Multiplicidade de soluções positivas de uma equação de Schrödinger não linear

❙❊❘❱■➬❖ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❉❖ ■❈▼❈✲❯❙P ❉❛t❛ ❞❡ ❉❡♣ós✐t♦✿ ✶✺✴✵✶✴✷✵✶✵

❆ss✐♥❛t✉r❛✿

▼✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ♣❛r❛ ✉♠❛

❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ♥ã♦ ❧✐♥❡❛r

▼♦r❡♥♦ P❡r❡✐r❛ ❇♦♥✉tt✐✶

❖r✐❡♥t❛❞♦r✿ ❙ér❣✐♦ ❍❡♥r✐q✉❡ ▼♦♥❛r✐ ❙♦❛r❡s

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ▼❛t❡♠át✐❝❛s ❡ ❞❡ ❈♦♠♣✉t❛çã♦ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❯❙P ✲ ❙ã♦ ❈❛r❧♦s ❏❛♥❡✐r♦✴✷✵✶✵

✏❯♠ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦ ❧❡✈❛ ✈♦❝ê ❞❡ ❆ ❛ ❇✳ ■♠❛❣✐♥❛çã♦ ❧❡✈❛ ✈♦❝ê ❛ q✉❛❧q✉❡r ❧✉❣❛r q✉❡ ✈♦❝ê q✉✐s❡r✳✑

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s ♣♦r t❡r ♠❡ ❞❛❞♦ ❢♦rç❛ ♣❛r❛ ❡♥❢r❡♥t❛r ❡ss❛ ❥♦r♥❛❞❛✳ ❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❙ér❣✐♦ ❍❡♥r✐q✉❡ ▼♦♥❛r✐ ❙♦❛r❡s✱ ✉♠ ♣r♦❢❡ss♦r ♠✉✐t♦ sá❜✐♦ ❡ ✐♥t❡❧✐❣❡♥t❡✱ ♣❡❧♦s ót✐♠♦s ❝♦♥s❡❧❤♦s ❡ ♣❡❧❛ ✐♠❡♥s❛ ♣❛❝✐ê♥❝✐❛✳ ❙❡♠ ❛ ❛❥✉❞❛ ❞♦ ♣r♦❢❡ss♦r ♥ã♦ t❡r✐❛ ❝♦♥s❡❣✉✐❞♦ ✈❡♥❝❡r ❡ss❡ ❞❡s❛✜♦✳

❖❜r✐❣❛❞♦ ❛ ❈❆P❊❙ ♣❡❧♦ ✜♥❛♥❝✐❛♠❡♥t♦ ❞♦s ♠❡✉s ❡st✉❞♦s✳

❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛♦s ♠❡✉ ❛♠✐❣♦s ❞❡ ♠❡str❛❞♦ q✉❡ ♠❡ ❛❥✉❞❛r❛♠ ♠✉✐t♦ ❞✉r❛♥t❡ t♦❞♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❛❣r❛❞❡ç♦ ♠✉✐t♦ ❛♦ ▼❛t❤❡✉s ❈✳ ❇♦rt♦❧❛♥ ❡ ❛ ❏✉❧✐❛♥❛ ❚✳ ❞❡ ▲✐♠❛ ♣♦r s❡r❡♠ ♠❡✉s ❛♠✐❣♦s ❞❡ t♦❞❛s ❛s ❤♦r❛s ❡ ♠❡✉s ♣r♦❢❡ss♦r❡s✱ ♥❛s ✐♥t❡r♠✐♥á✈❡✐s ❤♦r❛s ❞❡ ❡st✉❞♦✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛♦ ♠❡✉ ❛♠✐❣♦ ❘❛❢❛❡❧ ❘♦ss❛t♦✱ q✉❡ s❡♠♣r❡ ❛rr✉♠❛✈❛ ✉♠ t❡♠♣♦ ❡ ✧♠❛t❛✈❛✧ ❤♦r❛s ❞❡ ❡st✉❞♦s ♣❛r❛ ❝♦♥✈❡rs❛r s♦❜r❡ q✉❛❧q✉❡r ❛ss✉♥t♦ q✉❡ ♥ã♦ ❢♦ss❡ ❛ ♠❛t❡♠át✐❝❛✳

◆ã♦ ♣♦ss♦ ❡sq✉❡❝❡r ❞♦s ♠❡✉s ❣r❛♥❞❡s ❛♠✐❣♦s ❞❛ ❘❡♣ú❜❧✐❝❛ ■♥✜❧tr❛❞♦s✱ ▼♦sq✉✐t♦✱ ◆❛♥❞♦✱ ❋❛r♦❢❛✱ ❈❛①❛ ❡ ▲é♦ q✉❡ ♠❡ ❤♦s♣❡❞❛r❛♠ ❡ ♠❡ ❛❥✉❞❛r❛♠ ❛ ❞❡s❝♦♥tr❛✐r ❝♦♠ ♠✉✐t♦s ❝❤✉rr❛s❝♦s ❡ s❛♠❜❛s✳

❆❣r❛❞❡ç♦ ❛ ◆❛t❤❛❧✐❛ ❘♦ss❡tt✐✱ ♠✐♥❤❛ ❣r❛♥❞❡ ❝♦♠♣❛♥❤❡✐r❛ ❡ ♠❡✉ ❣r❛♥❞❡ ❛♠♦r✱ ♣♦r t❡r ♠❡ ❛❥✉❞❛❞♦ ❡♠ t♦❞❛s ❛s ❤♦r❛s ❞✐❢í❝❡✐s s❡♠ ❡❧❛ ❡✉ ♥ã♦ ❝♦♥s❡❣✉✐r✐❛ r❡❛❧✐③❛r ❡st❡ tr❛❜❛❧❤♦✳ ❖❜r✐❣❛❞♦ ♣♦r ♠❡ ❢❛③❡r ❢❡❧✐③✳

P♦r ✜♠ s♦✉ ❣r❛t♦ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛ ♣♦r t❡r ♠❡ ❞❛❞♦ ✉♠❛ ót✐♠❛ ❡❞✉❝❛çã♦ ❡ t❡r ♠❡ ❛❥✉❞❛❞♦ ❛ ♠❛♥t❡r ♦s ♠❡✉s ❡st✉❞♦s✳ ❱♦❝ês sã♦ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡s ♥❛ ♠✐♥❤❛ ✈✐❞❛✳

❘❡s✉♠♦

❊st❡ tr❛❜❛❧❤♦ é ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r

−∆u+ (λa(x) + 1)u=up, u >0 ❡♠ RN_{, (∗)}

♦♥❞❡a_≥0é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡p >1é ✉♠ ❡①♣♦❡♥t❡ s✉❜❝rít✐❝♦✳ ▼ét♦❞♦s ❱❛r✐❛❝✐♦♥❛✐s

sã♦ ❡♠♣r❡❣❛❞♦s ♣❛r❛ ♠♦str❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ λn → +∞ ❡ ❞❛ r❡s♣❡❝t✐✈❛ s❡q✉ê♥❝✐❛ ❞❡ s♦❧✉çõ❡s uλn ❝♦♥✈❡r❣✐♥❞♦ ♣❛r❛ ✉♠❛ s♦❧✉çã♦ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ❞♦ ♣r♦❜❧❡♠❛

❞❡ ❉✐r✐❝❤❧❡t

−∆u+u=up, u >0❡♠ Ω, u= 0 s♦❜r❡ ∂Ω,

❆❜str❛❝t

❚❤✐s ✇♦r❦ ✐s ❞❡✈♦t❡❞ t♦ st✉❞② t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥

−∆u+ (λa(x) + 1)u=up, u >0 ✐♥ RN_{, (∗)}

✇❤❡r❡ a ✐s ❛ ♥♦♥♥❡❣❛t✐✈❡ ❛♥❞ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ❛♥❞ p > 1 ✐s ❛ s✉❜❝r✐t✐❝❛❧ ❡①♣♦♥❡♥t✳

❱❛r✐❛t✐♦♥❛❧ ♠❡t❤♦❞s ❛r❡ ❡♠♣❧♦②❡❞ ✐♥ ♦r❞❡r t♦ s❤♦✇ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ s❡q✉❡♥❝❡λn→+∞ ❛♥❞ t❤❡ r❡s♣❡❝t✐✈❡ s❡q✉❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s ❝♦♥✈❡r❣✐♥❣ ✐♥ H1_(RN_{) t♦ ❛ ❧❡❛st ❡♥❡r❣② s♦❧✉t✐♦♥} ♦❢ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠

−∆u+u=up_{, u >0✐♥ Ω, u= 0 ♦♥∂Ω,}

✇❤❡r❡ Ω := inta−1_{(0)✳ ❋✉rt❤❡r♠♦r❡✱ ✐t ✐s st✉❞✐❡❞ t❤❡ ❡✛❡❝t ♦❢ t❤❡ t♦♣♦❧♦❣② ♦❢ t❤❡ s❡t}

Ω ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ ❡q✉❛t✐♦♥ (_∗) ❜② ✉s✐♥❣ t❤❡ ▲✉st❡r♥✐❦ ❛♥❞

❮♥❞✐❝❡

✶ ■♥tr♦❞✉çã♦ ✶

✷ ▼ú❧t✐♣❧❛s s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ✺

✷✳✶ ❈♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ❞♦ ❢✉♥❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✷ Pr♦✈❛ ❞♦s ❘❡s✉❧t❛❞♦s Pr✐♥❝✐♣❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

❆ ❆♣ê♥❞✐❝❡ ❆ ✷✾

❇ ❆♣ê♥❞✐❝❡ ❇ ✸✾

❈ ❆♣ê♥❞✐❝❡ ❈ ✹✶

❈❛♣ít✉❧♦

1

■♥tr♦❞✉çã♦

◆❡st❡ tr❛❜❛❧❤♦ t❡♠♦s ❝♦♠♦ ♦❜❥❡t✐✈♦ ❡st✉❞❛r ❛ ❡①✐stê♥❝✐❛ ❡ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ♣❛r❛ ❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦ ❞❡ ❙❝❤ö♥❞✐♥❣❡r ♥ã♦ ❧✐♥❡❛r✿

−∆u+ (λa(x) + 1)u=up, u > 0❡♠ RN_. ✭_Sλ,p✮

❆s s♦❧✉çõ❡s ♣♦s✐t✐✈❛s q✉❡ ❡♥❝♦♥tr❛r❡♠♦s ❞❡♣❡♥❞❡rã♦ ❞❡ λ✱ p❡ ❞♦ ♣♦t❡♥❝✐❛❧

bλ(x) = λa(x) + 1✳ ■r❡♠♦s tr❛❜❛❧❤❛r ❝♦♠ N > 3 ❡ ❝♦♠ ♦ ♣r♦❜❧❡♠❛ s✉❜❝rít✐❝♦✱ ✐st♦ é✱

1 < p < 2∗ ₋1✱ s❡♥❞♦ 2∗ = 2N/(N ₋2), N _≥ 3✳ ❖ ♣♦t❡♥❝✐❛❧ bλ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ❤✐♣ót❡s❡s✿

✭❆✶✮ a _∈ C(RN_,R₎✱ s❛t✐s❢❛③ _a > 0 ❡ Ω := int_{a−1_{(0)} é ♥ã♦ ✈❛③✐♦ ❝♦♠ ❢r♦♥t❡✐r❛ s✉❛✈❡✱} ❝♦♠ Ω = a−1_(0)✳

✭❆✷✮ ❊①✐st❡ M0 >0 t❛❧ q✉❡

µ(_{x_∈RN _:a(x)6M_0})<_∞,

♦♥❞❡ µ ❞❡♥♦t❛ ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❡♠ RN

P❛r❛ ❞❡s❝r❡✈❡r ♦s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s q✉❡ s❡rã♦ ❛q✉✐ ❛❜♦r❞❛❞♦s✱ ❝♦♥s✐❞❡r❡ ♦ ❢✉♥❝✐♦♥❛❧

Φλ.p(u) =

1 2

Z

RN

[_|∇u_|2_{+ (λa(x) + 1)u}2_]dx− 1

p+ 1

Z

RN|

u_|p+1_dx

❆ss♦❝✐❛❞♦ à ❡q✉❛çã♦ ✭Sλ,p✮✱ ♦ q✉❛❧ ❡stá ❞❡✜♥✐❞♦ ❡♠

E =_{u_∈H1(RN_{) :}

Z

RN

a(x)u2dx <+_∞}.

✷ ❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖

❉❡✜♥❛ ❛ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐

Mλ,p =

u_∈E_\{0_}:

Z

RN |∇

u_|2+ (λa(x) + 1)u2

dx=

Z

RN |

u_|p+1dx

❡ s❡❥❛

cλ,p = inf u∈Mλ,p

Φλ,p(u).

❊♠ ❬❇❲✵✵❪✱ ❇❛rts❝❤ ❡ ❲❛♥❣ ♣r♦✈❛r❛♠ ♦s s❡❣✉✐♥t❡s t❡♦r❡♠❛s✿

❚❡♦r❡♠❛ ✶✳✵✳✶✳ ❙✉♣♦♥❤❛ q✉❡ ❛s ❤✐♣ót❡s❡s ✭❆✶✮ ❡ ✭❆✷✮ ❡stã♦ s❛t✐s❢❡✐t❛s✳ ❊♥tã♦ ♣❛r❛ λ

s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ ♦ í♥✜♠♦ é ❛t✐♥❣✐❞♦✱ ❡ ❛ss✐♠ ✭Sλ,p✮ ♣♦ss✉✐ ✉♠❛ s♦❧✉çã♦ ❞❡ ♠❡♥♦r

❡♥❡r❣✐❛uλ,p ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ♥í✈❡❧cλ,p✳ ❆❧é♠ ❞✐ss♦✱ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛λn→+∞♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ t❛❧ q✉❡ uλn ❝♦♥✈❡r❣❡ ❡♠ H

1_(RN_{)✱ ❛ ♠❡♥♦s ❞❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱ ♣❛r❛} ✉♠❛ s♦❧✉çã♦ ❞❡ ♠❡♥♦r ❡♥❡r❣✐❛ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❉✐r✐❝❤❧❡t✱

−∆u+u=up, u >0 ❡♠ Ω. ✭Dp✮

❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ ♠♦str❛ ♦ ❡❢❡✐t♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞❡ Ω s♦❜r❡ ♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s

♣♦s✐t✐✈❛s ❞❡ ✭Sλ,p✮✳

❚❡♦r❡♠❛ ✶✳✵✳✷✳ ❙✉♣♦♥❤❛ q✉❡ ✭❆✶✮ ❡ ✭❆✷✮ ❡stã♦ s❛t✐s❢❡✐t❛s ❡ Ω é ❧✐♠✐t❛❞♦✳ ❊♥tã♦

❡①✐st❡♠ p0 <2∗ −1 ❡ ✉♠❛ ❢✉♥çã♦ Λ : (p0,2∗−1)→ R t❛❧ q✉❡ ✭Sλ,p✮ ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s

cat(Ω)✶ s♦❧✉çõ❡s ❞✐st✐♥t❛s ♣❛r❛ t♦❞♦ λ >Λ(p)✳

❈♦♠♦ ♥♦ ❝❛s♦ ❞❡ s♦❧✉çõ❡s ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛✱ ❛s s♦❧✉çõ❡s ♦❜t✐❞❛s ♣❡❧♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r ❝♦♥✈❡r❣❡♠ ♣❛r❛ ✉♠❛ s♦❧✉çã♦ ❞❡ ❡♥❡r❣✐❛ ♠í♥✐♠❛ ❞❡ ✭Dp✮✳

❚❡♦r❡♠❛ ✶✳✵✳✸✳ ❙✉♣♦♥❤❛ q✉❡ ✭❆✶✮ ❡ ✭❆✷✮ ❡stã♦ s❛t✐s❢❡✐t❛s✳ ❙❡❥❛ un ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❞❡ ✭Sλ,p✮✱ ❝♦♠ λn → +∞ ❡ lim supn→+∞Φλn,p(un) < ∞✳ ❊♥tã♦✱ ❛ ♠❡♥♦s ❞❡

✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱ un ❝♦♥✈❡r❣❡ ❢♦rt❡♠❡♥t❡✱ ❡♠ H1(RN) ♣❛r❛ ✉♠❛ s♦❧✉çã♦ ❞❡ ✭Dp✮✳ ❍á ✈ár✐♦s tr❛❜❛❧❤♦s q✉❡ ❡st✉❞❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ♣❛r❛ ♣r♦❜❧❡♠❛s r❡❧❛❝✐♦♥❛❞♦s à ❡q✉❛çã♦ ✭Sλ,p✮✳ P♦r ❡①❡♠♣❧♦✱ ❡♠ ❬❆❞▼❋❙✵✾❪✱ ❆❧✈❡s ❡t ❛❧❧ ❡st✉❞❛r❛♠ ♦

❝❛s♦ ❝♦♠ ❝r❡s❝✐♠❡♥t♦ ❝rít✐❝♦ ♣❛r❛ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛

−∆u+ (λV(x) +Z(x))u=βuq+u2∗−1✱ ♣❛r❛ u >0❡♠ RN_,

♦♥❞❡ λ, β _∈ (0,+_∞)✱ q _∈ (1,2∗ _{−1)✱ N > 3 ❡ V, Z : R}N _{→ R sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s} s❛t✐s❢❛③❡♥❞♦ ❛❧❣✉♠❛s ❤✐♣ót❡s❡s✳

✸

❊♠ ❬❆❙✵✽❪ ❆❧✈❡s ❡ ❙♦✉t♦ t❛♠❜é♠ r❡❛❧✐③❛r❛♠ ♦ ❡st✉❞♦ ❞❛s s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ❝♦♠ ❝r❡s❝✐♠❡♥t♦ ❝rít✐❝♦ ❡①♣♦♥❡♥❝✐❛❧ ♥♦ ❝❛s♦ N =R2 ♣❛r❛ ❛ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s

 



−∆u+ (λV(x) +Z(x))u=f(u), ❡♠ R2_,

u >0, ❡♠ R2_,

u_∈H1_(R2_),

♦♥❞❡f é ❞❡ ❝❧❛ss❡C1_{❝♦♠ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ❝rít✐❝♦✱λ∈(1,+∞)✱V, Z :R}2 _→Rsã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❝♦♠V(x)>0♣❛r❛ t♦❞♦x_∈R2 ❡_{Ω =intV}−1₍₀₎♣♦ss✉✐_k ❝♦♠♣♦♥❡♥t❡s

❝♦♥❡①❛s ❞❡♥♦t❛❞❛s ♣♦r Ωj✱ j _{∈ {}1,2, . . . k_}✱V−1_{({0}= Ω ❡ ∂Ω é s✉❛✈❡✳}

❖✉tr♦ tr❛❜❛❧❤♦ ✐♥t❡r❡ss❛♥t❡ é ❬❞❉✵✷❪✱ ♥♦ q✉❛❧ ❉❡ ❋✐❣✉❡✐r❡❞♦ ❡ ❉✐♥❣ ❡st✉❞❛r❛♠ ❛ ❡①✐stê♥❝✐❛ ❡ ❛ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❛s s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛

−∆u+ (λa(x) +a0(x))u=g(x, u)

u_∈H1_(RN₎

♣❛r❛ ♦ ❝❛s♦ s✉❜❝rít✐❝♦ ❡ ❡st✉❞❛r❛♠ ❛ ❡①✐stê♥❝✐❛ ❡ ❛ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛ ❝♦♠ ❝r❡s❝✐♠❡♥t♦ ❝rít✐❝♦

−∆u+ (λa(x) +a0(x))u=K(x)|u|2

∗₋₂

u+g(x, u)

u_∈H1_(RN_).

❊♠ ❬❉❙✵✼❪✱ ❉✐♥❣ ❡ ❙③✉❧❦✐♥ ❡st✉❞❛r❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r s❡♠✐❧✐♥❡❛r

−ǫ2∆u+V(x)u=g(x, u),

❝♦♠ ǫ > o ♣❡q✉❡♥♦ ❡

−∆u+λV(x)u=g(x, u),

❝♦♠ λ >0 ❣r❛♥❞❡✱ ♦♥❞❡ ♦ ♣♦t❡♥❝✐❛❧V ♣♦❞❡ ♠✉❞❛r ❞❡ s✐♥❛❧✳

❊♠ ❬❈❲✵✼❪✱ ❈❤❛❜r♦✇s❦✐ ❡ ❲❛♥❣ ❡st✉❞❛r❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❞♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ❞❡ ◆❡✉♠❛♥♥

−∆u+b(x)u=Q(x)_|u_|p−2_{u ❡♠ Ω}c ∂

∂νu(x) = 0 ❡♠ ∂Ω, u >0 ❡♠ Ω

c_.

◆❡st❡ tr❛❜❛❧❤♦✱ ♠♦t✐✈❛❞♦s ♣♦r ❬❇❲✵✵❪✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛s ❞❡♠♦♥str❛çõ❡s ❞♦s t❡♦r❡♠❛s ✶✳✵✳✶ ✲ ✶✳✵✳✸✳ P❛r❛ t❛♥t♦✱ ❞✐✈✐❞✐♠♦s ❡st❡ tr❛❜❛❧❤♦ ❡♠ ✉♠ ❝❛♣ít✉❧♦ ❡ três ❛♣ê♥❞✐❝❡s✳ ❖ ❈❛♣ít✉❧♦ ✷ é r❡s❡r✈❛❞♦ ♣❛r❛ ❡st✉❞❛r ❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ❞♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ✭Sλ,p✮ ❡ ♣r♦✈❛r ♦s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s ❞❡st❡ tr❛❜❛❧❤♦ s❡❣✉✐♥❞♦ ♦s ❛r❣✉♠❡♥t♦s

❈❛♣ít✉❧♦

2

▼ú❧t✐♣❧❛s s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ♣❛r❛ ❛

❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ♥ã♦ ❧✐♥❡❛r

✷✳✶ ❈♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ❞♦ ❢✉♥❝✐♦♥❛❧

◆❡st❛ s❡çã♦✱ ❝♦♥s✐❞❡r❛r❡♠♦s ✉♠❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ♣❛r❛ ♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ à ❡q✉❛çã♦

∆u+ (λa(x) + 1)u=f(x, u) ❡♠ RN_. ✭_Sλ✮

◆♦ q✉❡ s❡❣✉❡✱ ❛❞♠✐t✐r❡♠♦s q✉❡ ❛ ❢✉♥çã♦ f : RN _×R _→ R s❛t✐s❢❛ç❛ ❛s s❡❣✉✐♥t❡s

❤✐♣ót❡s❡s✿

✭❆✸✮ f _{∈ C}1_(RN _{×R,R)s❛t✐s❢❛③ f(x, u) =o(|u|)q✉❛♥❞♦ u→0 ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ x❀}

✭❆✹✮ ❊①✐st❡♠ ❝♦♥st❛♥t❡s a1, a2 >0 e 1< s <2∗−1t❛✐s q✉❡

|f(x, u)_|6a₁(1 +|u|s) ❡|f′(x, u)|6a₂(1 +|u|s−1) ♣❛r❛ t♦❞♦x∈RN_{, u∈}R_;

✭❆✺✮ ❊①✐st❡ q >2t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ x_∈RN✱_u∈R_\{0}✱ t❡♠♦s

0< qF(x, u)_≡q

Z u

0

f(x, v)dv6uf(x, u).

❈♦♥s✐❞❡r❡♠♦s ♦ ❝♦♥❥✉♥t♦E =

u_∈H1_(RN_{) :}

Z

RN

a(x)u2 dx <_∞

♠✉♥✐❞♦ ❞❛ ♥♦r♠❛

✻ ❈❆P❮❚❯▲❖ ✷✳ ▼Ú▲❚■P▲❆❙ ❙❖▲❯➬Õ❊❙ P❖❙■❚■❱❆❙

ku_k2_λ =

Z

RN

(_|∇u_|2+ (λa(x) + 1)u2)dx♣❛r❛ λ >0.

❖❜s❡r✈❡♠♦s q✉❡ ❛s ♥♦r♠❛s k · kλ ♣❛r❛ λ > 0 sã♦ ❡q✉✐✈❛❧❡♥t❡s ❡♠ E ❡ s❛❜❡♠♦s q✉❡ ❛s s♦❧✉çõ❡s ❞❡ ✭Sλ✮ sã♦ ♣♦♥t♦s ❝rít✐❝♦s ❞♦ ❢✉♥❝✐♦♥❛❧ Φλ :E →R✱ ♦♥❞❡

Φλ(u) =

1 2kuk

2 λ−

Z

RN

F(x, u)dx.

❖❜s❡r✈❡ t❛♠❜é♠ q✉❡ ♣♦r ✭❆✸✮ ❡ ✭❆✹✮ ♦ ❢✉♥❝✐♦♥❛❧Φλ ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ❡Φλ ∈C1(E,R) ♣❛r❛ t♦❞♦ λ>0 ✭✈❡r ✭❆✳✵✳✶✺✮✱ ✭❆✳✵✳✶✽✮ ❡ ✭❆✳✵✳✶✾✮ ♥♦ ❆♣ê♥❞✐❝❡ ❆✮✳

❆ s❡❣✉✐r ❡♥✉♥❝✐❛r❡♠♦s ♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡st❛ s❡çã♦✳

Pr♦♣♦s✐çã♦ ✷✳✶✳✶✳ ❙✉♣♦♥❤❛ q✉❡ ❛s ❤✐♣ót❡s❡s ✭❆✶✮✲✭❆✺✮ ❡st❡❥❛♠ s❛t✐s❢❡✐t❛s✳ ❊♥tã♦ ♣❛r❛ t♦❞♦ C0 >0 ❡①✐st❡ Λ0 >0 t❛❧ q✉❡ Φλ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ (P S)c ♣❛r❛ t♦❞♦ λ > Λ0 ❡ ♣❛r❛ t♦❞♦ c6C0✳

❆ ✜♠ ❞❡ ♣r♦✈❛r♠♦s ❛ Pr♦♣♦s✐çã♦ ✷✳✶✳✶✱ ❞❡♠♦♥str❛r❡♠♦s ❛❧❣✉♥s ❧❡♠❛s✳

▲❡♠❛ ✷✳✶✳✷✳ ❙❡❥❛ Kλ ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦s ♣♦♥t♦s ❝rít✐❝♦s ❞❡Φλ✳ ❊♥tã♦ ❡①✐st❡ σ >0 ✭✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ λ✮ t❛❧ q✉❡ _ku_kλ >kukH1 >σ ♣❛r❛ t♦❞♦ u∈K_λ\{0}✳

❉❡♠♦♥str❛çã♦✿ ❞❛ ❤✐♣ót❡s❡ ✭❆✸✮✱ ❞❛❞♦ǫ >0✱ ❡①✐st❡u0 =u0(ǫ)t❛❧ q✉❡|u|< u0✳ ❊♥tã♦

|f(x, u)_|6ǫ_|u_|

❡ ♣♦rt❛♥t♦

|f(x, u)u_|6ǫ|u|2.

P♦r ✭❆✹✮✱ ♣❛r❛ t♦❞♦ |u_|>u0✱

f(x, u)u6a1

1

|u_|s + 1

|u_|s+1 6a1

1

us 0

+ 1

|u_|s+1.

❉❡♥♦t❛♥❞♦ A(ǫ) =a1

1 us

0 + 1

✱ s❡❣✉❡ q✉❡

f(x, u)6ǫu2 +A(ǫ)_|u_|s+1✱ ♣❛r❛ t♦❞♦ x_∈RN_{, u∈}R_. ✭✷✳✶✮

P♦r ✜♠✱ ♣❛r❛ u_∈Kλ\{0} t❡♠♦s Φ′λ(u)(u) = 0✱ ♦✉ s❡❥❛✱

0 = Φ′_λ(u)(u) = _ku_k2_λ−

Z

RN

f(x, u)udx >_ku_k2_{λ −}

Z

RN

✷✳✶✳ ❈❖◆❉■➬➹❖ ❉❊ ❈❖▼P❆❈■❉❆❉❊ ❉❖ ❋❯◆❈■❖◆❆▲ ✼

❖❜s❡r✈❡ q✉❡

Z

RN

(ǫ_|u_|2+A(ǫ)_|u_|s+1)dx=ǫ_ku_kL22₍RN₎+A(ǫ)kuks+1_Ls+1₍RN₎.

❯s❛♥❞♦ ❛s ✐♠❡rsõ❡s ❞❡ ❙♦❜♦❧❡✈

0>_ku_k2_λ−ǫ_ku_k2_L2₍RN₎−A(ǫ)kuks+1_Ls+1₍RN₎

>_ku_k2 λ−C

h

ǫ_ku_k2

H1₍RN₎+A(ǫ)kuks+1_H1₍RN₎

i

>_ku_k2_H1₍RN₎−Cǫkuk2_H1₍RN₎−CA(ǫ)kuks+1_H1₍RN₎,

✉♠❛ ✈❡③ q✉❡ ku_k2

λ >kuk2H1₍RN₎✳ ❆ss✐♠

ku_k2_H1₍RN₎

1₋Cǫ₋K_ku_kHs−11₍RN₎

60✱ ♦♥❞❡ K =CA(ǫ).

❆✐♥❞❛✱ ❝♦♠♦ ku_k2

H1₍RN₎>0✱

ku_kH1₍RN₎>

1₋Cǫ K

_s−1₁

.

=σ.

▲❡♠❛ ✷✳✶✳✸✳ ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ c0 > 0✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ λ✱ t❛❧ q✉❡ s❡ (un) é ✉♠❛ s❡q✉ê♥❝✐❛ (P S)c ❞❡ Φλ✱ ❡♥tã♦

✭✐✮ lim sup

n→+∞ k

u_nk2 λ 6

2qc q₋2;

✭✐✐✮ c>c0 ♦✉ c= 0✳ ❉❡♠♦♥str❛çã♦✿ ❞❡ Φ′

λ(un) → 0 ❡ Φλ(un) → c s❡❣✉❡ q✉❡ ♣❛r❛ ǫ > 0 ❞❛❞♦✱ ❡①✐st❡♠

n0 ∈N ❡ ❝♦♥st❛♥t❡ C >0 t❛❧ q✉❡

Φλ(un)−

1

qΦ

′

λ(un)(un)6c+ǫ+Ckunkλ.

P♦r ♦✉tr♦ ❧❛❞♦✱ ♣♦r ✭❆✺✮✱

Φλ(un)−

1

qΦ

′

λ(un)(un) =

q₋2 2q kunk

2 λ+ Z RN 1

qf(x, un)un−F(x, un)

dx

> q−2

2q kunk

2 λ.

✭✷✳✷✮

P♦rt❛♥t♦✱

q₋2 2q kunk

2

✽ ❈❆P❮❚❯▲❖ ✷✳ ▼Ú▲❚■P▲❆❙ ❙❖▲❯➬Õ❊❙ P❖❙■❚■❱❆❙

❧♦❣♦✱ (un) é ❧✐♠✐t❛❞♦ ❡♠ E✳ ▼❛✐s ❛✐♥❞❛✱ ♣♦r ✭✷✳✷✮ ❡♥❝♦♥tr❛♠♦s

q₋2 2q kunk

2

λ 6Φλ(un)−

1

qΦ

′

λ(un)(un)6Φλ(un) +

1

qkΦ

′

λ(un)kkunkλ.

P♦rt❛♥t♦ ❞❛ ❧✐♠✐t❛çã♦ ❞❡ (un)❡♠ E ❡ ❞❡ Φλ(un)→cs❡❣✉❡ q✉❡

lim sup

n→+∞ k

unk2λ 6

2qc q₋2.

❆❣♦r❛✱ ♣♦r ✭❆✸✮✲✭❆✺✮ ❡①✐st❡ C > 0✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ λ>0 t❛❧ q✉❡

Φ′_λ(u)(u) =_ku_k2_λ−

Z

RN

f(x, u)udx> 1

2kuk

2

λ −Ckuks+1λ = (

1

2 −Ckuk

s−1 λ )kuk2λ.

❈♦♠♦ 1< s <2∗_{−1✱ ❡①✐st❡ σ}

1 >0t❛❧ q✉❡

1 4kuk

2

λ 6Φ′λ(u)(u) ♣❛r❛ kukλ < σ1. ✭✷✳✸✮ ❆❧é♠ ❞✐ss♦✱ s❡ c < σ12q2q−2 ❡(un)é (P S)c ♣❛r❛ Φλ✱ ❡♥tã♦

lim sup

n→∞ k

unk2λ 6

2qc q₋2 < σ

2 1.

❆ss✐♠✱ kunkλ < σ1 ♣❛r❛ n ❣r❛♥❞❡✳ ❊♥tã♦ ♣♦r ✭✷✳✸✮✱

1 4kunk

2

λ 6Φ′λ(un)(un) =o(1)kunkλ,

♦ q✉❡ ✐♠♣❧✐❝❛ kunkλ → 0 q✉❛♥❞♦ n → +∞✳ P♦rt❛♥t♦ Φ(un) → 0✱ ✐st♦ é✱ c = 0✳ ▲♦❣♦✱

c0 = σ

2 1(q−2)

2q ✳

❆ ❞❡♠♦♥str❛çã♦ ❞♦ ♣ró①✐♠♦ ❧❡♠❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ t❛♠❜é♠ ❡♠ ❬❇❲✾✺❪✳

▲❡♠❛ ✷✳✶✳✹✳ ❊①✐st❡ δ0 >0 t❛❧ q✉❡ ♣❛r❛ q✉❛❧q✉❡r s❡q✉ê♥❝✐❛ (un) q✉❡ é (P S)c ❞❡ Φλ ❝♦♠

λ >0 ❡ ❝❃✵✱ t❡♠♦s

lim inf

n→+∞kunk

s+1

Ls+1₍RN₎ >δ0c. ❉❡♠♦♥str❛çã♦✿ ❞♦ ▲❡♠❛ ✷✳✶✳✸ s❛❜❡♠♦s

kunk2λ 6

2q

q₋2c+o(1) q✉❛♥❞♦n →+∞. ✭✷✳✹✮

❆❣♦r❛✱ ❞♦ ▲❡♠❛ ✷✳✶✳✷✱ ❞❛ ❤✐♣ót❡s❡ ✭❆✸✮ ❡ ❞❛ ❤✐♣ót❡s❡ ✭❆✹✮✱ ♣❛r❛ t♦❞♦ ǫ > 0✱ ❡①✐st❡

A(ǫ)>0t❛❧ q✉❡ f(x, u)u6ǫ_|u_|2 _+A(ǫ)|u|s+1_{✱ ♦✉ s❡❥❛✱}

1

2f(x, u)u−F(x, u)6ǫ|u|

✷✳✶✳ ❈❖◆❉■➬➹❖ ❉❊ ❈❖▼P❆❈■❉❆❉❊ ❉❖ ❋❯◆❈■❖◆❆▲ ✾

♣❛r❛ t♦❞♦ x_∈RN ❡_u∈R✳ ❯s❛♥❞♦ q✉❡_(un)é ✉♠❛ s❡q✉ê♥❝✐❛ _{(P S)c} ❡ ✭✷✳✶✮✱

c= lim

n→+∞

Φλ(un)−

1 2Φ

′

λ(un)(un)

= lim

n→+∞

Z

RN

1

2f(x, un)un−F(x, un)

dx

6lim inf

n→+∞

Z

RN

(ǫ_|un|2+A(ǫ)|un|s+1)dx

6lim inf

n→+∞

2qǫ

q₋2c+A(ǫ)kunk

s+1 Ls+1₍RN₎

.

❊s❝♦❧❤❡♥❞♦ ǫ= q_4q−2 t❡♠♦s

lim inf

n→+∞ kunk

s+1

Ls+1₍RN₎ >

1 2A(ǫ)c.

❚♦♠❛♥❞♦ δ0 := _2A(ǫ)1 ✱ t❡♠♦s

lim inf

n→+∞kunk

s+1

Ls+1₍RN₎ >δ0c.

▲❡♠❛ ✷✳✶✳✺✳ ❙❡❥❛ C1 >0 ✉♠❛ ❝♦♥st❛♥t❡ ✜①❛❞❛✳ ❊♥tã♦ ♣❛r❛ t♦❞♦ ǫ >0✱ ❡①✐st❡♠ Λǫ >0 ❡ Rǫ >0 t❛❧ q✉❡ s❡(un) é ✉♠❛ s❡q✉ê♥❝✐❛ (P S)c ❞❡ Φλ ❝♦♠ c6C1 ❡ λ>Λǫ✱ ❡♥tã♦

lim sup

n→∞ kunk

s+1 Ls+1_(Bc

Rǫ)

6ǫ,

♦♥❞❡✱

B_Rc_ǫ =_{x_∈RN _{:|x|> Rǫ}.}

❉❡♠♦♥str❛çã♦✿ ♣❛r❛ R >0 ❞❛❞♦✱ ❝♦♥s✐❞❡r❡♠♦s ♦s s❡❣✉✐♥t❡s ❝♦♥❥✉♥t♦s✿

A1(R) = {x∈RN :|x|> R, a(x)>M0},

A2(R) = {x∈RN :|x|> R, a(x)6M0},

♦♥❞❡ M0 é ❞❛❞♦ ♣♦r (A2)✳ ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ✐♥t❡r♣♦❧❛çã♦ ❞❡ ❍ö❧❞❡r ♣❛r❛

2< s+ 1<2∗_✱

|un|Ls+1_(Bc

Rǫ) 6|un|

1−θ L2_(Bc

Rǫ)|un|

θ L2∗_(Bc

Rǫ).

P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❙♦❜♦❧❡✈✲●❛❣❧✐❛r❞♦✲◆✐r❡♥❜❡r❣✱

|un|L2∗_(Bc

✶✵ ❈❆P❮❚❯▲❖ ✷✳ ▼Ú▲❚■P▲❆❙ ❙❖▲❯➬Õ❊❙ P❖❙■❚■❱❆❙

▲♦❣♦✱

Z

Bc Rǫ

|un|s+1dx6C|un|1_L−2_(Bθ c

Rǫ)|∇un|

θ L2_(Bc

Rǫ).

❊s❝♦❧❤❡♥❞♦ θ= N(s_2(s+1)−1) ❡ ✉s❛♥❞♦ ♦ ▲❡♠❛ ✷✳✶✳✸ r❡s✉❧t❛ q✉❡ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛ t❡♠♦s

Z

Bc Rǫ

|un|s+1dx 6C

2q q₋2C1

θ(s+1)Z

A1(Rǫ)

u2ndx+

Z

A2(Rǫ) u2ndx

(1−θ)(₂s+1)

.

❆♥❛❧✐s❛r❡♠♦s ❛❣♦r❛ ❛s ✐♥t❡❣r❛✐s Z A1(Rǫ)

u2_ndx ❡

Z

A2(Rǫ) u2_ndx✳

❉♦ ▲❡♠❛ ✷✳✶✳✸✱

Z

A1(Rǫ)

u2_ndx6 1

λM0+ 1

Z

RN

(λa(x) + 1)u2_ndx

6 1

λM0+ 1k

unk2λ 6

1

λM0+ 1

2qC1

q₋2+o(1)

✱ q✉❛♥❞♦n _→+_∞.

◆♦✈❛♠❡♥t❡ ♣❡❧♦ ▲❡♠❛ ✷✳✶✳✸ ❡ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✱

Z

A2(Rǫ)

u2_ndx6

Z

RN|

un|2rdx

1_r Z

A2(Rǫ) dx

_r1′

6C_ku_nk2_H1₍RN₎µ(A₂(R))

1

r′

6C_kunk2λµ(A2(Rǫ))

1

r′

6C2qC1

q₋2µ(A2(Rǫ))

1

r′,

♣❛r❛ 1 < r < N N−2 ❡ r

′ ₌ r

r−1✱ ♦♥❞❡ C = C(N, r) é ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛✳ ❈♦♠♦

Z

A1(Rǫ)

u2_ndx❡

Z

A2(Rǫ)

u2_ndx♣♦❞❡♠ s❡r tã♦ ♣❡q✉❡♥♦s q✉❛♥t♦ s❡ q✉❡✐r❛ ❞❡s❞❡ q✉❡A1(Rǫ)✱λ❡

Rǫs❡❥❛♠ r❡s♣❡❝t✐✈❛♠❡♥t❡ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡s✱ ❡♥tã♦ ❞❡s❞❡ q✉❡λs❡❥❛ s✉✜❝✐❡♥t❡♠❡♥t❡

❣r❛♥❞❡

Z

Bc Rǫ

|un|s+1dx→0, q✉❛♥❞♦ n→ ∞. ▲♦❣♦✱ lim sup

n→+∞ kunk

s+1 Ls+1_(Bc

Rǫ) < ǫ.

▲❡♠❛ ✷✳✶✳✻✳ ❙❡❥❛ λ>0 ✜①❛❞♦ ❡ s❡❥❛ (un) ✉♠❛ s❡q✉ê♥❝✐❛ (P S)c ❞❡ Φλ✳ ❊♥tã♦✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ un ⇀ u ❡♠ E✱ ❝♦♠ u s❡♥❞♦ ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ (Sλu)✳ ❆❧é♠ ❞✐ss♦✱

vn=un−u é ✉♠❛ s❡q✉ê♥❝✐❛ (P S)c′✱ ❝♦♠ c′ =c−Φ_λ(u)✳

✷✳✶✳ ❈❖◆❉■➬➹❖ ❉❊ ❈❖▼P❆❈■❉❆❉❊ ❉❖ ❋❯◆❈■❖◆❆▲ ✶✶

n _→ +∞✳ ❱❛♠♦s ♠♦str❛r q✉❡ u é ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ Φλ✳ ❈♦♠♦ un é ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❡ ❝♦♥✈❡r❣❡ ❢r❛❝♦ ♣❛r❛ u t❡♠♦s✿

un →u em LpLoc(RN), para todo 26p < 2∗. ✭✷✳✺✮ P♦rt❛♥t♦✱ ♣❛r❛ t♦❞♦ w_∈E t❡♠♦s

Φ′_λ(u)(w) = lim

n→+∞Φ ′

λ(un)(w) = 0,

♣♦✐s (un) é ✉♠❛ s❡q✉ê♥❝✐❛ (P S)c✳ P♦rt❛♥t♦ u é ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ Φλ✱ ♦✉ s❡❥❛✱u é ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ (Sλ)✳ ❱❛♠♦s ❛❣♦r❛ ♠♦str❛r q✉❡ vn = un −u é ✉♠❛ s❡q✉ê♥❝✐❛ (P S)c′✱

❝♦♠ c′ _=c−Φ

λ(u)✱ ✐st♦ é✱

• Φλ(vn)→c−Φλ(u), n →+∞

• Φ′

λ(vn)→0, n→+∞

Pr✐♠❡✐r❛♠❡♥t❡✱ ✈❛♠♦s ♠♦str❛r q✉❡✱ Φλ(vn)→c−Φλ(u)✱ q✉❛♥❞♦ n →+∞✳ ❈♦♠♦

Φλ(vn) = Φλ(un)−Φλ(u)− hvn, uiE

+

Z

RN

[F(x, vn+u)−F(x, vn)−F(x, u)]dx

✭✷✳✻✮

❡ hvn, uiE →0✱ q✉❛♥❞♦ n→+∞✱ ❡♥tã♦ ❜❛st❛ ♠♦str❛r q✉❡

Z

RN

[F(x, vn+u)−F(x, vn)−F(x, u)]dx→0✱ q✉❛♥❞♦ n→+∞. ❉❛❞♦ ǫ >0✱ ❡s❝♦❧❤❛ R(ǫ)>0 t❛❧ q✉❡

Z

Bc R(ǫ)

|u_|s+1dx6ǫ ❡

Z

Bc R(ǫ)

|u_|2dx6ǫ, ✭✷✳✼✮

♦♥❞❡ Bc

R(ǫ) ={x∈RN :|x|>R(ǫ)}.❉❛í✱

Z

Bc R(ǫ)

|F(vn+u)|−|F(vn)|dx6

Z

Bc R(ǫ)

|f(vn+ξ(x, n)u)||u|dx

6C

Z

Bc R(ǫ)

(_|vn|+|u|+ (|vn|+|u|)s)|u|dx

6CkvnkL2_(Bc

R(ǫ))kukL2(B

c

R(ǫ))+Ckuk

2 L2_(Bc

R(ǫ))

+C

" Z

Bc R(ǫ)

(_|v_n|+_|u_|)s+1dx

#_s+1s " Z

Bc R(ǫ)

|u_|s+1dx

#_s+11

✶✷ ❈❆P❮❚❯▲❖ ✷✳ ▼Ú▲❚■P▲❆❙ ❙❖▲❯➬Õ❊❙ P❖❙■❚■❱❆❙

q✉❛♥❞♦ ǫ _→ 0 ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ n✱ ❝♦♠ ❈ s❡♥❞♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ n✳ P♦r ♦✉tr♦ ❧❛❞♦✱

✉s❛♥❞♦ ❛ ❝♦♠♣❛❝✐❞❛❞❡ ❞❛s ✐♠❡rsõ❡s ❞❡ ❙♦❜♦❧❡✈ t❡♠♦s

Z

B_Rǫ |

F(x, vn+u)−F(x, vn)−F(x, u)|dx→0✱ q✉❛♥❞♦n →+∞. ✭✷✳✽✮

❯s❛♥❞♦ ✭✷✳✺✮ ❡ ✭✷✳✽✮✱ s❡❣✉❡ q✉❡

Φλ(vn)→c−Φλ(u).

P♦r ✜♠✱ ✈❛♠♦s ♠♦str❛r q✉❡ Φ′

λ(vn)→0✳ ❙❡❥❛ w∈E✱ ❡♥tã♦ ❞❡ ✭✷✳✻✮ t❡♠♦s

Φ′λ(vn)(w) = Φ′λ(un)(w)−

Z

RN

[(f(x, vn) +f(x, un) +f(x, u))w]dx.

❈♦♠♦ Φ′

λ(un)→0t❡♠♦s q✉❡ ♠♦str❛r q✉❡

Z

Rn

(f(x, vn)−f(x, un) +f(x, u))wdx→0✱ q✉❛♥❞♦ n→+∞. ❉❛❞♦ ǫ >0✱ ❡s❝♦❧❤❛ R(ǫ)>0✱ ❝♦♠♦ ❡♠ ✭✷✳✼✮✱ ♣❛r❛ ♦❜t❡r

Z

Bc R(ǫ)

|f(x, u)w_|dx6C

Z

Bc R(ǫ)

(_|u_|+_|u_|s)_|w_|dx 6C√ǫ_kw_kλ+Cǫs+11 kwkλ.

❆❧é♠ ❞✐ss♦✱ ❞❛❞♦ x_∈Bc

R(ǫ)✱ ♣❛r❛ ❛❧❣✉♠0≤ξ(x)≤1✱

Z

Bc R(ǫ)

|f(x, vn)−f(x, vn+u)||w|dx6

Z

Bc R(ǫ)

|f′(x, vn+ξu)||u||w|dx

6C

Z

Bc R(ǫ)

(1 + (_|v_n|+_|u_|s−1_))|u||w|dx

6C_ku_kL2_(Bc

R(ǫ))kwkλ+CkwkλkukLs+1(BRc(ǫ))

=o(√ǫ) +o(ǫs+11 )kwk_λ.

❯s❛♥❞♦ ✭✷✳✺✮ t❡♠♦s Φ′

λ(vn)→0✳ ▲♦❣♦✱ vn=un−u é ✉♠❛ s❡q✉ê♥❝✐❛(P S)c′ ❞❡ Φλ✱ ♦♥❞❡

c′ _{=c−Φλ(u)✳}

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

❉❡♠♦♥str❛çã♦ ❞❛ Pr♦♣♦s✐çã♦ ✷✳✶✳✶✿ ✜①❡ 0< ǫ < δ0c0

✷✳✷✳ P❘❖❱❆ ❉❖❙ ❘❊❙❯▲❚❆❉❖❙ P❘■◆❈■P❆■❙ ✶✸

❙❡c′ _{>0✱ ❡♥tã♦ ♣❡❧♦ ▲❡♠❛ ✷✳✶✳✸ t❡♠♦s c}′ _>c

0 >0✳ ❉♦ ▲❡♠❛ ✷✳✶✳✹✱ (vn)s❛t✐s❢❛③

lim inf

n→+∞kvnk

s+1

Ls+1₍RN₎>δ0c′ >δ0c0. ❖❜s❡r✈❡ q✉❡

δ0c0 6lim inf n→+∞ kvnk

s+1

Ls+1₍RN₎ = lim inf

n→+∞kvnk

s+1 Ls+1_(B

Rǫ)+ lim infn→+∞kvnk

s+1 Ls+1_(Bc

Rǫ)

= lim inf

n→+∞kvnk

s+1 Ls+1_(Bc

Rǫ)

6lim sup

n→+∞ kvnk

s+1 Ls+1_(Bc

Rǫ),

✭✷✳✾✮

♣♦✐s lim inf

n→+∞ kvnk

s+1 Ls+1_(B

Rǫ) = 0✳ ❯s❛♥❞♦ ♦ ▲❡♠❛ ✷✳✶✳✺ t❡♠♦s

lim sup

n→+∞ k

vnks+1_Ls+1_(Bc Rǫ)

6ǫ < δ0c0

2 . ✭✷✳✶✵✮

▲♦❣♦✱ ❞❡ ✭✷✳✾✮ ❡ ✭✷✳✶✵✮ t❡♠♦s

δ0c0 <

δ0c0

2

❖ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ P♦rt❛♥t♦ c′ _{= 0 ❡ ❞❛í✱ v}

n → 0✱ ♦✉ s❡❥❛✱ un → u ❡ Φλ(u) =c✳ ▲♦❣♦✱ Φλ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ (P S)c ♣❛r❛ t♦❞♦ c6C0✳

✷✳✷ Pr♦✈❛ ❞♦s ❘❡s✉❧t❛❞♦s Pr✐♥❝✐♣❛✐s

◆❡st❛ s❡çã♦ ❞❡♠♦♥str❛r❡♠♦s ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❡st❡ tr❛❜❛❧❤♦✳ Pr✐♠❡✐r❛♠❡♥t❡ ❞❛❞♦ Ω _⊂ RN ✉♠ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦ ❝♦♠ ❢r♦♥t❡✐r❛ s✉❛✈❡✱ ❝♦♥s✐❞❡r❡♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡

❉✐r✐❝❤❧❡t✱

−∆u+u=up, u >0 ❡♠ Ω, u= 0 s♦❜r❡ ∂Ω, ✭Dp✮

❡ ❛ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ ❝♦rr❡s♣♦♥❞❡♥t❡✱

N(p,Ω) =

u_∈H₀1(Ω)_\{0_}:

Z

Ω

(_|∇u_|2+u2)dx=

Z

Ω|

u_|p+1dx

.

❈♦♥s✐❞❡r❡ t❛♠❜é♠

c(p,Ω) = inf

u∈N(p,Ω)Φp,Ω(u), ♦♥❞❡

Φp,Ω(u) =

1 2

Z

Ω

(_|∇u_|2+u2)dx₋ 1 p+ 1

Z

Ω|

u_|p+1dx✱ ♣❛r❛u_∈H₀1(Ω)

❡

m(p, λ,Ω) = inf

u∈Vp

Z

Ω |∇

u_|2+λu2

dx,

♦♥❞❡

Vp =Vp(Ω)=.

✶✹ ❈❆P❮❚❯▲❖ ✷✳ ▼Ú▲❚■P▲❆❙ ❙❖▲❯➬Õ❊❙ P❖❙■❚■❱❆❙

P❡❧❛ Pr♦♣♦s✐çã♦ ❆✳✵✳✶✺ ❞♦ ❆♣ê♥❞✐❝❡ ❆✱ ♦ ❢✉♥❝✐♦♥❛❧ Φp,Ω ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ❡♠H01(Ω) ❡ é ❞❡ ❝❧❛ss❡ C1_{✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ r >0✜①❡♠♦s B}

r=

x_∈RN _{:|x|< r} ❡ ❞❡✜♥✐♠♦s

c(p, r) :=c(p, Br).

P❛r❛u_∈N(p,Ω) ✈❛♠♦s ❞❡✜♥✐r ♦ ❝❡♥tr♦ ❞❡ ♠❛ss❛ ❞❡u✳

❉❡✜♥✐çã♦ ✷✳✷✳✶✳ ❙❡❥❛ u_∈N(p,Ω)✳ ❖ ❝❡♥tr♦ ❞❡ ♠❛ss❛ ❞❡ u é ❞❛❞❛ ♣♦r✿

β(u) =

Z

Ω|

u_|p+1xdx

Z

Ω|

u_|p+1dx .

◆♦✈❛♠❡♥t❡✱ ❛ ✜♠ ❞❡ ❞❡♠♦♥str❛r♠♦s ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❡st❡ tr❛❜❛❧❤♦✱ ♣r♦✈❛r❡✲ ♠♦s ❛❧❣✉♥s ❧❡♠❛s✳

▲❡♠❛ ✷✳✷✳✷✳ c(p,Ω)_{→ N}1SN2✱ q✉❛♥❞♦ p→2∗−1✱ ♦♥❞❡ S =. m(2∗, λ,Ω)✳

P❛r❛ ♣r♦✈❛r♠♦s ♦ ❧❡♠❛ ❛❝✐♠❛✱ ❞❡♠♦♥str❛r❡♠♦s ❞✉❛s ♣r♦♣♦s✐çõ❡s✳ Pr♦♣♦s✐çã♦ ✷✳✷✳✸✳ c(p,Ω) = _2(p+1)p−1 (m(p, λ,Ω))pp+1−1✳

❉❡♠♦♥str❛çã♦✿ ❛ ♣r♦✈❛ s❡rá ❢❡✐t❛ ❡♠ ❞♦✐s ♣❛ss♦s ✭✐✮ ❛ ✭✐✐✮ ❛ s❡❣✉✐r✿ ✭✐✮ c(p,Ω) 6 _2(p+1)p−1 (m(p, λ,Ω))pp+1−1✳

❙❡❥❛ u_∈Vp✱ ❛ss✐♠

Φp,Ω(tu) =

t2

2kuk

2 − t

p+1

p+ 1kukLp+1(Ω).

❈♦♥s✐❞❡r❡ t _∈Rt❛❧ q✉❡ _{tu ∈N(p,Ω)}✱ ❞❛í

t_ku_k2₋tp_ku_kp+1_Lp+1_(Ω) = 0,

✐♠♣❧✐❝❛♥❞♦

t= kuk

2

ku_kp+1_Lp+1_(Ω)

!_p−1₁

=_ku_kp−21, ♣♦✐s kukp+1

Lp+1_(Ω)= 1.

▲♦❣♦✱

Φp,Ω

ku_kp−21u

= p−1 2(p+ 1)

kuk

2

p−1u

2

= p−1 2(p+ 1)kuk

2(p+1)

p−1 .

❈♦♠♦ c(p,Ω) = inf

u∈N(p,Ω)Φp,Ω(u)✱ ❡♥tã♦

2(p+ 1)

p₋1 c(p,Ω)

p_p−₊₁1

6

2(p+ 1)

p₋1 Φp,Ω

ku_kp−21u

p−1

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

P♦rt❛♥t♦✱

2(p+ 1)

p₋1 c(p,Ω)

p_p−₊₁1

6_ku_k2✱ ♣❛r❛ t♦❞♦ u_∈Vp. ❆ss✐♠✱

2(p+ 1)

p₋1 c(p,Ω)

p_p−₊₁1

6m(p, λ,Ω),

✐st♦ é✱

c(p,Ω) 6 p−1

2(p+ 1)m(p, λ,Ω)

p+1

p−1

✭✐✐✮ c(p,Ω) > p−1

2(p+1)m(p, λ,Ω)

p+1

p−1.

❙❡❥❛ u_∈N(p,Ω)✳ ❖❜s❡r✈❡ q✉❡ ♣❛r❛u_∈N(p,Ω)✱ t❡♠♦s u

ku_kLp+1_(Ω) ∈

Vp✳ ❆ss✐♠✱

u

ku_kLp+1_(Ω)

= kuk

ku_kLp+1_(Ω)

= kuk

ku_kp+12

=_ku_k1−p+12 =kuk

p−1

p+1,

✉♠❛ ✈❡③ q✉❡ ku_k2 _=kukp+1

Lp+1_(Ω)✳ P♦rt❛♥t♦✱

u

ku_kLp+1_(Ω)

2

=_ku_k2(PP−+1)1 .

❆ss✐♠✱

m(p, λ,Ω) 6

u

ku_kLp+1(Ω)

2

=_ku_k2(pp+1−1) =

2(p+ 1)

p₋1 Φp,Ω(u)

p_p−₊₁1

,

♣♦✐s✱ ♥♦✈❛♠❡♥t❡ ❝♦♠♦ u_∈N(p,Ω)✱ _ku_k2 ₌ 2(p+1)

p−1 Φp,Ω(u)✳ ▲♦❣♦✱

p₋1

2(p+ 1)m(p, λ,Ω)

p+1

p−1 ₆Φ

p,Ω(u),

♣❛r❛ t♦❞♦ u_∈N(p,Ω)✳ P♦rt❛♥t♦✱

p₋1

2(p+ 1)m(p, λ,Ω)

p+1

p−1 ₆c(p,Ω).

❉❡st❡ ♠♦❞♦✱ ❞❡ ✭✐✮ ❡ ✭✐✐✮ s❡❣✉❡ q✉❡

c(p,Ω) = p−1

2(p+ 1)m(p,Ω)

p+1

p−1.

Pr♦♣♦s✐çã♦ ✷✳✷✳✹✳ P❛r❛ q✉❛❧q✉❡r ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ Ω_⊂RN ❡ ♣❛r❛ q✉❛❧q✉❡r _λ>0✱

lim

✶✻ ❈❆P❮❚❯▲❖ ✷✳ ▼Ú▲❚■P▲❆❙ ❙❖▲❯➬Õ❊❙ P❖❙■❚■❱❆❙

❉❡♠♦♥str❛çã♦✿ ♣r✐♠❡✐r❛♠❡♥t❡ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✱

ku_kLp_(Ω) =

Z

Ω|

u_|pdx

1_p

6_ku_kLq(Ω)|Ω| q−p

qp , ✭✷✳✶✶✮

♣❛r❛ t♦❞♦ p, q _∈[2,2∗_{] ❝♦♠ p < q ❡ ♣❛r❛ t♦❞♦ u∈H}1 0(Ω). ❉❡st❡ ♠♦❞♦✱

Z

Ω |∇

u_|2+λu2

dx

ku_k2 Lp_(Ω)

>_|Ω_|−2(qpq−p)

Z

Ω |∇

u_|2 +λu2

dx

ku_k2 Lq_(Ω)

,

✐♠♣❧✐❝❛♥❞♦

m(p, λ,Ω) >_|Ω_|−2(22∗∗−pp)S,

m(p, λ,Ω) 6_|Ω_|p−p2m(2, λ,Ω).

✭✷✳✶✷✮

P♦rt❛♥t♦ ♣❛r❛ q✉❛❧q✉❡rλ>0❡Ω✱m(p, λ,Ω)é ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡ ❡ ✐♥❢❡r✐♦r♠❡♥t❡

q✉❛♥❞♦ p_∈[2,2∗_{]✳ ❉❡✜♥❛}

m = lim inf

p→2∗ m(p, λ,Ω) ❡ M = lim sup

p→2∗

m(p, λ,Ω).

P❛r❛ ❝♦♥❝❧✉✐r ❛ ❞❡♠♦♥str❛çã♦ ❜❛st❛ ♠♦str❛r♠♦s q✉❡

m=S =M

▼♦str❛r❡♠♦s ♣r✐♠❡✐r♦ q✉❡ m =S✳

❉❡ ✭✷✳✶✷✮ s❡❣✉❡ q✉❡ m>S✳ ❙✉♣♦♥❤❛ m > S✳ P♦rt❛♥t♦ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ǫ_∈(0, m₋S)

❡ ❡s❝♦❧❤❡r ✉♠❛ ❢✉♥çã♦ u_∈H1

0(Ω)t❛❧ q✉❡

Z

Ω |∇

u_|2+λu2

dx

ku_k2 L2∗_(Ω)

< S + ǫ 2.

P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❛♣❧✐❝❛çã♦ p _{7→ k}u_kLp(Ω)✱ ❡①✐st❡ p ∈ (2,2∗) t❛❧ q✉❡

♣❛r❛ t♦❞♦ p_∈[p,2∗_{) t❡♠♦s}

Z Ω |∇

u_|2+λu2

dx

ku_k2 Lp_(Ω)

−

Z

Ω |∇

u_|2+λu2

dx

ku_k2 L2∗_(Ω)

< ǫ 2

▲♦❣♦✱ ♣❛r❛ t♦❞♦ p_∈[p,2∗_{] t❡♠♦s}

m(p, λ,Ω) 6

Z

Ω |∇

u_|2+λu2

dx

ku_k2 Lp_(Ω)

<

Z

Ω |∇

u_|2+λu2

dx

ku_k2 L2∗_(Ω)

+ ǫ

✷✳✷✳ P❘❖❱❆ ❉❖❙ ❘❊❙❯▲❚❆❉❖❙ P❘■◆❈■P❆■❙ ✶✼

♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✱ ♣♦✐s m = lim inf

p→2∗ m(p, λ,Ω). P♦rt❛♥t♦✱ m =S✳

P♦r ♦✉tr♦ ❧❛❞♦✱ ✉s❛♥❞♦ ♥♦✈❛♠❡♥t❡ q✉❡

m(p, λ,Ω) <

Z

Ω |∇

u_|2+λu2

dx

ku_k2 Lp_(Ω)

< S +ǫ

♣❛r❛ t♦❞♦ p_∈[p,2∗_{)✱ t❡♠♦s}

M = lim sup

p→2∗

m(p, λ,Ω)6S+ǫ

♣❛r❛ t♦❞♦ ǫ >0✳ P♦rt❛♥t♦✱ M 6S =m =S✱ ✐st♦ é✱ M =S =m✳

❋✐♥❛❧♠❡♥t❡ ❝♦♠ ❡ss❡s ❞♦✐s r❡s✉❧t❛❞♦s✱ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✷✳✷✳✷ é tr✐✈✐❛❧✱ ♥❛ ✈❡r❞❛❞❡ ❛ ♣r♦✈❛ é ❡①❛t❛♠❡♥t❡ ❛ ❞❡♠♦♥str❛çã♦ ❞❛s ❞✉❛s ♣r♦♣♦s✐çõ❡s ❛♥t❡r✐♦r❡s✳

◆♦ q✉❡ s❡❣✉❡✱ ❞❡♥♦t❡♠♦sS2 ❛ ♠❡❧❤♦r ❝♦♥st❛♥t❡ ❞❡ ❙♦❜♦❧❡✈ ❞❛ ✐♠❡rsã♦✶ D1,2(RN) ⊂

L2∗

(RN₎✱ ✐st♦ é✱

S2 = inf

Z

RN|∇

u_|2dx : u_∈D1,2(RN₎❡ _kukL2∗_(RN₎ = 1

. ✭✷✳✶✸✮

❉❡ ❛❣♦r❛ ❡♠ ❞✐❛♥t❡✱ ❛❞♠✐t✐r❡♠♦s q✉❡ Ω é ♦ ✐♥t❡r✐♦r ❞❡ a−1_{(0)✳ P❡❧❛ ❤✐♣ót❡s❡ ✭❆✶✮✱} ❡①✐st❡ r >0t❛❧ q✉❡ ❛s ✐♥❝❧✉sõ❡s Ω− _{֒→Ω֒→Ω}+ _{sã♦ ❤♦♠♦t♦♣✐❝❛♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s✱ ♦♥❞❡}

Ω+=_{x_∈RN _:dist(x,Ω) 6r_}

❡

Ω− =_{x_∈Ω :dist(x, ∂Ω)>r_}.

❆ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛ ❛ s❡❣✉✐r ❢♦✐ ❡①tr❛í❞❛ ❞♦ tr❛❜❛❧❤♦ ❬❇❈✾✶❪ ❡ ❢❛③ ✉s♦ ❞♦ ❧❡♠❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❡ ❝♦♠♣❛❝✐❞❛❞❡ ✭✈❡❥❛ ❆♣ê♥❞✐❝❡ ❇✮✳

Pr♦♣♦s✐çã♦ ✷✳✷✳✺✳ ❙❡ (un)⊂Vp(Ω) é t❛❧ q✉❡ kunk2 →S2✱ ❡♥tã♦ dist(β(un),Ω) →0✳ ❉❡♠♦♥str❛çã♦✿ s✉♣♦♥❤❛ ♣♦r ❛❜s✉r❞♦ q✉❡ dist(β(un),Ω) 9 0✳ ❆ss✐♠✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ ♣♦❞❡♠♦s ❛❞♠✐t✐r q✉❡

dist(β(un),Ω) > r >0

❡ _

  

  

un⇀ u ❡♠ D1,2(RN) |∇(un−u)|2 ⇀ µ ❡♠ M(RN) |un−u|2

∗

⇀ ν ❡♠ M(RN₎

un→u qt♣ ❡♠ Ω,

✶✽ ❈❆P❮❚❯▲❖ ✷✳ ▼Ú▲❚■P▲❆❙ ❙❖▲❯➬Õ❊❙ P❖❙■❚■❱❆❙

♦♥❞❡ µ ❡ν sã♦ ♠❡❞✐❞❛s ✜♥✐t❛s ❡ M(RN₎❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ❞❛s ♠❡❞✐❞❛s ✜♥✐t❛s ❡♠ RN✳

P❡❧❛ Pr♦♣♦s✐çã♦ ❇✳✵✳✷✶ ❞♦ ❆♣ê♥❞✐❝❡ ❇ r❡s✉❧t❛ q✉❡

S2 =k∇uk2L2_(Ω)+kµk+µ_∞ ✭✷✳✶✹✮

❡

1 =_ku_k2_L∗2∗_(Ω)+kνk+ν_∞, ✭✷✳✶✺✮

♦♥❞❡ µ∞=ν∞= 0✱ ♣♦✐s Ωé ❧✐♠✐t❛❞♦✳ ❆ss✐♠✱

kν_k22∗ ₆S−1

2 kµk

❡

ku_k2_L2∗_(Ω)6S₂−1k∇uk2_L2_(Ω). ✭✷✳✶✻✮

❯s❛♥❞♦ ❛s ❡q✉❛çõ❡s ❞❛❞❛s ❛❝✐♠❛ t❡♠♦s

S2 =k∇uk2L2_(Ω)+kµk>S2

ku_k2_L2∗_(Ω)+kνk

2 2∗

.

■♠♣❧✐❝❛♥❞♦ q✉❡✱

ku_k2_L∗2∗_(Ω)+kνk= 1>kuk2_L2∗_(Ω)+kνk

2

2∗ ✭✷✳✶✼✮

P♦r ✭✷✳✶✺✮✱ kν_{k ∈}[0,1]✳ ❙❡ _kν_{k ∈}(0,1)✱ ❡♥tã♦

kν_k22∗ >kνk. ✭✷✳✶✽✮

❉❡ ✭✷✳✶✼✮ ❡ ✭✷✳✶✽✮ s❡❣✉❡ q✉❡

ku_k2_L∗2∗_(Ω)+kνk>kuk2_L2∗_(Ω)+kνk,

▼❛s ✐st♦ é ✐♠♣♦ssí✈❡❧ ♣♦✐s✱ ♣♦r ✭✷✳✶✺✮✱ ku_k2∗

L2∗_(Ω) ∈ [0,1] ❡ 2∗ > 2✳ ▲♦❣♦✱ kνk = 0 ♦✉

kν_k= 1✳ ❙❡ _kν_k= 0✱ ❡♥tã♦✱ ♣♦r ✭✷✳✶✺✮✱

ku_kL2∗_(Ω)= 1. ✭✷✳✶✾✮

❯s❛♥❞♦ ✭✷✳✶✹✮ t❡♠♦s

S2 =k∇uk2L2_(Ω)+kµk>k∇uk2_L2_(Ω). ✭✷✳✷✵✮

❈♦♠❜✐♥❛♥❞♦ ✭✷✳✶✻✮✱ ✭✷✳✶✾✮ ❡ ✭✷✳✷✵✮

✷✳✷✳ P❘❖❱❆ ❉❖❙ ❘❊❙❯▲❚❆❉❖❙ P❘■◆❈■P❆■❙ ✶✾

▲♦❣♦✱ S2 é ❛t✐♥❣✐❞♦ ♣♦r ✉♠❛ ❢✉♥çã♦ u∈L2

∗

(Ω)✱ ♦ q✉❡ ❝♦♥tr❛❞✐③ ❛ Pr♦♣♦s✐çã♦ ❇✳✵✳✷✷ ❞♦

❆♣ê♥❞✐❝❡ ❇✳ P♦rt❛♥t♦✱ kν_k= 1✱ ♦✉ s❡❥❛✱ u= 0✳ ❆ss✐♠✱ ❞♦ ▲❡♠❛ ❇✳✵✳✷✶ ❝♦♥❝❧✉í♠♦s q✉❡ν

s❡ ❝♦♥❝❡♥tr❛ ❡♠ ✉♠ ú♥✐❝♦ ♣♦♥t♦ y _∈Ω✱ ♦✉ s❡❥❛✱

β(un)→

Z

Ω

xdν(x) =y_∈Ω,

❈♦♥tr❛❞✐③❡♥❞♦ dist(β(un),Ω) 90✳

▲❡♠❛ ✷✳✷✳✻✳ S =. m(2∗_{, λ,Ω) =m(2}∗_{, λ,R}N_{) = S} 2✳

❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ ♠♦str❛r❡♠♦s q✉❡ S2 = m(2∗, λ,RN)✳ P❛r❛ ✐ss♦✱ é s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡ m(2∗_{, λ,R}N_{)6S2✳ ❈♦♥s✐❞❡r❡ ❛ ❢❛♠í❧✐❛}

φǫ(x) = ǫ

2−N

2 φ(x ǫ), ♦♥❞❡ φ(x) = [N(N −2)]

N−2 4

[1 +_|x_|2_]N−22 ,

♣❛r❛ ǫ > 0✳ ➱ s❛❜✐❞♦ q✉❡ ❛ ♠❡❧❤♦r ❝♦♥st❛♥t❡ S2 ❞❛ ✐♠❡rsã♦ ❞❡ ❙♦❜♦❧❡✈ é ❛t✐♥❣✐❞❛ ♣❡❧❛ ❢❛♠í❧✐❛ φǫ✱ ✐st♦ é✱

R

Rn|∇φǫ|2dx

R

Rn|φǫ|2

∗

dx2/2∗ =S2.

P♦rt❛♥t♦✱ ♣❛r❛ ❝♦♥❝❧✉✐r ❡ss❡ ♣r✐♠❡✐r♦ ♣❛ss♦ ❞❛ ❞❡♠♦♥str❛çã♦✱ ❜❛st❛ ✈❡r✐✜❝❛r q✉❡

h(ǫ)=.

R

Rn|φǫ|2dx

R

Rn|φǫ|2

∗

dx2/2∗ →0, q✉❛♥❞♦ ǫ→0.

❉❡ ❢❛t♦✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❛ ❢✉♥çã♦ φǫ✱ ❛♣ós ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ♦❜t❡♠♦s

h(ǫ) =ǫ2

Z

Rn|

φ_|2dx_→0, q✉❛♥❞♦ ǫ_→0.

P❛r❛ ❝♦♥❝❧✉✐r ❛ ❞❡♠♦♥str❛çã♦✱ r❡st❛ ♣r♦✈❛r q✉❡m(2∗_{, λ,R}N_{) = m(2}∗_{, λ,Ω) ♣❛r❛ q✉❛❧q✉❡r}

q✉❡ s❡❥❛ Ω_⊂ RN✳ ➱ ✐♠❡❞✐❛t♦ ✈❡r✐✜❝❛r q✉❡ _m(2∗_{, λ,}RN₎6m(2∗_{, λ,Ω)✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱}

s❡❥❛ ψm ✉♠❛ s❡q✉ê♥❝✐❛ ♠✐♥✐♠✐③❛♥t❡ ❞❡ m(2∗, λ,RN)✳ P❡❧❛ ❞❡♥s✐❞❛❞❡ ❞❡ C0∞(RN) ❡♠

H1_(RN_{)✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ ψ}

m ∈C0∞(RN)✳ ❆♣ós ✉♠❛ tr❛♥s❧❛çã♦✱ ♣♦❞❡♠♦s ❛❞♠✐t✐r q✉❡

0_∈Ω✳ ❆❣♦r❛ ❞❡✜♥❛

vm(x) =R−N/2

∗

m ψm(x/Rm),

♣❛r❛ Rm > 0 s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ ❞❡ ♠♦❞♦ q✉❡ vm ∈ C0∞(Ω)✳ ◆♦✈❛♠❡♥t❡✱ ♣❛r❛ ❝♦♥❝❧✉✐r ❛ ❞❡♠♦♥str❛çã♦✱ ❜❛st❛ ♠♦str❛r q✉❡

g(m)=.

R

Ω|vm|2dx

R

Ω|vm|2

∗

✷✵ ❈❆P❮❚❯▲❖ ✷✳ ▼Ú▲❚■P▲❆❙ ❙❖▲❯➬Õ❊❙ P❖❙■❚■❱❆❙

❉❡ ❢❛t♦✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❛ ❢✉♥çã♦ vm✱ ❛♣ós ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ♦❜t❡♠♦s

g(m) = RNm(2∗−2)/2

Z

Ω/Rm

|ψm(y)|2dy→0, q✉❛♥❞♦ Rm →0.

Pr♦♣♦s✐çã♦ ✷✳✷✳✼✳ P❛r❛ q✉❛❧q✉❡r λ>0 ✜①❛❞♦✱ ❡①✐st❡ p_≡p(λ)✱ ❝♦♠ p_∈(2,2∗_{)✱ t❛❧ q✉❡}

♣❛r❛ t♦❞♦ p_∈[p,2∗_)✱

m(λ, p, r)<21−2pm(λ, p,Ω) ✭✷✳✷✶✮

❡

s❡ Φp,Ω(u)6m(λ, p, r)✱ ❡♥tã♦ β(u)∈Ω+r✱ ♣❛r❛ t♦❞♦ u∈Vp ✭✷✳✷✷✮ ❉❡♠♦♥str❛çã♦✿ ♣r✐♠❡✐r❛♠❡♥t❡ ♦❜s❡r✈❡ q✉❡ ♣❡❧❛ ❡s❝♦❧❤❛ ❞❡ r

m(λ, p, r)> m(λ, p,Ω).

❆ss✐♠✱ ♦ ❝♦♥❥✉♥t♦ Φc(p,r)_p,Ω é ♥ã♦ ✈❛③✐♦✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ✉s❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ ✷✳✷✳✹ ❡ ♦ ❢❛t♦

❞❡ q✉❡ lim

p→2∗2

1−2_p

= 2N2 t❡♠♦s

lim

p→2∗m(λ, p, r) = lim_p→2∗m(λ, p,Ω) =S.

▲♦❣♦✱ ❡①✐st❡ pˆ_∈(2,2∗_{)t❛❧ q✉❡ ✭✷✳✷✶✮ é s❛t✐s❢❡✐t❛ ♣❛r❛ t♦❞♦p∈[ˆp,2}∗_{)✳ ❆❣♦r❛ ♦❜s❡r✈❡ q✉❡}

♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✷✳✺ ❡ ♣❡❧♦ ▲❡♠❛ ✷✳✷✳✻✱ ❡①✐st❡ ǫ_∈(0,1) t❛❧ q✉❡

s❡

Z

Ω |∇

u_|2+λu2

dx

ku_k2 L2∗_(Ω)

< S+ǫ✱ ❡♥tã♦ β

u

ku_kL2∗_(Ω)

∈Ω+_r. ✭✷✳✷✸✮

❉❡s❞❡ ♠♦❞♦✱ ✜①❛♥❞♦ ǫ < ǫ

2(S+1)✱ ❡s❝♦❧❤❡r❡♠♦s p∈[ˆp,2

∗_{) t❛❧ q✉❡}

|m(λ, p, r)₋S_|< ǫ, _||Ω_|2(22∗−∗pp) −1|< ǫ ♣❛r❛ t♦❞♦p∈[p,2∗).

❆✜r♠❛♠♦s q✉❡ ♣❛r❛ t♦❞♦p_∈[p,2∗_{)✱ ✭✷✳✷✷✮ é s❛t✐s❢❡✐t❛✳ ❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❡u∈H}1

0(Ω) t❛❧ q✉❡ ku_kLp_(Ω) = 1 ❡

Z

Ω |∇

u_|2+λu2

dx6m(λ, p, r).

❉❡s❞❡ q✉❡ β

u

kuk_L2∗_(Ω)

=β(u)✱ ♦❜t❡♠♦s ✭✷✳✷✷✮ ❞❡ ✭✷✳✷✸✮ s❡ ♠♦str❛r♠♦s q✉❡

Z

Ω |∇

u_|2+λu2

dx

ku_k2 L2∗_(Ω)

✷✳✷✳ P❘❖❱❆ ❉❖❙ ❘❊❙❯▲❚❆❉❖❙ P❘■◆❈■P❆■❙ ✷✶

❉❡ ❢❛t♦✱ ♣♦r ✭✷✳✶✶✮✱ t❡♠♦s

Z

Ω |∇

u_|2_+λu2

dx

ku_k2 L2∗_(Ω)

6m(λ, p, r)_|Ω_|2(22∗−∗pp)

6(S+ǫ)(1 +ǫ)< S +ǫ.

▲❡♠❛ ✷✳✷✳✽✳ P❛r❛ t♦❞♦ r′ _{< r✱ ❡①✐st❡ p(r}′_)∈(1,2∗_{−1)t❛❧ q✉❡}

c(p, r′)<2c(p,Ω) ♣❛r❛ t♦❞♦ p_∈[p(r′),2∗₋1)

❡

β(u)_∈Ω+ ♣❛r❛ t♦❞♦ u_∈Np,Ω ❝♦♠ Φp,Ω(u)6c(p, r′).

❉❡♠♦♥str❛çã♦✿ ❝♦♠ ♦s três r❡s✉❧t❛❞♦s ❛♥t❡r✐♦r❡s ♠❛✐s ♦ ▲❡♠❛ ❇✳✵✳✷✶ ❞♦ ❆♣ê♥❞✐❝❡ ❇✱ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✷✳✷✳✽ é ✐♠❡❞✐❛t❛✳

❆❣♦r❛ ❝♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐

Mλ,p =

u_∈E_\{0_}:

Z

RN |∇

u_|2+ (λa(x) + 1)u2

dx=

Z

RN |

u_|p+1dx

❡ ❞❡✜♥❛ cλ,p = inf u∈Mλ,p

Φλ,p(u) ♦♥❞❡✱

Φλ,p(u) =

1 2

Z

RN |∇

u_|2+ (λa(x) + 1)u2

dx₋ 1 p+ 1

Z

RN|

u_|p+1dx.

❈♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ♦❜t❡r ♦s ♣♦♥t♦s ❝rít✐❝♦s ❞❡ Φλ,p s♦❜r❡ Mλ,p ✉s❛r❡♠♦s ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ q✉❡ ❢♦✐ ✉s❛❞♦ ❡♠ ❬✺❪ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❉✐r✐❝❤❧❡t ❡ ❡♠ ❬✷✷❪ ❝♦♠ ♦ ♣r♦❜❧❡♠❛ ❞❡ ◆❡✉♠❛♥♥✳ ❊s❝♦❧❤❡r❡♠♦s p ♣ró①✐♠♦ ❞❡ 2∗_{−1 ❞❡ ♠♦❞♦ q✉❡ ♦ ▲❡♠❛ ✶✳✸✳✷ s❡❥❛ s❛t✐s❢❡✐t♦✳}

P❛r❛ t❛❧ p✱ ❡s❝♦❧❤❡r❡♠♦s ǫ > 0 ♣❡q✉❡♥♦ t❛❧ q✉❡ ♣❛r❛ λ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ♣♦❞❡♠♦s

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

Φcλ,p+ǫ

λ,p ={u∈Mλ,p : Φλ,p(u)6cλ,p +ǫ},

❡♠ t❡r♠♦s ❞❛ ❝❛t❡❣♦r✐❛ ❞❡ Ω✳ ❋✐♥❛❧♠❡♥t❡ ♣❛r❛ ♦ ǫ > 0 ❡s❝♦❧❤✐❞♦ ♠♦str❛r❡♠♦s q✉❡

♦s ♣♦♥t♦s ❝rít✐❝♦s ♥❡st❡ ♥í✈❡❧ ♥ã♦ ♠✉❞❛♠ ❞❡ s✐♥❛❧✳ ❉❡s❞❡ q✉❡ H1

0(Ω) ⊂ E✱ ❡♥tã♦

cλ,p 6c(p,Ω)✳

❈♦r♦❧ár✐♦ ✷✳✷✳✾✳ P❛r❛ t♦❞♦ p_∈ (1,2∗_{−1)✱ ❡①✐st❡ Λ}

✷✷ ❈❆P❮❚❯▲❖ ✷✳ ▼Ú▲❚■P▲❆❙ ❙❖▲❯➬Õ❊❙ P❖❙■❚■❱❆❙

▲❡♠❛ ✷✳✷✳✶✵✳ cλ,p →c(p,Ω) q✉❛♥❞♦ λ →+∞✳

❉❡♠♦♥str❛çã♦✿ ♦❜s❡r✈❡ ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ cλ,p < c(p,Ω) é ❡str✐t❛✱ ♣♦✐s ❝❛s♦ ❝♦♥trár✐♦ ♣♦❞❡rí❛♠♦s ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦ ♥ã♦ ♥❡❣❛t✐✈❛ ❞❡ (Sλ,p) ❡♠ Ωc✳ ❖ q✉❡ é ✐♠♣♦ssí✈❡❧ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞♦ ▼á①✐♠♦ ♣❛r❛ ❛s ❡q✉❛çõ❡s ❡❧í♣t✐❝❛s✳ ❆❞♠✐t✐r❡♠♦s ♣❛r❛ ❛ s❡q✉ê♥❝✐❛ λn →+∞ ♦ s❡❣✉✐♥t❡ ❧✐♠✐t❡

lim

n→+∞cλn,p =A < c(p,Ω).

❖❜s❡r✈❡ q✉❡ A >0✳

❉❡ ❢❛t♦✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✷✳✷✳✾ ♣❛r❛ ❝❛❞❛ n✱ ❡①✐st❡ un∈Mλ,p t❛❧ q✉❡

cλn,p = Φλn,p(un) =

1

2kunkλn−

1

p+ 1kunk

p+1

Lp+1₍RN₎ =

p₋1 2(p+ 1)kunk

2 λn.

❆ss✐♠✱

kunk2λn,p =

2(p+ 1)

p₋1 cλn,p.

P❡❧♦ ▲❡♠❛ ✷✳✶✳✷✱ ❡①✐st❡ σ >0 t❛❧ q✉❡ σ2 _<kuk2

λn✳ ❉❛í✱

σ2 6lim sup

n→+∞ k

unk2λn =

2(p+ 1)

p₋1 A.

P♦rt❛♥t♦✱ ♣❡❧♦ ▲❡♠❛ ✷✳✶✳✸✱ A > c0 > 0✳ ❆❣♦r❛ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✷✳✷✳✾✱ cλ,p é ❛t✐♥❣✐❞♦ ♣❛r❛ λ ❣r❛♥❞❡✱ ❛ss✐♠ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ un∈Mλ,p q✉❡ sã♦ s♦❧✉çõ❡s ❞❡ (Sλn,p)t❛❧ q✉❡

Φλn,p(un) = cλn,p✳ ❉❡s❞❡ q✉❡ (un) é ❧✐♠✐t❛❞❛ ❡♠ H

1_(RN_{)✱ ♣♦❞❡♠♦s ❛❞♠✐t✐r u}

n ⇀ u ❡♠

H1_(RN_{) ❡ ❞❛í✱}

un →u❡♠ Lθ❧♦❝(RN)✱ ♣❛r❛ 26θ <2∗. ✭✷✳✷✹✮ ❆✜r♠❛♠♦s q✉❡ u_|Ωc = 0✳ ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛ u|_Ωc 6= 0✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦

F _⊂Ωc _{t❛❧ q✉❡ dist(F,Ω) >0✱ ❝♦♠ u|}

F 6= 0✳ ❊♥tã♦ ♣♦r ✭✷✳✷✹✮✱

Z

F

u2_ndx_→

Z

F

u2dx >0.

▼❛s✱ ❞❡s❞❡ q✉❡ a(x)>ǫ0 >0✱ ♣❛r❛ t♦❞♦ x∈F s❡❣✉❡ q✉❡

Φλn,p(un)>λn

Z

F

a(x)u2 n

dx >λ_nǫ₀

Z

F

u2

ndx→+∞,

q✉❛♥❞♦ n _→ +_∞ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✱ ♣♦✐s Φλn,p(un) → cλn,p✳ ▲♦❣♦✱ u|Ωc = 0. ❆❣♦r❛

♠♦str❛r❡♠♦s q✉❡ un → u ❡♠ Lp+1(RN)✳ ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛ q✉❡ ♥ã♦✱ ♦✉ s❡❥❛✱ un 9 u ❡♠ Lp+1_(RN_{)✳ ❉❛í✱ ♣❡❧♦ ❧❡♠❛ ❞❡ P✳ ▲✳ ▲✐♦♥s ❬✶✷❪✱ ❡①✐st❡ δ}

0 > 0, ρ > 0 xn ∈ RN✱ ❝♦♠ |xn| →+∞ t❛❧ q✉❡

lim inf

n→+∞

Z

Bρ(xn)

✷✳✷✳ P❘❖❱❆ ❉❖❙ ❘❊❙❯▲❚❆❉❖❙ P❘■◆❈■P❆■❙ ✷✸

P♦rt❛♥t♦✱

Φλn,p(un)>λn

Z

Bρ(xn)∩{x:a(x)>M0}

a(x)u2_ndx=λn

Z

Bρ(xn)T{x:a(x)>M0}

a(x)_|un−u|dx

>λ_nM₀

Z

Bρ(xn)

|un−u|2dx−M0

Z

Bρ(xn)T{x:a(x)>M0}

|un−u|dx

>λn M0

Z

Bρ(xn)

|un−u|2dx−o(1)

!

→+_∞,

q✉❛♥❞♦ λn →+∞✳ ❖ q✉❡ é ✉♠ ❛❜s✉r❞♦✳ ▲♦❣♦✱ un →u ❡♠ Lp+1(RN)✳ ❆❣♦r❛✱ ✉s❛♥❞♦ ♦ ❧❡♠❛ ❞❡ ❋❛t♦✉ s❡❣✉❡ q✉❡

Z

RN |∇

u_|2 _+u2

dx =

Z

RN |∇

u_|2_{+ (λa(x) + 1)u}2

dx,

♣♦✐s u_|Ωc = 0 ❡a(x) = 0 ❡♠ Ω✳ ❆ss✐♠✱

Z

RN |∇

u_|2(λa(x) + 1) +u2

dx6lim sup

n→+∞

Z

RN |∇

un|2(λa(x) + 1) +u2n

dx

= lim sup

n→+∞

Z

RN|

un|p+1dx=

Z

RN |

u_|p+1dx

P♦rt❛♥t♦✱

Z

RN |∇

u_|2+u2

dx6

Z

RN|

u_|p+1dx.

❈♦♠♦ u_|Ωc= 0✱ t❡♠♦s

Z

Ω |∇

u_|2+u2

dx6

Z

Ω|

u_|p+1dx.

❆ss✐♠✱ ❡①✐st❡ α_∈Mλ,p t❛❧ q✉❡

Z

Ω |∇

(αu)_|2_{+ (αu)}2

dx=

Z

Ω|

αu_|p+1_dx.

■♠♣❧✐❝❛♥❞♦ q✉❡

Φp,Ω(αu) =

p₋1 2(p+ 1)

Z

Ω

(_|∇(αu)_|2+ (αu)2)dx= p−1 2(p+ 1)

Z

RN

(_|∇(αu)_|2 + (αu)2)dx

6 p−1

2(p+ 1)

Z

❴RN_(|∇u|2_+u2_)dx

6lim inf

n→∞

p₋1 2(p+ 1)

Z

RN

(_|∇un|2+ (λa(x) + 1)u2n)dx=A.

▲♦❣♦✱ A>c(p,Ω)✳ P♦rt❛♥t♦ lim

λ→∞cλ,p =c(p,Ω)✳

▲❡♠❛ ✷✳✷✳✶✶✳ ❊①✐st❡p1 ∈(1,2∗−1)t❛❧ q✉❡ ♣❛r❛ t♦❞♦p∈[p1,2∗−1)✱ ❡①✐st❡Λ1(p)>Λ0(p) t❛❧ q✉❡