n −p p −1

  − R

  ¯ R

  n −p p −1

  = 0 s( ¯ R) = R ¯ R

  n −p p −1

  − R

  n −p p −1

  ¯ R

  n −p p −1

  R

  n −p p −1

  R

  n −p p −1

  n −p p −1

  −

n

−p

p

−1

  ¯ R

  n −p p −1

  = R ¯ R

  p

  n −p p −1

  R

  p

  −A

  = ¯ R ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❈♦♠♦ s(R) =

  = R ❡ ρ p

  ∈ [R, ¯ R], ♣♦rt❛♥t♦ s é ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡✱ ❧♦❣♦ ♦ ♠í♥✐♠♦ ❡ ♦ ♠á①✐♠♦ ❞❡ s ♦❝♦rr❡♠ q✉❛♥❞♦ ρ p

  −1 p > 0, ∀ρ p

  ✳ ❈♦♠ ✐ss♦ ♣♦❞❡♠♦s t❡r❡♠♦s✿

  β

  ′

  1−p p

  # = p − 1 n − p

  p−1

  .A

  1−p p

  ρ

  n−1 p

  |u

  (ρ

  p−1

  p

  )|

  p−2

  u

  ′

  (ρ

  p

  )

  ′ .

  .A

  p − 1 n − p

  p

  ′

  p−1

  .A

  

1−p

p

  . ❆ss✐♠✱ ✈♦❧t❛♥❞♦ ♥❛ ❡q✉❛çã♦ t❡r❡♠♦s

  (|¯ u

  ′

  (s)|

  p−2

  .¯ u

  (s))

  ′ (ρ p ).

  ′

  = d ds

  " ρ

  n−1 p

  |u

  ′

  (ρ p )|

  p−2

  u

  dρ

  ds = p − 1 n − p

  (p−1)(1−n) n −p

  r(ρ

  p−2

  .u

  ′

  (ρ

  p

  )

  ′

  = −λαρ

  n−1 p

  

p

  p

  )|u(ρ

  p

  )|

  α−2

  u(ρ

  p

  )|v(ρ

  p

  )|

  )|

  (ρ

  p−1

  p−2

  .A

  1−p p

  . dρ

  p

  ds ρ

  n−1 p

  |u

  ′

  (ρ p )|

  u

  ′

  ′

  (ρ p )

  ′ .

  ❙✉❜st✐t✉✐♥❞♦ ❛ ❡①♣r❡ssã♦ ❞❡ dρ

  p

  ds ❡ ♥♦t❛♥❞♦ q✉❡ ♥♦ s✐st❡♠❛ ❝♦♥s✐❞❡r❛❞♦

  ρ

  n−1 p

  .|u

  = p − 1 n − p

  (1−n)(p−1) n −p −

  ✺✾ ✹✳✸✳ ❊❙❚■▼❆❚■❱❆❙ P❆❘❆ ❆❯❚❖❱❆▲❖❘❊❙ ❊▼ ❆◆➱■❙ ❝♦♥❝❧✉✐r q✉❡ dρ

  p

  ′

  (ρ

  p ).

  dρ

  p

  ds

  p−2 .

  dρ

  ds !

  p−2

  ′

  = ρ

  n−1 p

  |u

  ′

  (ρ

  

p

  )|

  p−2

  .u

  )|

  ′

  ′

  p

  ds > 0

  ✱ ❞❛í (|¯ u

  ′

  (s)|

  p−2

  .¯ u

  ′

  (s))

  = u

  p

  ′ (ρ p ).

  dρ p ds

  p−2

  .u

  ′ (ρ p ).

  dρ p ds ! ′

  = |u

  ′

  (ρ

  u

  (ρ

  − s)

  p − 1 n − p (B

  p−1

  = A

  p

  B

  p

  − s

  (p−1)(1−n) n −p

  . A

  p −1 n −p p .

  p

  p

  − s)

  1−n n −p p−1

  = p − 1 n − p

  p−1

  .A

  (p−1)(1−n)

n

−p

  p

  . (B

  p

  ds

  . dρ

  p ).

  n−1 p

  1 ρ

  n−1 p

  . dρ

  p

  ds

  p−1

  ! ′ .

  ✭✹✳✶✸✮ ❆♥t❡s ❞❡ ❞❡r✐✈❛r♠♦s✱ ✈❛♠♦s s✐♠♣❧✐✜❝❛r ♦ t❡r♠♦

  1 ρ

  . dρ

  n−1 p

  p

  ds

  p−1

  ❡ ✈❡r q✉❡ ❡st❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ s✳ ❉❡ ❢❛t♦✱ ♣❡❧❛s ❞❡✜♥✐çõ❡s ❞❡ ρ

  p

  ❡ dρ

  p

  ds t❡♠♦s✿

  1 ρ

  • (p−1)2 n −p
  • (p−1)(n−1) n −p

  ✻✵ ✹✳✸✳ ❊❙❚■▼❆❚■❱❆❙ P❆❘❆ ❆❯❚❖❱❆▲❖❘❊❙ ❊▼ ❆◆➱■❙ (|¯ u

  (s)|

  p

  )|

  β

  ❡✱ ❝♦♠♦ ✜③❡♠♦s ❛ s✉❜st✐t✉✐çã♦ u(ρ p ) = ¯ u(s)

  ❡ v(ρ p ) = ¯ v(s)

  ❝♦♥❝❧✉í♠♦s q✉❡✿ |¯ u

  ′

  p−2

  p

  .¯ u

  ′

  (s)

  ′

  = −λαr(ρ p )q

  1 (s)|¯ u(s)| α−2

  ¯ u(s)|¯ v(s)|

  )|v(ρ

  u(ρ

  , q✉❡ é ❛ ❡q✉❛çã♦ q✉❡ q✉❡rí❛♠♦s ❝❤❡❣❛r✳ ❖❜s❡r✈❡ q✉❡ ♣❛r❛ ❛ s❡❣✉♥❞❛ ❡q✉❛çã♦ ♦s ❝á❧❝✉❧♦s s❡❣✉❡♠ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦

  (s)

  (s) ✳ ❆ss✐♠ ♦❜t❡♠♦s✿

  |¯ u

  ′

  (s)|

  p−2

  .¯ u

  ′

  ′

  α−2

  = −q

  1

  (s)λαr(ρ

  p

  )|u(ρ

  p

  )|

  β

  ❛♦s ❞❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦✳ P♦rt❛♥t♦ ❝♦♥s✐❞❡r❛♥❞♦ ❛ s❡❣✉♥❞❛ ❡q✉❛çã♦ ❞♦ s✐st❡♠❛ ❡ ❢❛③❡♥❞♦ ❛ s✉❜st✐t✉✐çã♦ t❡r❡♠♦s

  , ❡ ❡st❛ ❡①♣r❡ssã♦ é ❛ q✉❡ ❝❤❛♠❛♠♦s ❛♥t❡r✐♦r♠❡♥t❡ ❞❡ q

  − s > 0 ✳ ▲♦❣♦ q

  − s)

  − (n−1)p n

  −p −1

  , ❝♦♠♦ B

  p

  > 1 ❡ t❡♠♦s q✉❡ s ∈ [0, 1]✱ s❡❣✉❡ q✉❡ B

  p

  ′

  (p−1)p n −p p

  1

  (s) > 0 ❞❛í

  ❝♦♥❝❧✉í♠♦s q✉❡ q

  1

  (s) é ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡ ❡ ♣♦rt❛♥t♦ ♦❜t❡♠♦s q

  1

  (s) ≥ q

  (n − 1)p n − p (B p

  A

  |¯ v

  = −µβr(ρ q )q

  ′

  (s)|

  q−2

  .¯ v

  ′

  (s)

  ′

  2 (s)|¯ u(s)| α

  p

  ¯ v(s)|¯ v(s)|

  β−2 .

  ❉❛r❡♠♦s ✉♠ ❡♥❢♦q✉❡ ❛❣♦r❛ ♥❛ ❢✉♥çã♦ q

  

1

  (s) ✳ Pr✐♠❡✐r❛♠❡♥t❡ ♥♦t❡ q✉❡ q

  ′

  1

  (s) = p − 1 n − p

  1

  (n−1)p n −p

  ′

  )|

  p

  )|

  α−2

  u(ρ

  p

  )|v(ρ

  p

  β

  p

  = −

  p−1 n−p p−1

  A

  1−p+ p −1 n −p p ρ n−1 p p−1 n−p

  (B p − s)

  1−n n −p

  λαr(ρ p )|u(ρ p )|

  )|u(ρ

  r(ρ

  u(ρ p )|v(ρ p )|

  p−1 n−p p−1

  (s)|

  p−2

  .¯ u

  ′

  (s))

  ′

  = = −

  A

  n−1 p

  1−p p

  A

  p −1 n −p p p−1 n−p

  (B

  p

  − s)

  1−n n −p

  λαρ

  α−2

  β .

  − s)

  (p−1)p n −p p (B p

  − s)

  1−n n −p

  − (p−1)(n−1) n

  −p

  =

  p−1 n−p p

  A

  − s)

  (p−1)(−n+p+1) n −p

  (1−n)p n −p

  =

  p−1 n−p p

  A

  (p−1)p n −p p

  (B

  p

  p (B p

  A

  ◆♦t❡ q✉❡

  = =

  p−1 n−p p−1

  A

  1−p+ p −1 n −p p ρ

n−1

p p−1 n−p

  (B

  p

  − s)

  1−n n −p

  p−1 n−p p

  p−1 n−p p

  A

  (p−1)(−n+p+1) n −p p A p

  

B

p

  −s (p−1)(n−1) n

  −p

  (B p − s)

  1−n n −p

  =

  1 (0) ∀s ∈ [0, 1].

  ✻✶ ✹✳✸✳ ❊❙❚■▼❆❚■❱❆❙ P❆❘❆ ❆❯❚❖❱❆▲❖❘❊❙ ❊▼ ❆◆➱■❙ (s)

  (p−1)p n p −p

  p − 1 A p q (0) = .

  

1 (n−1)p

  n − p n

  −p

  B

  p

  P♦r s✐♠♣❧✐❝✐❞❛❞❡ ❝❤❛♠❛r❡♠♦s ❞❡ D ♦ ♥ú♠❡r♦ t❛❧ q✉❡

  p

  p − 1 D = . n − p

  R ◗✉❡r❡♠♦s ❡♥❝♦♥tr❛r ❝♦♥❞✐çõ❡s s♦❜r❡ R ❡ ¯ ❞❡ t❛❧ q✉❡ ❢♦r♠❛ q✉❡ q (0) ≥ 1,

  1

  ♣♦✐s ❛ss✐♠ ❡st❛r❡♠♦s ♥❛s ❝♦♥❞✐çõ❡s ❞♦ ❚❡♦r❡♠❛ q✉❛♥❞♦ p = q✳

  n −p p −1

  = R .B

  p p

  ❈♦♠♦ A t❡♠♦s q✉❡✱

  (p−1)p n −p n −p p −1 p

  (p−1)p (n−1)p

  R .B

  p − n n R −p −p p p −p q (0) = D. = DR B p = DR B = D .

  1

(n−1)p p

n

  B

  −p p

  B

  p p

  ❙✉❜st✐t✉✐♥❞♦ ❛ ❡①♣r❡ssã♦ ❞❡ B ♦❜t❡♠♦s q✉❡

  n −p n −p p

  " #

  p p −1 −1

  ( ¯ R − R ) q (0) = D R n .

  1 −p p −1

  ¯ R

  (0) ≥ 1

  1

  ❆ss✐♠✱ q s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ✈❛❧❡

  n n n n n 1 −p −p −p −p 1 1 −p

  • 1

  p p p p p p p p −1 −1 −1 −1 −1

  ¯ ¯ − R ≥ ¯ ⇔ ≥ D D .R R R R D .R − 1 .R .

  ❊♥tã♦ ✉♠❛ ♣r✐♠❡✐r❛ ❝♦♥❞✐çã♦ ❛♣❛r❡❝❡ s♦❜r❡ R✱ ✈✐st♦ q✉❡ ♦ t❡r♠♦ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ R

  ❞❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ é ♣♦s✐t✐✈♦ ❡ ¯ t❛♠❜é♠ é✱ ♦❜r✐❣❛r❡♠♦s q✉❡ ♦ t❡r♠♦

  1 p

  (D .R − 1) s❡❥❛ ♣♦s✐t✐✈♦✱ ♦✉ s❡❥❛✿

  1 R > .

  1 ✭✹✳✶✹✮ p

  D R ❈♦♥s✐❞❡r❛♥❞♦ ✐st♦✱ ❛❝❤❛r❡♠♦s ✉♠❛ ❝♦♥❞✐çã♦ ♣❛r❛ ¯ ❡♥✈♦❧✈❡♥❞♦ R✳ ❉❡ ❢❛t♦

  R ✈♦❧t❛♥❞♦ ♥❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡ ✐s♦❧❛♥❞♦ ¯ t❡r❡♠♦s

  p −1 n −p n −p 1 "

  1 # n −p

  • 1 +1

  

p p p p

−1 −1 n −p

  D .R D .R

  p −1

  ¯ ¯ R ≥ ⇔ R ≥ .

  1

  1

  ✭✹✳✶✺✮

  p p

  D .R − 1 D .R − 1 (t) ≥ 1

  1 R >

  1 p

  D

  ✻✷ ✹✳✸✳ ❊❙❚■▼❆❚■❱❆❙ P❆❘❆ ❆❯❚❖❱❆▲❖❘❊❙ ❊▼ ❆◆➱■❙ ❡

  p −1

   

  n

  "

  1 −p # n −p

  • 1

  p p −1

    D .R

  ¯ R > max R, .

  1

  ✭✹✳✶✻✮

  p

    D .R − 1

  ❈♦♠♦ ❥á ✈✐♠♦s✱ à ❡q✉❛çã♦

  p−2

  −∆ u = λr(x)|u| u

  p

  ✭✹✳✶✼✮ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❡q✉❛çã♦

  ′ n−1 ′ p−1 ′ n−1 p−2

  ρ |v (ρ )| v (ρ ) = −λr(ρ )ρ |v(ρ )| v(ρ )

  p p p p p p

  ♣♦r ♠❡✐♦ ❞❛ tr❛♥s❢♦r♠❛çã♦ r❛❞✐❛❧✳ ❙♦❜r❡ ❡st❛ ❡q✉❛çã♦ ❢❛③❡♥❞♦✱ ❛s ♠❡s♠❛s ♠✉❞❛♥ç❛s ❛♣r❡s❡♥t❛❞❛s ❛❝✐♠❛ ♦❜t❡♠♦s ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❡q✉✐✈❛❧❡♥t❡ ❞❛❞❛ ♣♦r

  ′ ′ p−2 ′ p−2

  − |¯ u (s)| .¯ u (s) = λr(ρ p )q

  1 (s)|¯ u(s)| u(s), s ∈ [0, 1]. ¯

  ✭✹✳✶✽✮ ❆❣♦r❛ ♣♦❞❡♠♦s ❡♥✉♥❝✐❛r ♦ ♣r✐♥❝✐♣❛❧ t❡♦r❡♠❛ ❞❡st❛ s❡çã♦✿

  ∞

  (Ω) ❚❡♦r❡♠❛ ✹✳✷✳ ❈♦♥s✐❞❡r❡ r ∈ L t❛❧ q✉❡ r(x) ≥ m > 0 ❡ r(x) = r(|x|)✱ ♦✉

  R s❡❥❛✱ r é ✉♠❛ ❢✉♥çã♦ r❛❞✐❛❧✳ ❈♦♥s✐❞❡r❡ R ❡ ¯ s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❡ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦✱ ♦ ❦✲és✐♠♦ ❛✉t♦✈❛❧♦r ✈❛r✐❛❝✐♦♥❛❧ ❞♦ ♣r♦❜❧❡♠❛ s♦❜r❡ ❛ r❡t❛

  µ = tλ, s❛t✐s❢❛③ Λ k Λ k

  λ + k (t) ≤ , p pt ♦♥❞❡ Λ k é ♦ ❦✲és✐♠♦ ❛✉t♦✈❛❧♦r ✈❛r✐❛❝✐♦♥❛❧ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❉✐r✐❝❤❧❡t

  p−2 n

  −∆ u = λr(x)|u| u; x ∈ Ω ⊂ R ,

  p

  ❡

  n Ω = {x ∈ R /R ≤ |x| ≤ ¯ R}.

  ❉❡♠♦♥str❛çã♦✳ ❆ ❞❡♠♦♥str❛çã♦ s❡❣✉❡ ♣♦r ♠❡✐♦ ❞❛s ♠✉❞❛♥ç❛s ❞❡ ✈❛r✐á✈❡✐s ❥á ❛♣r❡s❡♥t❛❞❛s✳ ❉❡ ❢❛t♦✱ ❥á ✈✐♠♦s ♣♦r ♠✉❞❛♥ç❛s ❞❡ ✈❛r✐á✈❡✐s q✉❡ ♦ s❡❣✉✐♥t❡ s✐st❡♠❛

  

  α−2 β

  −∆ u = λαr(x)|u| u|v| , x ∈ Ω

  p

      

  α β−2

  −∆ v = µβr(x)|u| v|v| , x ∈ Ω

  p

       u(x) = v(x) = 0, x ∈ ∂Ω

  ✻✸ ✹✳✸✳ ❊❙❚■▼❆❚■❱❆❙ P❆❘❆ ❆❯❚❖❱❆▲❖❘❊❙ ❊▼ ❆◆➱■❙ é ❡q✉✐✈❛❧❡♥t❡ ❛♦ s✐st❡♠❛

  

  ′ ′ p−2 ′ α−2 β

  − (|¯ u (s)| .¯ u (s)) = λαr(ρ )q (s)|¯ u(s)| ¯ u(s)|¯ v(s)| , s ∈ (0, 1)

  p

  1

      

  ′ ′ p−2 ′ α β−2

  − (|¯ v (s)| .¯ v (s)) = µβr(ρ )q (s)|¯ u(s)| |¯ v(s)| v(s), s ∈ (0, 1) ¯

  p

  1

       u(0) = ¯ ¯ u(1) = ¯ v(0) = ¯ v(1) = 0.

  P♦rt❛♥t♦✱ ♦s ❛✉t♦✈❛❧♦r❡s ❞❡st❡s s✐st❡♠❛s sã♦ ♦s ♠❡s♠♦s✳ ❏á ✈✐♠♦s t❛♠❜é♠ q✉❡ ❛ ❡q✉❛çã♦

  p−2 n

  −∆ u = λr(x)|u| u, x ∈ Ω ⊂ R ,

  p

  é ❡q✉✐✈❛❧❡♥t❡ ❛ ❡q✉❛çã♦

  ′ ′ p−2 ′ p−2

  − |¯ u (s)| .¯ u (s) = λr(ρ )q (s)|¯ u(s)| u(s), s ∈ [0, 1]. ¯

  p

  1 P♦rt❛♥t♦ ♥♦ss♦ t❡♦r❡♠❛ ❡stá ♥❛s ❝♦♥❞✐çõ❡s ❞♦ ❚❡♦r❡♠❛ q✉❛♥❞♦ p = q✳ ❉❡

  ❢❛t♦✱ ❝♦♥s✐❞❡r❡ µ = tλ,

  ❡ r(s) = r(ρ ¯ )q (s),

  p

  1

  ❛ss✐♠ t❡♠♦s ♦ s✐st❡♠❛ 

  ′ ′ p−2 ′ α−2 β

  − (|¯ u (s)| .¯ u (s)) = λα¯ r(s)|¯ u(s)| u(s)|¯ ¯ v(s)| , s ∈ (0, 1)     

  ′ ′ p−2 ′ α β−2

  − (|¯ v (s)| .¯ v (s)) = µβ¯ r(s)|¯ u(s)| |¯ v(s)| v(s), s ∈ (0, 1) ¯      u(0) = ¯ ¯ u(1) = ¯ v(0) = ¯ v(1) = 0.

  −A

  p

  = |x|, s = + B (s)

  p n −p p

  1

  ♦♥❞❡ ρ ✱ ❡ q é ❞❛❞❛ ♣♦r ❖❜s❡r✈❡ q✉❡

  p −1

  ρ p r(s) = r(ρ)q ¯

  1 (s) ≥ r(ρ) ≥ m,

  (s) ≥ 1 R

  1

  ♣♦✐s ❥á ♠♦str❛♠♦s q✉❡ q ♥❛s ❝♦♥❞✐çõ❡s ❞❡ R ❡ ¯ ✳ ▲♦❣♦✱ s♦❜ ❡st❛s ❝♦♥❞✐çõ❡s✱ ♥♦ss♦ s✐st❡♠❛ ❝♦♥s✐❞❡r❛❞♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠ s✐st❡♠❛ ❞♦ t✐♣♦

  

  

′ p−2 ′ ′ α−2 β

  −(|u (x)| u (x)) = λαr(x)|u| u|v| , ❡♠ Ω,

  

  

′ p−2 ′ ′ α β−2

   −(|v

  (x)| v (x)) = µβr(x)|u| v|v| , ❡♠ Ω,

  ❡ ❛ ❡q✉❛çã♦

  p−2 n

  −∆ u = λr(x)|u| u; x ∈ Ω ⊂ R ,

  p

  é ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❡q✉❛çã♦ ❞♦ t✐♣♦

  ′ p−2 ′ ′ p−2 −(|u (x)| u (x)) = λr(x)|u| u.

  • m
  • Λ

  1 (s)|¯ u(s)| α−2

  ′

  .¯ v

  p−2

  (s)|

  ′

  , s ∈ (0, 1) − (|¯ v

  β

  ¯ u(s)|¯ v(s)|

  = λαr(ρ p )q

  ′

  ′

  (s))

  ′

  .¯ u

  p−2

  (s)|

  ′

  − (|¯ u

  (s))

  = µβr(ρ

  ❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ ❥❛ ♠♦str❛♠♦s q✉❡ ♦ ♣r♦❜❧❡♠❛ é ❡q✉✐✈❛❧❡♥t❡ ❛♦ ♣r♦❜❧❡♠❛

  1

  = 2

  α+β

  2

  ❈♦♠♦ s ∈ [0, 1] ❡ ❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ p = q s❡❣✉❡ q✉❡

  1 (s).

  )q

  p

  (s) ❡ g(s) = µβr(ρ

  )q

  p

  p

  P♦❞❡♠♦s ✉s❛r ♦ ❚❡♦r❡♠❛ ♣❛r❛ ❛s ❢✉♥çõ❡s f (s) = λαr(ρ

  ¯ v(s), s ∈ (0, 1) ¯ u(0) = ¯ u(1) = ¯ v(0) = ¯ v(1) = 0.

  β−2

  |¯ v(t)|

  α

  (s)|¯ u(s)|

  

1

  )q

            

  (s) é ❞❛❞❛ ♣♦r

  ✻✹ ✹✳✸✳ ❊❙❚■▼❆❚■❱❆❙ P❆❘❆ ❆❯❚❖❱❆▲❖❘❊❙ ❊▼ ❆◆➱■❙ ❆ss✐♠✱ ♣❡❧♦ ❚❡♦r❡♠❛ q✉❛♥❞♦ p = q✱ ❝♦♥❝❧✉í♠♦s ♦ r❡s✉❧t❛❞♦✱ ♦✉ s❡❥❛✿

  k

  α p

  C λ

     

  1 β

  ❚❛♠❜é♠ ❝♦♥s❡❣✉✐♠♦s ✉♠❛ ❧✐♠✐t❛çã♦ ❞♦s ❛✉t♦✈❛❧♦r❡s ♣♦r ✉♠❛ ❝✉r✈❛ q✉❡ é ♦ s❡❣✉✐♥t❡ ❚❡♦r❡♠❛✿ ❚❡♦r❡♠❛ ✹✳✹✳ ❊①✐st❡ ✉♠❛ ❢✉♥çã♦ h(λ) t❛❧ q✉❡ µ ≥ h(λ) ♣❛r❛ ❝❛❞❛ ❛✉t♦✈❛❧♦r ❣❡♥❡r❛❧✐③❛❞♦ (λ, µ) ❞♦ ♣r♦❜❧❡♠❛ ♦♥❞❡ h(λ) é ❞❛❞❛ ♣♦r h(λ) =

  pt . ❖❜s❡r✈❛çã♦ ✹✳✸✳ ❖ ❝❛s♦ q✉❛♥❞♦ p > q✱ ❝♦♥t✐♥✉❛ ❡♠ ❛❜❡rt♦✳

  k

  p

  = Λ

  

1

  q/p k

  Λ

  q

  qt p q

  −1+q/p

  p

  k

  λ k (t) ≤ Λ

  Z

  r(ρ p (s))ds,    

  p ❡ B p sã♦ ❞❛❞❛s ♣♦r ❡ ρ p

  (1) = p − 1 n − p

  , A

  (n−1)p n −p

  − 1)

  p

  (B

  (p−1)p n −p p

  A

  

p

.

  1

  p β

  , q

  1 (1)

  q

  

α/p

  α

  p

  2

  C =

  , ♦♥❞❡

  p .

  ✻✺ ✹✳✸✳ ❊❙❚■▼❆❚■❱❆❙ P❆❘❆ ❆❯❚❖❱❆▲❖❘❊❙ ❊▼ ❆◆➱■❙ ❆ss✐♠

  α β

  Z Z

  1 p 1 p p

  2 ≤ λαr(ρ (s))q (s)ds µβr(ρ (s))q (s)ds

  p 1 p

  1 β α

  Z

  1 p Z 1 p

α β

p p

  ≤ (λαq

  1 (1)) r(ρ p (s))ds (µβq 1 (1)) r(ρ p (s))ds β α

  • 1 p p

  Z

  α β α β

  • p p p p

  = (λα) (µβ) (q (1)) r(ρ (s))ds

  1 p

  Z

  1 α β p p

  = (λα) (µβ) q (1) r(ρ (s))ds.

  1 p

  P♦rt❛♥t♦✱

  p

   

  β p

   2   

  µβ ≥ Z

  1 α

   

  p

  (λα) q (1) r(ρ (s))ds

  1 p p

   

  β

   C   

  = , Z

  1 α

   

  p

  λ r(ρ (s))ds

  p p

  2

  α

  ♦♥❞❡ C = ✳

  p

  α q

  1 (1)

  ❈♦♥❝❧✉sã♦✱

  q

   

  β

  1  C    µ ≥ .

  Z

  

1

α

    β

  p

  λ r(ρ (s))ds

  p

  ✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✲

  ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s

  ◆❡st❡ tr❛❜❛❧❤♦ ❝♦♥s✐❞❡r❛♠♦s ✉♠ s✐st❡♠❛ ❡❧í♣t✐❝♦ q✉❛s❡ ❧✐♥❡❛r✱ ❛ ♣r✐♥❝í♣✐♦ ♥♦

  n

  R ✳ P❛r❛ ♦ s✐st❡♠❛ ❡♠ R ♦❜t❡♠♦s ✉♠❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ t♦❞♦s ♦s s❡✉s ❛✉t♦✈❛❧♦r❡s✱

  ❡♠ q✉❡ Ω ⊂ R é q✉❛❧q✉❡r ❡ ❡st❛♠♦s s♦❜r❡ ❛ r❡t❛ µ = tλ✳ ❆✐♥❞❛ ❝♦♥s❡❣✉✐♠♦s ✉♠❛ ❝✉r✈❛ ❞♦ t✐♣♦ ❤✐♣ér❜♦❧❡ q✉❡ ❧✐♠✐t❛ ♦s ❛✉t♦✈❛❧♦r❡s ♣♦r ❜❛✐①♦✳ P❛r❛ ♦ s✐st❡♠❛

  n n

  ❡♠ R ❛ ♠❡s♠❛ ❡st✐♠❛t✐✈❛ é ✈á❧✐❞❛ s♦♠❡♥t❡ ♣❛r❛ ✉♠❛ ❝❧❛ss❡ s❡ ❝♦♥❥✉♥t♦s ❡♠ R ✱ ✐st♦ q✉❛♥❞♦ ❡st❛♠♦s s♦❜r❡ ✉♠❛ r❡t❛ µ = tλ✱ ❡ t❛♠❜é♠ ❡♥❝♦♥tr❛♠♦s ✉♠❛ ❝✉r✈❛ ❞♦ t✐♣♦ ❤✐♣ér❜♦❧❡ q✉❡ ❧✐♠✐t❛ ♣♦r ❜❛✐①♦ ♦s ❛✉t♦✈❛❧♦r❡s✳ ❊st❛s ❝♦♥❝❧✉sõ❡s s❡ ❞❡r❛♠ ♣♦r ♠❡✐♦ ❞❡ ♠✉❞❛♥ç❛s ❞❡ ✈❛r✐á✈❡✐s ♥♦ s✐st❡♠❛ ❝♦♥s✐❞❡r❛❞♦ ❡ ♦ ✉s♦ ❞❡ ♠ét♦❞♦s ✈❛r✐❛❝✐♦♥❛✐s ♣❛r❛ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞♦s ❛✉t♦✈❛❧♦r❡s ✈❛r✐❛❝✐♦♥❛✐s ❞♦ s✐st❡♠❛✳

  ◆❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✶✳✸ ❞❡ ❢♦✐ ❡♥❝♦♥tr❛❞♦ ✉♠ ❡rr♦ ❞❡ ❣r❛✜❛✳ ❉❡st❛ ❢♦r♠❛✱ ❣♦st❛rí❛♠♦s ❞❡ ❢❛③❡r ✉♠ ❛❣r❛❞❡❝✐♠❡♥t♦ ❛♦ Pr♦❢❡ss♦r ❏✉❛♥ P❛❜❧♦ P✐♥❛s❝♦ ♣♦r t❡r r❡s♣♦♥❞✐❞♦ ♥♦ss❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ s♦❜r❡ t❛❧ ❡rr♦ ❡ s✉❣❡r✐✉ ❛ ❝♦rr❡çã♦ ♥❛ ❡s❝♦❧❤❛

  k

  ❞♦ ❝♦♥❥✉♥t♦ C ✱ ✉s❛❞♦ ♥❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ❞❡st❛ ❞✐ss❡rt❛çã♦✳

  ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

  ❬✶❪ ❆♠❛♥♥✱ ❍✳ ▲✉st❡r♥✐❦✲❙❝❤♥✐r❡❧♠❛♥♥ t❤❡♦r② ❛♥❞ ♥♦♥❧✐♥❡❛r ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠s✳ ▼❛t❤✳ ❆♥♥✳ ✶✾✾ ✭✶✾✼✷✮ ✺✺✲✼✷✳

  ❬✷❪ ❆rrà③♦❧❛ ■r✐❛rt❡✱ ❊✳ ❆✳ ❙♦❜r❡ ✉♠ ♣❛r ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ♣❛r❛ ✉♠❛ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s ❡❧í♣t✐❝♦s ❡♥✈♦❧✈❡♥❞♦ ♦ ♣✲▲❛♣❧❛❝✐❛♥♦✳ ❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦✱ ❈❛♠♣✐♥❛s ❬❙✳P✳✿s✳♥✳❪✱ ✷✵✵✹

  ❬✸❪ ❇❆❘❚▲❊✱ ❘✳ ●✳ ❚❤❡ ❊❧❡♠❡♥ts ♦❢ ■♥t❡❣r❛t✐♦♥ ❛♥❞ ▲❡❜❡s❣✉❡ ▼❡❛s✉r❡✳ ◆❡✇ ❨♦r❦✿ ❲✐❧❡② ❈❧❛ss✐❝s✳ ❲✐❧❡②✲■♥t❡rs❝✐❡♥❝❡✳ ✶✾✾✺✳

  ❬✹❪ ❇❘❊❩■❙✱ ❍✳ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s✱ ❙♦❜♦❧❡✈ ❙♣❛❝❡s ❛♥❞ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ P❛r✐s✿ ❙♣✐♥❣❡r✱ ✷✵✶✶✳

  ❬✺❪ ❉❡❧ P✐♥♦✱ ▼✳❀ ❉r❛❜❡❦✱ P✱❀ ▼❛♥ás❡✈✐❝❤✱ ❘✳ ❚❤❡ ❋r❡❞❤♦❧♠ ❛❧t❡r♥❛t✐✈❡ ❛t t❤❡ ✜rst ❡✐❣❡♥✈❛❧✉❡ ❢♦r t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ♣✲▲❛♣❧❛❝✐❛♥✳ ❏✳ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ✶✺✶✭✷✮✱ ✭✶✾✾✾✮✱ ✸✽✻ ✲ ✹✶✾✳

  ❬✻❪ ❉❡ ◆á♣♦❧✐✱ P✳▲✳❀ P✐♥❛s❝♦✱ ❏✳P✳ ❊st✐♠❛t❡s ❢♦r ❡✐❣❡♥✈❛❧✉❡s ♦❢ q✉❛s✐❧✐♥❡❛r ❡❧❧✐♣t✐❝ s②st❡♠s✳ ❏✳ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✷✷✼ ✭✷✵✵✻✮ ✶✵✷✲ ✶✶✺

  ❬✼❪ ❉❡ ◆á♣♦❧✐✱ P✳❀ ▼❛r✐❛♥♥✐✱ ❈✳ ◗✉❛s✐❧✐♥❡❛r ❡❧❧✐♣t✐❝ s②st❡♠s ♦❢ r❡s♦♥❛♥t t②♣❡ ❛♥❞ ♥♦♥❧✐♥❡❛r ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠s✳ ❆❜str✳ ❆♣♣❧✳ ❆♥❛❧✳ ✼ ✭✸✮ ✭✷✵✵✷✮ ✶✺✺✲✶✻✼✳

  ❬✽❪ ❉♦s❧②✱ ❖✳❀ ❘❡❤❛❦✱ P✳ ❍❛❧❢✲▲✐♥❡❛r ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✳ ❱♦❧✉♠❡ ✷✵✷ ◆♦rt❤✲❍♦❧❧❛♥❞ ▼❛t❤❡♠❛t✐❝s ❙t✉❞✐❡s✳ ◆♦rt❤ ❍♦❧❧❛♥❞ ✭✷✵✵✺✮

  ❬✾❪ ❉r❛❜❡❦✱ P✳❀ ▼❛♥ás❡✈✐❝❤✱ ❘✳ ❖♥ t❤❡ ❝❧♦s❡❞ s♦❧✉t✐♦♥s t♦ s♦♠❡ ♥♦♥❤♦♠❡❣❡♥❡♦✉s ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠s ✇✐t❤ ♣✲ ▲❛♣❧❛❝✐❛♥✳ ❉✐✛❡r❡♥t✐❛❧ ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s ✶✷ ✭✻✮ ✭✶✾✾✾✮ ✼✼✸✲✼✽✽✳

  ❬✶✵❪ ▲■▼❆✱ ❊✳ ▲✳ ❆♥á❧✐s❡ ❘❡❛❧✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ■▼P❆✱ ❱♦❧✳ ✷✱ ✷✵✵✻✳ ❬✶✶❪ ▼❊❉❊■❘❖❙✱ ▲✳ ❆✳ ❞❛ ❏✳❀ ▼■❘❆◆❉❆✱ ▼✳ ❆✳ ▼✳ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ✿

  ■♥✐❝✐❛çã♦ ❛♦s ♣r♦❜❧❡♠❛s ❡❧ít✐❝♦s ♥ã♦ ❤♦♠♦❣ê♥❡♦s✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❯❋❘❏✴■▼✱ ✷✵✵✵

  ❬✶✷❪ ❖▲■❱❊■❘❆✱ ❈➱❙❆❘ ❘✳ ❉❊✳ ■♥tr♦❞✉çã♦ ➚ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ■▼P❆✱ ✷✵✶✷✳

  ✻✽ ❘❊❋❊❘✃◆❈■❆❙ ❇■❇▲■❖●❘➪❋■❈❆❙

  ❬✶✸❪ Pr♦tt❡r✱ ▼✳❚❤❡ ❣❡♥❡r❛❧✐③❡❞ s♣❡❝tr✉♠ ♦❢ s❡❝♦♥❞ ♦r❞❡r ❡❧❧✐♣t✐❝ s②st❡♠s✳ ❘♦❝❦② ▼♦✉♥t❛✐♥ ❏✳ ▼❛t❤✳ ✾ ✭✸✮ ✭✶✾✼✾✮ ✺✵✸✲✺✶✽✳ ❬✶✹❪ ❘❛❜✐♥♦✇✐t③✱ P✳ ❍✳ ▼✐♥✐♠❛① ♠❡t❤♦❞s ✐♥ ❝r✐t✐❝❛❧ ♣♦✐♥t t❤❡♦r②

  ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s t♦ ❞✐❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ▼✐❛♠✐✿ ❈❇▼❙ ❘❡❣✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡✱ ✶✾✽✹✳

  ❬✶✺❪ ❙■▼❖◆✱ ❏✳❘❡❣✉❧❛r✐té ❞❡ ❧❛ s♦❧✉t✐♦♥ ❞✬✉♥❡ éq✉❛t✐♦♥ ♥♦♥ ❧✐♥é❛✐r❡

  N

  ❞❛♥s R ✳ ❏✳ ❞✬❆♥❛❧②s❡ ♥♦♥ ❧✐♥é❛r✐❡✳ Pr♦❝❡❡❞✐♥❣s✱ ❇❡s❛♥ç♦♥✱ ❋r❛♥❝❡✱ ✶✾✼✼✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ✻✻✺✱ ❙♣r✐♥❣❡r✲❱❡r❧❛♥❣✱ ❇❡r❧✐♥✳

  ❬✶✻❪ ❲❡✐♥❜❡r❣❡r✱ ❍✳ ❯♣♣❡r ❛♥❞ ❧♦✇❡r ❜♦✉♥❞s ❢♦r ❡✐❣❡♥✈❛❧✉❡s ❜② ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞s✳ ❈♦♠♠✳ P✉r❡ ❆♣♣❧✳ ▼❛t❤✳ ✾ ✭✶✾✺✻✮ ✻✶✸✲✻✷✸✳

Novo documento