Equações elípticas com não lineradidades críticas e perturbações de ordem inferi...

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Equações elípticas com não linearidades críticas e

perturbações de ordem inferior

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Equações elípticas com não linearidades críticas e

perturbações de ordem inferior

Maycon Sullivan Santos Araújo

Orientador: Prof. Dr. Eugenio Tommaso Massa

Dissertação apresentada ao Instituto de Ciências Matemáticas e de Computação - ICMC-USP, como parte dos requisitos para obtenção do título de Mestre em Ciências – Matemática. EXEMPLAR DE DEFESA

USP – São Carlos Maio de 2015

SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP

Data de Depósito:

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▼❡✉ ❛♠♦r✱ ❞✐s❝✐♣❧✐♥❛ é ❧✐❜❡r❞❛❞❡ ❈♦♠♣❛✐①ã♦ é ❢♦rt❛❧❡③❛

❚❡r ❜♦♥❞❛❞❡ é t❡r ❝♦r❛❣❡♠

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛❝✐♠❛ ❞❡ t✉❞♦ ❛ ♠❡✉s ♣❛✐s✱ ❏♦sé ❈❛r❧♦s ❖❧✐✈❡✐r❛ ❆r❛ú❥♦ ❡ ❚❡r❡③✐♥❤❛ ▼❛r✐❛ ❞♦s ❙❛♥t♦s ❆r❛rú❥♦✱ ♣♦r t♦❞♦ ❡s❢♦rç♦ ❡ ❞❡❞✐❝❛çã♦ ❡♠ ❢❛③❡r ❞❡ ♠✐♠ ❛ ♣❡ss♦❛ q✉❡ s♦✉ ❤♦❥❡✳

❆❣r❛❞❡ç♦ ❛ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣r♦❢✳ ❞r✳ ❊✉❣❡♥✐♦ ❚♦♠♠❛ss♦ ▼❛ss❛✱ ♣♦r t✉❞♦ q✉❡ ❡❧❡ s❡ ❞✐s♣ôs ❛ ♠❡ ❡♥s✐♥❛r r❡❢❡r❡♥t❡ à ♣❡sq✉✐s❛ ♠❛t❡♠át✐❝❛✱ ♠❛s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡♠ ❢❛③ê✲❧♦✳

❆❣r❛❞❡ç♦ ❛ ♠❡✉ ♦r✐❡♥t❛❞♦r ❞❡ ❣r❛❞✉❛çã♦✱ ♣r♦❢✳ ❞r✳ P❛✉❧♦ ❞❡ ❙♦✉③❛ ❘❛❜❡❧♦✱ ♣♦r t❡r ♠❡ ✐♥✐❝✐❛❞♦ ♥❛s ❊❉P✬s ❊❧í♣t✐❝❛s✱ ♣♦r t❡r✱ ❡♠ ❛❧❣✉♥s ♠♦♠❡♥t♦s✱ ♠❡ ❡♥❝♦r❛❥❛❞♦ ❛ s✉♣❡r❛r ♠✐♥❤❛s ❧✐♠✐t❛çõ❡s ❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♣♦r ❛❝r❡❞✐t❛r ❡♠ ♠✐♠ ❛♦ ♣♦♥t♦ ❞❡ ♠❡ ✐♥❝❡♥t✐✈❛r ❛ tr✐❧❤❛r ❝❛♠✐♥❤♦s ❧♦♥❣íq✉♦s ♥❛ ❜✉s❝❛ ❞❡ ✐♥❝r❡♠❡♥t❛r ♠✐♥❤❛ ❢♦r♠❛çã♦ ❛❝❛❞ê♠✐❝❛✳

❆❣r❛❞❡ç♦ à ❈❆P❊❙ ♣❡❧♦ ❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦✳

❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❡ ❝♦❧❡❣❛s ❞❡ ❡st✉❞♦s q✉❡✱ ❞✉r❛♥t❡ t♦❞❛ ♠✐♥❤❛ ❝❛rr❡✐r❛ ❛❝❛❞ê♠✐❝❛ ❛té ❛q✉✐✱ ♠❡ ❛❥✉❞❛r❛♠ ❞❡ ❛❧❣✉♠❛ ♠♦❞♦ ❡♠ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❛❝❛❞ê♠✐❝❛✳

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❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦✱ t✐✈❡♠♦s ❝♦♠♦ ♦❜❥❡t✐✈♦ ❡st✉❞❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❢r❛❝❛s ♥ã♦ tr✐✈✐❛✐s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❡❧í♣t✐❝♦ ❝♦♠ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡ ❝rít✐❝❛

  

−∆u=λu+u2+∗−1+g(x, u+) +f(x), ❡♠ Ω

u= 0, s♦❜r❡ ∂Ω, ✭

P✮

♦♥❞❡ Ω é ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❝♦♠ ❢r♦♥t❡✐r❛ s✉❛✈❡ ❡♠ RN✱ ❝♦♠ _N _≥ ₃✱ ₂∗ ₌ 2N

N−2 é ♦

❡①♣♦❡♥t❡ ❝rít✐❝♦ ❞❡ ❙♦❜♦❧❡✈✱ u+ = max(u,0)✱ g ∈ C(Ω×R,R+)✱ λ > λ1✱ λ /∈ σ(−∆) ❡

f ∈Lr_(Ω)_{✱ ❝♦♠} _{r > N}_✳

❈♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ♦❜s❡r✈❛r ❛s ♠✉❞❛♥ç❛s q✉❡ ♦❝♦rr❡♠ ❞♦ ❝❛s♦ s✉❜❝rít✐❝♦ ♣❛r❛ ♦ ❝rít✐❝♦ ❡ ❛s ❞✐❢❡r❡♥t❡s té❝♥✐❝❛s ✈❛r✐❛❝✐♦♥❛✐s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❡❧í♣t✐❝♦s✱ ❡st✉❞❛♠♦s✱ ✐♥✐❝✐❛❧♠❡♥t❡✱ ✉♠ ♣r♦❜❧❡♠❛ ✉♠ ♣♦✉❝♦ ♠❛✐s ❛♥t✐❣♦ q✉❡ ✭P✮✱ q✉❡✱ ♣♦r s✉❛ ✈❡③✱ ♠♦t✐✈♦✉ s❡✉

❡st✉❞♦✳ ❚❛❧ ♣r♦❜❧❡♠❛ é

  

−∆u=λu+up₊+f, ❡♠ Ω

u= 0, s♦❜r❡∂Ω, ✭

P′_✮

♦♥❞❡ ❝♦♥s✐❞❡r❛♠♦s ♦ ❝❛s♦ s✉❜❝rít✐❝♦✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦p∈(1,2∗₋₁₎_✳

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❆❜str❛❝t

■♥ t❤✐s ✇♦r❦✱ ✇❡ ❛✐♠❡❞ t♦ st✉❞② t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♥♦♥tr✐✈✐❛❧ ✇❡❛❦ s♦❧✉t✐♦♥s ❢♦r t❤❡ ❡❧❧✐♣t✐❝ ♣r♦❜❧❡♠ ✇✐t❤ ❝r✐t✐❝❛❧ ♥♦♥✲❧✐♥❡❛r✐t②

  

−∆u=λu+u2+∗−1+g(x, u+) +f(x) ✐♥ Ω

u= 0 ♦♥ ∂Ω, ✭

P✮

✇❤❡r❡ Ω✐s ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✇✐t❤ s♠♦♦t❤ ❜♦✉♥❞❛r② ✐♥ RN✱ ✇✐t❤ _N _≥ ₃✱ ₂∗ ₌ 2N

N−2 ✐s t❤❡

❝r✐t✐❝❛❧ ❙♦❜♦❧❡✈ ❡①♣♦♥❡♥t✱ u+ = max(u,0)✱ g ∈ C(Ω×R,R+)✱ λ > λ1✱ λ /∈ σ(−∆) ❛♥❞

f ∈Lr_(Ω)_{✱ ✇✐t❤} _{r > N}_✳

■♥ ♦r❞❡r t♦ ♦❜s❡r✈❡ ❞✐✛❡r❡♥t ✈❛r✐❛t✐♦♥❛❧ t❡❝❤♥✐q✉❡s ❢♦r s♦❧✈✐♥❣ ❡❧❧✐♣t✐❝ ♣r♦❜❧❡♠s✱ ✇❡ st✉❞✐❡❞ ✐♥✐t✐❛❧❧② ❛ ♣r♦❜❧❡♠ ❛ ❧✐tt❧❡ ♦❧❞❡r t❤❛♥ ✭P✮✱ ✇❤✐❝❤✱ ✐♥ t✉r♥✱ ❧❡❞ t♦ ✐ts st✉❞②✳ ❚❤✐s ♣r♦❜❧❡♠ ✐s

  

−∆u=λu+up₊+f ✐♥ Ω

u= 0 ♦♥ ∂Ω, ✭

P′_✮

✇❤❡r❡ ✇❡ ❝♦♥s✐❞❡r t❤❡ s✉❜❝r✐t✐❝❛❧ ❝❛s❡✱ t❤❛t ✐s✱ ✇❤❡♥ p∈(1,2∗₋₁₎_✳

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◆♦t❛çõ❡s ❡ ❉❡✜♥✐çõ❡s

• Ω ❞❡♥♦t❛ ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❡♠ RN❀

• ∂Ω❞❡♥♦t❛ ❛ ❢r♦♥t❡✐r❛ ❞❡ Ω❀

• RN

+ ={x∈RN;xN >0} ❡ ∂RN+ ={x∈RN;xN = 0}❀

• ❉✐③❡♠♦s q✉❡ Ω t❡♠ ❢r♦♥t❡✐r❛ ❞❡ ❝❧❛ss❡ Ck_{✱ ❝♦♠} _k _∈_N_{✱ s❡ ♣❛r❛ ❝❛❞❛} _x _∈ _∂_Ω _{❡①✐st❡♠}

rx >0 ❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ Ψx : Bx =B(x, rx)→ D⊂ RN ❞❡ ❝❧❛ss❡ Ck(Bx)t❛✐s q✉❡

Ψx(Bx ∩Ω) ⊂ RN₊ ❡ Ψx(Bx∩∂Ω) ⊂ ∂RN₊✳ ❙❡ Ω t❡♠ ❢r♦♥t❡✐r❛ Ck ♣❛r❛ t♦❞♦ k ∈ N✱

❞✐③❡♠♦s q✉❡ s✉❛ ❢r♦♥t❡✐r❛ é s✉❛✈❡❀

• |Ω| ❞❡♥♦t❛ ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❞❡ Ω ❡ ωN ❞❡♥♦t❛ ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❞❛ ❡s❢❡r❛

✉♥✐tár✐❛ N✲❞✐♠❡♥s✐♦♥❛❧❀

• P❛r❛ ✉♠❛ ❢✉♥çã♦ u : Ω → R✱ ❞❡✜♥✐♠♦s s✉❛ ♣❛rt❡ ♣♦s✐t✐✈❛ ❝♦♠♦ s❡♥❞♦ _u₍_x₎₊ ₌ max(u(x),0)❀

• ∇u=_∂x∂u₁, ...,_∂x∂u

N

❞❡♥♦t❛ ♦ ❣r❛❞✐❡♥t❡ u❀

• ∆u=PN_i=1 ∂2_u ∂x2

i ❞❡♥♦t❛ ♦ ❧❛♣❧❛❝✐❛♥♦ ❞❡ u❀

• ❙❡ X é ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦✱|| · ||X ❞❡♥♦t❛ ❛ ♥♦r♠❛ ❞❡ X❀

• ❙❡ X é ✉♠ ❡s♣❛ç♦ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✱h·,·iX ❞❡♥♦t❛ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡X❀

• ❉❡♥♦t❛r❡♠♦s ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❢r❛❝❛ ❡♠ X ♣♦r ✏⇀✧❡ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❢♦rt❡ ❡♠ X ♣♦r

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①

• ❙❡❥❛♠ X ❡ Y ❡s♣❛ç♦ ✈❡tr♦r✐❛✐s ♥♦r♠❛❞♦s✳ ❙❡ X ⊂Y✱ ❞✐r❡♠♦s q✉❡ X ❡stá ❝♦♥t✐♥✉❛✲

♠❡♥t❡ ♠❡r❣✉❧❤❛❞♦ ❡♠ Y s❡ ❡①✐st❡ C > 0 t❛❧ q✉❡ ||x||Y ≤ C||x||X ♣❛r❛ t♦❞♦ x ∈ X✳

❉❡♥♦t❛♠♦s ✐ss♦ ♣♦r

X ֒→Y .

❙❡✱ ❛❧é♠ ❞✐ss♦✱ t♦❞❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❡♠ X ❛❞♠✐t✐r s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ ❡♠ Y✱ ❞✐r❡♠♦s q✉❡ X ❡stá ❝♦♠♣❛❝t❛♠❡♥t❡ ♠❡r❣✉❧❤❛❞♦ ❡♠Y✳ ❉❡♥♦t❛♠♦s ✐ss♦ ♣♦r

X ֒→→Y;

• ❙❡p∈[1,∞)✱ Lp_{(Ω) =}_{_u_{: Ω}_→_R _{♠❡♥s✉rá✈❡❧}_:R

Ω|u|

p_{dx <}_∞} _{❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ❞❡}

▲❡❜❡s❣✉❡ ❝♦♠ ❛ ♥♦r♠❛ ❞❛❞❛ ♣♦r

||u||p =

Z

Ω

|u(x)|p

dx

1

p

;

• L∞_(Ω) _{❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s q✉❡ sã♦ ❧✐♠✐t❛❞❛s q✉❛s❡ t♦❞❛ ♣❛rt❡} ✭q✳t✳♣✳✮ ❡♠ Ω ❝♦♠ ♥♦r♠❛ ❞❛❞❛ ♣♦r

||u||∞= inf{C >0 :|u(x)| ≤C ♣❛r❛ q✉❛s❡ t♦❞♦ x∈Ω};

• C∞

0 (Ω)❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ✐♥✜♥✐t❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡✐s ❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦

❡♠ Ω❀

• ❙❡ p∈[1,∞) ❡ k∈ N_, ❞❡✜♥✐♠♦s ♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ _Wk,p_{(Ω) =}_{_u_∈_Lp_{(Ω) :} _Dα_u _∈

Lp_(Ω)_, _{∀ |}_α_{| ≤} _k_}_{✱ ♦♥❞❡} _α _{é ✉♠ ♠✉❧t✐✲í♥❞✐❝❡ ❡} _Dα_u _{é ❛ ❞❡r✐✈❛❞❛ ❞❡} _u _{♥♦ s❡♥t✐❞♦}

❢r❛❝♦✳ ❊♠ Wk,p_(Ω) _{❝♦♥s✐❞❡r❛♠♦s ❛ ♥♦r♠❛ ❞❛❞❛ ♣♦r}

||u||k,p =

 Z

Ω

X

|α|≤k

|Dα_u_|p_dx

 

1

p

; ✭✵✳✶✮

• W₀k,p(Ω) é ♦ ❢❡❝❤♦ ❞♦ ❡s♣❛ç♦ C∞

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①✐

• ❈♦♥s✐❞❡r❛r❡♠♦s E =W₀1,2(Ω) ❡q✉✐♣❛❞♦ ❝♦♠ ❛ ♥♦r♠❛ ❞❡ ❉✐r✐❝❤❧❡t

||u||E =

Z

Ω

|∇u|2 1 2

,

q✉❡ é ♣r♦✈❡♥✐❡♥t❡ ❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦

hu, viE =

Z

Ω

∇u· ∇v

❡ é ❡q✉✐✈❛❧❡♥t❡ à ♥♦r♠❛ ✭✵✳✶✮ ♣❛r❛ k = 1 ❡ p= 2❀

• ❉❡♥♦t❛r❡♠♦s ♣♦r 0< λ1 < λ2 ≤λ3 ≤...≤λk ≤... ♦s ❛✉t♦✈❛❧♦r❡s ❞❡ −∆❡♠ E ❡ ♣♦r

σ(−∆) = {λk : k ∈ N} s❡✉ ❡s♣❡❝tr♦✳ ❉❡♥♦t❛r❡♠♦s ♣♦r {φk : k ∈ N} ❛s ❛✉t♦❢✉♥çõ❡s

❝♦rr❡s♣♦♥❞❡♥t❡s✳ ▲❡♠❜r❛♠♦s q✉❡ {φk}k∈N ❝♦♥st✐t✉✐ ✉♠ ❝♦♥❥✉♥t♦ ♦rt♦❣♦♥❛❧ ❡♠ E ❡

❡♠ L2_(Ω)_{✱ q✉❡}_φ

(16)

(17)

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶

✷ ❋❡rr❛♠❡♥t❛s ❜ás✐❝❛s ✺

✸ ❆ ♣r✐♠❡✐r❛ s♦❧✉çã♦ ✾

✹ ❆ s❡❣✉♥❞❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ s✉❜❝rít✐❝♦ ✶✺ ✹✳✶ ❆♣r❡s❡♥t❛çã♦ ❡ r❡❢♦r♠✉❧❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✹✳✷ ❆ ❡str✉t✉r❛ ❞❡ ❡♥❧❛❝❡ ❞♦ ❢✉♥❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✹✳✸ ❆ ❝♦♥❞✐çã♦(P S) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✹✳✹ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✹✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✺ ❆ s❡❣✉♥❞❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❝rít✐❝♦ ✷✼ ✺✳✶ ❆♣r❡s❡♥t❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✺✳✷ ❘❡❢♦r♠✉❧❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✺✳✸ ❊st✐♠❛♥❞♦ ❛ ♣❡rt✉❜❛çã♦f −fm ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✺✳✹ ❋✉♥çõ❡s ❡①tr❡♠❛✐s ♣❛r❛ ♦ ♠❡r❣✉❧❤♦ ❞❡ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✺✳✺ ❆s ❛✉t♦❢✉♥çõ❡s ❛♣r♦①✐♠❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✺✳✻ ❖❜s❡r✈❛çã♦ s♦❜r❡ ❛s ❢✉♥çõ❡s ❝♦rt❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✺✳✼ ❆ ❡str✉t✉r❛ ❞❡ ❡♥❧❛❝❡ ❞♦ ❢✉♥❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✺✳✽ ❆ s♦❧✉çã♦ ❢r❛❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✺✳✾ ❊st✐♠❛♥❞♦ ♦s ♥í✈❡✐s ❞❡ ♠✐♥✐♠❛① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✺✳✶✵ Pr♦✈❛ ❞♦s ❚❡♦r❡♠❛s ✺✳✶ ❡ ✺✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸

(18)

①✐✈ ❙❯▼➪❘■❖

❇ ❖♣❡r❛❞♦r ❞❡ ◆❡♠②ts❦✐✐ ✽✺

(19)

❈❛♣ít✉❧♦

1

■♥tr♦❞✉çã♦

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

  

−∆u=λu+u2+∗−1+g(x, u+) +f(x), ❡♠ Ω

u= 0, s♦❜r❡ ∂Ω ✭

P✮

❛❜♦r❞❛❞♦ ♣♦r ❬❈❘✵✷❪✳

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

  

−∆u=λu+u₊p +f(x), ❡♠ Ω

u= 0, s♦❜r❡ ∂Ω ✭

P′_✮

❛❜♦r❞❛❞♦ ♣♦r ❬❘❙✽✻❪✳

❊♠ t♦❞♦ ❡st❡ tr❛❜❛❧❤♦Ω⊂RN _{é ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❝♦♠ ❢r♦♥t❡✐r❛ s✉❛✈❡✱}_{λ > λ} 1✱ ❝♦♠

λ /∈σ(−∆)✱ ❡f ∈Lr_(Ω)_{✱ ❝♦♠}_{r > N}_✳

❊♠ ✭P✮ ❝♦♥s✐❞❡r❛♠♦sN ≥3✱2∗ ₌ 2N

N−2 é ♦ ❡①♣♦❡♥t❡ ❝rít✐❝♦ ♣❛r❛ ♦ ♠❡r❣✉❧❤♦ ❞❡ ❙♦❜♦❧❡✈✱

g : Ω×R _→R₊ é ❝♦♥tí♥✉❛✱ _g₍_·_{, s}₎_≡ ₀ ♣❛r❛ t♦❞♦ _s _≤ ₀ ❡ _g t❡♠ ❝r❡s❝✐♠❡♥t♦ s✉❜❝rít✐❝♦ ♥♦

✐♥✜♥✐t♦✳ ❊♠ ✭P′_{✮ ❝♦♥s✐❞❡r❛♠♦s} _N _≥₂_{✱ s❡♥❞♦}_p_∈₍₁_,_∞₎ _♣❛r❛ _N _{= 2}_{✱ ❡} _p_∈₍₁_,₂∗₋₁₎ _♣❛r❛

N ≥3✳

(20)

✷ ❈❛♣ít✉❧♦ ✶ ✖ ■♥tr♦❞✉çã♦

❆s ❝♦♥❞✐çõ❡s s♦❜r❡ λ ❡ g✱ ♥♦s ♣r♦❜❧❡♠❛s ✭P✮ ❡ ✭P′_{✮✱ ✐♠♣❧✐❝❛♠ q✉❡ ❛ ❢✉♥çã♦ r❡❛❧ ❞❡} ✈❛r✐á✈❡❧ r❡❛❧ k(s) = λs+s2+∗−1+g(x, s+) ✭♦✉ k(s) = λs+sp+ ♥♦ ❝❛s♦ ❞♦ ♣r♦❜❧❡♠❛ ✭P′✮✮

✐♥t❡r❛❣❡ ❝♦♠ t♦❞♦s ♦s ❛✉t♦✈❛❧♦r❡s ❞♦ ❧❛♣❧❛❝✐❛♥♦ ❡①❝❡t♦ ♦s ♣r✐♠❡✐r♦s✱ ♥♦ s❡❣✉✐♥t❡ s❡♥t✐❞♦

λ1 < λ= lim

s→−∞

k(s)

s <slim→∞

k(s)

s =∞.

❚❛❧ ❢❛t♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❧✐t❡r❛t✉r❛✱ ❡♥q✉❛❞r❛ ❡st❡s ♣r♦❜❧❡♠❛s ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ♣r♦❜❧❡♠❛s ❝❤❛♠❛❞♦s ❞❡ t✐♣♦ ❆♠❜r♦s❡tt✐✲Pr♦❞✐✱ ✈❡r✱ ♣♦r ❡①❡♠♣❧♦✱ ❬❉❋❏✾✾❪✳

❆ ♣r✐♠❡✐r❛ s♦❧✉çã♦ ❢r❛❝❛ s❡rá s❡♠♣r❡ ✉♠❛ s♦❧✉çã♦ ♥❡❣❛t✐✈❛uneg ❝♦♠✉♠ ❛♦s ♣r♦❜❧❡♠❛s

✭P✮ ❡ ✭P′_{✮✳ ❊❧❛ s❡rá ♦❜t✐❞❛ ♥♦ ❝❛♣ít✉❧♦ ✸ ❡①♣❧♦r❛♥❞♦ ♦ ❢❛t♦ ❞❡ ❛ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡ s❡r s❡♠♣r❡} ♥✉❧❛ ♣❛r❛ u ♥❡❣❛t✐✈❛s✳

❆ s❡❣✉♥❞❛ s♦❧✉çã♦ ❢r❛❝❛ s❡rá ♦❜t✐❞❛ ❛tr❛✈és ❞♦ ❚❊❖❘❊▼❆ ❉❊ ❊◆▲❆❈❊✳

❆♦ ❡st✉❞❛r♠♦s ♦ ♣r♦❜❧❡♠❛ ✭P′_{✮✱ ✈❡r❡♠♦s ❝♦♠♦ ❝♦♥str✉✐r ✉♠❛ ❣❡♦♠❡tr✐❛ ❞❡ ❡♥❧❛❝❡ ❡} ❝♦♠♦ ✈❡r✐✜❝❛r ❛ ❝♦♥❞✐çã♦ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ✭❝♦♥❞✐çã♦ (P S)✮ ❡♠ ✉♠ ♣r♦❜❧❡♠❛ s✉❜❝rít✐❝♦✳ ❆♦

❡st✉❞❛r♠♦s ♦ ♣r♦❜❧❡♠❛ ✭P✮✱ ♥♦t❛r❡♠♦s ❛s ❞✐✜❝✉❧❞❛❞❡s ❛❞✐❝✐♦♥❛✐s ❞❡ r❡s♦❧✈❡r ✉♠ ♣r♦❜❧❡♠❛

♣❛r❡❝✐❞♦✱ ♠❛s ♥♦ ❝❛s♦ ❝rít✐❝♦✳ ❈♦♠♦ E = W₀1,2(Ω) ♥ã♦ ❡stá ❝♦♠♣❛❝t❛♠❡♥t❡ ❝♦♥t✐❞♦ ❡♠

L2∗

(Ω)✱ ♦s ❛r❣✉♠❡♥t♦s ✈❛r✐❛❝✐♦♥❛✐s ❝❧áss✐❝♦s ♥ã♦ s❡ ❛♣❧✐❝❛♠ ❛♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦✲

❜❧❡♠❛❀ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♥❡st❛s ❝♦♥❞✐çõ❡s✱ ♦ ❢✉♥❝✐♦♥❛❧ ♥ã♦ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ (P S)✳ ◗✉❛♥❞♦

✐ss♦ ♦❝♦rr❡✱ ❞✐③❡♠♦s q✉❡ ❡st❛♠♦s ❧✐❞❛♥❞♦ ❝♦♠ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❢❛❧t❛ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡✳ ❊ss❡ t✐♣♦ ❞❡ s✐t✉❛çã♦ ❢♦✐ ✐♥✐❝✐❛❧♠❡♥t❡ ❡st✉❞❛❞❛ ♣♦r ❇r❡③✐s ❡ ◆✐r❡♥❜❡r❣ ❡♠ ❬❇◆✽✸❪✳ ❊❧❡s ❝♦♥s✐❞❡r❛r❛♠ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿

        

−∆u=u2∗₋₁

+g(x, u), ❡♠ Ω

u >0, ❡♠ Ω

u= 0, s♦❜r❡ ∂Ω

✭✶✳✶✮

♦♥❞❡ g é ✉♠❛ ♣❡rt✉❜çã♦ ❞❡ ♦r❞❡♠ ✐♥❢❡r✐♦r ❞❡ u2∗₋₁

✱ ♦✉ s❡❥❛✱ g(x,s) s2∗−1 −−−→

s→∞ 0✳ ❈♦♠ ❤✐♣ót❡s❡s ❛♣r♦♣r✐❛❞❛s s♦❜r❡ ❛ ❢✉♥çã♦g✱ ♦s ❛✉t♦r❡s ♠♦str❛r❛♠ q✉❡✱ ♠❡s♠♦ tr❛❜❛❧❤❛♥❞♦ ♥✉♠ ♣r♦❜❧❡♠❛

♦♥❞❡ ♦❝♦rr❡ ❢❛❧t❛ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡✱ ❛✐♥❞❛ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦ tr✐✈✐❛❧✳ ❊❧❡s ♠♦str❛r❛♠ q✉❡ ❡♠ ❝❡rt♦s ♥í✈❡✐s ♠✐♥✐♠❛① c é ♣♦ssí✈❡❧ ♦❜t❡r ✉♠❛ s❡q✉ê♥❝✐❛ (P S) ♥♦

♥í✈❡❧ c ❡ q✉❡ s❡ c ∈ (0, 1

NS

N

2 )✱ ♦♥❞❡ S é ❛ ♠❡❧❤♦r ❝♦♥st❛♥t❡ ♣❛r❛ ♦ ♠❡r❣✉❧❤♦ ❞❡ ❙♦❜♦❧❡✈

E ֒→ L2∗

(Ω)✱ ❡♥tã♦ ❛ s❡q✉ê♥❝✐❛ ♦❜t✐❞❛ ❛❞♠✐t❡ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ ❡♠ E ❝♦♠ ❧✐♠✐t❡

(21)

✸

➱ ✐♠♣♦rt❛♥t❡ ♦❜s❡r✈❛r q✉❡ s❡g ≡0♥♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮✱ ❡♥tã♦ ♥ã♦ ❡①✐st❡ s♦❧✉çã♦ q✉❛♥❞♦ Ω é ✉♠ ❝♦♥❥✉♥t♦ ❡str❡❧❛❞♦✱ ✈❡❥❛ ❬P♦❤✻✺❪✳ ■ss♦ ♣❛r❡❝❡ s✉❣❡r✐r q✉❡ ❛ ♣❡rt✉❜❛çã♦ ❞❡ ♦r❞❡♠

✐♥❢❡r✐♦r g ❡①❡r❝❡ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♣❛r❛ r❡✈❡rt❡r ❛ s✐t✉❛çã♦ ❞❡ ❢❛❧t❛ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡✳

❙❡ ♦❜s❡r✈❛r♠♦s ❜❡♠✱ ♦ ♣r♦❜❧❡♠❛ ✭P✮ ♣♦ss✉✐ ✉♠❛ ♣❡rt✉❜❛çã♦ ❞❡ ♦r❞❡♠ ✐♥❢❡r✐♦r ❞❡ u2∗₋₁

✱ ❧♦❣♦✱ t♦♠❛♥❞♦ ❬❇◆✽✸❪ ❝♦♠♦ ❢♦♥t❡ ❞❡ ✐♥s♣✐r❛çã♦✱ ♣❛r❡❝❡ s✉❣❡st✐✈♦ ❡s♣❡r❛r q✉❡ s❡❥❛ ♣♦ssí✈❡❧ ❝♦♥t♦r♥❛r ❛ ❢❛❧t❛ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ❡ ♦❜t❡r ✉♠❛ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ❞❡ ✭P✮✳

P❛r❛ ♦❜t❡r ✐ss♦✱ s❡ t♦r♥❛ ✐♠♣♦rt❛♥t❡ ❡st✐♠❛r ❝♦♠ ♣r❡❝✐sã♦ ♦ ♥í✈❡❧ ♠✐♥✐♠❛① ❞♦ ❢✉♥❝✐♦♥❛❧✱ ♣❛r❛ ♣♦❞❡r ♠♦str❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ❝♦rr❡s♣♦♥❞❡♥t❡✳ P❛r❛ ❡st✐♠❛r ❡st❡ ♥í✈❡❧ é✱ ❡♠ ❣❡r❛❧✱ ♥❡❝❡ssár✐♦ ❡①♣❧♦r❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ❢✉♥çõ❡s ❡①tr❡♠❛✐s Φǫ ❞♦ ♠❡r❣✉❧❤♦ ❞❡

❙♦❜♦❧❡✈✳ ❙❡❣✉✐♥❞♦ ❡ss❛s ✐❞❡✐❛s✱ ❞❡ ❋✐❣✉❡✐r❡❞♦ ❡ ❨❛♥❣✱ ❡♠ ❬❉❋❏✾✾❪✱ ♣r♦✈❛r❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s❡❣✉♥❞❛ s♦❧✉çã♦ ❢r❛❝❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭P✮ ♥♦ ❝❛s♦ g ≡ 0 ❡ N ≥7✱ ✐st♦ é✱ q✉❛♥❞♦

❛ ♣❡rt✉r❜❛çã♦ ❞❡ ♦r❞❡♠ ✐♥❢❡r✐♦r é ❞❛❞❛ ❛♣❡♥❛s ♣❡❧♦ t❡r♠♦λu✳

◆♦ ✐♥t✉✐t♦ ❞❡ ♠❡❧❤♦r❛r ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❡♠ ❬❉❋❏✾✾❪✱ ❈❛❧❛♥❝❤✐ ❡ ❘✉❢✱ ❡♠ ❬❈❘✵✷❪✱ ✜③❡r❛♠ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❞✐❢❡r❡♥t❡✳ ❆ ✐❞❡✐❛✱ ✐♥s♣✐r❛❞❛ ❡♠ tr❛❜❛❧❤♦s ❛♥t❡r✐♦r❡s✱ ♣♦r ❡①❡♠♣❧♦ ❬●❘✾✼❪✱ ❝♦♥s✐st❡ ❡♠ ✉s❛r ♦♣♦rt✉♥❛s ❢✉♥çõ❡s ❞❡ ❵❝✉t ♦✛✬✱ ❛ ✜♠ ❞❡ ✏❝❛✈❛r ❜✉r❛❝♦s✑ ♥❛ s♦❧✉çã♦ ♥❡❣❛t✐✈❛uneg ❡ ♥❛s ❛✉t♦❢✉♥çõ❡s ❞♦ ▲❛♣❧❛❝✐❛♥♦✱ ❡ ❞❡ ❝♦♥❝❡♥tr❛r ❛s ❢✉♥çõ❡s ❡①tr❡♠❛✐sΦǫ ❡♠

❜♦❧❛s ♣❡q✉❡♥❛s✱ ❞❡ ♠♦❞♦ ❛ ♦❜t❡r♠♦s ❢✉♥çõ❡s ❝♦♠ ♣r♦♣r✐❡❞❛❞❡s s✐♠✐❧❛r❡s às ❞❛s ♦r✐❣✐♥❛✐s✱ ♠❛s ❝♦♠ s✉♣♦rt❡s ❞✐s❥✉♥t♦s✳ ❊ss❛s ♠❛♥✐♣✉❧❛çõ❡s ❝r✐❛♠ ❛❧❣✉♥s ❡rr♦s✱ ♠❛s ❡❧❡s sã♦ ♠❛✐s ❢á❝❡✐s ❞❡ ❡st✐♠❛r q✉❡ ♦s t❡r♠♦s ♠✐st♦s q✉❡ s✉r❣❡♠ ❡♠ ❡①♣r❡ssõ❡s ❝♦♠♦ (v +uneg+ Φǫ)2

∗

+

q✉❛♥❞♦ s❡✉s s✉♣♦rt❡s ♥ã♦ sã♦ ❞✐s❥✉♥t♦s✳ ❆❧é♠ ❞✐ss♦✱ ♠♦✈❡r ❡ss❡s ✏♣❡q✉❡♥♦s ❜✉r❛❝♦s✑ ♣❛r❛ ♣ró①✐♠♦ ❞❛ ❢r♦♥t❡✐r❛✱ ♦♥❞❡ ❛ ❢✉♥çã♦|uneg|é ♣❡q✉❡♥❛✱ ♣♦ss✐❜✐❧✐t❛ ♦❜t❡r ❡st✐♠❛t✐✈❛s ♠❡❧❤♦r❡s

q✉❡ ❛s ❡♥❝♦♥tr❛❞❛s ❡♠ ❬❉❋❏✾✾❪✳ ❈♦♠ ❡ss❛ ❛❜♦r❞❛❣❡♠ ❥á é ♣♦ssí✈❡❧ ❡st❡♥❞❡r ♦ r❡s✉❧t❛❞♦ ❡♠ ❬❉❋❏✾✾❪ ♣❛r❛ ♦ ❝❛s♦ N = 6✳ P❛r❛ ♦s ❝❛s♦s N = 3,4,5✱ ❛✐♥❞❛ é ♥❡❝❡ssár✐♦ ❛ss✉♠✐r q✉❡

❛ ❢✉♥çã♦ g ❢♦r♥❡ç❛ ✉♠❛ ✉❧t❡r✐♦r ♣❡rt✉r❜❛çã♦ ❞❡ ♦r❞❡♠ ✐♥❢❡r✐♦r ❛ u2∗₋₁

✱ ♣❛r❛ r❡❝✉♣❡r❛r ❛ ❝♦♠♣❛❝✐❞❛❞❡✳

◆♦ ❝❛♣ít✉❧♦ ✷✱ ❡♥✉♥❝✐❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♥❡❝❡ssár✐♦s✱ ♣❛r❛ ♦ ❡st✉❞♦ ❞♦s ♣r♦❜❧❡♠❛s ❝✐t❛❞♦s✱ ❝♦♠ ❛s ❞❡✈✐❞❛s r❡❢❡rê♥❝✐❛s✳ ◆♦ ❝❛♣ít✉❧♦ ✸✱ ♠♦str❛r❡♠♦s ❝♦♠♦ ♦❜t❡r ❛ ♣r✐♠❡✐r❛ s♦❧✉çã♦ ❢r❛❝❛✱ ❝♦♠✉♠ ❛♦s ♣r♦❜❧❡♠❛s ✭P✮ ❡ ✭P′_{✮✳ ◆♦ ❝❛♣ít✉❧♦ ✹✱ ❡♥❝♦♥tr❛r❡♠♦s ❛ s❡❣✉♥❞❛} s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✭P′✮✳ P♦r ✜♠✱ ♥♦ ❝❛♣ít✉❧♦ ✺✱ ❛❜♦r❞❛r❡♠♦s ♦ ♣r♦❜❧❡♠❛ ✭P✮✱ ♠♦str❛♥❞♦ ❝♦♠♦

(22)

(23)

❈❛♣ít✉❧♦

2

❋❡rr❛♠❡♥t❛s ❜ás✐❝❛s

❈♦♠♦ é r♦t✐♥❡✐r♦ ❡♠ ♠ét♦❞♦s ✈❛r✐❛❝✐♦♥❛✐s✱ ✐r❡♠♦s ❛ss♦❝✐❛r ❝❛❞❛ ❡q✉❛çã♦ ❛ ✉♠ ❢✉♥❝✐♦✲ ♥❛❧ I : E → R ❞❡✜♥✐❞♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ _E✱ ❡ ✈❡r✐✜❝❛r❡♠♦s s❡ t❛❧ ❢✉♥❝✐♦♥❛❧ ❛❞♠✐t❡

❛❧❣✉♠ ♣♦♥t♦ ❝rít✐❝♦ ♥ã♦ tr✐✈✐❛❧❀ t❛❧ ♣♦♥t♦ ❝rít✐❝♦ s❡rá ❛ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦ tr✐✈✐❛❧ q✉❡ ♣r♦❝✉✲ r❛♠♦s✳ P❛r❛ ❡ss❡ ✜♠ ❡♥✉♥❝✐❛r❡♠♦s✱ ❛ s❡❣✉✐r✱ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ ✉♠ t❡♦r❡♠❛ q✉❡ s❡rã♦ ❛s ❢❡rr❛♠❡♥t❛s ❜ás✐❝❛s ♣❛r❛ ❛ ♣r♦❝✉r❛ ❞❡ t❛✐s ♣♦♥t♦s ❝rít✐❝♦s✳

❉❡✜♥✐çã♦ ✷✳✶ ✭❙❊◗❯✃◆❈■❆ ❉❊ P❆▲❆■❙✲❙▼❆▲❊✮✳ ❙❡❥❛♠I ∈C1₍_E,_R₎ _❡ _c_∈_R_{❀ ❞✐r❡♠♦s q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛} _{_v

n}n∈N ⊂E é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡

P❛❧❛✐s✲❙♠❛❧❡ ♣❛r❛I ♥♦ ♥í✈❡❧ c s❡

I(vn)−−−→

n→∞ c ❡ I

′₍_v

n)−−−→

n→∞ 0.

◆♦s r❡❢❡r✐r❡♠♦s ❛ t❛✐s s❡q✉ê♥❝✐❛s ❛♣❡♥❛s ❝♦♠♦ s❡q✉ê♥❝✐❛s (P S) ♣❛r❛ I ♥♦ ♥í✈❡❧ c✳

❉❡✜♥✐çã♦ ✷✳✷ ✭❈❖◆❉■➬➹❖ ❉❊ P❆▲❆■❙✲❙▼❆▲❊✮✳

❙❡❥❛♠ I ∈ C1₍_E,_R₎ _❡ _c _∈ _R_{❀ ❞✐r❡♠♦s q✉❡ ♦ ❢✉♥❝✐♦♥❛❧} _I _{s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ P❛❧❛✐s✲}

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

❝♦♥✈❡r❣❡♥t❡✳ ◆♦s r❡❢❡r✐r❡♠♦s ❛ t❛❧ ❝♦♥❞✐çã♦ ❝♦♠♦ ❝♦♥❞✐çã♦(P S)c✳ ❆❧é♠ ❞✐ss♦✱ ❞✐r❡♠♦s q✉❡

♦ ❢✉♥❝✐♦♥❛❧ I s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ✭❝♦♥❞✐çã♦ (P S)✮ s❡ I s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦

(P S)c ♣❛r❛ t♦❞♦ c∈R✳

❉❡✜♥✐çã♦ ✷✳✸ ✭❊◆▲❆❈❊✮✳

(24)

✻ ❈❛♣ít✉❧♦ ✷ ✖ ❋❡rr❛♠❡♥t❛s ❜ás✐❝❛s

✉♥✐tár✐❛ ❢❡❝❤❛❞❛ B1(0) ❞❡Rk ♣❛r❛ ❛❧❣✉♠ ♥❛t✉r❛❧k ≥2✳ ❉❡♥♦t❡ ♣♦r ∂Q ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡Q

❝♦rr❡s♣♦♥❞❡♥t❡ à ❢r♦♥t❡✐r❛ ❞❡B1(0)s❡❣✉♥❞♦ ♦ ❤♦♠❡♦♠♦r✜s♠♦✳ ❉✐③❡♠♦s q✉❡ S ❡Q❡♥❧❛ç❛♠

s❡

✭❛✮ S∩∂Q=∅❀

✭❜✮ ❞❛❞♦ h∈C(E, E)✱ ❝♦♠ h|∂Q =Id✱ ✈❛❧❡ h(Q)∩S 6=∅✳

❊①❡♠♣❧♦ ✷✳✹✳

❈♦♥s✐❞❡r❡ E =W⊕X✱ ❝♦♠ dimW =k−1✱ ♦♥❞❡ k≥2✳ ❙❡❥❛♠ Bρ={u∈E :||u||E < ρ}✱

♦♥❞❡ ρ >0✱ ❡ e∈X✱ ❝♦♠ ||e||E = 1✳ ❙❡❥❛♠ r >0 ❡ R > ρ❀ ❞❡✜♥❛ ♦s s✉❜❝♦♥❥✉♥t♦s

Sρ=∂Bρ∩X ❡ QR,r = (Br∩W)⊕ {se :s∈[0, R]}. ✭✷✳✶✮

➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ QR,r é ❤♦♠❡♦♠♦r❢♦ à ❜♦❧❛ ✉♥✐tár✐❛ ❢❡❝❤❛❞❛ ❞❡ Rk✳ ❆❧é♠ ❞✐ss♦✱ ♦

❡①❡♠♣❧♦ 8.3❞❡ ❬❙tr✵✽❪ ♠♦str❛ q✉❡ ❡ss❡s ❝♦♥❥✉♥t♦s ❡♥❧❛ç❛♠✳ ◆❡st❡ ❝❛s♦✱ t❡♠♦s q✉❡ ∂QR,r =

Γ1∪Γ2∪Γ3 ♦♥❞❡

Γ1 ={u∈E :u∈W, ||u||E ≤r},

Γ2 ={u∈E :u=w+se, w∈W, ||w||E =r, s∈(0, R)},

Γ3 ={u∈E :u=w+Re, w∈W, ||w||E ≤r}.

❉✉r❛♥t❡ ❛ ♣r♦✈❛ ❞❛ ❡①✐stê♥❝✐❛ ❞❛ s❡❣✉♥❞❛ s♦❧✉çã♦ ❢r❛❝❛ ♣❛r❛ ♦s ♣r♦❜❧❡♠❛s ✭P✮ ❡ ✭P′_✮ ♥❡❝❡ss✐t❛r❡♠♦s ❝♦♥str✉✐r ❝♦♥❥✉♥t♦s ❝♦♠♦ ❡♠ ✭✷✳✶✮✳

❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r é ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❊❧❡ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ❚❡♦r❡♠❛ ✹✳✸✱ ❝♦♠❜✐♥❛❞♦ ❝♦♠ ♦s ❝♦r♦❧ár✐♦s ✹✳✷ ❡ ✹✳✸ ❞❡ ❬▼❲✽✾❪✳

❚❡♦r❡♠❛ ✷✳✺ ✭❚❊❖❘❊▼❆ ❉❊ ❊◆▲❆❈❊✮✳

❙❡❥❛ I ∈C1₍_E,_R₎ _{❡ s❡❥❛♠} _S _❡ _Q _{s✉❜❝♦♥❥✉♥t♦s ❞❡} _E _{q✉❡ ❡♥❧❛ç❛♠✱ ❝♦♠♦ ♥❛ ❉❡✜♥✐çã♦ ✷✳✸✳}

❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛♠ ❞✉❛s ❝♦♥st❛♥t❡s β < α t❛✐s q✉❡

❛✮ I|S ≥α❀

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✼

❊♥tã♦ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ (P S) ♣❛r❛ I ♥♦ ♥í✈❡❧ c ♦♥❞❡ c ≥ α❀ ❛❧é♠ ❞✐ss♦✱ c ♣♦❞❡ s❡r

❝❛r❛❝t❡r✐③❛❞♦ ❝♦♠♦ c= inf

h∈Γmaxu∈Q I(h(u))✱ ♦♥❞❡

Γ ={h∈C(E, E) : h|∂Q =Id}.

❖❜s❡r✈❡ q✉❡✱ ✉♠❛ ✈❡③ ♦❜t✐❞❛ ❛ s❡q✉ê♥❝✐❛(P S)❢♦r♥❡❝✐❞❛ ♣❡❧♦ ❚❊❖❘❊▼❆ ❉❊ ❊◆▲❆❈❊✱

❝❛s♦ ♦ ❢✉♥❝✐♦♥❛❧ s❛t✐s❢❛ç❛ ❛ ❝♦♥❞✐çã♦ (P S) ♦✉(P S)c✱ ❥á t❡r❡♠♦s ♦❜t✐❞♦ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠

♣♦♥t♦ ❝rít✐❝♦ ♥♦ ♥í✈❡❧ c✳ ❊♠ ❝❛s♦ ❝♦♥trár✐♦✱ ❛✐♥❞❛ ❢❛❧t❛rá ♠♦str❛r q✉❡ é ♣♦ssí✈❡❧ ♦❜t❡r

(26)

(27)

❈❛♣ít✉❧♦

3

❆ ♣r✐♠❡✐r❛ s♦❧✉çã♦

◆❡st❡ ❝❛♣ít✉❧♦ ♠♦str❛r❡♠♦s ❝♦♠♦ ♦❜t❡r ❛ ♣r✐♠❡✐r❛ s♦❧✉çã♦ ❢r❛❝❛ ❝♦♠✉♠ ❛♦s ♣r♦❜❧❡♠❛s ✭P✮ ❡ ✭P′_{✮✳ P❛r❛ ✐ss♦✱ ❝♦♥s✐❞❡r❛r❡♠♦s ♦ ♣r♦❜❧❡♠❛ ♠❛✐s ❣❡r❛❧}

  

−∆u=λu+up₊+g(x, u+) +f(x), ❡♠ Ω

u= 0, s♦❜r❡ ∂Ω. ✭

e

P✮

♦♥❞❡ N ≥ 2✱ s❡♥❞♦ p ∈ (1,2∗_] _s❡ _N _≥ ₃ _❡ _p _∈ ₍₁_,_∞₎ _s❡ _N _{= 2}_{❀ ❛❧é♠ ❞✐ss♦✱} _Ω _⊂ _RN _é

✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❝♦♠ ❢r♦♥t❡✐r❛ s✉❛✈❡✱g : Ω×R_→ R₊ é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ t❛❧ q✉❡

g(·, s)≡0 ♣❛r❛ t♦❞♦s ≤0✱ λ > λ1✱ ❝♦♠ λ /∈σ(−∆) ❡ f ∈Lr(Ω)✱ ❝♦♠ r > N✳

❉❛q✉✐ ❡♠ ❞✐❛♥t❡ ❝♦♥s✐❞❡r❛r❡♠♦s

f =h+tφ1 ✭✸✳✶✮

♦♥❞❡t ∈R✱_φ₁ _>₀ é ❛ ♣r✐♠❡✐r❛ ❛✉t♦❢✉♥çã♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦ ❡♠_E ❡ _h_∈_Lr_(Ω)✳

❚❡♦r❡♠❛ ✸✳✶✳

❉❛❞♦h ∈Lr_(Ω)_{❡①✐st❡}_T ₌_T₍_h₎_{t❛❧ q✉❡ ♣❛r❛ t♦❞♦}_{t > T} _{♦ ♣r♦❜❧❡♠❛ ✭}_Pe_{✮ ❛❞♠✐t❡ ✉♠❛ s♦❧✉çã♦}

❢r❛❝❛ ♥❡❣❛t✐✈❛ ut∈E =W₀1,2 t❛❧ q✉❡ ut∈W₀1,r(Ω)∩W2,2(Ω)✳ ❆❧é♠ ❞✐ss♦✱ ut∈C1,1− N

r(Ω)✳

Pr♦✈❛✳

❈♦♠♦2≤N < r✱ t❡♠♦s q✉❡h∈L2_(Ω)_{✳ P❛r❛ ♦❜t❡r♠♦s ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✱ r❡s♦❧✈❡r❡♠♦s✱}

(28)

✶✵ ❈❛♣ít✉❧♦ ✸ ✖ ❆ ♣r✐♠❡✐r❛ s♦❧✉çã♦

✶✳ ❉❛❞♦ h∈L2_(Ω)_{✱ ❝♦♥s✐❞❡r❡}

  

−∆u=λu+h, ❡♠ Ω

u= 0, s♦❜r❡ ∂Ω; ✭✸✳✷✮

✷✳ ❉❛❞♦ t∈R✱ ❝♦♥s✐❞❡r❡

  

−∆w=λw+tφ1, ❡♠ Ω

w= 0, s♦❜r❡ ∂Ω. ✭✸✳✸✮

▼♦str❛r❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ♣❛r❛ ✭✸✳✷✮ ❡ ♣❛r❛ ✭✸✳✸✮✱ ♥❡st❡ ú❧t✐♠♦ ❝❛s♦ ♣❛r❛t

✜①♦✳

❈♦♠❡❝❡♠♦s ❝♦♠ ♦ ♣r♦❜❧❡♠❛ ✭✸✳✷✮✳ P❛r❛ r❡s♦❧✈ê✲❧♦✱ ❝♦♥s✐❞❡r❛r❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛

  

−∆u=h, ❡♠ Ω

u= 0, s♦❜r❡ ∂Ω. ✭✸✳✹✮

❉❡✜♥❛♠♦s ❛s ❛♣❧✐❝❛çõ❡s A :E ×E →R ❡ _H _:_E _→Rr❡s♣❡❝t✐✈❛♠❡♥t❡ ♣♦r

A(u, v) =

Z

Ω

∇u· ∇vdx ❡ H(v) =

Z

Ω

hvdx. ✭✸✳✺✮

❯t✐❧✐③❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r ♦❜s❡r✈❛♠♦s ❢❛❝✐❧♠❡♥t❡ q✉❡Aé ✉♠❛ ❢♦r♠❛ ❜✐❧✐♥❡❛r ❝♦♥✲

tí♥✉❛ ❡ ❝♦❡r❝✐✈❛✳ ❚❛♠❜é♠ é ❢á❝✐❧ ♦❜s❡r✈❛r q✉❡ H ∈E′

✱ ♣❛r❛ ✐ss♦ ✉t✐❧✐③❛♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r ❥✉♥t♦ ❝♦♠ ♦ ✐t❡♠ ✭✐✮ ❞♦ ❚❊❖❘❊▼❆ ❉❊ ▼❊❘●❯▲❍❖ ❉❊ ❙❖❇❖▲❊❱ ✭❚❡♦r❡♠❛ ❆✳✷✮✳ ▲♦❣♦✱ ♣❡❧♦ ❚❊❖❘❊▼❆ ❉❊ ▲❆❳ ✲ ▼■▲●❘❆◆ ✭❚❡♦r❡♠❛ ❆✳✺✮✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ u∈E

t❛❧ q✉❡ A(u, v) = H(v)♣❛r❛ t♦❞♦ v ∈E✱ ♦✉ s❡❥❛✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ u∈E q✉❡

Z

Ω

∇u· ∇vdx=

Z

Ω

hvdx ∀v ∈E. ✭✸✳✻✮

■ss♦ ✐♠♣❧✐❝❛ q✉❡ ué ❛ ú♥✐❝❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✭✸✳✹✮✳ P♦❞❡♠♦s ❡♥tã♦ ❞❡✜♥✐r ♦ ♦♣❡r❛❞♦r ❧✐♥❡❛r

e

T :L2_(Ω)_→_E _{q✉❡ ❛ss♦❝✐❛ ❝❛❞❛} _h_∈_L2_(Ω) _{à ú♥✐❝❛ s♦❧✉çã♦ ❢r❛❝❛}_u₌_T_e₍_h₎_∈_E _{❞❡ ✭✸✳✹✮✳}

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✶✶

❚❊❖❘❊▼❆ ❉❊ ▼❊❘●❯▲❍❖ ❉❊ ❙❖❇❖▲❊❱✱ s❡❣✉❡ q✉❡

||u||2_E =

Z

Ω

|∇u|2dx

=

Z

Ω

hudx

≤ ||h||2||u||2

≤C||h||2||u||E;

❧♦❣♦✱ s❡ u ∈ E \ {0}✱ ❡♥tã♦ ||Te(h)||E = ||u||E ≤ C||h||2 < ∞ ❡ ✐ss♦ ♥♦s ♠♦str❛ q✉❡ Te é

❝♦♥tí♥✉♦✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ ❛ ✐♠❡rsã♦I :E →L2_(Ω) _{é ✉♠ ♦♣❡r❛❞♦r ❝♦♠♣❛❝t♦✱ t❡♠♦s q✉❡}

♦ ♦♣❡r❛❞♦r T :L2_(Ω) _→_L2_(Ω) _{❞❛❞♦ ♣♦r}

T(h) = (I◦Te)(h)

t❛♠❜é♠ ♦ é✳

P❡❧❛ ♣❛ss❛❣❡♠ ❛♥t❡r✐♦r✱ t❡♠♦s q✉❡ u0 ∈ E s❡rá ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✭✸✳✷✮ s❡✱ ❡ só s❡✱

u0 = T(λu0 +h)✳ ◆♦ ❡♥t❛♥t♦✱ ♥♦s ♣❡r❣✉♥t❛♠♦s s❡ ❞❛❞♦ h ∈ L2(Ω) ❡①✐st❡ u0 ∈ E t❛❧

q✉❡ u0 = T(λu0 +h)✳ ❈❛s♦ ❡①✐st❛✱ u0 é ú♥✐❝♦❄ ❙❡ ❛ r❡s♣♦st❛ ❛ ✉♠❛ ❞❡ss❛s ♣❡r❣✉♥t❛s ❢♦r

♥❡❣❛t✐✈❛✱ ♥ã♦ ♣♦❞❡r❡♠♦s ❝❤❡❣❛r à ❝♦♥❝❧✉sã♦ q✉❡ q✉❡r❡♠♦s✳ ❆ ❛✜r♠❛çã♦ ❛ s❡❣✉✐r ♥♦s ❞á ❛ r❡s♣♦st❛ ❞❡s❡❥❛❞❛✿

❆❋■❘▼❆➬➹❖✳

❉❛❞♦ h∈L2_(Ω) _{❡①✐st❡ ✉♠ ú♥✐❝♦} _u

h ∈E s❛t✐s❢❛③❡♥❞♦

uh =T(λuh+h).

Pr♦✈❛ ❞❛ ❆❋■❘▼❆➬➹❖✳

❈♦♠♦T é ✉♠ ♦♣❡r❛❞♦r ❝♦♠♣❛❝t♦✱ ♣❡❧♦ ❚❊❖❘❊▼❆ ❉❆ ❆▲❚❊❘◆❆❚■❱❆ ❉❊ ❋❘❊❉❍❖▲▼

✭❚❡♦r❡♠❛ ❆✳✹✮✱ t❡♠♦s q✉❡ ❛ ❛♣❧✐❝❛çã♦Id−λT :L2_(Ω) _→_L2_(Ω) _{é ✐♥✈❡rsí✈❡❧ ♦✉ t❡♠ ♥ú❝❧❡♦}

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✶✷ ❈❛♣ít✉❧♦ ✸ ✖ ❆ ♣r✐♠❡✐r❛ s♦❧✉çã♦

t❛❧ q✉❡ λT(w) = w✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡ w é ❛ ú♥✐❝❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✭✸✳✹✮✱ ♦✉ s❡❥❛✱

  

−∆w=λw, ❡♠ Ω

w= 0, s♦❜r❡ ∂Ω.

▼❛s ✐ss♦ ❝♦♥tr❛r✐❛ ❛ ❤✐♣ót❡s❡ ❞❡ λ /∈σ(−∆)✳

❈♦♥❝❧✉í♠♦s✱ ❡♥tã♦✱ q✉❡ Id−λT é ✐♥✈❡rsí✈❡❧❀ s❡♥❞♦ ❛ss✐♠✱ ❞❛❞♦ h ∈ L2_(Ω)_{✱ ❡①✐st❡ ✉♠}

ú♥✐❝♦ uh ∈L2(Ω) t❛❧ q✉❡

(Id−λT)(uh) =T(h),

❡q✉✐✈❛❧❡♥t❡♠❡♥t❡

uh =T(λuh+h).

P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ T✱ uh ∈E❀ ❛ss✐♠ ✜❝❛ ♣r♦✈❛❞❛ ❛ ❛✜r♠❛çã♦✳

❉❛ ❛✜r♠❛çã♦✱ s❡❣✉❡ q✉❡u0 ∈E é ❞❡ ❢❛t♦ ❛ ú♥✐❝❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✭✸✳✷✮ ♣❛r❛ h∈L2(Ω)

✜①♦✳

❈♦♠♦ h ∈ Lr_(Ω)_{✱ t❡♠♦s✱ ♣❡❧♦ ❚❊❖❘❊▼❆ ❉❊ ❘❊●❯▲❆❘■❉❆❉❊ ✭❚❡♦r❡♠❛ ❆✳✻✮✱ q✉❡}

u0 ∈ W01,r(Ω)∩W2,r(Ω)✳ ❉♦ ✐t❡♠ iii ❞♦ ❚❊❖❘❊▼❆ ❉❊ ▼❊❘●❯▲❍❖ ❉❊ ❙❖❇❖▲❊❱✱

❝♦♠ k = 2✱ s❡❣✉❡ q✉❡ u0 ∈ C1,β(Ω)✱ ♦♥❞❡ 0 < β ≤ 1− N_r❀ ❡♠ ♣❛rt✐❝✉❧❛r u0 ∈ C1,1− N

r(Ω)✳

▲♦❣♦✱ u0 ❡ s✉❛s ❞❡r✐✈❛❞❛s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ sã♦ ❧✐♠✐t❛❞❛s ❡♠ Ω✳

❈♦♠♦ φ1 ∈ C∞(Ω)✱ ♥ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡ wt = _λ₁t₋_λφ1 é ❛ ú♥✐❝❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡

✭✸✳✸✮ ♣❛r❛ ❝❛❞❛ t∈R✳

❈♦♥s✐❞❡r❡ut =u0+wt✳ P❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡u0 ❡φ1✱ t❡♠♦s q✉❡ut∈W01,r(Ω)∩W2,r(Ω)

❡ ut∈C1,1− N

r(Ω)✳ ❆❧é♠ ❞✐ss♦✱ é s✐♠♣❧❡s ✈❡r✐✜❝❛r q✉❡ ♦ ❢❛t♦ ❞❡u₀ s❡r s♦❧✉çã♦ ❞❡ ✭✸✳✷✮ ❡ w_t

❞❡ ✭✸✳✸✮ ♥♦s ❞á

−∆ut−λut =f. ✭✸✳✼✮

❖❜s❡r✈❡ ❛❣♦r❛ q✉❡ s❡ ♣✉❞❡r♠♦s ❛✜r♠❛r q✉❡ut≤0 ❡♠ Ω♣❛r❛ ❛❧❣✉♠ t∈R✱ ❡♥tã♦ ♣♦❞❡♠♦s

❞✐③❡r q✉❡ ✭✸✳✼✮ é ❡q✉✐✈❛❧❡♥t❡ ❛ ✭Pe✮ ✉♠❛ ✈❡③ q✉❡✱ ♥❡st❡ ❝❛s♦✱ (ut)+ ≡ g(·,(ut)+) ≡ 0✳

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✶✸

❞❡✈❡♠♦s ✈❡r✐✜❝❛r s❡ é ♣♦ssí✈❡❧ t❡r♠♦s ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡

u0(x)

φ1(x)

≤ t

λ−λ1

∀x∈Ω, ✭✸✳✽✮

♣❛r❛ ❛❧❣✉♠t∈R✳

❈♦♠♦ φ1 >0❡♠ Ω✱ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡ é ❝♦♥tí♥✉❛ ❛ ❢✉♥çã♦ U : Ω→R ❞❛❞❛ ♣♦r

U(x) = u0(x)

φ1(x)

.

◗✉❡r❡♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡U é ❧✐♠✐t❛❞❛ ❡♠Ω✳ ❙✉♣♦♥❤❛♠♦s ❡♥tã♦ q✉❡U ♥ã♦ s❡❥❛ ❧✐♠✐t❛❞❛

❡♠ Ω❀ ❧♦❣♦✱ ❡①✐st❡♠ {xn}N ⊂Ω❡ {c_n}N ⊂R t❛✐s q✉❡ |u₀(x_n)|=c_nφ₁(x_n)❝♦♠ c_n −−−→

n→∞ ∞✳

❖❜s❡r✈❡ q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱xn−−−→

n→∞ x0 ♣❛r❛ ❛❧❣✉♠ x0 ∈∂Ω✱ ♣♦✐s✱ ❡♠ q✉❛❧q✉❡r ❝♦♠♣❛❝t♦ ❝♦♥t✐❞♦ ❡♠ Ω✱ t❡♠♦s φ > δ >0 ❡u0 é ❧✐♠✐t❛❞❛✳

P❛r❛ ❝❛❞❛ n∈N s❡❥❛ _y_n_∈_∂_Ωt❛❧ q✉❡ _||_x_n₋_y_n_||₌_dist₍_x_n_{, ∂}_Ω)✳ ◆♦t❡ q✉❡ _y_n _−−−→

n→∞ x0✳

❈♦♠♦Ωt❡♠ ❢r♦♥t❡✐r❛ s✉❛✈❡✱ t❡♠♦s q✉❡ ♦ ✈❡t♦ryn−xn é ✉♠❛ ♥♦r♠❛❧ ❡①t❡r✐♦r ❛ ∂Ω❡♠ yn✱

❧♦❣♦ ♦ ✈❡t♦r yn−xn

||yn−xn|| ❝♦♥✈❡r❣❡ ❛♦ ✈❡t♦r ~nq✉❡ é ❛ ♥♦r♠❛❧ ❡①t❡r✐♦r ✉♥✐tár✐❛ ❛ ∂Ω ❡♠ x0✳

❈♦♠♦ u0 ∈C1(Ω)✱ t❡♠♦s q✉❡ max x∈Ω

|∇u0(x)| ≤C✱ ♦♥❞❡ C >0✱ ❧♦❣♦

φ1(xn)−φ1(yn)

||xn−yn||

= φ1(xn)

||xn−yn||

= |u0(xn)|

cn||xn−yn|| ✭✸✳✾✮

≤ C||xn−yn||

cn||xn−yn||

= C

cn

−−−→

n→∞ 0.

P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦φ1 ∈C1(Ω)✱ ❡①✐st❡♠ξn∈[0,1]t❛✐s q✉❡

lim

n→∞

φ1(xn)−φ1(yn)

||xn−yn||

= lim

n→∞

∇φ1(ξnxn+ (1−ξn)yn)·(xn−yn)

||xn−yn||

=−∇φ1(x0)·~n; ✭✸✳✶✵✮

❞❡ ✭✸✳✾✮ ❡ ❞❡ ✭✸✳✶✵✮ s❡❣✉❡ q✉❡∇φ1(x0)·~n= 0✱ ♠❛s ✐ss♦ ❝♦♥tr❛r✐❛ ♦ ▲❡♠❛ ✸✳✹ ❞❡ ❬●❚✵✶❪✳

P♦rt❛♥t♦✱ U ❞❡✈❡ s❡r ❧✐♠✐t❛❞❛ ❡♠ Ω✳

P♦❞❡♠♦s ❡♥tã♦ ❝♦♥s✐❞❡r❛rT :=T(h) = (λ−λ1)max x∈Ω

u0(x)

φ1(x)

❡ ♦❜s❡r✈❛r q✉❡ s❡ t♦♠❛r✲ ♠♦s q✉❛❧q✉❡rt > T✱ ♦❜t❡♠♦s ✭✸✳✽✮ ♦ q✉❡ ✐♠♣❧✐❝❛ut <0 ❡♠ Ω✳

(32)

(33)

❈❛♣ít✉❧♦

4

❆ s❡❣✉♥❞❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛

s✉❜❝rít✐❝♦

✹✳✶ ❆♣r❡s❡♥t❛çã♦ ❡ r❡❢♦r♠✉❧❛çã♦ ❞♦ ♣r♦❜❧❡♠❛

❊st✉❞❛♠♦s ♥❡st❡ ❝❛♣ít✉❧♦ ♦ ♣r♦❜❧❡♠❛ ❡❧í♣t✐❝♦ s✉❜❝rít✐❝♦ ❛❜♦r❞❛❞♦ ❡♠ ❬❘❙✽✻❪✳

❙❡❥❛ Ω ⊂ RN _✭_N _≥ ₂_{✮ ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❝♦♠ ❢r♦♥t❡✐r❛ s✉❛✈❡✳ ❈♦♥s✐❞❡r❛r❡♠♦s ♦}

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

  

−∆u=λu+u+p +f(x), ❡♠ Ω

u= 0, s♦❜r❡ ∂Ω. ✭

P′_✮

♦♥❞❡ p ∈ (1,2∗ ₋₁₎_{✱ s❡} _N _≥ ₃_{✱ ❡} _p _∈ ₍₁_,_∞₎_{✱ s❡} _N _{= 2}_❀ _{λ > λ}

1 é t❛❧ q✉❡ λ 6∈ σ(−∆) ❡

f =h+tφ1 ♦♥❞❡ h∈Lr(Ω)✱ ❝♦♠ r > N ❝♦♠♦ ❡♠ ✭✸✳✶✮✳

❖❜s❡r✈❡ q✉❡ ❛ ❢✉♥çã♦ r❡❛❧ ❞❡ ✈❛r✐á✈❡❧ r❡❛❧k(s) =λs+sp₊ ✐♥t❡r❛❣❡ ❝♦♠ t♦❞♦s ♦s ❛✉t♦✈❛✲

❧♦r❡s ❞♦ ▲❛♣❧❛❝✐❛♥♦ ❡①❝❡t♦ ♦s ♣r✐♠❡✐r♦s ♥♦ s❡❣✉✐♥t❡ s❡♥t✐❞♦

λ1 < λ= lim

s→−∞

k(s)

s <slim→∞

k(s)

s =∞;

t❛❧ ❢❛t♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❧✐t❡r❛t✉r❛✱ ❡♥q✉❛❞r❛ ♦ ♣r♦❜❧❡♠❛ ✭P✮ ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ♣r♦❜❧❡♠❛s

(34)

✶✻ ❈❛♣ít✉❧♦ ✹ ✖ ❆ s❡❣✉♥❞❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ s✉❜❝rít✐❝♦

❚❡♦r❡♠❛ ✹✳✶✳

❙❡ λ > λ1 ❡ λ /∈σ(−∆)✱ ❡♥tã♦ ❞❛❞❛ h∈Lr(Ω) ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ T =T(h) t❛❧ q✉❡ ♣❛r❛

t > T ♦ ♣r♦❜❧❡♠❛ ✭P′_{✮ ❛❞♠✐t❡ ♣❡❧♦ ♠❡♥♦s ❞✉❛s s♦❧✉çõ❡s ❢r❛❝❛s✳}

❆ ♣r✐♠❡✐r❛ s♦❧✉çã♦ ❢r❛❝❛ é ❛ s♦❧✉çã♦ ♥❡❣❛t✐✈❛ut ♦❜t✐❞❛ ❞♦ ❚❡♦r❡♠❛ ✸✳✶✱ ♦♥❞❡ t > T(h)

é ❞❛❞♦✳

P❛r❛ ❡♥❝♦♥tr❛r♠♦s ✉♠❛ s❡❣✉♥❞❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✭P′_{✮✱ ❝♦♥s✐❞❡r❛r❡♠♦s} _u ₌ _v ₊_u

t ❡ ♦

♣r♦❜❧❡♠❛

  

−∆v =λv+ (v+ut)p+, ❡♠ Ω

v = 0, s♦❜r❡ ∂Ω. ✭

P′

t✮

❖❜s❡r✈❡ q✉❡ u s❡rá s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✭P′✮ s❡ ❡ só s❡ v ❢♦r s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✭Pt′✮✱ ❞❡ ❢❛t♦

−∆u=λu+up₊+f ⇐⇒ −∆v−∆ut=λv+ (v+ut)p++λut+f

⇐⇒ −∆v =λv+ (v+ut)p+.

❖❜s❡r✈❡ t❛♠❜é♠ q✉❡ v = 0é ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ tr✐✈✐❛❧ ♣❛r❛ ✭P′

t✮ ❝♦rr❡s♣♦♥❞❡♥t❡ à s♦❧✉çã♦

♥❡❣❛t✐✈❛ ut ❞❡ ✭P′✮✳ ▲♦❣♦✱ ❡♥❝♦♥tr❛r ✉♠❛ s❡❣✉♥❞❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✭P′✮ é ❡q✉✐✈❛❧❡♥t❡ ❛

❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦ tr✐✈✐❛❧ ❞❡ ✭P′

t✮✳

◆♦ss♦ ♦❜❥❡t✐✈♦ ❛❣♦r❛ é ♣r♦❝✉r❛r ♣♦♥t♦s ❝rít✐❝♦s ♥ã♦ ♥✉❧♦s ❞♦ ❢✉♥❝✐♦♥❛❧ I ∈ C1₍_E,_R₎

❞❛❞♦ ♣♦r

I(v) = 1 2

Z

Ω

(|∇v|2₋_λv2₎_dx₋ 1

p+ 1

Z

Ω

(v+ut)p+1+ dx; ✭✹✳✶✮

❝✉❥❛ ❞❡r✐✈❛❞❛ é ❞❛❞❛ ♣♦r

I′(v)·ϕ =

Z

Ω

∇v· ∇ϕdx−λ

Z

Ω

vϕdx−

Z

Ω

(v+ut)p+ϕdx ∀ϕ ∈E. ✭✹✳✷✮

❯♠ t❛❧ ♣♦♥t♦ ❝rít✐❝♦✱ s❡ ❡①✐st✐r✱ s❡rá ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦ tr✐✈✐❛❧ ❞❡ ✭P′

t✮✳

✹✳✷ ❆ ❡str✉t✉r❛ ❞❡ ❡♥❧❛❝❡ ❞♦ ❢✉♥❝✐♦♥❛❧

❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❡①♣♦st♦ ♥♦ ❝❛♣ít✉❧♦ ✷✱ ✉♠❛ ❢♦r♠❛ ❞❡ ♣r♦❝✉r❛r♠♦s ♣♦♥t♦s ❝rít✐❝♦s ❞♦ ❢✉♥❝✐♦♥❛❧ I ❞❛❞♦ ♣♦r ✭✹✳✶✮ é ✈❡r✐✜❝❛r s❡ I s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ (P S) ♦✉ (P S)c ❡ ✈❡r✐✜❝❛r s❡

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✹✳✷ ❆ ❡str✉t✉r❛ ❞❡ ❡♥❧❛❝❡ ❞♦ ❢✉♥❝✐♦♥❛❧ ✶✼

◆❡st❛ s❡çã♦ ♠♦str❛r❡♠♦s q✉❡ I t❡♠ ❡str✉t✉r❛ ❞❡ ❡♥❧❛❝❡✱ ♦✉ s❡❥❛✱ ♠♦str❛r❡♠♦s q✉❡ ❡①✐s✲

t❡♠ s✉❜❝♦♥❥✉♥t♦s ❢❡❝❤❛❞♦sS ❡ Q ❞❡E q✉❡ ❡♥❧❛ç❛♠ ❡ ❝♦♥st❛♥t❡s β < α ♣❛r❛ ♦s q✉❛✐s sã♦

✈á❧✐❞❛s ❛s ❤✐♣ót❡s❡s ❞♦ ❚❊❖❘❊▼❆ ❉❊ ❊◆▲❆❈❊ ♣❛r❛ ♦ ❢✉♥❝✐♦♥❛❧I✳ P❛r❛ ✐ss♦✱ r❡❝♦r❞❛♠♦s

q✉❡λ1 < λ ❡ q✉❡ λ6∈σ(−∆)✱ ❧♦❣♦ ❞❡✈❡ ❡①✐st✐r ✉♠ ♥❛t✉r❛❧k ≥1 t❛❧ q✉❡

λk < λ < λk+1.

❈♦♥s✐❞❡r❡✱ ❡♥tã♦✱ Wk :=hφiiki=1 ❡ Xk :=Wk⊥✱ ♦✉ s❡❥❛✱ Wk é ♦ s✉❜❡s♣❛ç♦ ❞❡ E ❣❡r❛❞♦ ♣❡❧❛s

k♣r✐♠❡✐r❛s ❛✉t♦❢✉♥çõ❡s ❞♦ ▲❛♣❧❛❝✐❛♥♦ ❡Xks❡✉ ❝♦♠♣❧❡♠❡♥t♦ ♦rt♦❣♦♥❛❧ ❡♠E✳ ❖❜s❡r✈❡ q✉❡

❞❡ss❡ ♠♦❞♦ t❡♠♦s E =Wk⊕Xk✳

❖s ❧❡♠❛s ❛ s❡❣✉✐r sã♦ ❢❡rr❛♠❡♥t❛s ❡ss❡♥❝✐❛✐s ♣❛r❛ ♠♦str❛r q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ I✱ ❞❡ ❢❛t♦✱

t❡♠ ❡str✉t✉r❛ ❞❡ ❡♥❧❛❝❡✳

▲❡♠❛ ✹✳✷✳

❉❛❞♦ ε >0✱ ❡①✐st❡ e∈Xk✱ ❝♦♠ ||e||E =ε✱ t❛❧ q✉❡ ♦ ❝♦♥❥✉♥t♦

{x∈Ω;w(x) +e(x)>1,∀w∈Wk,||w||E ≤1}

t❡♠ ♠❡❞✐❞❛ ♣♦s✐t✐✈❛✳

Pr♦✈❛✳

❖❜s❡r✈❡ q✉❡ s❡ w ∈ Wk✱ ❡♥tã♦ w ∈ L∞(Ω) ✉♠❛ ✈❡③ q✉❡ φi ∈ C∞(Ω) ❡ w = Pk_i=1wiφi✳

❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ Wk é ✉♠ ❡s♣❛ç♦ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛✱ t♦❞❛s ❛s ♥♦r♠❛s sã♦ ❡q✉✐✈❛❧❡♥t❡s❀

❧♦❣♦ ❡①✐st❡M >0 t❛❧ q✉❡ ||w||∞≤M ♣❛r❛ t♦❞♦ w∈Wk ❝♦♠ ||w||E ≤1✳

P❡❧❛ ♦❜s❡r✈❛çã♦ ❆✳✸ ♥♦ ❛♣ê♥❞✐❝❡ ❆✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡Xk6⊂L∞(Ω)❀ ❧♦❣♦ ❡①✐st❡e∈Xk✱

❝✉❥❛ ♥♦r♠❛ ❡♠E ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞❛ ✐❣✉❛❧ ❛ ε✱ t❛❧ q✉❡||e||∞=∞✳ ■ss♦ ✐♠♣❧✐❝❛ q✉❡ ❞❛❞♦

C > 0✱ ❡①✐st❡ AC ⊂ Ω✱ ❝♦♠ |AC| > 0✱ t❛❧ q✉❡ e|AC > C✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❝♦♥s✐❞❡r❛♥❞♦

C=M + 1✱ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ❡♠ AM+1

e+w > M+ 1−M = 1 ♣❛r❛ t♦❞♦w∈Wk✱ ❝♦♠||w||E ≤1.

■ss♦ ♣r♦✈❛ ♦ ❧❡♠❛✳

▲❡♠❛ ✹✳✸✳

(36)

✶✽ ❈❛♣ít✉❧♦ ✹ ✖ ❆ s❡❣✉♥❞❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ s✉❜❝rít✐❝♦

[1,2∗₋₁₎_{é ❞❛❞♦✱ ❡♥tã♦ ❡①✐st❡♠} _{η, r}

0 >0 t❛✐s q✉❡

Z

Ω

w+e+ f

r

q+1

+

dx≥η >0 ✭✹✳✸✮

♣❛r❛ t♦❞♦ w∈Wk ❝♦♠ ||w||E ≤1 ❡ t♦❞♦ r ≥r0✳

Pr♦✈❛✳

P❡❧♦ ▲❡♠❛ ✹✳✷✱ ♦ ❝♦♥❥✉♥t♦

Ae={x∈Ω :w(x) +e(x)>1,∀w∈Wk,||w||E ≤1}

t❡♠ ♠❡❞✐❞❛ ♣♦s✐t✐✈❛✱ ♦✉ s❡❥❛✱ ❡①✐st❡ η > 0 t❛❧ q✉❡ |Ae| ≥ 2q+1η > 0✳ ◆♦t❡ q✉❡ η ❞❡♣❡♥❞❡

❛♣❡♥❛s ❞❡ e✳

P♦r ❤✐♣ót❡s❡✱ ❡①✐st❡ K > 0 t❛❧ q✉❡ −K ≤ f ≤ 0✳ ❉❛í✱ ♣♦r ✉♠ ❧❛❞♦ t❡♠♦s q✉❡ ♣❛r❛

q✉❛❧q✉❡r r >0

Z

Ω

w+e+f

r

q+1

+

dx≤

Z

Ω

(w+e)q+1₊ dx <∞.

P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ ❝♦♥s✐❞❡r❛r♠♦s r0 >2K✱ s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ Ae ♦❜t❡r❡♠♦s

w+e+ f

r0

≥1 + f

r0

>1−K

r0

> 1

2,

❛ss✐♠ t❡♠♦s

Z

Ω

w+e+ f

r0

q+1

+

dx≥

Z

Ae

w+e+ f

r0

q+1

+

dx≥ 1

2q+1|Ae| ≥η >0.

❈♦♠♦ ♣❛r❛ t♦❞♦r≥r0 t❡♠♦s

w+e+_rf₀

+ ≤ w+e+

f r

+✱ ♦ ❧❡♠❛ ❡stá ♣r♦✈❛❞♦✳

❈♦♠ ♦ ❛✉①í❧✐♦ ❞❛ Pr♦♣♦s✐çã♦ ❆✳✶ ❡ ❞♦s ▲❡♠❛s ✹✳✷ ❡ ✹✳✸✱ ♠♦str❛r❡♠♦s q✉❡ ❛s ❤✐♣ót❡s❡s ❞♦ ❚❊❖❘❊▼❆ ❉❊ ❊◆▲❆❈❊ ✈❛❧❡♠ ♣❛r❛ ♥♦ss♦ ❢✉♥❝✐♦♥❛❧ I✳

❘❡❝♦r❞❡♠♦s q✉❡ ❞❛❞♦ρ >0✱ Bρ={v ∈E :||v||< ρ}✳

▲❡♠❛ ✹✳✹✳

❊①✐st❡♠ ρ, α >0 t❛✐s q✉❡ s❡ Sρ=∂Bρ∩Xk✱ ❡♥tã♦

(37)

✹✳✷ ❆ ❡str✉t✉r❛ ❞❡ ❡♥❧❛❝❡ ❞♦ ❢✉♥❝✐♦♥❛❧ ✶✾

Pr♦✈❛✳

◗✉❡r❡♠♦s ♠♦str❛r q✉❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡sρ✱α >0t❛✐s q✉❡ ♣❛r❛ t♦❞♦v ∈Sρ✈❛❧❡I(v)≥α✳

❉❡ ❢❛t♦✱ s❡ v ∈ Sρ✱ ✉s❛♥❞♦ ♦ ✐t❡♠ ✭❜✮ ❞❛ Pr♦♣♦s✐çã♦ ❆✳✶ ❡ ♦ ✐t❡♠ ✭✐✮ ❞♦ ❚❊❖❘❊▼❆ ❉❊

▼❊❘●❯▲❍❖ ❉❊ ❙❖❇❖▲❊❱✱ ♦❜t❡♠♦s

I(v) = 1 2

Z

Ω

|∇v|2dx− λ

2

Z

Ω

v2dx− 1

p+ 1

Z

Ω

(v+ut)p+1+ dx

≥ 1

2

Z

Ω

|∇v|2dx− λ

2λk+1

Z

Ω

|∇v|2dx− 1

p+ 1

Z

Ω

(v)p+1₊ dx

≥ 1

2

1− λ

λk+1

Z

Ω

|∇v|2dx− 1

p+ 1

Z

Ω

|v|p+1dx

≥ 1

2

1− λ

λk+1

||v||2_E −C||v||p+1_E

=

1 2

1− λ

λk+1

−C||v||p_E−1

||v||2_E

❈♦♠♦λ∈(λk, λk+1)✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r

||v||E =ρ=

1 4C

1− λ

λk+1

1

p−1

❡ α=

1 4

1− λ

λk+1

p+1

p−1 1

C

2

p−1

❡ ♦❜t❡r♠♦s ♦ r❡s✉❧t❛❞♦ q✉❡ q✉❡rí❛♠♦s✳

▲❡♠❛ ✹✳✺✳

❊①✐st❡♠ ǫ >0✱ R > ρ_ǫ ❡ _ee∈Xk✱ ❝♦♠ ||ee||E = 1✱ t❛✐s q✉❡ s❡

QR,ǫ = (BR∩Wk)⊕ {see:s∈[0, ǫR]}, ✭✹✳✹✮

❡♥tã♦

I|∂QR,ǫ ≤0.

Pr♦✈❛✳

❊s❝♦❧❤❛ ǫ >0 ❞❡ ♠♦❞♦ q✉❡ ǫ2 _< λ

λk −1✳ ❙❡❥❛ e ∈Xk ❛ ❢✉♥çã♦ ♦❜t✐❞❛ ❞♦ ▲❡♠❛ ✹✳✷ t❛❧ q✉❡

||e||E =ǫ ❡ ❝♦♥s✐❞❡r❡ ee= e_ǫ✳

◗✉❡r❡♠♦s ♠♦str❛r q✉❡ ❡①✐st❡R > ρ_ǫ t❛❧ q✉❡ s❡ v ∈∂QR,ǫ✱ ❡♥tã♦ I(v)≤0✳

(38)

✷✵ ❈❛♣ít✉❧♦ ✹ ✖ ❆ s❡❣✉♥❞❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ s✉❜❝rít✐❝♦

♦ ❡①❡♠♣❧♦ ✷✳✹✱ t❡♠♦s q✉❡ ∂QR,ǫ =∪3i=1Γi ♦♥❞❡

Γ1 ={v ∈E :v ∈Wk, ||v||E ≤R},

Γ2 ={v ∈E :v =w+see, w∈Wk, ||w||E =R, s ∈(0, ǫR)},

Γ3 ={v ∈E :v =w+ǫRee, w ∈Wk ||w||E ≤R}.

❚❡♠♦s ❡♥tã♦ q✉❡✿

• ❙❡ v ∈Γ1✱ ✉s❛♥❞♦ ♦ ✐t❡♠ ✭❛✮ ❞❛ Pr♦♣♦s✐çã♦ ❆✳✶✱ t❡♠♦s

I(v) = 1 2

Z

Ω

|∇v|2dx−λ

2

Z

Ω

|v|2dx− 1

p+ 1

Z

Ω

(v+ut)p+1+ dx

≤ 1

2

Z

Ω

|∇v|2_dx₋λ

2

Z

Ω

|v|2_dx

≤ 1

2(λk−λ)

Z

Ω

|v|2dx;

❝♦♠♦ λk < λ✱ t❡♠♦s q✉❡

I(v)≤0 ∀v ∈Γ1.

• ❙❡ v =w+s_ee∈Γ2✱ ❞❛❞♦ R >0 t❡♠♦s

I(v) = I(w+s_ee) = 1

2

Z

Ω

|∇(w+s_ee)|2dx− λ

2

Z

Ω

|w+s_ee|2dx

− 1

p+ 1

Z

Ω

(w+s_ee+ut)p+1+ dx

≤ 1

2||w+see||

2

E−

λ

2||w+see||

2 2.

❈♦♠♦ w ❡_ee sã♦ ❢✉♥çõ❡s ♦rt♦❣♦♥❛✐s t❛♥t♦ ❡♠ E q✉❛♥t♦ ❡♠L2_(Ω)_{✱ t❡♠♦s}

I(v) ≤ 1

2||w||

2

E −

λ

2||w||

2 2+ 1 2s 2_|| e

e||2E −

λ

2s

2_||

e

e||2₂ ≤ 1

2||w||

2

E −

λ

2λk

||w||2E +

1 2ǫ

2_R2_||

e

(39)

✹✳✷ ❆ ❡str✉t✉r❛ ❞❡ ❡♥❧❛❝❡ ❞♦ ❢✉♥❝✐♦♥❛❧ ✷✶

= 1 2R

2 ₋ λ

2λk

R2+ 1 2ǫ

2_R2

= 1 2R

2₍₁₋ λ

λk

+ǫ2);

❝♦♠♦ ❡s❝♦❧❤❡♠♦s ǫ2 _< λ

λk −1✱ t❡♠♦s q✉❡ I(v)<0✳

• P♦r ✜♠✱ s❡ v =w+ǫR_ee∈Γ3✱

I(v) = I(w+ǫRe_e) = I(w+Re) = 1

2||w||

2

E−

λ

2||w||

2

2+

1 2R

2_||_e_||2

E−

λ

2R

2_||_e_||2 2

−R

p+1

p+ 1

Z

Ω

_w

R +e+ ut

R

p+1

+ dx

≤ 1

2(1−

λ λk

)||w||2_E + 1 2ǫ

2_R2₋ Rp+1

p+ 1

Z

Ω

_w

R +e+ ut R p+1 + dx ≤ 1 2ǫ

2_R2₋ Rp+1

p+ 1

Z

Ω

_w

R +e+ ut

R

p+1

+ dx.

◆♦t❡ q✉❡ w

R ∈ Wk ❡ q✉❡ || w

R||E ≤ 1✳ ❈♦♠♦ ut ∈ C(Ω) ❡ é ♥❡❣❛t✐✈❛✱ ♣❡❧♦ ❧❡♠❛ ✹✳✸

❡①✐st❡♠ R0, η >0t❛✐s q✉❡ ♣❛r❛ t♦❞♦ R≥R0 ✈❛❧❡

Z

Ω

_w

R +e+ ut

R

p+1

+ dx≥η

❛ss✐♠✱ t❡♠♦s q✉❡

I(v) ≤ 1

2R

2_ǫ2₋ η

p+ 1R

p+1_.

❈♦♠♦ ǫ > 0 ❡stá ✜①♦ ❡ p∈ (1,2∗₋₁₎_{✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r} _{R >}_max_{_R

0,ρ_ǫ} s✉✜❝✐❡♥t❡✲

♠❡♥t❡ ❣r❛♥❞❡ ❞❡ ♠♦❞♦ ❛ t❡r♠♦s I(v)≤0✳

❖❜s❡r✈❡ q✉❡ ♦s ❝♦♥❥✉♥t♦s Sρ ❡ QR,ǫ✱ ♦❜t✐❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♥♦s ▲❡♠❛s ✹✳✹ ❡ ✹✳✺ sã♦

t❛✐s ❝♦♠♦ ♦s ❝♦♥❥✉♥t♦s ❡♠ ✭✷✳✶✮ ♥♦ ❡①❡♠♣❧♦ ✷✳✹❀ ❧♦❣♦✱Sρ ❡QR,ǫ ❡♥❧❛ç❛♠✳

(40)

✷✷ ❈❛♣ít✉❧♦ ✹ ✖ ❆ s❡❣✉♥❞❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ s✉❜❝rít✐❝♦

❊◆▲❆❈❊✳ ▲♦❣♦ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ❡♠ ✉♠ ♥í✈❡❧ c≥α ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ (P S) ♣❛r❛ I✱

♦♥❞❡ α >0 ✈❡♠ ❞♦ ▲❡♠❛ ✹✳✹✳ ❆❧é♠ ❞✐ss♦✱ c♣♦❞❡ s❡r ❝❛r❛❝t❡r✐③❛❞♦ ❝♦♠♦

c= inf

h∈Γumax∈QR,ǫ

I(h(u)) ✭✹✳✺✮

♦♥❞❡ Γ ={h∈C(E, E) :h|∂QR,ǫ=Id}❀ ❧❡♠❜r❛♥❞♦ q✉❡ R, ǫ >0sã♦ ❡s❝♦❧❤✐❞♦s ❞❡ ❛❝♦r❞♦ ❝♦♠

♦ ▲❡♠❛ ✹✳✺✳

✹✳✸ ❆ ❝♦♥❞✐çã♦

(

P S

)

❙❡ ❞❡♥♦t❛r♠♦s ♣♦r {vn}n∈N ❛ s❡q✉ê♥❝✐❛ (P S) ♣❛r❛ I ♥♦ ♥í✈❡❧c ♦❜t✐❞❛ ♥❛ s❡çã♦ ✹✳✷ ♣♦r

♠❡✐♦ ❞♦ ❚❊❖❘❊▼❆ ❉❊ ❊◆▲❆❈❊✱ t❡♠♦s q✉❡ {vn}n∈N s❛t✐s❢❛③

I(vn)−−−→

n→∞ c ❡ I

′₍_v

n)−−−→

n→∞ 0.

❙❡ ♣✉❞❡r♠♦s ♠♦str❛r q✉❡ ❡①✐st❡♠v0 ∈E ❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❞❡{vn}n∈N✱ q✉❡ ❝♦♥t✐♥✉❛r❡♠♦s

❞❡♥♦t❛♥❞♦ ♣♦r vn✱ t❛✐s q✉❡ vn −−−→

n→∞ v0 ❡♠ E✱ ♣♦❞❡r❡♠♦s✱ ❡♥tã♦✱ ❞✐③❡r q✉❡ I(v0) = c ❡

I′₍_v

0) = 0❀ ✐ss♦✱ ♣♦r s✉❛ ✈❡③✱ s✐❣♥✐✜❝❛r✐❛ q✉❡ v0 é ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ♥ã♦ ♥✉❧♦ ❞♦ ❢✉♥❝✐♦♥❛❧ I✱

♦ q✉❡ ❡q✉✐✈❛❧❡r✐❛ ❛ ❞✐③❡r q✉❡ v0 é ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦ tr✐✈✐❛❧ ❞♦ ♣r♦❜❧❡♠❛ ✭Pt′✮✳

◆❡st❛ s❡çã♦ ♠♦str❛r❡♠♦s q✉❡ ♦ ❢✉♥❝✐♦♥❛❧I ❞❛❞♦ ♣♦r ✭✹✳✶✮ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ (P S)✳

▲❡♠❛ ✹✳✻✳

❖ ❢✉♥❝✐♦♥❛❧ I :E →R ❞❡✜♥✐❞♦ ❡♠ ✭✹✳✶✮ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✭P❙✮✳

Pr♦✈❛✳

❉❛❞♦ c∈R✱ s❡❥❛ _{_v_n_}_n_∈N ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ E s❛t✐s❢❛③❡♥❞♦

I(vn)−−−→

n→∞ c ❡ I

′

(vn)−−−→

n→∞ 0.

Pr✐♠❡✐r♦✱ ♠♦str❛r❡♠♦s q✉❡ (vn) é ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❡♠ E❀ ♣❛r❛ ✐ss♦✱ s✉♣♦r❡♠♦s

q✉❡ ||vn||E −−−→

n→∞ ∞ ❡ ❜✉s❝❛r❡♠♦s ❝❤❡❣❛r ♥✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❉❡ ✭✹✳✷✮✱ ♣❛r❛ ❝❛❞❛n ∈N✱ t❡♠♦s

I′(vn)·vn =

Z

Ω

|∇vn|2dx−λ

Z

Ω

v_n2dx−

Z

Ω

(vn+ut)p+vndx

=

Z

Ω

|∇vn|2dx−λ

Z

Ω

vn2dx−

Z

Ω

(vn+ut)p+1+ dx+

Z

Ω

(41)

✹✳✸ ❆ ❝♦♥❞✐çã♦(P S) ✷✸

❉✐ss♦ ❡ ❞❡(4.1)♦❜t❡♠♦s

I(vn)−

1 2I

′₍_v

n)·vn=

1 2−

1

p+ 1

Z

Ω

(vn+ut)p+1+ dx+

1 2

Z

Ω

(vn+ut)p+(−ut)dx.

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

❙❡❥❛M > 0 t❛❧ q✉❡ |I(vn)| ≤M✳ P❛r❛n s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ t❡♠♦s

|I(vn)−

1 2I

′₍_v

n)·vn| ≤M +

1 2kI

′₍_v

n)kE′kvnkE ≤M +kvnkE

❛ss✐♠

1 2−

1

p+ 1

Z

Ω

(vn+ut)+p+1dx≤ |I(vn)−

1 2I

′₍_v

n)·vn| ≤M +||vn||E.

❙❡❥❛ 1 C = 1 2 − 1 p+1

❡ ♠✉❧t✐♣❧✐q✉❡♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ♣♦r C

||vn|| p+1 p E ♣❛r❛ ♦❜t❡r♠♦s Z Ω

(vn+ut)p+

||vn||E

p+1

p

dx≤ CM

||vn|| p+1

p E

+ C

||vn|| 1

p E

.

❈♦♠♦ ❡st❛♠♦s ❛ss✉♠✐♥❞♦ q✉❡||vn||E −−−→

n→∞ ∞✱ t❡♠♦s q✉❡

(vn+ut)p₊

||vn||E −−−→_n_→∞ 0❡♠ L p+1

p (Ω)✳

❉❛❞♦ϕ∈E✱ ♣❡❧♦ ✐t❡♠ ✭✐✮ ❞♦ ❚❊❖❘❊▼❆ ❉❊ ▼❊❘●❯▲❍❖ ❉❊ ❙❖❇❖▲❊❱✱ t❡♠♦s q✉❡ ϕ∈Lp+1_(Ω)_{❀ s❡♥❞♦ ❛ss✐♠✱ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✱ ♦❜t❡♠♦s}

Z Ω

(vn+ut)p+

||vn||E

ϕ dx ≤ "Z Ω

(vn+ut)p+

||vn||E

p+1

p

dx

# p p+1

||ϕ||p+1

≤



 CM

||vn|| p+1

p E

+ C

||vn|| 1 p E   p p+1

||ϕ||p+1 −−−→

n→∞ 0 ✭✹✳✻✮

❡ ✐ss♦ ♥♦s ♠♦str❛ q✉❡ (vn+ut)p₊

||vn||E −−−→_n_→∞ 0❡♠ E

′ ₌_H−1_✳

❆❣♦r❛ ♦❜s❡r✈❡ q✉❡ ♣❛r❛ ❝❛❞❛ϕ ∈E✱ t❡♠♦s

Z Ω

∇vn· ∇ϕdx−λ

Z

Ω

vnϕdx−

Z

Ω

(vn+ut) p

+ϕdx