Existência e concentração de soluções para uma equação de schrödinger estacionária

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❊①✐stê♥❝✐❛ ❡ ❈♦♥❝❡♥tr❛çã♦ ❞❡ ❙♦❧✉çõ❡s ♣❛r❛

✉♠❛ ❊q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❊st❛❝✐♦♥ár✐❛

❏♦♥❛s ❆♥t♦♥✐♦ P❛❞♦✈❛♥✐ ❊❞❡r❧✐

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼❛r❝♦s ❚❛❞❡✉ ❞❡ ❖❧✐✈❡✐r❛ P✐♠❡♥t❛ ❈♦♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❙✉❡tô♥✐♦ ❞❡ ❆❧♠❡✐❞❛ ▼❡✐r❛

Pr♦❣r❛♠❛✿ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧

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❯◆■❱❊❘❙■❉❆❉❊ ❊❙❚❆❉❯❆▲ P❆❯▲■❙❚❆

❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ❡ ❚❡❝♥♦❧♦❣✐❛ ❞❡ Pr❡s✐❞❡♥t❡ Pr✉❞❡♥t❡

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧

❊①✐stê♥❝✐❛ ❡ ❈♦♥❝❡♥tr❛çã♦ ❞❡ ❙♦❧✉çõ❡s ♣❛r❛

✉♠❛ ❊q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❊st❛❝✐♦♥ár✐❛

❏♦♥❛s ❆♥t♦♥✐♦ P❛❞♦✈❛♥✐ ❊❞❡r❧✐

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼❛r❝♦s ❚❛❞❡✉ ❞❡ ❖❧✐✈❡✐r❛ P✐♠❡♥t❛ ❈♦♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❙✉❡tô♥✐♦ ❞❡ ❆❧♠❡✐❞❛ ▼❡✐r❛

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠á✲ t✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧ ❞❛ ❋❛❝✉❧✲ ❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ❡ ❚❡❝♥♦❧♦❣✐❛ ❞❛ ❯◆❊❙P ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧✳

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FICHA CATALOGRÁFICA

Padovani Ederli, Jonas Antonio.

P138e Existência e concentração de soluções para uma Equação de Schrödinger Estacionária / Jonas Antonio Padovani Ederli. - Presidente Prudente : [s.n], 2015

46 f. : il.

Orientador: Marcos Tadeu de Oliveira Pimenta

Dissertação (mestrado) - Universidade Estadual Paulista, Faculdade de Ciências e Tecnologia

Inclui bibliografia

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s ♣♦r t❡r ♠❡ ❞❛❞♦ ❛ ❣r❛ç❛ ❞❡ ❝♦♥❝❧✉✐r ♠❛✐s ❡st❛ ❡t❛♣❛ ❞❛ ♠✐♥❤❛ ✈✐❞❛ ❡ ♣♦r t❡r ♣❡r♠✐t✐❞♦ ❝♦♥❤❡❝❡r ♣❡ss♦❛s ❡s♣❡❝✐❛✐s q✉❡ ♠❡ ❛❥✉❞❛r❛♠ ♠✉✐t♦ ❞✉r❛♥t❡ ❡st❡ ♣❡rí♦❞♦✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♣❛✐s ❆♥t♦♥✐♦ ❡ ▼❛r✐❛ ❙✉❡❧✐ ♣❡❧♦ ❛♠♦r q✉❡ s❡♠♣r❡ ❞❡♠♦♥str❛r❛♠ ♣♦r ♠✐♠ ❡ ♣❡❧❛ ❡❞✉❝❛çã♦ só❧✐❞❛ ❡ ❝r✐stã q✉❡ ♠❡ ❞❡r❛♠✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ✐r♠ã♦s ❡ à ♠✐♥❤❛ ❢❛♠í❧✐❛ ♥♦ ❣❡r❛❧ q✉❡ s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❛♠ ❡♠ ❝❛❞❛ ♣❛ss♦ ❞❡st❛ ❝♦♥q✉✐st❛✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦ ●✉✐❧❤❡r♠❡✱ ❋❡r♥❛♥❞♦✱ ❉❛♥✐❧♦ ✭❑✉rt✮✱ ❊❧t♦♥✱ ▲❡♦♥❛r❞♦✱ ●✉st❛✈♦✱ ❱✐♥í❝✐✉s✱ ❉♦✉❣❧❛s ✭❨✉❣✐✮✱ ❏✉♥✐♦r✱ ❍❡❧♦ís❛✱ ❈r✐s❧❛✐♥❡✱ ❆❞r✐❛♥♦✱ ❈✐♥t✐❛✱ ❘❛❢❛❡❧ ✭❈❛st❛♥❤❛✮✱ ❘❛❢❛❡❧ ✭Pã♦✮✱ ■r✐♥❡✉ ✭P♦✇❡r❢❡r❛✮✱ ❏♦sé ❱❛♥t❡r❧❡r ✭P❛♥❝❛❞❛✮ ❡ t♦❞♦s ❛q✉❡❧❡s q✉❡ ❞❡ ❛❧❣✉♠❛ ♠❛♥❡✐r❛ ❝♦♥tr✐❜✉✐r❛♠ ♣❛r❛ ♦ ❜♦♠ ❛♥❞❛♠❡♥t♦ ❞❡st❡ tr❛✲ ❜❛❧❤♦✱ ♠❡ ❛❥✉❞❛♥❞♦ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡✳ ❙♦✉ ❣r❛t♦✱ s♦❜r❡t✉❞♦✱ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦ ♣♦r ❡❧❡s ♣r♦♣♦r❝✐♦♥❛❞♦s✳

❆❣r❛❞❡ç♦ ♣❡❧♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ❞❛ ❣r❛❞✉❛çã♦ ❡ ❞♦ ♠❡str❛❞♦ q✉❡ ♥ã♦ ♠❡❞✐r❛♠ ❡s❢♦r✲ ç♦s ♣❛r❛ ♠❡ ❡♥s✐♥❛r✳ ❚❡♥❤♦ ♣❧❡♥❛ ❝❡rt❡③❛ ❞❡ q✉❡ s❡♠ ❡❧❡s ♥❛❞❛ ❞✐ss♦ s❡r✐❛ ♣♦ssí✈❡❧✳

❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ▼❛r❝♦s ❚❛❞❡✉ ❞❡ ❖❧✐✈❡✐r❛ P✐♠❡♥t❛ ♣❡❧❛ s✉❛ ✐♥✜♥✐t❛ ♣❛❝✐✲ ê♥❝✐❛ ❡ ♣r❡♦❝✉♣❛çã♦ ❝♦♠✐❣♦✱ ♣❡❧❛ ❞❡❞✐❝❛çã♦ ✐♥t❡❣r❛❧ ❡♠ ♠❡ ❛t❡♥❞❡r ❡ t✐r❛r ♠✐♥❤❛s ❞ú✈✐❞❛s ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣♦r t❡r ♠❡ ❞❛❞♦ ♦ ❡①❡♠♣❧♦ ❞♦ q✉❡ é s❡r ✉♠ ❡①❝❡❧❡♥t❡ ♣r♦✜ss✐♦♥❛❧✳

❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s ❙✉❡tô♥✐♦ ❞❡ ❆❧♠❡✐❞❛ ▼❡✐r❛ ✭❝♦♦r✐❡♥t❛❞♦r✮ ❡ ❘♦❜❡rt♦ ❞❡ ❆❧✲ ♠❡✐❞❛ Pr❛❞♦ q✉❡ ❝♦♥tr✐❜✉✐r❛♠ s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡ ❝♦♠ ❛s ❞✐❝❛s ❡ ❝♦♠ ❛s ❝♦rr❡çõ❡s ✈❛❧✐♦s❛s✳

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❘❡s✉♠♦

◆❡ss❡ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ♣❛r❛ ✉♠❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❡st❛❝✐♦♥ár✐❛ ♥ã♦✲❧✐♥❡❛r✱ q✉❛♥❞♦ ✉♠ ♣❛râ♠❡tr♦ t❡♥❞❡ ❛ ③❡r♦✳ ▼❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ♣r♦✈❛♠♦s q✉❡ q✉❛♥❞♦ ♦ ♣❛râ♠❡tr♦ t❡♥❞❡ ❛ ③❡r♦✱ ❛ s❡q✉ê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♦❜t✐❞❛s ♣♦ss✉✐ ✉♠ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ q✉❡ t❡♥❞❡ ❛ s❡ ❝♦♥❝❡♥tr❛r ❡♠ t♦r♥♦ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❣❧♦❜❛❧ ❞♦ ♣♦t❡♥❝✐❛❧✳ ❆ té❝♥✐❝❛ ✉t✐❧✐③❛❞❛ ❝♦♥s✐st❡ ♥❛ ✉t✐❧✐③❛çã♦ ❞❡ ♠ét♦❞♦s ✈❛r✐❛❝✐♦♥❛✐s ♣❛r❛ ❝♦♠♣❛r❛r ❛s s♦❧✉çõ❡s ♦❜t✐❞❛s ❝♦♠ ❛ s♦❧✉çã♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❧✐♠✐t❡ q✉❡ ❡♥✈♦❧✈❡ ♦ ✈❛❧♦r ❞❡ ♠í♥✐♠♦ ❞♦ ♣♦t❡♥❝✐❛❧✳

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❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② s♦♠❡ r❡s✉❧ts ❛❜♦✉t ❡①✐st❡♥❝❡ ❛♥❞ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ❢♦r ❛ ♥♦♥❧✐♥❡❛r st❛t✐♦♥❛r② ✈❡rs✐♦♥ ♦❢ t❤❡ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥✱ ❛s ❛ ♣❛r❛♠❡t❡r ❣♦❡s t♦ ③❡r♦✳ ▼♦r❡ s♣❡❝✐✜❝❛❧❧②✱ ✇❡ ♣r♦✈❡ t❤❛t t❤❡ s❡q✉❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s ❤❛✈❡ ❛ ♠❛①✐♠✉♠ ♣♦✐♥ts ✇❤✐❝❤ ❝♦♥❝❡♥tr❛t❡ ❛r♦✉♥❞ t❤❡ ❣❧♦❜❛❧ ♠✐♥✐♠✉♠ ♦❢ t❤❡ ♣♦t❡♥t✐❛❧✱ ❛s ❛ ♣❛r❛♠❡t❡r ❣♦❡s t♦ ③❡r♦✳ ❚❤❡ t❡❝❤♥✐q✉❡ ✉s❡❞ r❡❧✐❡s ♦♥ ✈❛r✐❛t✐♦♥❛❧ ♠❡t❤♦❞s t♦ ❝♦♠♣❛r❡ t❤❡ s♦❧✉t✐♦♥s ✇✐t❤ t❤❡ s♦❧✉t✐♦♥ ♦❢ ❛ ❧✐♠✐t ♣r♦❜❧❡♠ ✇❤✐❝❤ ❤❛✈❡ ✐♥❢♦r♠❛t✐♦♥ ♦♥ t❤❡ ♠✐♥✐♠✉♠ ♦❢ t❤❡ ♣♦t❡♥t✐❛❧✳

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❙✉♠ár✐♦

❘❡s✉♠♦ ✺

❆❜str❛❝t ✼

❈❛♣ít✉❧♦s

✶ ■♥tr♦❞✉çã♦ ✶✶

✷ Pr❡❧✐♠✐♥❛r❡s ✶✸

✷✳✶ ❖s ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ❖ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✸ ❘❡s✉❧t❛❞♦s ❞❡ ❊①✐stê♥❝✐❛ ✶✼

✹ ❘❡s✉❧t❛❞♦s ❞❡ ❈♦♥❝❡♥tr❛çã♦ ✸✺

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❈❛♣ít✉❧♦

✶

■♥tr♦❞✉çã♦

❉❡ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ♥❛ ❢ís✐❝❛✲♠❛t❡♠át✐❝❛ é ❛ ✈❡rsã♦ ❡st❛❝✐♦♥ár✐❛ ❞❛ ❡q✉❛çã♦ ❞❡ ❙❝❤rö✲ ❞✐♥❣❡r ♥ã♦✲❧✐♥❡❛r✱ ♦✉ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱

  

−ǫ2_∆_u₊_V₍_x₎_u₌_f₍_u₎ _❡♠ _RN

u∈H1₍_RN₎

u >0.

✭✶✳✶✮

❆ ❡q✉❛çã♦ ✭✶✳✶✮ ❢♦✐ ❡st✉❞❛❞❛ ♣♦r ❘❛❜✐♥♦✇✐t③ ❡♠ [✶✸]✱ ♦♥❞❡ ❢♦✐ ♣r♦✈❛❞♦ ✉♠ r❡s✉❧t❛❞♦

❞❡ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ✉s❛♥❞♦ ♣✐♦♥❡✐r❛♠❡♥t❡ ♠ét♦❞♦s ♣✉r❛♠❡♥t❡ ✈❛r✐❛❝✐♦♥❛✐s✱ s✉♣♦♥❞♦ q✉❡ ❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡f s❡ ❝♦♠♣♦rt❛ ❝♦♠♦ ✉♠❛ ♣♦tê♥❝✐❛ s✉❜❝rít✐❝❛ ❡ ♦ ♣♦t❡♥❝✐❛❧V s❛t✐s❢❛③

✉♠❛ ❝♦♥❞✐çã♦ ❣❧♦❜❛❧✳ ❆♣ós ✐ss♦✱ ❲❛♥❣ ❡♠ [✶✺]✱ ♣r♦✈♦✉ q✉❡ ♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮ ❝♦♠ ❛ ♥ã♦✲

❧✐♥❡❛r✐❞❛❞❡f(u) = |u|p−1_u_{✱ ❛❞♠✐t❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ s♦❧✉çõ❡s q✉❡ s❡ ❝♦♥❝❡♥tr❛♠ ❡♠ t♦r♥♦}

❞♦ ♠í♥✐♠♦ ❣❧♦❜❛❧ ❞♦ ♣♦t❡♥❝✐❛❧ V✳ ❆♥t❡s ❞❡ss❡s✱ r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❡ ❝♦♥❝❡♥tr❛çã♦

❞❡ s♦❧✉çõ❡s ♣❛r❛ ✭✶✳✶✮ ❤❛✈✐❛♠ s✐❞♦ ♣r♦✈❛❞♦s ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♣♦r ❋❧♦❡r ❡ ❲❡✐♥st❡✐♥ ❡♠ ❬✺❪ ♣❛r❛ ♦ ❝❛s♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ❡ ❞❡♣♦✐s ❣❡♥❡r❛❧✐③❛❞♦s ♣❛r❛ ❞✐♠❡♥sõ❡s ♠❛✐s ❛❧t❛s ♣♦r ❖❤ ❡♠ ❬✶✵❪✳

❖✉tr♦ tr❛❜❛❧❤♦ ❜❛st❛♥t❡ ✐♠♣♦rt❛♥t❡ ♥♦ ❡st✉❞♦ ❞❡ss❡ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛ é ♦ ❛rt✐❣♦ ❞❡ ❉❡❧ P✐♥♦ ❡ ❋❡❧♠❡r ❬✹❪✱ ♦♥❞❡ ♦s ❛✉t♦r❡s ❛❜♦r❞❛♠ ♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮ ❝♦♠ ✉♠❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ ❞♦ t✐♣♦ ♣♦tê♥❝✐❛ s✉❜❝rít✐❝❛ ❡ ♣♦t❡♥❝✐❛❧ s❛t✐s❢❛③❡♥❞♦ ✉♠❛ ❝♦♥❞✐çã♦ q✉❡ ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦ ✉♠❛ ✈❡rsã♦ ❧♦❝❛❧ ❞❛ ❝♦♥❞✐çã♦ s✉♣♦st❛ ♣♦r ❲❛♥❣ ❡♠ ❬✶✺❪✳ ◆❡ss❡ tr❛❜❛❧❤♦✱ ♦s ❛✉t♦r❡s ✐♥✲ tr♦❞✉③❡♠ ✉♠❛ té❝♥✐❝❛ q✉❡ ✜❝♦✉ ❝♦♥❤❡❝✐❞❛ ♣♦r ▼ét♦❞♦ ❞❡ P❡♥❛❧✐③❛çã♦✱ ❛ q✉❛❧ ✈❡♠ s❡♥❞♦ ❧❛r❣❛♠❡♥t❡ ✉t✐❧✐③❛❞❛ ❛té ♦s ❞✐❛s ❛t✉❛✐s✳

●❡♥❡r❛❧✐③❛çõ❡s ❛❝❡r❝❛ ❞♦s r❡s✉❧t❛❞♦s ❞❡ ❘❛❜✐♥♦✇✐t③✱ ❲❛♥❣✱ ❉❡❧ P✐♥♦ ❡ ❋❡❧♠❡r ❡ ♦✉tr♦s✱ ✈ê♠ s❡♥❞♦ ❞❡s❡♥✈♦❧✈✐❞♦s ♣♦r ✈ár✐♦s ❛✉t♦r❡s✱ ❡♥✈♦❧✈❡♥❞♦ ❤✐♣ót❡s❡s ♠❛✐s ❣❡r❛✐s s♦❜r❡ ❛ ♥ã♦✲ ❧✐♥❡❛r✐❞❛❞❡f ❡ ♣♦t❡♥❝✐❛❧V✱ ❜❡♠ ❝♦♠♦ t❛♠❜é♠ ♣❛r❛ ♦✉tr♦s ♦♣❡r❛❞♦r❡s ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦

♦ p−❧❛♣❧❛❝✐❛♥♦✱ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❆❧✈❡s ❡ ❋✐❣✉❡✐r❡❞♦ ❡♠ ❬✷❪ ❡ ❜✐❤❛r♠ô♥✐❝♦ ❞❡ P✐♠❡♥t❛ ❡ ❙♦❛r❡s ❬✶✶✱ ✶✷❪✱ ❡♥tr❡ ♦✉tr♦s✳

◆❡st❡ tr❛❜❛❧❤♦✱ ❢❛r❡♠♦s ✉♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ ❞♦s tr❛❜❛❧❤♦s[✶✸]❡[✶✺]✱ ♦♥❞❡ s❡ ❡st✉❞❛

♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮ ❝♦♠ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡f ❡ ♣♦t❡♥❝✐❛❧V s❛t✐s❢❛③❡♥❞♦ ♦ s❡❣✉✐♥t❡ ❝♦♥❥✉♥t♦ ❞❡

❤✐♣ót❡s❡s✳

(V1) V ∈C0(RN)❀

(V2) 0< V0 = inf

RNV <lim inf_|_x_|→₊_∞V❀

(17)

✶✳ ■♥tr♦❞✉çã♦ ✶✷

(f1) f ∈C1(R)❀

(f2) f(0) =f′(0) = 0❀

(f3) ❡①✐st❡♠ ❝♦♥st❛♥t❡s c1, c2 >0 ❡ p ∈(1,2∗−1)✱ t❛✐s q✉❡ |f(s)| ≤c1|s|+c2|s|p✱ ♣❛r❛

t♦❞♦s ∈R✱ ♦♥❞❡₂∗ ₌ 2N

N−2✱

(f4) ❊①✐st❡θ > 2t❛❧ q✉❡

0< θF(s)≤f(s)s,

♣❛r❛ t♦❞♦s ∈R_\{₀_}✱ ♦♥❞❡ _F₍_s_{) =}

Z s

0

f(t)dt❀

(f5)

f(s)

s é ❝r❡s❝❡♥t❡ ♣❛r❛ s >0✳

❖s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❡ss❡ tr❛❜❛❧❤♦ sã♦ ♦s s❡❣✉✐♥t❡s t❡♦r❡♠❛s✳

❚❡♦r❡♠❛ ✶ ❙✉♣♦♥❤❛ q✉❡ (f1)−(f5)✱ (V1) ❡ (V2) ✈❛❧❤❛♠✳ ❊♥tã♦ ❡①✐st❡ ǫ0 > 0 t❛❧ q✉❡

♣❛r❛ 0< ǫ < ǫ0✱ ❡①✐st❡ uǫ s♦❧✉çã♦ ❞❡ ✭✶✳✶✮ t❛❧ q✉❡ Iǫ(uǫ) =cǫ✳

❚❡♦r❡♠❛ ✷ ❙❡❥❛♠ V s❛t✐s❢❛③❡♥❞♦(V1) ❡ (V2)❡ f(s) =|s|p−1s ♦♥❞❡ 1< p < N_N+2₋₂✳ ❊♥tã♦

♣❛r❛ t♦❞❛ s❡q✉ê♥❝✐❛ ǫm → 0✱ ❡①✐st❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ q✉❡ ❝♦♥t✐♥✉❛r❡♠♦s ❛ ❞❡♥♦t❛r ♣♦r

(ǫm) t❛❧ q✉❡ ✭✶✳✶✮ ✭❝♦♠ ǫm ♥♦ ❧✉❣❛r ❞❡ ǫ✮ ♣♦ss✉✐ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ um ∈ H1(RN)

❡ um s❡ ❝♦♥❝❡♥tr❛ ❡♠ ✉♠ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❣❧♦❜❛❧ x0 ❞❡ V ♥♦ s❡❣✉✐♥t❡ s❡♥t✐❞♦✿ P❛r❛

❝❛❞❛ m > 0 s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ um ♣♦ss✉✐ s♦♠❡♥t❡ ✉♠ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❧♦❝❛❧ xm

✭♣♦rt❛♥t♦✱ ❣❧♦❜❛❧✮✱ ❝♦♠xm →x0✱ q✉❛♥❞♦ m→ ∞✱ ❡ ♣❛r❛ t♦❞♦ δ >0❡ m s✉✜❝✐❡♥t❡♠❡♥t❡

❣r❛♥❞❡✱

max

|x−x0|≤δ

um(x)>(V0) 1

1−p. ✭✶✳✷✮

◆❛ ❞❡♠♦♥str❛çã♦ ❞❡ ❛♠❜♦s✱ ❡♠♣r❡❣❛♠♦s ♠ét♦❞♦s ✈❛r✐❛❝✐♦♥❛✐s ❡ ✉t✐❧✐③❛♠♦s ♦s ❛r❣✉✲ ♠❡♥t♦s ❞❡ ❘❛❜✐♥♦✇✐t③ ♣❛r❛ ❛ ♣r♦✈❛ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮✱ ♣❛r❛ ✈❛❧♦r❡s ❞❡ ǫ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦s✳

❯♠❛ ✈❡③ ♦❜t✐❞❛s ❛s s♦❧✉çõ❡s✱ ♠♦str❛♠♦s q✉❡ ♦s ♥í✈❡✐s ♠✐♥✐♠❛① ❛ss♦❝✐❛❞♦s ❛♦ ♣r♦❜❧❡♠❛

(1.1)❝♦♥✈❡r❣❡♠ ♣❛r❛ ♦ ♥í✈❡❧ ♠✐♥✐♠❛① ❞♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ❧✐♠✐t❡

  

−∆u+V0u = f(u) ❡♠ RN

u∈H1₍_RN

)

u >0.

■st♦✱ ♣♦r s✉❛ ✈❡③✱ ♥♦s ♣❡r♠✐t❡ ♠♦str❛r ❛ ❝♦♥❝❡♥tr❛çã♦ ❞❛s s♦❧✉çõ❡s ❡♠ t♦r♥♦ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❞❡ V ✉t✐❧✐③❛♥❞♦✲s❡ ❞♦s ❛r❣✉♠❡♥t♦s ❞❡ ❲❛♥❣✳

◆♦ss❛ ♣r✐♥❝✐♣❛❧ ❝♦♥tr✐❜✉✐çã♦ é ❞❡ ❝❛rát❡r ❡str✐t❛♠❡♥t❡ ♣❡❞❛❣ó❣✐❝♦ ♥♦ s❡♥t✐❞♦ ❞❡ ❢❛✲ ❝✐❧✐t❛r ❛ ❧❡✐t✉r❛ ❞♦s ❛rt✐❣♦s [✶✸] ❡ [✶✺]✱ ♣❛r❛ ✐♥✐❝✐❛♥t❡s ♥❛ ár❡❛ ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s

(18)

❈❛♣ít✉❧♦

✷

Pr❡❧✐♠✐♥❛r❡s

✷✳✶ ❖s ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈

❙❡❥❛ Ω ⊂ RN ✉♠ ❞♦♠í♥✐♦ q✉❛❧q✉❡r✱ ❧✐♠✐t❛❞♦ ♦✉ ♥ã♦✳ ❈♦♠❡ç❛r❡♠♦s ❡st❡ ❝❛♣ít✉❧♦

❞❡✜♥✐♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡r✐✈❛❞❛ ❢r❛❝❛ ❞❡ ✉♠❛ ❢✉♥çã♦✳

❉❡✜♥✐çã♦ ✶ ❯♠ ♠✉❧t✐✲í♥❞✐❝❡ α é ✉♠❛ ♥✲✉♣❧❛ (α1, ..., αN)✱ ♦♥❞❡ αi ∈ N✱ ♣❛r❛ t♦❞♦ 0 <

i≤n✳ ❚❡♠♦s ❛ss♦❝✐❛❞♦ ❛♦ ♠✉❧t✐✲í♥❞✐❝❡ α ❛❧❣✉♥s sí♠❜♦❧♦s✱ ✉♠ ❞❡❧❡s é Dα₌ ∂|α|

∂α1

x₁...∂αN_xN ✱

♦♥❞❡ |α|=α1+α2+...+αN✱ ❝❤❛♠❛❞♦ ❞❡ ♦r❞❡♠ ❞♦ ♠✉❧t✐✲í♥❞✐❝❡ α✳

❉❡✜♥✐çã♦ ✷ ❙❡❥❛u ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ ❡♠Ω✱ ♦✉ s❡❥❛✱ ♣❛r❛ ❝❛❞❛ s✉❜❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦

K ⊂ Ω✱ t❡♠♦s q✉❡

Z

K

|u|dx < ∞ ❡ ❝♦♥s✐❞❡r❡ α ✉♠ ♠✉❧t✐✲í♥❞✐❝❡ q✉❛❧q✉❡r✳ ❊♥tã♦ ✉♠❛

❢✉♥çã♦v ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ é ❝❤❛♠❛❞❛ ❞❡ α✲és✐♠❛ ❞❡r✐✈❛❞❛ ❢r❛❝❛ ❞❡ u s❡ s❛t✐s❢❛③ Z

Ω

ϕvdx= (−1)|α|Z

Ω

uDα_ϕdx _{✭✷✳✶✮}

♣❛r❛ t♦❞❛ϕ ∈C₀|α|(Ω)✳ ◆❡ss❡ ❝❛s♦ ❞❡♥♦t❛♠♦s v =Dα_u_✳

❉✐③❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ é ❢r❛❝❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡❧ s❡ ❛ s✉❛ ❞❡r✐✈❛❞❛ ❢r❛❝❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❡①✐st❡ ❡ ❞✐r❡♠♦s q✉❡ ❡❧❛ é k ✈❡③❡s ❢r❛❝❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡❧ s❡ s✉❛ ❞❡r✐✈❛❞❛ ❢r❛❝❛

❛té ❛ ♦r❞❡♠ k ❡①✐st❡✳ ❱❛♠♦s ❞❡♥♦t❛r ♦ ❡s♣❛ç♦ ❧✐♥❡❛r ❞❛s ❢✉♥çõ❡s k ✈❡③❡s ❢r❛❝❛♠❡♥t❡

❞✐❢❡r❡♥❝✐á✈❡✐s ❡♠ Ω ♣♦r Wk_(Ω)_{✳ ◆♦t❡ q✉❡} _Ck_(Ω) _⊂ _Wk_(Ω) _{❡ q✉❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡r✐✈❛❞❛}

❢r❛❝❛ é ✉♠❛ ❡①t❡♥sã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡r✐✈❛❞❛ ❝❧áss✐❝❛ q✉❡ ♣r❡s❡r✈❛ ❛ ✈❛❧✐❞❛❞❡ ❞❛ ✐♥t❡❣r❛çã♦ ♣♦r ♣❛rt❡s ✭✷✳✶✮✳

❉❡✜♥✐çã♦ ✸ ❙❡❥❛♠ Ω ⊂ RN _{✉♠ ❛❜❡rt♦✱} ₁_≤ _p _{≤ ∞} _❡ _k _∈ _N_{✳ ❉❡✜♥✐♠♦s ♦s ❡s♣❛ç♦s ❞❡}

❙♦❜♦❧❡✈Wk,p_(Ω) _{❝♦♠♦ s❡♥❞♦}

Wk,p(Ω) :={u∈Lp(Ω); Dαu∈Lp(Ω), para 0≤ |α| ≤k} ❖❜s❡r✈❛çã♦ ✶ ❖ ❡s♣❛ç♦Wk,p_(Ω) _{é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱ ❞♦t❛❞♦ ❞❛ ♥♦r♠❛}

kukk,p=



 X

0≤|α|≤k

kDα_u_kp p

 

1

p

, se 1≤p < ∞ ✭✷✳✷✮

kukk,∞= max

0≤|α|≤kkD α

uk∞, se p=∞. ✭✷✳✸✮

(19)

✷✳ Pr❡❧✐♠✐♥❛r❡s ✶✹

❖❜s❡r✈❛çã♦ ✷ ◗✉❛♥❞♦ p= 2✱ ❞❡♥♦t❛♠♦s Wk,p_(Ω) _{s✐♠♣❧❡s♠❡♥t❡ ♣♦r} _Hk_(Ω)_{✳ ❊♠ ♣❛rt✐✲}

❝✉❧❛r✱ s❡ k= 1✱ t❡♠♦s ♦ ❡s♣❛ç♦

H1(Ω) =W1,2(Ω) ={u∈L2(Ω); ∂u

∂xi

∈L2(Ω), ♣❛r❛ 1≤i≤n}.

❉❡✜♥✐♠♦s t❛♠❜é♠✱ ♦ ❡s♣❛ç♦ W₀k,p(Ω)✱ ❝♦♠♦ s❡♥❞♦

W0k,p(Ω) :=C0∞(Ω)

k.kk,p

.

❚❡♦r❡♠❛ ✸ ❖ s✉❜❡s♣❛ç♦ C∞_(Ω)_∩_Wk,p_(Ω) _{é ❞❡♥s♦ ❡♠} _Wk,p_(Ω)_✳

❱❡r ❞❡♠♦♥str❛çã♦ ❡♠ [✶]✳

❚❡♦r❡♠❛ ✹ ❙❡ Ω s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞♦ ❝♦♥❡ ✐♥t❡r✐♦r ✉♥✐❢♦r♠❡✱ ✐st♦ é✱ ❡①✐st❡ ✉♠ ❝♦♥❡

✜①♦ KΩ t❛❧ q✉❡ ❝❛❞❛ x∈Ω é ♦ ✈ért✐❝❡ ❞❡ ✉♠ ❝♦♥❡ KΩ(x)⊂Ω ❡ ❝♦♥❣r✉❡♥t❡ ❛ KΩ✱ ❡♥tã♦

❡①✐st❡ ✉♠❛ ✐♠❡rsã♦ ❝♦♥tí♥✉❛

Wk,p_(Ω)_֒_→_Lq_(Ω)_{, para} ₁_≤_q_≤ N p

N −kp, onde kp < N, ✭✷✳✹✮

✐st♦ é✱ ❛ ❛♣❧✐❝❛çã♦ ❞❡ ✐♥❝❧✉sã♦ i:Wk,p_(Ω)_→_Lq_(Ω) _{é ❝♦♥tí♥✉❛✳}

❱❡r ❞❡♠♦♥str❛çã♦ ❡♠ [✶]✳

✷✳✷ ❖ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛

◆❡st❛ s❡çã♦ ✈❛♠♦s ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❡ ♣❛r❛ ✐ss♦✱ ❞❡✜♥✐r❡♠♦s ♦❜❥❡t♦s q✉❡ s❡r✈✐rã♦ ❞❡ ♣ré✲r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♣r♦✈❛ ❞❡ss❡✳

❙❡❥❛ E ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ r❡❛❧✳ ❯♠❛ ❛♣❧✐❝❛çã♦I :E →Ré ❝❤❛♠❛❞❛ ❞❡ ❢✉♥❝✐♦♥❛❧✳

P❛r❛ ❢❛③❡r s❡♥t✐❞♦ ♦ q✉❡ ✈❛♠♦s ❡♥t❡♥❞❡r ♣♦r ♣♦♥t♦ ❝rít✐❝♦ ❞❡I✱ ✈❛♠♦s ❞❡✜♥✐r ♦ q✉❡ ✈❡♠

❛ s❡r ✉♠ ❢✉♥❝✐♦♥❛❧ s❡r ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ s❡♥t✐❞♦ ❞❡ ❋ré❝❤❡t✳

❉❡✜♥✐çã♦ ✹ ❉✐③❡♠♦s q✉❡ Ié ❋ré❝❤❡t ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ u ∈ E s❡ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦

❧✐♥❡❛r ❝♦♥tí♥✉❛ L= L(u) : E →R q✉❡ ❝✉♠♣r❡ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦✿ ♣❛r❛ q✉❛❧q✉❡r _{ǫ >} ₀

❞❛❞♦✱ ❡①✐st❡ ✉♠ δ =δ(ǫ, u)>0 t❛❧ q✉❡

|I(u+v)−I(u)−Lv| ≤ǫkvk✱

♣❛r❛ t♦❞♦ v ∈E✱ ❝♦♠ kvk ≤δ✳ ❆ ❛♣❧✐❝❛çã♦ L s❡rá ❞❡♥♦t❛❞❛ ♣♦r I′(u)✳

◆♦t❡ q✉❡ I′₍_u₎_∈_E∗_{✱ ♦♥❞❡}_E∗ _{é ♦ ❡s♣❛ç♦ ❞✉❛❧ ❞❡} _E_✳

❉❡✜♥✐çã♦ ✺ ❯♠ ♣♦♥t♦ ❝rít✐❝♦ u ❞❡ I é ✉♠ ♣♦♥t♦ ❡♠ q✉❡ I′₍_u_{) = 0}_{✱ ♦✉ s❡❥❛✱} _I′₍_u₎_ψ _{= 0} ♣❛r❛ t♦❞❛ ψ ∈E✳ ❖ ✈❛❧♦r ❞❡ I ❡♠ u é ❡♥tã♦ ❝❤❛♠❛❞♦ ❞❡ ✈❛❧♦r ❝rít✐❝♦ ❞❡ I✳

■r❡♠♦s ♣r♦✈❛r ❛❣♦r❛ ♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❡ ♣❛r❛ ✐ss♦ ✉s❛r❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

▲❡♠❛ ✶ ✭▲❡♠❛ ❞❛ ❉❡❢♦r♠❛çã♦✮ ❙❡❥❛ ϕd _:= _ϕ−1_(]_{− ∞}_{, d}_])_✱ _X _{✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✱}

(20)

✷✳ Pr❡❧✐♠✐♥❛r❡s ✶✺

✭✐✮ η(u) =u✱ ♣❛r❛ t♦❞♦ u /∈ϕ−1_([_c₋₂_{ǫ, c}_{+ 2}_ǫ_])_❀

✭✐✐✮ η(ϕc+ǫ₎_⊂_ϕc−ǫ_✳

❚❡♦r❡♠❛ ✺ ✭❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✮ ❙❡❥❛X✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✱ϕ ∈C2₍_X,_R₎_✱

e ∈ X ❡r > 0 t❛✐s q✉❡ kek > r✳ ❈♦♥s✐❞❡r❡ b := inf

kuk=rϕ(u) > ϕ(0) ≥ ϕ(e)✳ ❊♥tã♦✱ ♣❛r❛

❝❛❞❛ ǫ >0✱ ❡①✐st❡ u∈X t❛❧ q✉❡

✭a✮ c−2ǫ≤ϕ(u)≤c+ 2ǫ❀

✭a✮ kϕ′₍_u₎_k_<₂_ǫ_✳ ♦♥❞❡

c= inf

γ∈Γtmax∈[0,1]ϕ(γ(u)) ✭✷✳✺✮

❡ Γ ={γ ∈C([0,1], X); γ(0) = 0 e γ(1) =e}✳

❉❡♠♦♥str❛çã♦✳ ◆♦t❡ q✉❡ b ≤ max

0≤t≤1ϕ(γ(t))✱ ❡ ❡♥tã♦ b ≤ c ≤ 0max≤t≤1ϕ(γ(te))✳ ❙✉♣♦♥❤❛

q✉❡✱ ♣❛r❛ ❛❧❣✉♠ ǫ >0✱ ❛ ❝♦♥❝❧✉sã♦ ❞♦ ❚❡♦r❡♠❛ ♥ã♦ s❡❥❛ ✈á❧✐❞❛✳ P♦❞❡♠♦s ❛ss✉♠✐r q✉❡

c−2ǫ≥ϕ(0)≥ϕ(e). ✭✷✳✻✮

P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ c✱ ❡①✐st❡ γ ∈Γ t❛❧ q✉❡

max

0≤t≤1ϕ(γ(t))≤c+ǫ. ✭✷✳✼✮

❈♦♥s✐❞❡r❡ β:=η◦γ✱ ♦♥❞❡ηé ❞❛❞♦ ❝♦♠♦ ♥♦ ❧❡♠❛ ❛♥t❡r✐♦r✳ P❡❧♦ ✐t❡♠ (i) ❞♦ ▲❡♠❛ ❞❛

❉❡❢♦r♠❛çã♦ ❡ ♣♦r ✷✳✻ t❡♠♦s q✉❡

β(0) =η(γ(0)) =η(0) = 0

❡ q✉❡ β(1) = e✳ ▲♦❣♦✱ t❡♠♦s q✉❡β ∈Γ✳ ❙❡❣✉❡ ❞♦ ✐t❡♠(ii) ❞♦ ▲❡♠❛ ❞❛ ❉❡❢♦r♠❛çã♦ ❡ ❞❡

✷✳✼ q✉❡

c≤ max

0≤t≤1ϕ(β(t))≤c−ǫ✱

(21)

(22)

❈❛♣ít✉❧♦

✸

❘❡s✉❧t❛❞♦s ❞❡ ❊①✐stê♥❝✐❛

◆❡st❛ s❡çã♦✱ ♦ ♥♦ss♦ ♦❜❥❡t✐✈♦ é ♣r♦✈❛r q✉❡ ♦ ♣r♦❜❧❡♠❛

  

−ǫ2_∆_u₊_V₍_x₎_u₌_f₍_u₎ _❡♠ _RN

u∈H1₍_RN₎

u >0.

✭✸✳✶✮

♣♦ss✉✐ s♦❧✉çã♦✱ ♦♥❞❡ N ≥ 3 ❡ f ❡ V s❛t✐s❢❛③❡♠ ❛s ❝♦♥❞✐çõ❡s (f1)−(f5)✱ (V1) ❡ (V2) sã♦

s❛t✐s❢❡✐t❛s✳

❆ ❛❜♦r❞❛❣❡♠ ❝♦♠❡ç❛ ♦❜s❡r✈❛♥❞♦ q✉❡ ♦ ♣r♦❜❧❡♠❛

−ǫ2_∆_u₊_V₍_x₎_u₌_f₍_u₎ _❡♠ _RN

é ❡q✉✐✈❛❧❡♥t❡ ❛♦ ♣r♦❜❧❡♠❛

−∆v+V(ǫx)v =f(v) ❡♠ RN_, ✭✸✳✷✮

♦♥❞❡ ❛s s♦❧✉çõ❡s uǫ ❞❡ ✭✸✳✶✮ ❡ vǫ ❞❡ ✭✸✳✷✮ sã♦ r❡❧❛❝✐♦♥❛❞❛s ♣♦r vǫ(x) = uǫ(ǫx)✳ ❆ss✐♠

❡st✉❞❡♠♦s ♦ ♣r♦❜❧❡♠❛ ✭✸✳✷✮✳

P❛r❛ ❝❛❞❛ ǫ >0 ❞❡✜♥✐♠♦s ♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rtHǫ ⊂H1(RN) ❝♦♠♦ s❡♥❞♦

Hǫ ={u∈H1(RN); kukǫ <∞}✱

♦♥❞❡

k.kǫ :Hǫ 7−→R

❞❡✜♥❡ ✉♠❛ ♥♦r♠❛ ❡♠Hǫ ❞❛❞❛ ♣♦r

kukǫ =

Z

RN

(|∇u|2+V(ǫx)u2dx 1

2

✱

q✉❡ ✈❡♠ ❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦

hu, viǫ =

Z

RN

(∇u∇v+V(ǫx)uv)dx✳

P♦r (V1)✱ ♣❛r❛ N >2 t❡♠♦s ❛s ✐♠❡rsõ❡s ❝♦♥tí♥✉❛s✿

Hǫ ֒→H1(RN)֒→Lp(RN)✱ ♣❛r❛ 2≤p≤2∗✳

❚❡♠♦s ❛ss♦❝✐❛❞♦ à ❡q✉❛çã♦ ✭✸✳✷✮✱ ♦ ❢✉♥❝✐♦♥❛❧ ❞❛❞♦ ♣♦r

(23)

✸✳ ❘❡s✉❧t❛❞♦s ❞❡ ❊①✐stê♥❝✐❛ ✶✽

Iǫ(u) =

1 2

Z

RN

(|∇u|2+V(ǫx)u2)dx−

Z

RN

F(u)dx

♣❛r❛ u∈Hǫ✳

Pr♦✈❛✲s❡ q✉❡ Iǫ ∈C1(Hǫ,R)✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s q✉❡Iǫ(0) = 0✳

▲❡♠❛ ✷ ❖ ❢✉♥❝✐♦♥❛❧ Iǫ s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s ❣❡♦♠étr✐❝❛s ❞♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥✲

t❛♥❤❛✱ ♦✉ s❡❥❛✿

✭✐✮ ❡①✐st❡♠ ❝♦♥st❛♥t❡s ρ✱ α >0 t❛✐s q✉❡ Iǫ|∂Bρ > α✱ ❡

✭✐✐✮ ❡①✐st❡ ✉♠ e∈Hǫ\Bρ t❛❧ q✉❡ Iǫ(e)<0✳

❉❡♠♦♥str❛çã♦✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ♣r♦✈❡♠♦s ♦ ✐t❡♠ (ii)✳ ◆♦t❡ q✉❡✱ ♣❛r❛ u ∈ Hǫ \ {0} ❡

t >0✱ ❡①✐st❡ r >0 t❛❧ q✉❡

|{x∈RN_; _|_tu₍_x₎_|_{> r}_}|_>₀ ✭✸✳✸✮

♦♥❞❡ |X| ❞❡♥♦t❛ ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❞♦ ❝♦♥❥✉♥t♦X✳

❉❡ ❢❛t♦✱ s❡ |{x ∈ RN_; _|_tu₍_x₎_| _{> r}_}| _{= 0} ♣❛r❛t♦❞♦ _{r >} ₀✱ t❡rí❛♠♦s q✉❡ _|_tu₍_x₎_| _{= 0}

q✳t✳♣✱ ♦ q✉❡ ❝♦♥tr❛r✐❛ ♦ ❢❛t♦ ❞❡ u6= 0 ❡♠ Hǫ✳ ❊♥tã♦✱

Z

RN

F(tu)dx≥

Z

{x∈RN_;_|_tu₍_x₎_|_>r_}

F(tu)dx✳

▲♦❣♦✱ ♣♦r ✭✸✳✸✮ ❡ ♣❡❧❛ ❝♦♥❞✐çã♦ (f4)✱ s❡❣✉❡ q✉❡

Iǫ(tu) =

t2

2kuk

2

ǫ −

Z

RN

F(tu)dx≤ t

2

2kuk

2

ǫ −

Z

{x∈RN_;_|_tu₍_x₎_|_>r_}

F(tu)dx

≤ t

2

2kuk

2

ǫ −a3tµ

Z

{x∈RN_;_|_tu₍_x₎_|_>r_}

|u|µ_dx_{−→ −∞}_✱

q✉❛♥❞♦ t−→ ∞ ❡ ❞❡ss❛ ❢♦r♠❛✱ ♦ ✐t❡♠(ii) ❡stá ♣r♦✈❛❞♦✳

❆❣♦r❛✱ ♣r♦✈❡♠♦s ♦ ✐t❡♠ (i)✳

P♦r (f2)✱ t❡♠♦s q✉❡ ♣❛r❛ t♦❞♦η >0✱ ❡①✐st❡ δ >0 t❛❧ q✉❡ s❡|t|< δ✱

|F(t)| ≤ η

2|t|

2 _{✭✸✳✹✮}

◆♦t❡ q✉❡✱ ♣♦r (f4)✱ ❡①✐st❡ A=A(η)>0 t❛❧ q✉❡✱ s❡|t| ≥δ✱

|F(t)| ≤A(η)|t|p+1 _{✭✸✳✺✮}

❉❡ ❢❛t♦✱ ❝♦♠♦

|F(t)| ≤ |t||f(t)| ≤ |t|(c1|t|+c2|t|p) =c1|t|2+c2|t|p+1✱

t❡♠♦s q✉❡

|F(t)| |t|p+1 ≤

c1|t|2+c2|t|p+1

|t|p+1 =

c1

|t|p−1 +c2 ≤

c1

(24)

✸✳ ❘❡s✉❧t❛❞♦s ❞❡ ❊①✐stê♥❝✐❛ ✶✾

❉❡ ✭✸✳✹✮ ❡ ✭✸✳✺✮✱ s❡❣✉❡ q✉❡

|F(t)| ≤ η

2|t|

2₊_A_|_t_|p+1_✳

♣❛r❛ t♦❞♦t≥0✳

❋❛③❡♥❞♦J(u) =

Z

RN

F(t)dx✱ ♣❡❧❛s ✐♠❡rsõ❡s ❝♦♥tí♥✉❛s ❞❡ ❙♦❜♦❧❡✈✱ t❡♠♦s q✉❡

|J(u)|=

Z RN

F(u)

dx≤ Z RN _η

2|u|

2₊_A_|_u_|p+1_dx₌ η

2kuk

2

L2 +Akuk

p+1

Lp+1

≤C η₂kuk2

ǫ +Akukpǫ+1

=Ckuk2

ǫ η

2 +Akuk

p−1

ǫ

_✳

❊♥tã♦✱ t♦♠❛♥❞♦kukǫ < ₂η_A

1

p−1✱ t❡♠♦s

|J(u)| ≤ηCkuk2

ǫ ✭✸✳✻✮

❈♦♠♦

Iǫ(u) = 1₂kuk2ǫ −J(u)✱

♣♦r ✭✸✳✻✮ s❡❣✉❡ q✉❡

Iǫ(u) = 1₂kuk2ǫ −J(u)≥ 1₂kuk2ǫ −Cηkuk2ǫ =kuk2ǫ 1₂ −Cη

✳

▲♦❣♦✱ ❡s❝♦❧❤❡♥❞♦ η >0❞❡ ♠♦❞♦ q✉❡ 1₂ −Cη >0✱ s❡ kukǫ =ρ✱ t❡♠♦s q✉❡

Iǫ(u)≥α✱

♦♥❞❡α :=ρ2 1

2 −Cη

✳

❆ s❡❣✉✐r✱ ❛❧é♠ ❞❡ ❞❡✜♥✐r ♦ q✉❡ ✈❡♠ ❛ s❡r ❛ ❱❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐✱ ✈❛♠♦s t❛♠❜é♠ ❛♣r❡✲ s❡♥t❛r ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ♠✉✐t♦ ✐♥t❡r❡s❛♥t❡ ❛ r❡s♣❡✐t♦ ❞❡❧❛✳ ▼♦str❛r❡♠♦s q✉❡ ❛ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐✱ ❞❡♥♦t❛❞❛ ♣♦r Nǫ✱ é r❛❞✐❛❧♠❡♥t❡ ❤♦♠❡♦♠♦r❢❛ à ❡s❢❡r❛ ✉♥✐tár✐❛ S1 ❡♠ Hǫ✳

❉❡✜♥✐çã♦ ✻ ❉❡✜♥✐♠♦s ❝♦♠♦ s❡♥❞♦ ❛ ❱❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ ♦ ❝♦♥❥✉♥t♦Nǫ ❞❛❞♦ ♣♦r

Nǫ =

u∈Hǫ\ {0};

Z

RN

(|∇u|2₊_V₍_ǫx₎_u2₎_dx₌

Z

RN

f(u)udx

. ✭✸✳✼✮

➱ ✐♥t❡r❡ss❛♥t❡ ♥♦t❛r q✉❡ Nǫ é ✉♠ ❝♦♥❥✉♥t♦ q✉❡ ❝♦♥té♠ t♦❞❛s ❛s s♦❧✉çõ❡s ❢r❛❝❛s ♥ã♦✲

tr✐✈✐❛✐s ❞♦ ♣r♦❜❧❡♠❛ ✭✸✳✶✮✳

❆♥t❡s ❞❡ ♣r♦✈❛r ♦ ♣ró①✐♠♦ ❧❡♠❛✱ ❝♦♥s✐❞❡r❡ ♣❛r❛ t♦❞♦u∈Hǫ\ {0} ❡t >0❛ ❛♣❧✐❝❛çã♦

ψǫ(t) = Iǫ(tu). ✭✸✳✽✮

◆♦t❡ q✉❡ψǫ(0) = 0❡ ✉s❛♥❞♦ ❛r❣✉♠❡♥t♦s s✐♠✐❧❛r❡s ❛♦ ❞♦ ▲❡♠❛ ✷✱ t❡♠♦s q✉❡ψǫ(t)>0

♣❛r❛ t s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ ❡ ψǫ(t) < 0 ♣❛r❛ t s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳ P♦rt❛♥t♦✱ ♦

max

t≥0 ψǫ(t)❡①✐st❡ ❡ é ❛ss✉♠✐❞♦ ❡♠ ✉♠ ❝❡rt♦ t=ϕǫ(u)>0✳ ❉❡r✐✈❛♥❞♦ ψǫ✱ t❡♠♦s q✉❡

ψ′

ǫ(t) =Iǫ′(tu)u=tkuk2ǫ −

Z

RN

f(tu)udx=t2kuk2_ǫ −

Z

RN

(25)

✸✳ ❘❡s✉❧t❛❞♦s ❞❡ ❊①✐stê♥❝✐❛ ✷✵

❆♣❧✐❝❛♥❞♦ ❡♠ t=ϕǫ(u)✱ t❡♠♦s

ψ′(ϕǫ(u)) = (ϕǫ(u))kuk2ǫ −

Z

RN

f(ϕǫ(u)u)udx. ✭✸✳✾✮

❈♦♠♦ ϕǫ(u) é ♦ ♣♦♥t♦ ♦♥❞❡ ψ ❛ss✉♠❡ ♦ s❡✉ ♠á①✐♠♦✱ ❛ ❞❡r✐✈❛❞❛ ♥❡ss❡ ♣♦♥t♦ é ♥✉❧❛✱

♦✉ s❡❥❛✱ ψ′₍_ϕ

ǫ(u)) = 0✳ ❊♥tã♦✱ ✉s❛♥❞♦ ❡ss❡ ❢❛t♦ ❡♠ ✭✸✳✾✮✱ s❡❣✉❡ q✉❡

(ϕǫ(u))2kuk2ǫ =

Z

RN

f(ϕǫ(u)u)ϕǫ(u)udx

P♦rt❛♥t♦✱ ϕǫ(u)u∈ Nǫ✳

▲❡♠❛ ✸ ❖ ♥ú♠❡r♦ ϕǫ(u)>0 é ♦ ú♥✐❝♦ ✈❛❧♦r ❞❡ t t❛❧ q✉❡ tu ∈ Nǫ✳

❉❡♠♦♥str❛çã♦✳ P❛r❛ ♣r♦✈❛r ❛ ✉♥✐❝✐❞❛❞❡ ❞❡ ϕǫ(u) ✱ ✈❛♠♦s s✉♣♦r q✉❡ ❡①✐st❡♠ ❞♦✐s

✈❛❧♦r❡s ❞✐❢❡r❡♥t❡s ❡ ♠♦str❛r q✉❡ ❡❧❡s sã♦ ♦s ♠❡s♠♦s✳ P❛r❛ ✐ss♦✱ t♦♠❡♠♦s ✉♠ ϕbǫ(u)✱ t❛❧

q✉❡ 0<ϕ_bǫ(u)6=ϕǫ(u) ❡ϕbǫ(u)u∈ Nǫ ✳ ❙❡♥❞♦ ❛ss✐♠✱ t❡♠♦s q✉❡

I′

ǫ(ϕbǫ(u)u)ϕbǫ(u)u= 0✱

♦✉ s❡❥❛✱

(ϕ_bǫ(u))2kuk2ǫ =

Z

RN

f(ϕ_bǫ(u)u)ϕbǫ(u)udx✳

❊♥tã♦

kuk2

ǫ =

Z

RN

f(ϕ_bǫ(u)u)u

b ϕǫ(u)

dx. ✭✸✳✶✵✮

P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ ϕǫ(u)u∈ Nǫ✱

kuk2_ǫ =

Z

RN

f(ϕǫ(u)u)u

ϕǫ(u)

dx. ✭✸✳✶✶✮

▲♦❣♦✱ ❞❡ ✭✸✳✶✵✮ ❡ ✭✸✳✶✶✮ s❡❣✉❡ q✉❡

Z

RN

f(ϕ_bǫ(u)u)u

b ϕǫ(u)

dx=

Z

RN

f(ϕǫ(u)u)u

ϕǫ(u)

dx. ✭✸✳✶✷✮

❈♦♠♦ ❡s❝♦❧❤❡♠♦s ϕbǫ(u) 6= ϕǫ(u)✱ ♣♦❞❡♠♦s s✉♣♦r✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ q✉❡

b

ϕǫ(u)< ϕǫ(u)✳ P❡❧❛ ❤✐♣ót❡s❡ (f5)✱ t❡♠♦s q✉❡

b

ϕǫ(u)< ϕǫ(u)⇒

f(ϕ_bǫ(u)u(x))

b ϕǫ(u)

< f(ϕǫ(u)u(x)) ϕǫ(u) ✱

♣❛r❛ t♦❞♦x∈RN_{✳ ❆ss✐♠✱}

Z

RN

f(ϕ_bǫ(u)u)

b ϕǫ(u)

− f(ϕǫ(u)u)

ϕǫ(u)

udx6= 0✱

❝♦♥tr❛❞✐③❡♥❞♦ ✭✸✳✶✷✮✳

▲❡♠❛ ✹ ❆ ❛♣❧✐❝❛çã♦ T : S1 _{→ N}

ǫ ❞❡✜♥✐❞❛ ♣♦r T(u) = ϕǫ(u)u é ❜✐❥❡t♦r❛ ❡ s✉❛ ✐♥✈❡rs❛

T−1 _:_N

ǫ → S1 é ❞❛❞❛ ♣♦r T−1(u) =

u

kukǫ

(26)

✸✳ ❘❡s✉❧t❛❞♦s ❞❡ ❊①✐stê♥❝✐❛ ✷✶

❉❡♠♦♥str❛çã♦✳ P❛r❛ ♣r♦✈❛r ❡ss❡ ❧❡♠❛✱ ❜❛st❛ ✈❡r✐✜❝❛r q✉❡T ◦T−1 ₌_T−1 _◦_T ₌_u_✳

❉❡ ❢❛t♦✱ ♣r✐♠❡✐r❛♠❡♥t❡ ♥♦t❡ q✉❡ ♣❛r❛ t♦❞♦ u∈ S1_✱

T−1_◦_T₍_u_{) =}_T−1₍_T₍_u_{)) =} ϕǫ(u)u

kϕǫ(u)kǫkukǫ

=u✳

◆♦t❡ ❛✐♥❞❛ q✉❡✱ ♣❛r❛ t♦❞♦u∈ Nǫ✱ t❡♠♦s q✉❡ϕǫ

u

kukǫ

=kukǫ✳ ❊♥tã♦✱ ♣❛r❛u∈ Nǫ✱

T ◦T−1₍_u_{) =} _T₍_T−1₍_u_{)) =}_ϕ

ǫ

u

kukǫ

u

kukǫ

=u✳

❆ss✐♠✱ ♣❛r❛ ❝♦♥❝❧✉✐r♠♦s q✉❡ Nǫ é r❛❞✐❛❧♠❡♥t❡ ❤♦♠❡♦♠♦r❢❛ à ❡s❢❡r❛ S1 ❡♠ Hǫ✱ ❜❛st❛

♠♦str❛r q✉❡ ❛ ❛♣❧✐❝❛çã♦u7→ϕǫ(u) é ❝♦♥tí♥✉❛ ❡♠ Hǫ\ {0}✳

Pr♦♣♦s✐çã♦ ✶ ❆ ❛♣❧✐❝❛çã♦ Λ :Hǫ\ {0} −→R+✱ ❞❛❞❛ ♣♦r Λ(u) =ϕǫ(u) é ❝♦♥tí♥✉❛✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛um −→u ❡♠ Hǫ\{0}✳ ❈♦♠♦ ϕǫ(um)um ∈ Nǫ✱ t❡♠♦s q✉❡

Iǫ′(ϕǫ(um)um)ϕǫ(um)um = 0✱

♦✉ s❡❥❛✱

(ϕǫ(um))2kumk2ǫ =

Z

RN

f(ϕǫ(um)um)ϕǫ(um)umdx. ✭✸✳✶✸✮

▼♦str❡♠♦s q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ (ϕǫ(um))é ❧✐♠✐t❛❞❛✳

❈♦♠♦ ϕǫ(um) > 0✱ t❡♠♦s q✉❡ ❛♥❛❧✐s❛r ❞♦✐s ❝❛s♦s✳ ❙❡ ϕǫ(um) ≤ 1 ❛♦ ❧♦♥❣♦ ❞❡

✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱ ♥ã♦ ❤á ♦ q✉❡ ♣r♦✈❛r✳ ❈♦♥s✐❞❡r❡♠♦s ❡♥tã♦ ϕǫ(um) > 1 ❡ ♥♦t❡ q✉❡✱

ϕǫ(um)|um|>|um|✳ ❆ss✐♠✱ ♣♦r (f4)✱ s❡ um(x)>0✱ ❡♥tã♦

Z ϕǫ(um)um(x)

um(x)

µ sds≤

Z ϕǫ(um)um(x)

um(x)

f(s)

F(s)ds

⇒F(ϕǫ(um)um(x))≥(ϕǫ(um))µF(um) ✭✸✳✶✹✮

❆♥❛❧♦❣❛♠❡♥t❡✱ s❡um(x)<0✱ ♣r♦✈❛✲s❡ q✉❡

F(ϕǫ(um)um(x))≥(ϕǫ(um))µF(um) ✭✸✳✶✺✮

❆ss✐♠✱ ♣♦r (f4) t❡♠♦s q✉❡

Z

RN

f(ϕǫ(um)um)ϕǫ(um)umdx≥µ

Z

RN

F(ϕǫ(um)um)dx≥µ

Z

RN

(ϕǫ(um))µF(um)dx

❆ss✐♠✱ ❡♠ ✭✸✳✶✸✮

(ϕǫ(um))2kumk2ǫ =

Z

RN

f(ϕǫ(um)um)ϕǫ(um)umdx ≥µ

Z

RN

(ϕǫ(um))µF(um)dx

=⇒ ϕǫ(um)

2

ϕǫ(um)µ

≥

µ Z

RN

F(um)dx

(27)

✸✳ ❘❡s✉❧t❛❞♦s ❞❡ ❊①✐stê♥❝✐❛ ✷✷

=⇒ϕǫ(um)µ−2 ≤

1

µ

kumk2ǫ

Z

RN

F(um)dx

✭✸✳✶✻✮

❉❡✈❡♠♦s ❡♥❝♦♥tr❛r ✉♠ ❧✐♠✐t❡ s✉♣❡r✐♦r ♣❛r❛ ϕǫ(um) ❡ ♣❛r❛ ✐ss♦ é s✉✜❝✐❡♥t❡ q✉❡

kumkǫ

Z

RN

F(um)dx

→ Z kukǫ

RN

F(u)dx

✳

❈♦♠♦ um →u ❡♠ Hǫ✱ ♣❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ♥♦r♠❛✱ t❡♠♦s q✉❡

kumk2ǫ −→ kuk2ǫ ✭✸✳✶✼✮

❙❡♥❞♦ ❛ss✐♠✱ ✈❛♠♦s ♣r♦✈❛r q✉❡

Z

RN

F(um)dx−→

Z

RN

F(u)dx. ✭✸✳✶✽✮

❆ ✐❞❡✐❛ é ✉t✐❧✐③❛r ♦ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ❉♦♠✐♥❛❞❛ ●❡♥❡r❛❧✐③❛❞❛✱ ❡♥tã♦ ✈❛♠♦s ✈❡r✐✜❝❛r q✉❡ s✉❛s ❤✐♣ót❡s❡s sã♦ s❛t✐s❢❡✐t❛s✳ ❉❡ ❢❛t♦✱ ♦❜s❡r✈❡ q✉❡

(i)❈♦♠♦F é ❝♦♥tí♥✉❛ ❡um →uq✳t✳♣ ❡♠RN✱ s❡❣✉❡ q✉❡F(um)−→F(u)q✳t✳♣ ❡♠RN✳

(ii) ❚❡♠♦s t❛♠❜é♠ q✉❡

|F(s)| ≤a1|s|2+a2|s|p+1✳

❊♥tã♦✱

|F(um)| ≤a1|um|2+a2|um|p+1 −→a1|u|2+a2|u|p+1

❆❧é♠ ❞✐ss♦✱ ♣❡❧❛s ✐♠❡rsõ❡s ❝♦♥tí♥✉❛s ❞❡ ❙♦❜♦❧❡✈

(iii✮ Z

RN

(a1|um|2+a2|um|p+1)dx−→

Z

RN

(a3|u|2+a4|u|p+1)dx✳

▲♦❣♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ❉♦♠✐♥❛❞❛ ●❡♥❡r❛❧✐③❛❞❛✱

Z

RN

F(um)dx−→

Z

RN

F(u)dx.

❙❡♥❞♦ ❛ss✐♠✱ ❡♠ ✭✸✳✶✻✮✱ ♣♦r ✭✸✳✶✼✮ ❡ ✭✸✳✶✽✮ t❡♠♦s q✉❡

ϕǫ(um)µ−2 ≤

1

µ

kumk2ǫ

Z

RN

F(um)dx

−→ kuk

2

ǫ

Z

RN

F(u)dx

q✉❛♥❞♦ m−→ ∞✳

▲♦❣♦ ϕǫ(um) é ❧✐♠✐t❛❞❛✱ ❡♥tã♦ ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ ✉♠ϕ ≥0✳

❆✜r♠❛çã♦✿ ϕ6= 0✳

❉❡ ❢❛t♦✱ s❡ ϕ = 0✱ t❡♠♦s ❡♠ ✭✸✳✶✸✮ q✉❡

kumk2ǫ =

Z

RN

f(ϕǫ(um)um)u2m

ϕǫ(um)um

dx ✭✸✳✶✾✮

(28)

✸✳ ❘❡s✉❧t❛❞♦s ❞❡ ❊①✐stê♥❝✐❛ ✷✸

lim

ϕǫ(um)um→0

f(ϕǫ(um)um)um

ϕǫ(um)

= 0✳

❆❧é♠ ❞✐ss♦✱ (um) é ❧✐♠✐t❛❞❛✱ ❡♥tã♦ s❡❣✉❡ q✉❡ ❛ ✐♥t❡❣r❛❧ ❞♦ ú❧t✐♠♦ ♠❡♠❜r♦ ❞❡ ✭✸✳✶✾✮

❝♦♥✈❡r❣❡ ♣❛r❛ ③❡r♦✳ ▲♦❣♦✱ t❡r❡♠♦s q✉❡ kukǫ = 0✱ ❝♦♥tr❛❞✐③❡♥❞♦ ♦ ❢❛t♦ ❞❡ q✉❡ u ∈

Hǫ\ {0}✳ ❉❡ss❛ ❢♦r♠❛✱ ϕ >0✳ ❊♥tã♦✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱ ϕǫ(um)−→ϕ >0✳ P❡❧❛

✉♥✐❝✐❞❛❞❡ ❞❡ϕǫ(u)✱ s❡❣✉❡ q✉❡ ϕ =ϕǫ(u)✳ P♦rt❛♥t♦✱ ϕǫ(um)−→ϕǫ(u) ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡

Λ(um)−→Λ(u)✱ ♠♦str❛♥❞♦ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ Λ✳

❆ss✐♠✱ ❝♦♥❝❧✉í♠♦s q✉❡ Nǫ é r❛❞✐❛❧♠❡♥t❡ ❤♦♠❡♦♠♦r❢❛ à ❡s❢❡r❛ S1 ❡♠ Hǫ✳

❙❡❥❛♠ cǫ ❡ c∗ǫ ❞❡✜♥✐❞♦s ♣♦r

cǫ = inf g∈Γǫ

max

0≤t≤1Iǫ(g(t)) ✭✸✳✷✵✮

❡

c∗_ǫ = inf

u∈Hǫ\{0}

max

t≥0 Iǫ(tu) ✭✸✳✷✶✮

♦♥❞❡ Γǫ é ❞❛❞♦ ♣♦r

Γǫ ={g ∈C([0,1], Hǫ); g(0) = 0 e Iǫ(g(1))<0}, ✭✸✳✷✷✮

Pr♦♣♦s✐çã♦ ✷ c∗

ǫ =cǫ = inf

Nǫ

Iǫ✳

❉❡♠♦♥str❛çã♦✳ P❛r❛ ❝❛❞❛u∈Hǫ✱ ❝♦♠♦ ψǫ ❛ss✉♠❡ ♦ s❡✉ ♠á①✐♠♦ ❡♠ ϕǫ(u)>0✱ t❡♠♦s

q✉❡

max

t≥0 ϕǫ(tu) = maxt≥0 Iǫ(tu) = Iǫ(ϕǫ(u)u)✳

♦♥❞❡ ❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ s❡❣✉❡ ❞❛ ✉♥✐❝✐❞❛❞❡ ❞❡ϕǫ(u)✳ ▲♦❣♦✱

c∗_ǫ = inf

u∈Hǫ\{0}

max

t≥0 Iǫ(tu) = u∈Hinfǫ\{0}

Iǫ(ϕǫ(u)u) = inf

Nǫ

Iǫ ✭✸✳✷✸✮

❆✜r♠❛çã♦✿ P❛r❛ t♦❞♦ g ∈Γǫ✱ g([0,1])∩ Nǫ 6=∅✳

❉❡ ❢❛t♦✱ t♦♠❡♠♦s u ∈Hǫ\ {0}✱ ❞❡ ❢♦r♠❛ q✉❡ ♦✉ u∈ Nǫ ♦✉ u ❡stá ♥♦ ✐♥t❡r✐♦r ❞❡Nǫ✳

❙❡u ❡stá ♥♦ ✐♥t❡r✐♦r ❞❡ Nǫ✱ t❡♠♦s q✉❡ ϕǫ(u)>1 ❡ ❡♥tã♦ ϕ′ǫ(1) ≥0✳ ◆♦t❡ q✉❡✱

ϕ′

ǫ(1) ≥0 =⇒Iǫ′(u)u≥0.

❆ss✐♠✱

kuk2_ǫ ≥

Z

RN

f(u)udx ✭✸✳✷✹✮

◆♦t❡ q✉❡✱ ♣♦r (f4)✱ t❡♠♦s q✉❡✱ ♣❛r❛ t♦❞♦ s∈R\ {0}✱

µF(s)≤f(s)s =⇒µ Z

RN

F(s)dx≤

Z

RN

f(s)sdx.

▲♦❣♦✱

µ

2

Z

RN

F(s)dx≤ 1

2

Z

RN

f(s)sdx. ✭✸✳✷✺✮

(29)

✸✳ ❘❡s✉❧t❛❞♦s ❞❡ ❊①✐stê♥❝✐❛ ✷✹

Iǫ(u) = 1₂kuk2ǫ −

Z

RN

F(u)dx≥ 1

2

Z

RN

f(u)udx−

Z

RN

F(u)dx

≥ µ

2

Z

RN

F(u)dx−

Z

RN

F(u)dx=µ 2 −1

Z

RN

F(u)dx.

❈♦♠♦ µ >2✱ t❡♠♦s q✉❡ µ₂ −1>0✳ ❆ss✐♠✱

_µ

2 −1

Z

RN

F(u)dx >0

♦ q✉❡ ✐♠♣❧✐❝❛

Iǫ(u)>0✳

❚❡♠♦s q✉❡g(1) ❡stá ♥♦ ❡①t❡r✐♦r ❞❡Nǫ✱ ♣♦✐sIǫ(g(1))<0✳ P♦r ♦✉tr♦ ❧❛❞♦✱ g(0) ❡stá ♥♦

✐♥t❡r✐♦r ❞❡ Nǫ✳ ▲♦❣♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❛ ❆❧❢â♥❞❡❣❛✱ g([0,1])∩ Nǫ 6= ∅ ❡ ❛ ❛✜r♠❛çã♦ ❡stá

♣r♦✈❛❞❛✳ ❙❡♥❞♦ ❛ss✐♠✱

max

0≤t≤1Iǫ(g(t))≥infNǫ

Iǫ =c∗ǫ✳

P♦rt❛♥t♦✱

cǫ ≥c∗ǫ. ✭✸✳✷✻✮

P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ u ∈ Hǫ \ {0} ✜①♦✱ Iǫ(tu) < 0✱ ♣❛r❛ t s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳

❆ss✐♠✱ ❝❛❞❛ r❛✐♦ {tu; t ≥ 0} ♣♦❞❡ s❡r ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ❢✉♥çã♦ gu ∈ Γǫ ❛ ♠❡♥♦s ❞❡ ✉♠

r❡❡s❝❛❧♦♥❛♠❡♥t♦✳ ❆ss✐♠✱

c∗_ǫ = inf

u∈Hǫ\{0}

max

t≥0 Iǫ(tu) =u∈Hinfǫ\{0}

max

0≤t≤1Iǫ(gu(t))≥ginf∈Γǫ

max

0≤t≤1Iǫ(g(t)) =cǫ ✭✸✳✷✼✮

P♦rt❛♥t♦✱ ❞❡ ✭✸✳✷✻✮ ❡ ✭✸✳✷✼✮ ♦❜té♠✲s❡ q✉❡ cǫ=c∗ǫ✳

❖❜s❡r✈❛çã♦ ✸ ❈♦♠♦ Nǫ ❤♦♠❡♦♠♦r❢♦ à ❡s❢❡r❛ ✉♥✐tár✐❛✱ ❡st❡ ❞✐✈✐❞❡ Hǫ ❡♠ ❞✉❛s ❝♦♠✲

♣♦♥❡♥t❡s ❝♦♥❡①❛s✳ ◆❛ ♣r♦✈❛ ❛♥t❡r✐♦r✱ ♦s t❡r♠♦s ✧✐♥t❡r✐♦r✧❡ ✧❡①t❡r✐♦r✧❞❡ Nǫ s❡ r❡❢❡r❡♠✱

r❡s♣❡❝t✐✈❛♠❡♥t❡✱ à ❝♦♠♣♦♥❡♥t❡ ❝♦♥❡①❛ q✉❡ ❝♦♥té♠ ❛ ♦r✐❣❡♠ ❡ ❛ q✉❡ ♥ã♦ ❝♦♥té♠✳

❖❜s❡r✈❛çã♦ ✹ ❈♦♠♦ cǫ = inf

Nǫ

Iǫ ❡ q✉❛❧q✉❡r ♣♦♥t♦ ❝rít✐❝♦ ♥ã♦ tr✐✈✐❛❧ ❞❡ Iǫ ♣❡rt❡♥❝❡ ❛ Nǫ✱

s❡ cǫ é ✉♠ ✈❛❧♦r ❝rít✐❝♦ ❞❡ Iǫ✱ ❡♥tã♦ é ♦ ♠❡♥♦r ✈❛❧♦r ❝rít✐❝♦ ♣♦s✐t✐✈♦ ❞❡ Iǫ✳

❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ ♥♦s ♠♦str❛ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ♠♦♥ót♦♥❛ ❞❡ cǫ ❝♦♠ r❡❧❛çã♦ ❛ V✳

❈♦♥s✐❞❡r❡ ♣❛r❛ ❝❛❞❛ j = 1, 2✱ ♦ ♣r♦❜❧❡♠❛

−∆u+aj(x)u=f(u)❡♠ RN ✭✸✳✷✽✮

♦♥❞❡ ♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛ ✭✸✳✷✽✮ é ❞❛❞♦ ♣♦r

Ij(u) =

1 2

Z

RN

(|∇u|2+aj(x)u2)dx−

Z

RN

F(u)dx✱

❡ ❝♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦ Γj ❞❛❞♦ ♣♦r

(30)

✸✳ ❘❡s✉❧t❛❞♦s ❞❡ ❊①✐stê♥❝✐❛ ✷✺

Pr♦♣♦s✐çã♦ ✸ ❙❡❥❛ f s❛t✐s❢❛③❡♥❞♦ ❛s ❤✐♣ót❡s❡s (f1)−(f5) ❡ a1 ❡ a2 ∈C0(RN) ❞❡ ♠♦❞♦

q✉❡ ❡①✐st❡ d >0 t❛❧ q✉❡ a1✱a2 ≥ d ❡♠ RN✳ ❙❡ a2 ≥a1 ❡♠ RN✱ ❡♥tã♦ c2 ≥c1✱ ♦♥❞❡ ♦s cj

sã♦ ♦s r❡s♣❡❝t✐✈♦s ♥í✈❡✐s ♠✐♥✐♠❛① ❛ss♦❝✐❛❞♦s ❛♦ ♣r♦❜❧❡♠❛ ✭✸✳✷✽✮ ❝♦♠ aj ✐❣✉❛❧ ❛a1 ❡ a2✳

❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡

a2 ≥a1 =⇒I2(u)≥ I1(u), ✭✸✳✷✾✮

♣❛r❛ t♦❞♦u∈H1₍_RN₎_{✳ ❊♥tã♦✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡} _Γ

j✱ t❡♠♦s q✉❡ g ∈Γ2 =⇒g ∈Γ1✳

P♦r ✭✸✳✷✾✮✱

I2(u)≥I1(u) =⇒ max

0≤t≤1I2(g(t))≥0max≤t≤1I1(g(t))✳

▲♦❣♦✱

c2 = inf

g∈Γ20max≤t≤1I2(g(t))≥ginf∈Γ20max≤t≤1I1(g(t))≥ginf∈Γ10max≤t≤1I1(g(t)) =c1✳

P❛r❛ ♣r♦✈❛r ❛ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦✱ ❝♦♥s✐❞❡r❡ ♦ ❢✉♥❝✐♦♥❛❧ IV0 ❞❡✜♥✐❞♦ ♣♦r

IV0(u) =

1 2

Z

RN

(|∇u|2+V0u2)dx−

Z

RN

F(u)dx✳

❆❧é♠ ❞✐ss♦✱ s❡❥❛♠ ΓV0 ❡cV0 ❞❡✜♥✐❞♦s ❝♦♠♦

ΓV0 ={g ∈C([0,1], H

1₍_RN₎₎_❀ _g_{(0) = 0}_❡ _I

V0(g(1))<0}

❡

cV0 = inf

g∈ΓV0

max

0≤t≤1IV0(g(t))✳

Pr♦♣♦s✐çã♦ ✹ ❙❡ ❛s ❤✐♣ót❡s❡s(V1)−(V2)❡ (f1)−(f5)sã♦ s❛t✐s❢❡✐t❛s✱ ❡♥tã♦ ♦✉cǫ é ♥í✈❡❧

❝rít✐❝♦ ❞❡Iǫ ♦✉ cǫ ≥cV0.

❉❡♠♦♥str❛çã♦✳ P♦r ✭✷✳✺✮✱ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛(wm)⊂Hǫ t❛❧ q✉❡kwmkǫ = 1❡ q✉❛♥❞♦

m→ ∞✱

max

θ≥0 Iǫ(θwm)→cǫ. ✭✸✳✸✵✮

❊♥tã♦✱ ❛ss♦❝✐❛♥❞♦ ❛ ❝❛❞❛wm✉♠❛ ❢✉♥çã♦gm ∈Γǫ✱ ❞❡ ♠♦❞♦ q✉❡ max

0≤t≤1gm(t) = maxθ≥0 Iǫ(θwm)✱

t❡♠♦s q✉❡ ♣❡❧♦ ❚❡♦r❡♠❛ ✷✳✹ ❞❡ [✶✻]✱ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ (um) ⊂ Hǫ✱ 0 < δm → 0 ❡

0≤tm ≤1 t❛✐s q✉❡✱

kum−gm(tm)kǫ ≤δ

1 2

m ✭✸✳✸✶✮

cǫ−δm < Iǫ(um)< cǫ ✭✸✳✸✷✮

❡

kIǫ′(um)kǫ ≤δ

1 2

(31)

✸✳ ❘❡s✉❧t❛❞♦s ❞❡ ❊①✐stê♥❝✐❛ ✷✻

❙❡♥❞♦ ❛ss✐♠✱ ✭✸✳✸✷✮ ❡ ✭✸✳✸✸✮ ✐♠♣❧✐❝❛♠ q✉❡ (um) é ❧✐♠✐t❛❞❛ ❡♠ Hǫ✳ ▲♦❣♦✱ ❛ ♠❡♥♦s ❞❡

✉♠❛ s✉❜s❡q✉ê♥❝✐❛ um ⇀ uǫ ❡♠ Hǫ ❡ um → uǫ ❡♠ Lp_loc₍RN₎✱ ♣❛r❛ 1 ≤ p < 2∗✱ ♦♥❞❡ uǫ é

✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✭✸✳✶✮✳ ❊♥tã♦✱ ❡①✐st❡♠(ym)⊂RN✱β >0 ❡R >0✱ t❛✐s q✉❡

lim inf

m→∞

Z

BR(ym)

u2

mdx > β. ✭✸✳✸✹✮

❉❡ ❢❛t♦✱ ♣♦✐s ❝❛s♦ ❝♦♥trár✐♦✱ ♣❛r❛ t♦❞♦ R >0✱ t❡rí❛♠♦s q✉❡

lim inf

m→∞ _ysup_∈_RN

Z

BR(y)

u2_mdx= 0✳

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♣❡❧♦ ▲❡♠❛ ■✳✶ ❞❡ [✽]✱ s❡❣✉❡ q✉❡

um −→0❡♠ Lp, ♣❛r❛ 2≤p < 2∗. ✭✸✳✸✺✮

▼❛s✱ ✉s❛♥❞♦ ✭✸✳✸✷✮✱✭✸✳✸✸✮ ❡ ♦ ❢❛t♦ ❞❡ kumkǫ s❡r ❧✐♠✐t❛❞❛✱ ♦❜té♠✲s❡

Iǫ(um)−

1 2I

′

ǫ(um)um −→cǫ >0 ✭✸✳✸✻✮

P♦r ♦✉tr♦ ❧❛❞♦✱ ✭✸✳✸✺✮ ❡ ❛s ❤✐♣ót❡s❡s (f2) ❡ (f3) ♠♦str❛♠ q✉❡

Iǫ(um)−1₂Iǫ′(um)um =

Z

RN

1

2umf(um)−F(um)

dx−→0✱

♦ q✉❡ ❝♦♥tr❛r✐❛ ✭✸✳✸✻✮✳

❙❡(ym)❝♦♥té♠ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛✱ ♣♦r ✭✸✳✸✹✮✱uǫ 6= 0✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ ❝❛❞❛

ρ >0✱ ❝♦♠♦ ♣♦r (f4)✱ t❡♠♦s q✉❡✱ ♣❛r❛ s∈R\{0}✱

0< µF(s)≤sf(s)⇒F(s)≤ sf(s)

µ < sf(s)

2

♦ q✉❡ ✐♠♣❧✐❝❛

1

2sf(s)−F(s)>0.

▲♦❣♦✱ ♣❡❧❛s ✐♠❡rsõ❡s ❝♦♠♣❛❝t❛s ❞❡ ❙♦❜♦❧❡✈✱

Iǫ(um)−1₂Iǫ′(um)um ≥

Z

Bρ(0)

1

2f(um)um−F(um)

dx→

Z

Bρ(0)

1

2f(uǫ)uǫ−F(uǫ)

dx.

P♦r ♦✉tr♦ ❧❛❞♦✱

Iǫ(um)−1₂Iǫ′(um)um −→cǫ✳

❊♥tã♦✱ q✉❛♥❞♦ m→ ∞✱ t❡♠♦s q✉❡

cǫ≥

Z

Bρ(0)

1

2f(uǫ)uǫ−F(uǫ)

dx.

❈♦♠♦ρé q✉❛❧q✉❡r ❡ ♦ ✐♥t❡❣r❛♥❞♦ é ♣♦s✐t✐✈♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ▼♦♥ót♦♥❛✱

s❡❣✉❡ q✉❡

cǫ ≥

Z

RN

1

2f(uǫ)uǫ−F(uǫ)

dx. ✭✸✳✸✼✮

(32)

✸✳ ❘❡s✉❧t❛❞♦s ❞❡ ❊①✐stê♥❝✐❛ ✷✼

cǫ ≥

Z

RN

1

2f(uǫ)uǫ−F(uǫ)

dx=Iǫ(uǫ)✳

▲♦❣♦✱ ♣❡❧❛ ❖❜s❡r✈❛çã♦ ✺✱Iǫ(uǫ) =cǫ ❡ ♦ r❡s✉❧t❛❞♦ ❡stá ♣r♦✈❛❞♦ ♣❛r❛ ❡st❡ ❝❛s♦✳

❆❣♦r❛✱ s✉♣♦♥❤❛♠♦s q✉❡(ym)♥ã♦ s❡❥❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛✱ ❡♥tã♦✱ ♣❛r❛ t♦❞♦α >0

❡ρ >0✱

max

θ≥0 Iǫ(θwm)≥Iǫ(αwm) =IV0(αwm) +

Z

RN

1

2(V(ǫx)−V0)|αwm|

2_dx

=IV0(αwm) +

Z

Bρ(0)

1

2_dx₊

Z

RN_\_B ρ(0)

1

2_dx_✳

❈♦♠♦ ♣♦r (V2) ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ρ ❞❡ ♠♦❞♦ q✉❡ V(x)≥ V0✱ ♣❛r❛ t♦❞♦ x ∈(Bρ(0))c✱

s❡❣✉❡ q✉❡

max

θ≥0 Iǫ(θwm)≥IV0(αwm) +

Z

Bρ(0)

1

2_dx. _{✭✸✳✸✽✮}

❈♦♠♦ ✭✸✳✸✽✮ ✈❛❧❡ ♣❛r❛ t♦❞♦α >0✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ❡♠ ♣❛rt✐❝✉❧❛r α=ϕV0(wm)✳

▲♦❣♦✱ s❡❣✉❡ q✉❡✱

max

θ≥0 Iǫ(θwm)≥IV0(ϕV0(wm)wm) +

Z

Bρ(0)

1

2(V(ǫx)−V0)|ϕV0(wm)wm| 2_dx

≥inf

Nǫ

IV0 +

Z

Bρ(0)

1

2(V(ǫx)−V0)|ϕV0(wm)wm| 2_dx_✳

=cV0 +

Z

Bρ(0)

1

2(V(ǫx)−V0)|ϕV0(wm)wm| 2_dx_✱

♦✉ s❡❥❛✱ t❡♠♦s q✉❡

max

θ≥0 Iǫ(θwm)≥cV0 +

Z

Bρ(0)

1

2(V(ǫx)−V0)|ϕV0(wm)wm|

2_dx. _{✭✸✳✸✾✮}

❆✜r♠❛çã♦✿ ❆ s❡q✉ê♥❝✐❛ (ϕV0(wm)) é ❧✐♠✐t❛❞❛✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✳

❝♦♠ ❡❢❡✐t♦✱ ❝♦♠♦ ϕV0(wm)>0✱ t❡♠♦s ❞♦✐s ❝❛s♦s ❛ ❝♦♥s✐❞❡r❛r✳ ❆ s❛❜❡r✿

(i) ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ϕV0(wm)≤1♦✉

(ii) ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ϕV0(wm)>1 ♣❛r❛ m s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳

◆♦ ❝❛s♦ (i)♥ã♦ ❤á ♦ q✉❡ ♣r♦✈❛r✳

P❛r❛ ♦ ❝❛s♦(ii)✱ ❝♦♠♦ ϕV0(wm)>1✱ ♣♦r(f4) t❡♠♦s q✉❡

ϕV0(wm)

2 _≥_µ

Z

RN

F(ϕV0(wm)wm)dx≥µϕV0(wm)

µZ

RN

F(wm)dx

▲♦❣♦✱ t❡♠♦s q✉❡

ϕV0(wm)

µ−2 _≤ 1

µ

1

Z

RN

F(wm)dx

(33)

✸✳ ❘❡s✉❧t❛❞♦s ❞❡ ❊①✐stê♥❝✐❛ ✷✽

❙❡ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ♦ t❡r♠♦ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡ ✭✸✳✹✵✮ é ❧✐♠✐t❛❞♦✱ ❡♥❝♦♥✲ tr❛♠♦s ✉♠ ❧✐♠✐t❡ s✉♣❡r✐♦r ♣❛r❛ϕV0(wm)✳ ❈❛s♦ ❝♦♥trár✐♦✱ ♣❛r❛ m→ ∞✱

Z

RN

F(wm)dx−→0. ✭✸✳✹✶✮

❆✜r♠❛çã♦✿Z

RN

F(wm)dx90.

❉❡ ❢❛t♦✱ ♥♦t❡ q✉❡ ❝♦♠♦ ❡♠ ✭✸✳✸✶✮

gm(tm)≡ξmwm ✭✸✳✹✷✮

♦♥❞❡ gm ∈Γǫ ❡ ξm ∈R+✱ ♣❛r❛ t♦❞♦ m∈N✳ P♦r ✭✸✳✸✶✮✱

kξmwm−umkǫ ≤δ

1 2

m ✭✸✳✹✸✮

❡ ❝♦♠♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ♣♦r ✭✸✳✸✷✮ ❡ ✭✸✳✸✸✮✱ t❡♠♦s q✉❡ (um)é ❧✐♠✐t❛❞❛ ❡♠ Hǫ✳

▲♦❣♦✱ ❝♦♠♦ ✈❛❧❡ ✭✸✳✹✸✮✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ K >0✱ q✉❡ ✐♥❞❡♣❡♥❞❡ ❞❡ m✱ t❛❧ q✉❡

ξm ≤δ

1 2

m+kumkǫ ≤K✳

❙❡♥❞♦ ❛ss✐♠✱ ♣❛r❛ q✉❛❧q✉❡r r >0 ❡y ∈RN✱

kwmkL2₍_B

r(y)) =

1

ξmkξmwmkL

2₍_B

r(y)) ≥

1

KkξmwmkL2(Br(y))✳ ▲♦❣♦✱ t❡♠♦s q✉❡

kwmkL2₍_B

r(y))≥

1

KkξmwmkL2(Br(y)) ≥

1

K kumkL2(Br(y))− kum−ξmwmkL2(Br(y))

.

✭✸✳✹✹✮ ❙❡❣✉❡ ❞❡ ✭✸✳✹✸✮ ❡ ❞❛s ✐♠❡rsõ❡s ❞❡ ❙♦❜♦❧❡✈ q✉❡

kwmkL2₍_B

r(y)) ≥

1

K

kumkL2₍_B

r(y))−Cδ

1 2

m

. ✭✸✳✹✺✮

▲♦❣♦✱ ♣❡❧♦ ▲❡♠❛ ■✳✶ ❞❡ ❬✽❪ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛(ym)⊂Rn❡ ❝♦♥st❛♥t❡sβ >0✱ R >0✱

t❛✐s q✉❡

lim inf

m→∞

Z

BR(ym)

wm2dx≥β✳

❊♥tã♦ ❡♠ ✭✸✳✹✺✮✱ ❡s❝♦❧❤❡♥❞♦ y=ym ❡r =R✱ ♣❛r❛ m ❣r❛♥❞❡ ♦❜t❡♠♦s

kwmkL2₍_B

R(ym))≥

1 K β 2 1 2 ✭✸✳✹✻✮

P❛r❛ ♣r♦✈❛r q✉❡ Z

RN

F(wm)dx90✱ ❜❛st❛ ♠♦str❛r ❡♥tã♦ q✉❡ ❡①✐st❡ β1 >0 t❛❧ q✉❡✱ ❛

♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱

Z

BR(ym)

(34)

✸✳ ❘❡s✉❧t❛❞♦s ❞❡ ❊①✐stê♥❝✐❛ ✷✾

P♦r (f4)✱ t❡♠♦s q✉❡ F(s) > 0 ♣❛r❛ t♦❞♦ s ∈ R t❛❧ q✉❡ |s| ≥ 1 ❡ F(s) ≥ K1|s|µ ❝♦♠

K1 >0✳ ▲♦❣♦✱ ♣❛r❛ t♦❞♦ γ >0✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡Aγ>0 ❞❡ ♠♦❞♦ q✉❡

|s|2 _≤_γ₊_A

γF(s), ✭✸✳✹✼✮

♣❛r❛ t♦❞♦s∈R✳

P❛r❛ ♠♦str❛r q✉❡ ✭✸✳✹✼✮ ✈❛❧❡✱ ❝♦♥s✐❞❡r❡ γ > 0 ♣❡q✉❡♥♦✱ ♦✉ s❡❥❛✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡

γ <1✳ ❚❡♠♦s q✉❡ ❛♥❛❧✐s❛r ❞♦✐s ❝❛s♦s✳

❈❛s♦ ✶✿ ❙❡ |s|>1✱ ❡♥tã♦ ❝♦♠♦ µ >2✱

F(s)≥K1|s|µ> K1|s|2✳

❈❛s♦ ✷✿ ❙❡ |s| ≤γ✱ ❡♥tã♦ t❡♠♦s ♥♦✈❛♠❡♥t❡ q✉❡ ❝♦♥s✐❞❡r❛r ❞♦✐s ❝❛s♦s✳

✭✐✮ ❙❡ |s| ≤1✱ ❝♦♠♦AγF(s)>0✱ s❡❣✉❡ q✉❡

|s|2 _{≤ |}_s_{| ≤}_γ _≤_γ₊_A

γF(s)✳

✭✐✐✮ ❙❡ γ ≤ |s|<1✱ ♣❛r❛ Aγ s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱

|s|2 _≤_γ₊_A

γ min γ≤s≤1F(s)✳

▲♦❣♦✱

Z

BR(ym)

|wm|2dx≤

Z

BR(ym)

(γ +AγF(wm))dx=γ|BR|+Aγ

Z

BR(ym)

F(wm)dx✱

♦♥❞❡|BR| é ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❞❡BR✳ ❙❡

Z

RN

F(wm)dx−→0

q✉❛♥❞♦m → ∞✱ ❝♦♠♦γ é ❛r❜✐trár✐♦✱ s❡❣✉❡ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛♥t❡r✐♦r q✉❡ Z

BR(ym)

w2_mdx→0

q✉❛♥❞♦m → ∞✱ ♦ q✉❡ é ✐♠♣♦ssí✈❡❧ ❡♠ ✈✐rt✉❞❡ ❞❡ ✭✸✳✹✻✮✳

P♦rt❛♥t♦✱ ϕV0(wm) é ❧✐♠✐t❛❞❛✳

❉❛♥❞♦ ❝♦♥t✐♥✉✐❞❛❞❡ ♥❛ ❞❡♠♦♥str❛çã♦ ❞❛ Pr♦♣♦s✐çã♦ ✹✱ ✈❛♠♦s s✉♣♦r q✉❡ ❡①✐st❡ ✉♠

η1 >0✱ t❛❧ q✉❡

kwmkL2₍_B

ρ(0)) ≥η1. ✭✸✳✹✽✮

❊ss❛ s✉♣♦s✐çã♦ s❡rá ♣r♦✈❛❞❛ ❧♦❣♦ ❛❜❛✐①♦✳

❈♦♠♦ gm(tm) = ξmwm✱ ♦♥❞❡ξm ∈R+ ♣❛r❛ t♦❞♦ m∈N✱ ♣♦r ✭✸✳✸✶✮✱ t❡♠♦s q✉❡

kξmwm−umkǫ≤δ

1 2

m. ✭✸✳✹✾✮

(35)

✸✳ ❘❡s✉❧t❛❞♦s ❞❡ ❊①✐stê♥❝✐❛ ✸✵

kumkL2₍_B

ρ(0)) ≥ kξmwmkL2(Bρ(0))− kξmwm−umkL2(Bρ(0)). ✭✸✳✺✵✮ P♦r ✭✸✳✹✾✮✱ ♦ ú❧t✐♠♦ t❡r♠♦ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡ ✭✸✳✺✵✮ t❡♥❞❡ à ③❡r♦✱ q✉❛♥❞♦ m → ∞✳ ❆ss✐♠✱ s❡ ξm → 0 ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱ ❝♦♠♦ wm é ❧✐♠✐t❛❞❛✱ ξmwm → 0✳ ❊

♣❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ Iǫ✱ Iǫ(ξmwm) → 0✱ ♦ q✉❡ ❝♦♥tr❛r✐❛ ✭✸✳✸✵✮✱ ♦✉ s❡❥❛✱ (ξm) é ❧✐♠✐t❛❞❛

✐♥❢❡r✐♦r♠❡♥t❡ ♣♦r ✉♠ M > 0✳ ■ss♦ s✐❣♥✐✜❝❛ q✉❡ ♣❛r❛ t♦❞♦ m ∈ N✱ _|_ξ_m_{| ≥} _M✳ ❙❡♥❞♦

❛ss✐♠✱ ❞❡ ✭✸✳✺✵✮

kumkL2₍_B

ρ(0)) ≥ kξmwmkL2(Bρ(0))− kξmwm−umkL2(Bρ(0))

≥ |ξm|η1−Ckξmwm−umkǫ ≥M η1−Cδ 1 2

m ✭✸✳✺✶✮

P♦rt❛♥t♦✱ ✭✸✳✺✶✮ ♥♦s ♠♦str❛ q✉❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡η2 >0❞❛❞❛ ♣♦rη2 =M η1−Cδ 1 2

m

t❛❧ q✉❡✱

kumkL2₍_B

ρ(0)) ≥η2✳

▲♦❣♦✱ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱ um ⇀ uǫ ❡♠ Hǫ✱ q✉❡ é s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦ tr✐✈✐❛❧

❞❡ ✭✸✳✶✮✱ ❝♦♠ Iǫ(uǫ) =cǫ✳

❆❣♦r❛ ♥♦s r❡st❛ ✈❡r✐✜❝❛r q✉❡ ✭✸✳✹✽✮ é ✈á❧✐❞♦✳

❉❡ ❢❛t♦✱ s❡ ✭✸✳✹✽✮ ♥ã♦ ✈❛❧❡ss❡✱ t❡rí❛♠♦s q✉❡✱ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❞❡ m′s→ ∞✱

kwmkL2₍_B

ρ(0)) →0. ✭✸✳✺✷✮

❆ss✐♠✱ ♣♦r ✭✸✳✸✵✮✱ ✭✸✳✺✷✮ ❡ ♣❡❧♦ ❢❛t♦ ❞❛ s❡q✉ê♥❝✐❛ (ϕV0(wm)) s❡r ❧✐♠✐t❛❞❛✱ s❡❣✉❡ ❞❛

r❡❧❛çã♦ ✭✸✳✸✾✮ ❞❡s❝r✐t❛ ♣♦r

max

θ≥0 Iǫ(θwm)≥cV0+

Z

Bρ(0)

1

2(V(ǫx)−V0)|ϕV0(wm)wm| 2_dx_✱

q✉❡ cǫ≥cV0✳ ❉❡st❛ ❢♦r♠❛✱ ❛ ♣r♦✈❛ ❞❛ Pr♦♣♦s✐çã♦ ✹ ❡stá ❝♦♠♣❧❡t❛✳

▲❡♠❛ ✺ ❊①✐st❡ w∈H1₍_RN₎ _{t❛❧ q✉❡}

−∆w+V0w=f(w) em RN ✭✸✳✺✸✮

❡ IV0(w) =cV0✱ ♦♥❞❡ IV0(u) =

1 2

Z

RN

(|∇u|2₊_V 0u2)−

Z

RN

F(u)dx✱ u∈Hǫ ❡ cV0 é ♦ ♥í✈❡❧

♠✐♥✐♠❛① ❛ss♦❝✐❛❞♦ ❛ IV0✳

❉❡♠♦♥str❛çã♦✳ P❡❧♦ ❚❡♦r❡♠❛ ✽✳✺ ❞❡[✶✻]✱ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛(wm)⊂H1(RN)t❛❧ q✉❡

IV0(wm)→cV0 ❡ I

′

V0(wm)→0✳ ◆♦t❡ q✉❡✱ ♣♦r(f4)✱ ♣♦r ✉♠ ❧❛❞♦

IV0(wm)− 1

µI

′

V0(wm)wm =

1 2 − 1 µ

kwmk2V0 +

Z

RN

1

µf(wm)wm−F(wm)

dx

≥1₂ −_µ1kwmk2V0✱