CENTRO DE CIÊNCIAS EXATAS E DE TECNOLOGIA PROGRAMA DE PÓS GRADUAÇÃO EM MATEMÁTICA

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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙

❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

❘❡s✉❧t❛❞♦s ❞♦ t✐♣♦ ❆♠❜r♦s❡tt✐✲Pr♦❞✐

♣❛r❛ ♣r♦❜❧❡♠❛s q✉❛s✐❧✐♥❡❛r❡s

▼♦✐sés ❆♣❛r❡❝✐❞♦ ❞♦ ◆❛s❝✐♠❡♥t♦

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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙

❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

▼♦✐sés ❆♣❛r❡❝✐❞♦ ❞♦ ◆❛s❝✐♠❡♥t♦

❖r✐❡♥t❛❞♦r✿ Pr♦❢ ❉r✳ ❋r❛♥❝✐s❝♦ ❖❞❛✐r ❱✐❡✐r❛ ❞❡ P❛✐✈❛

❘❡s✉❧t❛❞♦s ❞♦ t✐♣♦ ❆♠❜r♦s❡tt✐✲Pr♦❞✐

♣❛r❛ ♣r♦❜❧❡♠❛s q✉❛s✐❧✐♥❡❛r❡s

❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐✲ ❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ▼❛t❡♠át✐❝❛✱ ár❡❛ ❞❡ ❝♦♥❝❡♥tr❛✲ çã♦✿ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s P❛r❝✐❛✐s

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Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária UFSCar Processamento Técnico

com os dados fornecidos pelo(a) autor(a)

N244rt Nascimento, Moisés Aparecido do Resultados do tipo Ambrosetti-Prodi para problemas quasilineares / Moisés Aparecido do Nascimento. -- São Carlos : UFSCar, 2015. 65 p.

Tese (Doutorado) -- Universidade Federal de São Carlos, 2015.

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❆●❘❆❉❊❈■▼❊◆❚❖❙

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❘❊❙❯▼❖

◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s r❡s✉❧t❛❞♦s ❞♦ t✐♣♦ ❆♠❜r♦ss❡t✐✲Pr♦❞✐ ♣❛r❛ ♣r♦❜❧❡♠❛s q✉❛s✐❧✐♥❡❛r❡s ❡♥✈♦❧✈❡♥❞♦ ♦ ♦♣❡r❛❞♦r ♣✲▲❛♣❧❛❝✐❛♥♦✳ ❈♦♥s✐❞❡r❛♠♦s ♦ ❝❛s♦ ❡s❝❛❧❛r ❡ ✉♠ ♣r♦✲ ❜❧❡♠❛ ❝♦♠ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s✳ P❛r❛ ♦s ❝❛s♦s ❡s❝❛❧❛r❡s✱ tr❛❜❛❧❤❛♠♦s ❝♦♠ ❛s ❝♦♥❞✐çõ❡s ❞❡ ◆❡✉♠❛♥♥ ❡ ❉✐r✐❝❤❧❡t✱ ❥á ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❡♥✈♦❧✈❡♥❞♦ s✐st❡♠❛✱ ❝♦♥s✐❞❡r❛♠♦s ❛ ❝♦♥❞✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t✳ P❛r❛ ♦❜t❡r t❛✐s r❡s✉❧t❛❞♦s ✉s❛♠♦s ❛ t❡♦r✐❛ ❞♦ ❣r❛✉ ❞❡ ▲❡r❛②✲❙❝❤❛✉❞❡r ❡ ❡st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐✳

P❛❧❛✈r❛s ❈❤❛✈❡✿ ❣r❛✉ ❞❡ ▲❡r❛②✲❙❝❤❛✉❞❡r✱ ❡st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐✱ p✲▲❛♣❧❛❝✐❛♥♦✱ ♣r♦✲

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❆❇❙❚❘❆❈❚

❲❡ ♣r❡s❡♥t r❡s✉❧ts ♦❢ ❆♠❜r♦ss❡t✐✲Pr♦❞✐ t②♣❡ t♦ q✉❛s✐❧✐♥❡❛r ♣r♦❜❧❡♠s ✐♥✈♦❧✈✐♥❣ t❤❡ ♣✲ ▲❛♣❧❛❝❡ ♦♣❡r❛t♦r✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ s❝❛❧❛r ❝❛s❡ ❛♥❞ ❛ ❛ ♣r♦❜❧❡♠ ✇✐t❤ s②st❡♠s ♦❢ ❡q✉❛t✐♦♥s✳ ■♥ t❤❡ s❝❛❧❛r ❝❛s❡✱ ✇❡ ✇♦r❦ ✇✐t❤ t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ ◆❡✉♠❛♥♥ ❛♥❞ ❉✐r✐❝❤❧❡t✳ ■♥ t❤❡ ♣r♦❜❧❡♠ ✐♥✈♦❧✈✐♥❣ s②st❡♠✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ ❉✐r✐❝❤❧❡t✳ ■♥ ♦r❞❡r t♦ ❣❡t t❤❡ r❡s✉❧ts ✇❡ ✉s❡ t❤❡ t❤❡♦r② ♦❢ ▲❡r❛②✲❙❝❤❛✉❞❡r ❞❡❣r❡❡ ❛♥❞ ❛ ♣r✐♦r✐ ❡st✐♠❛t❡s✳

❑❡② ❲♦r❞s✿ ▲❡r❛②✲❙❝❤❛✉❞❡r ❉❡❣r❡❡✱ ❛ ♣r✐♦r✐ ❡st✐♠❛t❡s✱ p✲▲❛♣❧❛❝✐❛♥✱ ◆❡✉♠❛♥♥ ♣r♦✲

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❙❯▼➪❘■❖

❆❣r❛❞❡❝✐♠❡♥t♦s ✐

❘❡s✉♠♦ ✐✐

❆❜str❛❝t ✐✐✐

■♥tr♦❞✉çã♦ ✶

Pr❡❧✐♠✐♥❛r❡s ✶

✵✳✶ Pr✐♥❝í♣✐♦s ❞❡ ❝♦♠♣❛r❛çã♦ ❡ ❞♦ ▼á①✐♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✵✳✷ ❊st✐♠❛t✐✈❛s ❛ Pr✐♦r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✵✳✸ ❆❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ ❘❡❣✉❧❛r✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✵✳✹❖ ❖♣❡r❛❞♦r H ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹

✶ Pr♦❜❧❡♠❛s ❝♦♠ ❝♦♥❞✐çã♦ ❞❡ ◆❡✉♠❛♥♥ ✼

✶✳✶ ❘❡s✉❧t❛❞♦s Prí♥❝✐♣❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ Pr✐♠❡✐r❛ ❙♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✶✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸✳✶ ◆ã♦ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ t s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✶✳✸✳✷ ❈♦♥❝❧✉sã♦ ❞❛ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✶✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

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❙❯▼➪❘■❖ ✈

✶✳✹ ❊st✐♠❛t✐✈❛ ❛ ♣r✐♦r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✺ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✶✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✷ Pr♦❜❧❡♠❛s ❝♦♠ ❝♦♥❞✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t ✷✶

✷✳✶ ❘❡s✉❧t❛❞♦s Prí♥❝✐♣❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ Pr✐♠❡✐r❛ ❙♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸ ❉❡♠♦♥str❛çã♦ ❞♦s ❚❡♦r❡♠❛s ✷✳✶ ❡ ✷✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✸✳✶ ❊st✐♠❛t✐✈❛ ❛✲♣r✐♦r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸✳✷ ❈♦♥❝❧✉sã♦ ❞❛ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✷✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✸✳✸ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✷✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✸ ❙✐st❡♠❛s ❝♦♠ ❝♦♥❞✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t ✸✺

✸✳✶ ❆♣r❡s❡♥t❛çã♦ ❞♦ Pr♦❜❧❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷ Pr❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✷✳✶ Pr✐♥❝í♣✐♦ ❞♦ ▼á①✐♠♦ ♣❛r❛ ❙✐st❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✷✳✷ ❖ ❖♣❡r❛❞♦r H ♥❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✸✳✸ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✸✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✸✳✶ ❖❜s❡r✈❛çõ❡s s♦❜r❡ ❙♦❧✉çõ❡s ❞❡ ❱✐s❝♦s✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✸✳✷ ❘❡s✉❧t❛❞♦s ❆✉①✐❧✐❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✸✳✸ ❈♦♥❝❧✉sã♦ ❞❛ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✸✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✹ ❊①✐stê♥❝✐❛ ❞❡ s✉❜s♦❧✉çã♦ ♣❛r❛ (St) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

✸✳✺ ❊st✐♠❛t✐✈❛s ❛ Pr✐♦r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✻ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✸✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

❆ ■❞❡♥t✐❞❛❞❡ ❞❡ P✐❝♦♥❡ ✺✽

❇ ●r❛✉ ❞❡ ▲❡r❛②✲❙❝❤❛✉❞❡r ✻✵

(11)

■◆❚❘❖❉❯➬➹❖

❖ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦✱ é ♦ ❡st✉❞♦ ❞❡ ♣r♦❜❧❡♠❛s ❞♦ t✐♣♦ ❆♠❜r♦s❡tt✐✲Pr♦❞✐ ❡♥✈♦❧✈❡♥❞♦ ♦ ♦♣❡r❛❞♦r ♣✲▲❛♣❧❛❝✐❛♥♦✳ ❆♣r❡s❡♥t❛r❡♠♦s r❡s✉❧t❛❞♦s ♣❛r❛ ♦ ❝❛s♦ ❡s❝❛❧❛r ❡ ♣❛r❛ s✐st❡♠❛s✱ ♥♦ ❝❛s♦ ❡s❝❛❧❛r ❝♦♥s✐❞❡r❛♠♦s ❝♦♥❞✐çõ❡s ❞❡ ◆❡✉♠❛♥♥ ❡ ❉✐r✐❝❤❧❡t ♥❛ ❢r♦♥t❡✐r❛✱ ❥á ♣❛r❛ ♦ ❝❛s♦ ❞❡ s✐st❡♠❛s ❛ ❝♦♥❞✐çã♦ ❞❡ ❢r♦♥t❡✐r❛ s❡rá❞❡ ❉✐r✐❝❤❧❡t✳

❖ ❡st✉❞♦ ❞❡ ♣r♦❜❧❡♠❛s ❞♦ t✐♣♦ ❆♠❜r♦ss❡t✐✲Pr♦❞✐ ❢♦✐ ✐♥✐❝✐❛❞♦ ❝♦♠ ♦ tr❛❜❛❧❤♦ ♣✐♦♥❡✐r♦ ❞❡ ❆✳ ❆♠❜r♦s❡tt✐ ❡ ●✳ Pr♦❞✐✱ q✉❡ ❡♠ ❬✷❪ ❝♦♥s✐❞❡r❛r❛♠ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ s❡♠✐❧✐♥❡❛r✿

(PD)

⎧ ⎨ ⎩

−∆u=f(u) +v(x) ;x∈Ω

u= 0 ;x∈∂Ω

♦♥❞❡ v ∈C0,α_(Ω) _❡ _f _∈_C2₍_R₎ _{s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s✿}

✭■✮ f′′

(s)>0 ∀s∈R✱

✭■■✮ 0< lim

s→−∞

f(s)

s < λ1 <s→lim₊∞

f(s)

s < λ2✳

❯t✐❧✐③❛♥❞♦ t❡♦r❡♠❛s ❞❡ ✐♥✈❡rsã♦ ♣❛r❛ ❛♣❧✐❝❛çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❝♦♠ s✐♥❣✉❧❛r✐❞❛❞❡s ❡♠ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✱ ❡❧❡s ♣r♦✈❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ Γ ❝♦♥❡①❛ ❡ ❢❡❝❤❛❞❛✱ ❞❡

❝❧❛ss❡C1 ❡♠ C0,α_(Ω) _{q✉❡ ❞✐✈✐❞❡ ♦ ❡s♣❛ç♦ ❡♠ ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s} _A

0 ❡A1 ❞❡ ♠♦❞♦

q✉❡ ♦ ♣r♦❜❧❡♠❛(PD)t❡♠ ❡①❛t❛♠❡♥t❡ ✉♠❛ s♦❧✉çã♦✱ ♥❡♥❤✉♠❛ s♦❧✉çã♦ ♦✉ ❡①❛t❛♠❡♥t❡ ❞✉❛s

s♦❧✉çõ❡s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ s❡ v ❢♦r ❝♦♥s✐❞❡r❛❞♦ ❡♠ Γ✱ A0 ❡ A1✳ ❆ ❝♦♥❞✐çã♦ (II) s✐❣♥✐✜❝❛

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■♥tr♦❞✉çã♦ ✷

q✉❡ ❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ f ❝r✉③❛ ♦ ♣r✐♠❡✐r♦ ❛✉t♦✲✈❛❧♦r λ1 ❞♦ ♣r♦❜❧❡♠❛✱

(Pλ)

⎧ ⎨ ⎩

−∆u=λu em Ω

u= 0 sobre ∂Ω

q✉❛♥❞♦s ✈❛r✐❛ ❞❡−∞ á +∞✳

❯♠❛ r❡♣r❡s❡♥t❛çã♦ ❝❛rt❡s✐❛♥❛ ❞❡ Γ ❢♦✐ ✐♥tr♦❞✉③✐❞❛ ♣♦r ❇❡r❣❡r ❡ P♦❞♦❧❛❦ ❡♠ ❬✽❪✱ ♦♥❞❡

♦s ❛✉t♦r❡s ❝♦♥s✐❞❡r❛r❛♠ ❛ s❡❣✉✐♥t❡ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ v✿ v(x) = tφ(x) +h(x)✱ s❡♥❞♦ φ(x)❛

♣r✐♠❡✐r❛ ❛✉t♦❢✉♥çã♦ ♣♦s✐t✐✈❛ ❞❡ (−∆, W₀1,2(Ω)) ❡ h(x) ∈ {s♣❛♥(φ)}⊥✳ ❆ss✐♠ ♦ ♣r♦❜❧❡♠❛

(PD) ♣♦❞❡ s❡r r❡❡s❝r✐t♦ ❝♦♠♦

(PDt)

⎧ ⎨ ⎩

−∆u=f(u) +tφ(x) +h(x) ;x∈Ω

u= 0 ;x∈∂Ω

❯s❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡ r❡❞✉çã♦ ❞❡ ▲✐❛♣✉♥♦✈✲❙❝❤♠✐❞t✱ ❡❧❡s ♠♦str❛r❛♠ ♣r❡❝✐s❛♠❡♥t❡ ♦ ♠❡s♠♦ r❡s✉❧t❛❞♦ q✉❡ ❆♠❜r♦s❡tt✐ ❡ Pr♦❞✐✱ ❛♣r❡s❡♥t❛❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ❊①✐st❡ t1 ∈ R t❛❧

q✉❡✱ (PD) t❡♠ ❡①❛t❛♠❡♥t❡ ③❡r♦✱ ✉♠❛ ♦✉ ❞✉❛s s♦❧✉çõ❡s s❡ t > t1, t = t1 ♦✉ t < t1✱

r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❑❛③❞❛♥ ❡ ❲❛r♥❡r ❡♠ ❬✷✸❪ ❝♦♥s✐❞❡r❛r❛♠ ❢✉♥çõ❡s f ♠❛✐s ❣❡r❛✐s✱ ❡♥❢r❛q✉❡❝❡♥❞♦ ❞❡ss❛

❢♦r♠❛ ❛s ❤✐♣ót❡s❡s✭❡ ❛s ❝♦♥❝❧✉sõ❡s✮ ❞❡ ❇❡r❣❡r✲P♦❞♦❧❛❦✱ s✉♣♦♥❞♦ q✉❡ f s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦

−∞ ≤lim sup

s→−∞

f(s)

s < λ1 <lim infs→+∞

f(s)

s ≤ ∞.

✭✶✮

❊♠ ❬✷✸❪✱ ♦s ❛✉t♦r❡s ❝♦♥str♦❡♠ ✉♠❛ ❛♣r♦♣r✐❛❞❛ s✉♣❡rs♦❧✉çã♦ ❡ ♣r♦✈❛♠ q✉❡ ❡①✐st❡ t1 t❛❧

q✉❡ ♦ ♣r♦❜❧❡♠❛ (PDt)✱ ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦ s❡ t < t1 ❡ ♥ã♦ ♣♦ss✉✐ s♦❧✉çã♦ s❡

t > t1✳ ❉❛♥❝❡r ❬✶✸❪ ❡st❡♥❞❡ ♦s r❡s✉❧t❛❞♦s ❞❡ ❬✷✸❪ ♣❛r❛ ♦♣❡r❛❞♦r❡s ❞✐❢❡r❡♥❝✐❛✐s ♥❛ ❢♦r♠❛

❞✐✈❡r❣❡♥t❡✳ ❆❧é♠ ❞✐ss♦✱ ❧✐♠✐t❛♥❞♦ ♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ f ♣❛r❛t≥0✭t❛❧ ❝r❡s❝✐♠❡♥t♦ ♣♦❞❡ s❡r

s✉♣❡r❧✐♥❡❛r✮ ♦ ❛✉t♦r ♦❜té♠ ❡st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐ ♣❛r❛ ❛s s♦❧✉çõ❡s✱ ❞♦♥❞❡ s❡❣✉❡ ❡①✐stê♥❝✐❛ ❞❡ ♣❡❧♦ ♠❡♥♦s ❞✉❛s s♦❧✉çõ❡s ♣❛r❛ t < t1 ❡ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ t =t1✳

❊♠ ❬✷✶❪✱ ❍❡ss ❡♥❢r❛q✉❡❝❡ ❛s ❤✐♣ót❡s❡s s♦❜r❡ φ✿ ❙✉♣♦♥❞♦ φs✉❛✈❡✱ φ ≥ 0 ❛♥❞ φ = 0✳

❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛✱ ❛ ❝♦♥str✉çã♦ ❞❛ s✉♣❡rs♦❧✉çã♦ ❢❡✐t❛ ❡♠ ❬✷✸❪ ♥ã♦ é ♠❛✐s ♣♦ssí✈❡❧✳ ❊♠ ❬✷✶❪✱ ✉♠❛ ♣r✐♠❡✐r❛ s♦❧✉çã♦ ♣❛r❛ t <<−1é ❡♥❝♦♥tr❛❞❛ ❝♦♠ ❛r❣✉♠❡♥t♦s ❞✐st✐♥t♦s ❞♦s q✉❡

❡st❛✈❛♠ s❡♥❞♦ ❝♦♥s✐❞❡r❛❞♦s ❛té ❡♥tã♦✱ ❡ ❛ t❡♦r✐❛ ❞♦ ❣r❛✉✳ ❊♠ ❬✼❪✱ ❇❡r❡st②❝❦✐ ❡ ▲✐♦♥s ❝♦♥s✐❞❡r❛r❛♠ ✉♠ ♣r♦❜❧❡♠❛ s✐♠✐❧❛r ❝♦♠ f ♣♦❞❡♥❞♦ s❡r s✉♣❡r❧✐♥❡❛r ❡ ❝♦♠ ❝♦♥❞✐çã♦ ❞❡

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■♥tr♦❞✉çã♦ ✸

❊♠ ❬✷✹❪✱ ❑♦✐③✉♠✐ ❡ ❙❝❤♠✐❞t ❝♦♥s✐❞❡r❛r❛♠ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ♣❛r❛ ♦ ♦♣❡r❛❞♦r p✲

▲❛♣❧❛❝✐❛♥♦✱ p > 1✱

(Pp)

⎧ ⎨ ⎩

−∆pu=f(u) +tφ+h ;x∈Ω

u= 0 ;x∈∂Ω

♦♥❞❡✱ φ, h ∈C(Ω) ❝♦♠ φ≥0 ❡f ∈C1 _{s❛t✐s❢❛③❡♥❞♦}

α= lim

t→−∞

f(t)

|t|p−2_t < λ1 < β= lim_t→∞

f(t)

|t|p−2_t,

❡♠ q✉❡λ1 é♦ ♠❡♥♦r ❛✉t♦✈❛❧♦r ❞♦ ♣r♦❜❧❡♠❛

(Pa)

⎧ ⎨ ⎩

−∆pu−λ|u|p−2u= 0 ;x∈Ω

u= 0 ;x∈∂Ω.

❆ ♣r✐♠❡✐r❛ s♦❧✉çã♦ é♦❜t✐❞❛ ❝♦♠♣❛r❛♥❞♦ (Pp) ❝♦♠ ♦ ♣r♦❜❧❡♠❛ ❧✐♠✐t❡

⎧ ⎨ ⎩

−∆pu=α|u+|p−1−β|u−|p−1+h ;x∈Ω

u= 0 ;x∈∂Ω.

❊♠ ❬✷✹❪✱ ♦s ❛✉t♦r❡s ♣r♦✈❛♠ q✉❡ ❡①✐st❡ t(h)<<−1t❛❧ q✉❡ ♣❛r❛ t♦❞♦ t ≤t(h)♦ ♣r♦❜❧❡♠❛ (Pp) t❡♠ ✉♠❛ s♦❧✉çã♦ ♥❡❣❛t✐✈❛✳ ❊♠ s❡❣✉✐❞❛✱ s✉♣♦♥❞♦ φ > 0 ✐♥ Ω✱ ❡❧❡s ♠♦str❛♠ q✉❡

❡①✐st❡♠ t1, t2✱ t1 ≤ t2 t❛❧ q✉❡ (Pp) t❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦ ♣❛r❛ t ≤ t2✱ ♥ã♦ ♣♦ss✉✐

s♦❧✉çã♦ s❡t > t2 ❡ ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ❞✉❛s s♦❧✉çõ❡s s❡ t < t1✳ P♦r ✜♠✱ ♦s ❛✉t♦r❡s ♠♦str❛♠

♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s q✉❛♥❞♦ φ ≥0✱ ✉s❛♥❞♦ ❛r❣✉♠❡♥t♦s ❞❡ ❧✐♠✐t❡✳

❆r❝♦②❛ ❡ ❘✉✐③✱ ❡♠ ❬✹❪✱ s✉♣õ❡♠ q✉❡ f é❝♦♥tí♥✉❛ ❡ s❛t✐s❢❛③

lim sup

s→−∞

f(s)

|s|p−2_s < λ1 <lim inf_s→+∞

f(s)

|s|p−2_s ≤lim sup

s→₊∞

f(s)

|s|p−2_s =λ

′

<∞

❡ q✉❡ ♣❛r❛ t♦❞♦ M > 0✱ ❡①✐st❡ ξ >0t❛❧ q✉❡

f(s) +ξ|s|p−2_s _{é♥ã♦ ❞❡❝r❡s❝❡♥t❡ ❡♠} _s_∈_[_{−M, M}_]_.

❖s ❛✉t♦r❡s ♦❜t❡♠✱ ❡♥tr❡ ♦✉tr♦s r❡s✉❧t❛❞♦s✱ q✉❡t1 =t2q✉❛♥❞♦p >2❡φs❛t✐s❢❛③✿ φ >0❡♠

Ω❡ ∂φ_∂ν <0❡♠∂Ω✳ ❆s ♣r✐♥❝✐♣❛✐s té❝♥✐❝❛s ✉t✐❧✐③❛❞❛s ❢♦r❛♠ s✉❜✲s✉♣❡rs♦❧✉çã♦✱ ♣rí♥❝✐♣✐♦s ❞❡

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■♥tr♦❞✉çã♦ ✹

❖ ♣r♦❜❧❡♠❛ ♣❛r❛ ♦ p✲▲❛♣❧❛❝✐❛♥♦ ❝♦♠ ❝♦♥❞✐çã♦ ❞❡ ◆❡✉♠❛♥♥ ❢♦✐ ❝♦♥s✐❞❡r❛❞♦ ♣♦r ❉❡

P❛✐✈❛ ❡ ▼♦♥t❡♥❡❣r♦ ❬✶✻❪✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❡♠ ❬✶✻❪✱ ♦s ❛✉t♦r❡s ❝♦♥s✐❞❡r❛♠ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛

(PN t)

⎧ ⎨ ⎩

−∆pu=f(x, u) +t ;x∈Ω

|∇u|p−₂∂u

∂ν = 0 ;x∈∂Ω

❝♦♠ f : Ω×R → R ✉♠❛ ❢✉♥çã♦ ❞❡ ❈❛r❛t❤é♦❞♦r② s❛t✐s❢❛③❡♥❞♦ ❝♦♥❞✐çõ❡s ❝♦♠♦ ❡♠ ❬✹❪✳

❖s ❛✉t♦r❡s ♣r♦✈❛♠ q✉❡ ❡①✐st❡ t0 ∈ R t❛❧ q✉❡ (PN t) ♥ã♦ t❡♠ s♦❧✉çõ❡s s❡ t > t0✱ ❡ (PN t)

t❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦ ♠✐♥✐♠❛❧ s❡ t < t0✳ ❙❡ ❡♠ ❛❞✐çã♦ f ❢♦r ❧♦❝❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③

❝♦♥tí♥✉❛ ❡♠ s ✉♥✐❢♦r♠❡♠❡♥t❡ q✳t✳♣ x ∈ Ω✱ ❡♥tã♦ ❡①✐st❡ t1 ≤ t0 t❛❧ q✉❡ ♣❛r❛ t < t1 ♦

♣r♦❜❧❡♠❛ (PN t)t❡♠ ♣❡❧♦ ♠❡♥♦s ❞✉❛s s♦❧✉çõ❡s ❞✐st✐♥t❛s✳ ❆❧é♠ ❞✐ss♦✱ ❛ ✐❣✉❛❧❞❛❞❡ t1 =t0

♦❝♦rr❡ s❡f ∈C(Ω×_R)✳

❆❣♦r❛ ❛♣r❡s❡♥t❛r❡♠♦s ♥♦ss♦s r❡s✉❧t❛❞♦s✳ ◆♦ ❈❛♣ít✉❧♦ ✶✱ ❡st✉❞❛♠♦s ♦ s❡❣✉✐♥t❡ ♣r♦✲ ❜❧❡♠❛ ❞❡ ◆❡✉♠❛♥♥✿

(Pt)

⎧ ⎨ ⎩

−∆pu=f(x, u) +tφ(x) +h(x) ;x∈Ω

|∇u|p−₂∂u

∂ν = 0 ;x∈∂Ω

♦♥❞❡ φ(x)≥0✱φ(x)≡0❡φ, h∈L∞

(Ω)✱Ω⊂RN _{✉♠ ❛❜❡rt♦✱ ❧✐♠✐t❛❞♦ ❡ ❝♦♠ ❢r♦♥t❡✐r❛} _∂_Ω

s✉❛✈❡ ❡ f : Ω×_R→_R ✉♠❛ ❢✉♥çã♦ ❞❡ ❈❛r❛t❤❡♦❞♦r② s❛t✐s❢❛③❡♥❞♦ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿

lim sup

s→−∞

f(x, s)

|s|p−2_s <0<lim inf_s→₊∞

f(x, s)

|s|p−2_s ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ x∈Ω✳

❙✉♣♦♠♦s t❛♠❜é♠ q✉❡ ∀M >0✱ ∃λ >0t❛❧ q✉❡✱

g(x, u) =f(x, u) +λ|u|p−2_u _{é ♥ã♦ ❞❡❝r❡s❝❡♥t❡} _∀u_∈_[_{−M, M}_]_✱

❡ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦

|f(x, s)| ≤c(1 +|s|p−1_); _∀₍_{x, s}₎_∈_Ω_×

R.

Pr♦✈❛♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❞♦ t✐♣♦ ❆♠❜r♦s❡tt✐✲Pr♦❞✐✿ ❡①✐st❡♠ t1 ≤t0 ∈R✱ t❛✐s q✉❡

✭✐✮ ❙❡ t < t1✱ ❡♥tã♦ (Pt)♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ❞✉❛s s♦❧✉çõ❡s✳

✭✐✐✮ ❙❡ t≤t0✱ ❡♥tã♦ ♦ ♣r♦❜❧❡♠❛ ♣♦ss✉✐ (Pt) ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦✳

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■♥tr♦❞✉çã♦ ✺

◆ã♦ ❝♦♥❤❡❝❡♠♦s ❛té ♦ ♠♦♠❡♥t♦ ✉♠ r❡s✉❧t❛❞♦ ♥❡st❡ s❡♥t✐❞♦✱ ✐st♦ é✱ ❡♥✈♦❧✈❡♥❞♦ ♦ ♦♣❡r❛❞♦r p✲▲❛♣❧❛❝✐❛♥♦✱ ❝♦♠ ❝♦♥❞✐çã♦ ❞❡ ◆❡✉♠❛♥♥ ♥❛ ❢r♦♥t❡✐r❛ ❡ φ ≥ 0✱ φ ≡ 0✳ ❖ q✉❡

t♦r♥❛ ❡st❡ r❡s✉❧t❛❞♦ r❡❧❡✈❛♥t❡ é ♦ ❢❛t♦ ❞❛ ♥ã♦ ❡①✐stê♥❝✐❛ ❞❡ s✉♣❡rs♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛

(Pt)✱ s❡♥❞♦ ❛ss✐♠✱ ♥ã♦ ♣♦❞❡♠♦s ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❞❛ ♣r✐♠❡✐r❛ s♦❧✉çã♦ t❛❧ ❝♦♠♦ ❢♦✐ ❢❡✐t♦

♥♦s tr❛❜❛❧❤♦s ❞❡ ❉❡ P❛✐✈❛✲▼♦♥t❡♥❡❣r♦ ❬✶✻❪✳ ❊♠ t❛❧ tr❛❜❛❧❤♦✱ ♣❛r❛ ❣❛r❛♥t✐r ❛ ❡①✐stê♥❝✐❛ ❞❡ s✉♣❡rs♦❧✉çã♦ ♦s ❛✉t♦r❡s ♣r❡❝✐s❛r❛♠ ❞❛ ❤✐♣ót❡s❡ φ ≡ 1✳ ❊♠ ✉♠ ❝❡rt♦ s❡♥t✐❞♦✱ ♥♦ss♦

r❡s✉❧t❛❞♦ ❡st❡♥❞❡ ♦ r❡s✉❧t❛❞♦ ❞❡ ❇❡r❡st②❝❦✐✲▲✐♦♥s ❬✼❪ ♣❛r❛ ♦ p✲▲❛♣❧❛❝✐❛♥♦✳

◆♦ ❈❛♣ít✉❧♦ ✷ ❝♦♥s✐❞❡r❛♠♦s ♦ ♣r♦❜❧❡♠❛ (Pp) ❝♦♠ φ(x) ≥ 0 ❡♠ Ω¯✱ φ, h ∈ L∞(Ω)✱

1< p < ∞✳ ❖❜t❡♠♦s ♦s ♠❡s♠♦s r❡s✉❧t❛❞♦s ❞♦ ❈❛♣ít✉❧♦ ✶✱ ❡♥tr❡t❛♥t♦ ❛❧❣✉♥s r❡s✉❧t❛❞♦s

❛✉①✐❧✐❛r❡s ♣r❡❝✐s❛r❛♠ ❞❡ ❞❡♠♦♥str❛çõ❡s ❞✐❢❡r❡♥t❡s ❞❛s q✉❡ ❢♦r❛♠ ❢❡✐t❛s ♥♦ ♣r♦❜❧❡♠❛ ❞❡ ◆❡✉♠❛♥♥✳ ◆♦ss♦s r❡s✉❧t❛❞♦s t❛♠❜é♠ ♣♦❞❡♠ s❡r ❝♦♠♣❛r❛❞♦s ❛♦s ♦❜t✐❞♦s ♣♦r ❑♦✐③✉♠✐✲ ❙❝❤♠✐❞t ❬✷✹❪ ❡ ♣♦r ❆r❝♦②❛✲❘✉✐③ ❡♠ ❬✹❪✳ ▼❛s ❛s té❝♥✐❝❛s ✉t✐❧✐③❛❞❛s ♥❡st❡s tr❛❜❛❧❤♦s ♥ã♦ ♣♦❞❡♠ s❡r ❛♣❧✐❝❛❞❛s ♥❛ ♥♦ss❛ s✐t✉❛çã♦✳ ◆♦ss♦s r❡s✉❧t❛❞♦s ❝♦♠♣❧❡t❛♠ ♦s ♦❜t✐❞♦s ♣♦r ❡st❡s ❛✉t♦r❡s✳

◆♦ q✉❡ ❝♦♥❝❡r♥❡ ♣r♦❜❧❡♠❛s ❞♦ t✐♣♦ ❆♠❜r♦ss❡t✐✲Pr♦❞✐ ❡♥✈♦❧✈❡♥❞♦ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❝♦♠ ♦ ♦♣❡r❛❞♦r ▲❛♣❧❛❝✐❛♥♦✱ ♣♦❞❡♠♦s ❝✐t❛r ♣♦r ❡①❡♠♣❧♦ ❬✶✶❪✱ ❬✶✹❪✱ ❬✸✵❪ ❡ ❬✶✺❪✳ ❉❡ ❋✐❣✉❡✐r❡❞♦ ❡ ❙✐r❛❦♦✈✱ ❡♠ ❬✶✹❪✱ ❡st✉❞❛♠ ❝♦♠ ❛✉①✐❧✐♦ ❞❛ t❡♦r✐❛ ❞❡ s♦❧✉çõ❡s ❞❡ ✈✐s❝♦s✐❞❛❞❡✱ ✉♠ ♣r♦❜❧❡♠❛ ❞♦ t✐♣♦ ❆♠❜r♦s❡tt✐✲Pr♦❞✐ ♣❛r❛ ♦♣❡r❛❞♦r❡s ✉♥✐❢♦r♠❡♠❡♥t❡ ❡❧ít✐❝♦s ♥❛ ❢♦r♠❛ ♥ã♦✲❞✐✈❡r❣❡♥t❡ ❡ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ♥ã♦ s✉❛✈❡s✳ ❘❡s✉❧t❛❞♦s s✐♠✐❧❛r❡s ❡♥✈♦❧✈❡♥❞♦ ♦ ♦♣❡r❛❞♦r p✲▲❛♣❧❛❝✐❛♥♦

❢♦r❛♠ ♦❜t✐❞♦s ♣♦r ▼✐♦tt♦ ❡♠ ❬✷✽❪✳ ❉❡ ❢❛t♦✱ ♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❡♠ ❬✷✽❪ é ✉♠❛ ✈❡rsã♦ ♣❛r❛ s✐st❡♠❛ ❞❡ ❬✹✱ ❚❡♦r❡♠❛ ✸✳✻❪✳

❖ ❈❛♣ít✉❧♦ ✸ é ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦ ❞♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛

(St)

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

−∆pu1 =f1(x, u1, u2) +t1φ1 +h1, ;x∈Ω

−∆pu2 =f2(x, u1, u2) +t2φ2 +h2, ;x∈Ω

u1 =u2 = 0 ;x∈∂Ω

♦♥❞❡ Ω ⊂ _RN _{é ✉♠ ❞♦♠í♥✐♦ s✉❛✈❡ ❧✐♠✐t❛❞♦✱} _φ

i, hi ∈ L∞(Ω)✱ ❝♦♠ φi ≥ 0✱ i = 1,2✱

t = (t1, t2) ∈ R2 é ✉♠ ♣❛râ♠❡tr♦ ❡ fi : Ω×R×R → R✱ i = 1,2✱ sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s

(16)

P❘❊▲■▼■◆❆❘❊❙

✵✳✶ Pr✐♥❝í♣✐♦s ❞❡ ❝♦♠♣❛r❛çã♦ ❡ ❞♦ ▼á①✐♠♦

❖s r❡s✉❧t❛❞♦s q✉❡ ❡♥✉♥❝✐❛r❡♠♦s ❛q✉✐ s❡rã♦ ✉s❛❞♦s ♥♦ ❞❡❝♦rr❡r ❞❡st❡ tr❛❜❛❧❤♦✳ ❈♦♠❡✲ ç❛r❡♠♦s ❡♥✉♥❝✐❛♥❞♦ ✉♠ ♣r✐♥❝í♣✐♦ ❞❡ ❝♦♠♣❛r❛çã♦ ❞❡✈✐❞♦ ❛ ❬✷✵❪ ❡ ♦✉tr♦ ❞❡✈✐❞♦ ❛ ❬✸✶❪ ❡ ❡♠ s❡❣✉✐❞❛✱ ❞♦✐s ♣rí♥❝✐♣✐♦s ❞♦ ♠á①✐♠♦ ❜❡♠ ❝♦♥❤❡❝✐❞♦s ♣❛r❛ ♦ ❝❛s♦ ❡s❝❛❧❛r✳ P❛r❛ ❛ ❞❡✲ ♠♦♥str❛çã♦ ❞❛ ♣r✐♠❡✐r❛ ♣r♦♣♦s✐çã♦✱ ♣♦❞❡♠♦s ❝✐t❛r ♣♦r ❡①❡♠♣❧♦ ✭❬✶✼❪✱ ❚❡♦r❡♠❛ ✺✮✳ P❛r❛ ❛ s❡❣✉♥❞❛ ♣r♦♣♦s✐çã♦ ✈❡r ✭❬✸✷❪✱ ❚❡♦r❡♠❛ ✺✮✳

▲❡♠❛ ✵✳✶✳ ❙❡❥❛♠ u, v ∈W1,p_(Ω) _{❢✉♥çõ❡s ♥ã♦ ♥❡❣❛t✐✈❛s s❛t✐s❢❛③❡♥❞♦}

⎧ ⎨ ⎩

−∆pu+up−1 ≤ −∆pv+vp−1 ;em Ω

|∇u|p−₂∂u

∂ν ≤ |∇v|

p−₂∂v

∂ν ;sobre ∂Ω

❊♥tã♦✱ u≤v ❡♠ Ω✳

▲❡♠❛ ✵✳✷✳ ❙❡❥❛♠ u, v ∈W1,p_(Ω) _{❢✉♥çõ❡s s❛t✐s❢❛③❡♥❞♦}

⎧ ⎨ ⎩

−∆pu+λ|u|p−2u≤ −∆pv+λ|v|p−1v ;em Ω

u≤v ;sobre ∂Ω

❝♦♠ λ >0✳ ❊♥tã♦✱ u≤v ❡♠ Ω✳

Pr♦♣♦s✐çã♦ ✵✳✸✳ ❈♦♥s✐❞❡r❡ ♦ ♣r♦❜❧❡♠❛

(17)

Pr❡❧✐♠✐♥❛r❡s ✷

⎧ ⎨ ⎩

−∆pu=a|u|p−1u+g(x) ;x∈Ω

u= 0 ;x∈∂Ω

P❛r❛ g ∈ Lp′_(Ω)_{✱ ♦ ♣r✐♥❝í♣✐♦ ❞♦ ♠á①✐♠♦ ♦❝♦rr❡ ♣❛r❛ ❡ss❡ ♣r♦❜❧❡♠❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱}

a < λ1✱ s❡♥❞♦ λ1 ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ❞❡ (−∆p, W01,p(Ω))✳

❖ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ Pr✐♥❝í♣✐♦ ❞♦ ▼á①✐♠♦ ❞❡ ❱á③q✉❡③✳

Pr♦♣♦s✐çã♦ ✵✳✹✳ ❙❡❥❛ u ∈ C1_(Ω) _{t❛❧ q✉❡} _∆

pu ∈ L2loc(Ω) ❝♦♠ u ≥ 0 q✳t✳♣ ❡♠ Ω ❡

∆pu ≤ ϑ(u) q✳t✳♣ ❡♠ Ω✱ s❡♥❞♦ ϑ : [0,+∞] → R ❝♦♥tí♥✉❛✱ ♥ã♦ ❞❡❝r❡s❝❡♥t❡✱ ϑ(0) = 0 ❡

❛✐♥❞❛ ♦✉ ϑ(s) = 0 ♣❛r❛ ❛❧❣✉♠ s > 0 ♦✉ ϑ(s) > 0 ♣❛r❛ t♦❞♦ s > 0 ❡ ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦

♦❝♦rr❡

1 0

(ϑ(s)s)−p1 ds=∞.

❊♥tã♦ s❡ u ♥ã♦ é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛ s♦❜r❡ Ω✱ t❡♠♦s q✉❡ u é ♣♦s✐t✐✈❛ ❡♠ t♦❞♦ Ω✳ ❆❧é♠

❞✐ss♦✱ s❡u∈C1_(Ω_{∪ {x}

0}) ♣❛r❛ ❛❧❣✉♠ x0 ∈∂Ωq✉❡ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❛ ❡s❢❡r❛ ✐♥t❡r✐♦r

❡ u(x0) = 0 ❡♥tã♦

∂u

∂ν(x0)>0 ♦♥❞❡ ν é ✉♠ ✈❡t♦r ♥♦r♠❛❧ ❡①t❡r✐♦r ❛ x0✳

✵✳✷ ❊st✐♠❛t✐✈❛s ❛ Pr✐♦r✐

P❛r❛ ♣r♦✈❛r ❛ ❧✐♠✐t❛çã♦ ❞❛ ♣❛rt❡ ♥❡❣❛t✐✈❛ ❞❡ ✉♠❛ ❡✈❡♥t✉❛❧ s♦❧✉çã♦ ❞❡ (PDt)✱❢♦✐ ✉s❛❞♦

♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❞❡✈✐❞♦ ❛ ▲❛❞②③❤❡♥s❦❛②❛ ❡ ❯r❛❧✬ts❡✈❛✱♣❛r❛ ♠❛✐s ❞❡t❛❧❤❡s s♦❜r❡ ❛ ❞❡♠♦♥str❛çã♦ ✈❡❥❛ ♣♦r ❡①❡♠♣❧♦ ✭❬✷✺❪✱▲❡♠❛ ✺✳✶✮✳

▲❡♠❛ ✵✳✺✳ ([✷✺],▲❡♠❛ 5.1) ❙❡❥❛ u(x) ✉♠❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ ❡♠ Ω✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛

k0 >0 t❛❧ q✉❡ ♣❛r❛ t♦❞♦ k≥k0 >0✱

Ak

(u−k)dx≤γkα₍_m₍_A k))1+ǫ

✭✷✮

♦♥❞❡ Ak ={x∈Ω :u(x)> k}✱ m(Ak) é ❛ ♠❡❞✐❞❛ ❞❡ Ak✱ γ✱ ǫ✱α sã♦ ❝♦♥st❛♥t❡s t❛✐s q✉❡

ǫ > 0 ❡ 0 ≤ α ≤ 1 +ǫ✳ ◆❡st❛s ❝♦♥❞✐çõ❡s✱ ❡①✐st❡ C = Cγ, α, ǫ, k0,uL1₍_A k₀)

t❛❧ q✉❡

(18)

Pr❡❧✐♠✐♥❛r❡s ✸

✵✳✸ ❆❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ ❘❡❣✉❧❛r✐❞❛❞❡

❖s ♣ró①✐♠♦s r❡s✉❧t❛❞♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬✹❪

▲❡♠❛ ✵✳✻✳ ❖ ♦♣❡r❛❞♦r p✲▲❛♣❧❛❝✐❛♥♦ ❞❡✜♥✐❞♦ ♣♦r❀

−∆p :W1 ,p

0 (Ω) →W

−1,p′_(Ω)

✭✸✮

−∆pu, v=

Ω

|∇u|p−2_∇u_{· ∇v dx}

✭✹✮

é ❧✐♠✐t❛❞♦ ❡ ❝♦♥tí♥✉♦✳ ❆❧é♠ ❞✐ss♦✱ −∆p é ❜✐❥❡t✐✈♦ ❡ s❡✉ ✐♥✈❡rs♦✱ ❞❡♥♦t❛❞♦ ♣♦r K✱ é

t❛♠❜é♠ ❧✐♠✐t❛❞♦ ❡ ❝♦♥tí♥✉♦✳

▲❡♠❛ ✵✳✼✳ ❙❡❥❛♠ fn, f ∈ L∞(Ω) ❝♦♠ fnL∞ < C ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ C > 0 ❡ t❛❧

q✉❡ fn →f ❡♠ W−1,p

′

(Ω)✳ ❈♦♥s✐❞❡r❡ un =K(fn)✱ u=K(f)✱ ❡♥tã♦ un → u ❡♠ C1,β(Ω)

♣❛r❛ t♦❞♦ 0 ≤ β < α✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦ ♦♣❡r❛❞♦r K : L∞

(Ω) → C₀1,β(Ω) é ❝♦♥tí♥✉♦ ❡

❝♦♠♣❛❝t♦✳

❖ ♣ró①✐♠♦s ❧❡♠❛s sã♦ ❝♦♠❜✐♥❛çõ❡s ❞❡ ✉♠ r❡s✉❧t❛❞♦ ❞❡ ▲❛❞②③❤❡♥s❦❛②❛ ❡ ❯r❛❧✬ts❡✈❛

([✷✺],❚❡♦r❡♠❛7.1)❡ ❡st✐♠❛t✐✈❛s C1 _{❞❡ ❬✸✶❪}

▲❡♠❛ ✵✳✽✳ ❙❡❥❛u∈W₀1,p(Ω) ✉♠❛ s♦❧✉çã♦ ❞♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿ (Pg)

⎧ ⎨ ⎩

−∆pu=f(x, u) ;x∈Ω

u= 0 ;x∈∂Ω

♦♥❞❡g é ✉♠❛ ❢✉♥çã♦ ❞❡ ❈❛r❛t❤❡♦❞♦r② s❛t✐s❢❛③❡♥❞♦ sgn[u].g(x, u)≤M(1+|u|q₎_{♣❛r❛ ❛❧❣✉♠}

1 < q < _NN p−p✳ ❊♥tã♦✱ u ∈ C

1,α_(Ω) _{♣❛r❛ ❛❧❣✉♠} ₀ _{< α <} ₁_{✱ ❡} _u

C1,α_(Ω) ≤ C(M)✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦ ♦♣❡r❛❞♦r K:L∞

(Ω) →C₀1,α(Ω) é ❧✐♠✐t❛❞♦✱ ✐st♦ é✱ K(f)_C1,α ≤C(f_L∞)✱

♣❛r❛ t♦❞❛ f ∈L∞

(Ω)✳

▲❡♠❛ ✵✳✾✳ ❙❡❥❛u∈W1,p_(Ω) _{✉♠❛ s♦❧✉çã♦ ❞♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿}

(Pg)

⎧ ⎨ ⎩

−∆pu=g(x, u) ;x∈Ω

|∇u|p−2∂u

(19)

Pr❡❧✐♠✐♥❛r❡s ✹

♦♥❞❡g é ✉♠❛ ❢✉♥çã♦ ❞❡ ❈❛r❛t❤❡♦❞♦r② s❛t✐s❢❛③❡♥❞♦ sgn[u].g(x, u)≤M(1+|u|q₎_{♣❛r❛ ❛❧❣✉♠}

1 < q < _NN p−p✳ ❊♥tã♦✱ u ∈ C

1,α_(Ω) _{♣❛r❛ ❛❧❣✉♠} ₀ _{< α <} ₁_{✱ ❡} _u

C1,α_(Ω) ≤ C(M)✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦ ♦♣❡r❛❞♦r K :L∞

(Ω) → C1,α_(Ω) _{é ❧✐♠✐t❛❞♦✱ ✐st♦ é✱} _K₍_g₎

C1,α ≤ C(g_L∞)✱

♣❛r❛ t♦❞❛ g ∈L∞

(Ω)✳

✵✳✹ ❖ ❖♣❡r❛❞♦r

H

❖s ♣ró①✐♠♦s ❧❡♠❛s ❢♦r❛♠ ♠♦t✐✈❛❞♦s ♣❡❧♦s r❡s✉❧t❛❞♦s ❡♠ [✷✶] ❡[✷✷]✳ ❆ ❞❡♠♦♥str❛çã♦

s❡❣✉❡ ❛s ♠❡s♠❛s ✐❞é✐❛s ❞❡ ([✷✷],Pr♦♣♦s✐çã♦ (a))✱ ♣❛r❛ ♦ ♣✲▲❛♣❧❛❝✐❛♥♦✳

▲❡♠❛ ✵✳✶✵✳ ❖ ♣r♦❜❧❡♠❛

(P∗)

⎧ ⎨ ⎩

−∆pu+c|u|p−2u=g(x) ;x∈Ω

|∇u|p−2∂u

∂ν = 0 ;x∈∂Ω

❛❞♠✐t❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ❡♠ W1,p_(Ω)_{✱ ♣❛r❛ t♦❞❛} _g _∈_L∞

(Ω)✳ ❆❧é♠ ❞✐ss♦✱ ♦ ♦♣❡r❛❞♦r H :

L∞

(Ω)→C1_(Ω) _{❞❛❞♦ ♣♦r}_H₍_g_{) =}_u _{s❡✱ ❡ s♦♠❡♥t❡ s❡✱} _u_{é s♦❧✉çã♦ ❞❡} ₍_P

∗)✱ é ❡str✐t❛♠❡♥t❡

❝r❡s❝❡♥t❡ ❡ ❝♦♠♣❛❝t♦✳ ❙❡ S ⊂ L∞

(Ω) é L∞

(Ω)✲❧✐♠✐t❛❞♦ ♥❛ t♦♣♦❧♦❣✐❛ ❞❡ Lp_(Ω) _{✐♥❞✉③✐❞❛}

♣❡❧♦ ✐♠❡rsã♦ L∞

(Ω) ֒→Lp_(Ω)_{✱ ❡♥tã♦ ♦ ♦♣❡r❛❞♦r} _H

S :S →C1(Ω) :g → H(g) é ❝♦♥tí♥✉♦✳

❉❡♠♦♥str❛çã♦✳ ◆♦t❡ q✉❡✱ ♣❡❧♦ ❢❛t♦ ❞❡ c > 0 ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❝♦♥st❛♥t❡s M1 < 0 ❡

M2 >0 t❛❧ q✉❡ ❛s ❢✉♥çõ❡s φ1(x) =M1 ❡ φ2(x) =M2✱ sã♦ s✉❜ ❡ s✉♣❡rs♦❧✉çõ❡s ❡str✐t❛s ❞♦

♣r♦❜❧❡♠❛ (P∗)✳ ❙❡❥❛ A = [φ₁, φ₂] ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ϕ ∈C1(Ω) t❛✐s q✉❡ φ₁ ≤ ϕ ≤φ₂✳

❙✉♣♦♥❤❛ q✉❡ u, v ∈W1,p_(Ω) _{s❛t✐s❢❛③}

−∆pu+c|u|p

−2_u_{≥ −}_∆

pv+c|v|p

−2_v _;_x_∈_Ω_, _|∇u|p−2∂u

∂ν ≥ |∇v|

p−2∂v

∂ν ;x∈∂Ω

❯s❛♥❞♦ ♦ ▲❡♠❛ ✵✳✶ ❞❡ ([✷✵],▲❡♠❛ 3.1)✐♠❡❞✐❛t❛♠❡♥t❡ ❝♦♥❝❧✉✐♠♦s q✉❡ ✉♠❛ s♦❧✉çã♦ ❞❡ (P∗)

é ú♥✐❝❛ ❡ q✉❡ ♦ ♦♣❡r❛❞♦r s♦❧✉çã♦✱ ❝❛s♦ ❡①✐st❛✱ é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡✳ ❙❡❥❛ AR =A∩BR✱

♦♥❞❡ BR é ❛ ❜♦❧❛ ❞❡ r❛✐♦ R ❡♠ C1(Ω)✳ P❡❧♦ ❢❛t♦ ❞❡ φ1(x) = M1 ❡ φ2(x) = M2 s❡r❡♠

s✉❜ ❡ s✉♣❡rs♦❧✉çõ❡s ❡str✐t❛s✱ ✉♠❛ s♦❧✉çã♦ u ❞❡ (P∗) ❡stá ♥♦ ✐♥t❡r✐♦r ❞❡ A✳ ▲♦❣♦ ♣❛r❛ R

s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ♦❜t❡♠♦s q✉❡

(20)

Pr❡❧✐♠✐♥❛r❡s ✺

♦♥❞❡ P : AR → C1(Ω) é ❞❡✜♥✐❞♦ ♣♦r P u = K(g−c|u|p−2u) ❡ K é ♦ ♦♣❡r❛❞♦r ❞❛❞♦

♥♦ ▲❡♠❛ ✵✳✾✳ P♦rt❛♥t♦✱ ♦ ♣r♦❜❧❡♠❛ (P∗) é ✉♥✐❝❛♠❡♥t❡ r❡s♦❧✉✈❡❧ ♣❛r❛ ❝❛❞❛ g ∈ L∞(Ω)

❡ ♦ ♦♣❡r❛❞♦r s♦❧✉çã♦ H : L∞

(Ω) → W1,p_(Ω) _{é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡✳ ❙❡❥❛} _{g

n} ✉♠❛

s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❡♠ L∞

(Ω) ❡ wn =Hgn✱ ❧♦❣♦

−∆pwn+c|wn|p

−2_w

n=gn.

P❡❧♦ ▲❡♠❛ ✵✳✾ wn ∈ C1,α(Ω) ❝♦♠ wn_C1,α_(Ω) < C✳ ❈♦♠♦ ❛ ✐♠❡rsã♦ C1,α ֒→ C1,β✱

0≤β < αé ❝♦♠♣❛❝t❛✱ ♣❛ss❛♥❞♦ ❛ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ s❡ ♥❡❝❡ssár✐♦✱ Hgn→w ❢♦rt❡♠❡♥t❡

❡♠ C1,β_(Ω)_{✱ ♣♦rt❛♥t♦} _H _: _L∞

(Ω) → C1_(Ω) _{é ❝♦♠♣❛❝t♦✳ ❙❡❥❛} _S _⊂ _L∞

(Ω) é L∞

(Ω)✲

❧✐♠✐t❛❞♦ ♥❛ t♦♣♦❧♦❣✐❛ ❞❡ Lp_(Ω) _{✐♥❞✉③✐❞❛ ♣❡❧♦ ✐♠❡rsã♦} _L∞

(Ω) ֒→ Lp_(Ω)_{✳ ❙✉♣♦♥❤❛ q✉❡}

HS :S →C1(Ω) ♥ã♦ s❡❥❛ ❝♦♥tí♥✉♦✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ (gn)⊂S ❝♦♥✈❡r❣✐♥❞♦ ❡♠

Lp _{♣❛r❛ ❛❧❣✉♠❛} _g _∈_S _{❡ t❛❧ q✉❡} _Hg

n− HgC1_(Ω) ≥δ ♣❛r❛ ✉♠ δ >0 ❝♦♥✈❡♥✐❡♥t❡✳ ❉❡s❞❡ q✉❡ (Hgn) é ❧✐♠✐t❛❞❛ ❡♠ C1,α(Ω) ♣❛r❛ ❛❧❣✉♠ 0 < α < 1✱ ❡ C1,α ֒→ C1,β✱ 0 ≤ β < α é

❝♦♠♣❛❝t♦✱ ♣❛ss❛♥❞♦ ❛ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ s❡ ♥❡❝❡ssár✐♦✱ Hgn → u ❢♦rt❡♠❡♥t❡ ❡♠ C1(Ω)✳

P❛ss❛♥❞♦ ❛♦ ❧✐♠✐t❡ ❡♠

−∆pHgn+c|Hgn|p−2Hgn=gn(x) ;x∈Ω |∇Hgn|p−2

∂Hgn

∂ν = 0 ;x∈∂Ω

❝♦♥❝❧✉✐♠♦s q✉❡ u s❛t✐s❢❛③

−∆pu+c|u|p

−2_u₌_g₍_x_{) ;}_x_∈_Ω _|∇u|p−2∂u

∂ν = 0 ;x∈∂Ω

❡ ♣❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞❛s s♦❧✉çõ❡s✱ s❡❣✉❡ q✉❡ u=Hg✳ ❖ q✉❡ ❣❡r❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦ ❝♦♠ ♦ ❢❛t♦

❞❡ q✉❡ 0< δ ≤lim infHgn− HgC1_(Ω) = 0✳

❖ ♣ró①✐♠♦ ❧❡♠❛ é ✉♠ ❛♥á❧♦❣♦ ❛♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ ❢❡✐t♦ ❝♦♠ ❝♦♥❞✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t✳

▲❡♠❛ ✵✳✶✶✳ ❖ ♣r♦❜❧❡♠❛

(P∗

)

⎧ ⎨ ⎩

−∆pu+c|u|p

−₂

u=g(x) ;x∈Ω

u= 0 ;x∈∂Ω

❛❞♠✐t❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ❡♠ W₀1,p(Ω)✱♣❛r❛ t♦❞❛ g ∈ L∞

(Ω)✳ ❆❧é♠ ❞✐ss♦✱ ♦ ♦♣❡r❛❞♦r

H : L∞

(Ω) → C1

0(Ω) é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ❡ ❝♦♠♣❛❝t♦✳ ❙❡ S ⊂ L

∞

(Ω) é L∞

(Ω)✲

❧✐♠✐t❛❞♦ ♥❛ t♦♣♦❧♦❣✐❛ ❞❡ Lp_(Ω) _{✐♥❞✉③✐❞❛ ♣❡❧♦ ✐♠❡rsã♦} _L∞

(Ω)֒→Lp_(Ω)_{✱ ❡♥tã♦ ♦ ♦♣❡r❛❞♦r}

(21)

Pr❡❧✐♠✐♥❛r❡s ✻

❉❡♠♦♥str❛çã♦✳ ◆♦t❡ q✉❡✱ ♣❡❧♦ ❢❛t♦ ❞❡ c > 0 ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❝♦♥st❛♥t❡s M1 < 0 ❡

M2 >0 t❛❧ q✉❡ ❛s ❢✉♥çõ❡s φ1(x) =M1 ❡ φ2(x) =M2✱ sã♦ s✉❜ ❡ s✉♣❡rs♦❧✉çõ❡s ❡str✐t❛s ❞♦

♣r♦❜❧❡♠❛ (P∗

)✳ ❙❡❥❛ A = [φ1, φ2] ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ϕ∈C01(Ω)✱ t❛✐s q✉❡ φ1 ≤ϕ≤φ2✳

❙✉♣♦♥❤❛ q✉❡ u, v ∈W₀1,p(Ω) s❛t✐s❢❛③

⎧ ⎨ ⎩

−∆pu+c|u|p−2u≥ −∆pv+c|v|p−2v ;x∈Ω

u≥v ;x∈∂Ω

❯s❛♥❞♦ ♦ ▲❡♠❛ ✵✳✷❞❡ ❬✸✶❪✱ ❝♦♥❝❧✉✐♠♦s q✉❡ ✉♠❛ s♦❧✉çã♦ ❞❡ (P∗

)é ú♥✐❝❛ ❡ q✉❡ ♦ ♦♣❡r❛❞♦r

s♦❧✉çã♦✱ ❝❛s♦ ❡①✐st❛✱ é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡✳ ❙❡❥❛AR =A∩BR✱ ♦♥❞❡BR é ❛ ❜♦❧❛ ❞❡ r❛✐♦

R ❡♠ C1

0(Ω)✳ P❡❧♦ ❢❛t♦ ❞❡ φ1(x) = M1 ❡ φ2(x) =M2 s❡r❡♠ s✉❜ ❡ s✉♣❡rs♦❧✉çõ❡s ❡str✐t❛s✱

✉♠❛ s♦❧✉çã♦u❞❡(P∗

)❡stá ♥♦ ✐♥t❡r✐♦r ❞❡A✳ ▲♦❣♦ ♣❛r❛Rs✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ❞❡✈❡♠♦s

t❡r

deg (I−P, AR,0) = 1,

♦♥❞❡ P : AR → C1(Ω) é ❞❡✜♥✐❞♦ ♣♦r P u = K(g−c|u|p−2u) ❡ K é ♦ ♦♣❡r❛❞♦r ❞❛❞♦

♥♦ ▲❡♠❛ ✵✳✽✳ P♦rt❛♥t♦✱ ♦ ♣r♦❜❧❡♠❛ (P∗

) é ✉♥✐❝❛♠❡♥t❡ r❡s♦❧✉✈❡❧ ♣❛r❛ ❝❛❞❛ g ∈ L∞

(Ω)

❡ ♦ ♦♣❡r❛❞♦r s♦❧✉çã♦ H : L∞

(Ω) → W₀1,p(Ω) é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡✳ ❙❡❥❛ {gn} ✉♠❛

s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❡♠ L∞

(Ω) ❡ wn =Hgn✱ ❧♦❣♦

−∆pwn+c|wn|p

−₂

wn=gn.

P❡❧♦ ❧❡♠❛ ✵✳✽ wn ∈ C₀1,α(Ω) ❝♦♠ wn_C1,α

0 (Ω) < C✳ ❈♦♠♦ ❛ ✐♠❡rsã♦ C

1,α

0 ֒→ C 1,β

0 ✱

0 ≤ β < α é ❝♦♠♣❛❝t❛✱ ♣❛ss❛♥❞♦ ❛ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ s❡ ♥❡❝❡ssár✐♦✱ Hgn → w ❡♠

C₀1,β(Ω)✱ ♣♦rt❛♥t♦ H : L∞

(Ω) → C1

0(Ω) é ❝♦♠♣❛❝t♦✳ ❙❡❥❛ S ⊂ L

∞

(Ω)✱ L∞

(Ω)✲❧✐♠✐t❛❞♦

♥❛ t♦♣♦❧♦❣✐❛ ❞❡ Lp_(Ω) _{✐♥❞✉③✐❞❛ ♣❡❧♦ ✐♠❡rsã♦} _L∞

(Ω)֒→Lp_(Ω) _❡_{g

n} ⊂S t❛❧ q✉❡ gn →g

❡♠ W−1,p′_(Ω)_{✱ ❡♥tã♦ ♣❡❧♦ ❧❡♠❛ ✵✳✻ t❡♠♦s q✉❡} _Hg

n→ Hg ❡♠ W1 ,p

0 (Ω) ❡ ♣❛ss❛♥❞♦ ❛ ✉♠❛

s✉❜s❡q✉ê♥❝✐❛ s❡ ♥❡❝❡ssár✐♦✱ t❡♠♦s q✉❡Hgn → Hg❡♠C01,β(Ω)❡ ✐ss♦ ❝♦♥❝❧✉✐ ❛ ❝♦♥t✐♥✉✐❞❛❞❡

(22)

❈❆P❮❚❯▲❖ ✶

P❘❖❇▲❊▼❆❙ ❈❖▼ ❈❖◆❉■➬➹❖ ❉❊

◆❊❯▼❆◆◆

✶✳✶ ❘❡s✉❧t❛❞♦s Prí♥❝✐♣❛✐s

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

❙❡❥❛♠ Ω ⊂RN _{✉♠ ❛❜❡rt♦✱ ❧✐♠✐t❛❞♦ ❡ ❝♦♠ ❢r♦♥t❡✐r❛} _∂_Ω _{s✉❛✈❡ ❡} _f _{: Ω}_×_R _→_R _✉♠❛

❢✉♥çã♦ ❞❡ ❈❛r❛t❤❡♦❞♦r② s❛t✐s❢❛③❡♥❞♦ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿

lim sup

s→−∞

f(x, s)

|s|p−2_s <0

✭✶✳✶✮

❡

lim inf

s→₊∞

f(x, s)

|s|p−2_s >0

✭✶✳✷✮

✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ x ∈ Ω✳ ◆♦ q✉❡ s❡❣✉❡✱ t ∈ _R ❡ 1 < p < ∞✳ ❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡

♣r♦❜❧❡♠❛✱

(Pt)

⎧ ⎨ ⎩

−∆pu=f(x, u) +tφ(x) +h(x) ;x∈Ω

|∇u|p−₂∂u

∂ν = 0 ;x∈∂Ω

(23)

❙❊➬➹❖ ✶✳✷ ∗ Pr✐♠❡✐r❛ ❙♦❧✉çã♦ ✽

♦♥❞❡ φ(x)≥0✱ φ(x)≡0 ❡ φ, h ∈L∞

(Ω)✳ ❙✉♣♦♥❤❛♠♦s t❛♠❜é♠ q✉❡ ∀M >0✱ ∃λ >0 t❛❧

q✉❡✱

g(x, u) =f(x, u) +λ|u|p−2_u

✭✶✳✸✮

é ♥ã♦ ❞❡❝r❡s❝❡♥t❡ ∀u∈[−M, M]✳

❚❡♦r❡♠❛ ✶✳✶✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛s ❝♦♥❞✐çõ❡s ✭✶✳✶✮✱ ✭✶✳✷✮ ❡ ✭✶✳✸✮ ❡st❡❥❛♠ s❛t✐s❢❡✐t❛s ❡ q✉❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ c t❛❧ q✉❡✱

|f(x, s)| ≤c(1 +|s|p−₁

); ∀s≤0

✭✶✳✹✮

❡ ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ x∈Ω ✳ ❊♥tã♦✱ ❡①✐st❡ t0 ∈R✱ t❛❧ q✉❡

✭✐✮ ❙❡ t < t0✱ ❡♥tã♦ (Pt) ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦✳

✭✐✐✮ ❙❡ t > t0✱ ❡♥tã♦ ♦ ♣r♦❜❧❡♠❛ (Pt) ♥ã♦ t❡♠ s♦❧✉çã♦✳

❘❡str✐♥❣✐♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭✶✳✹✮ ♣❛r❛ s → +∞ ♦❜t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ r❡❧❛t✐✈♦ ❛

♠✉❧t✐♣❧✐❝✐❞❛❞❡✳

❚❡♦r❡♠❛ ✶✳✷✳ ❙✉♣♦♥❞♦ ❛s ❝♦♥❞✐çõ❡s ✭✶✳✶✮✱ ✭✶✳✷✮ ❡ ✭✶✳✸✮ ❡ q✉❡ ❡①✐st❡ ✉♠❛ ❝♦♥t❛♥t❡ ct❛❧

q✉❡✱

|f(x, s)| ≤c(1 +|s|p−1_); _∀s_∈

R

✭✶✳✺✮

❡ ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ x∈Ω✳ ❊♥tã♦ ❡①✐st❡ t1 ∈R ❝♦♠ t1 ≤t0✱ t❛❧ q✉❡

✭✐✮ ❙❡ t=t0✱ ❡♥tã♦ (Pt) ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦✳

✭✐✐✮ ❙❡ t < t1✱ ❡♥tã♦ ♦ ♣r♦❜❧❡♠❛ (Pt) ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ❞✉❛s s♦❧✉çõ❡s ❞✐st✐♥t❛s✳

✶✳✷ Pr✐♠❡✐r❛ ❙♦❧✉çã♦

(24)

❙❊➬➹❖ ✶✳✷ ∗ Pr✐♠❡✐r❛ ❙♦❧✉çã♦ ✾

◆♦t❡ q✉❡ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ (Pt)é ❡q✉✐✈❛❧❡♥t❡ ❛ ♠♦str❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦

♣❛r❛ ❛ ❡q✉❛çã♦

u=H f(x, u) +c|u|p−2_u₊_tφ₊_h

✭✶✳✻✮

♦♥❞❡✱H é ♦ ♦♣❡r❛❞♦r ❞❛❞♦ ♥♦ ▲❡♠❛ ✵✳✶✵✳

▼♦str❛r❡♠♦s q✉❡ ❛ ❡q✉❛çã♦ ✭✶✳✻✮ t❡♠ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ❛❧❣✉♠ t∈_R❡ ♣❛r❛ ✐ss♦✱ ♣r❡❝✐✲

s❛r❡♠♦s ❞♦s ♣ró①✐♠♦s ❧❡♠❛s✱ q✉❡ ❢♦r❛♠ ♠♦t✐✈❛❞♦s ♣❡❧♦s r❡s✉❧t❛❞♦s ❡♠([✷✶],▲❡♠❛s ✶ ❡ ✷)✳

▲❡♠❛ ✶✳✸✳ P❛r❛ ❝❛❞❛ R1 >0 ❞❛❞♦✱ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ T =T(R1) t❛❧ q✉❡

v =H τ f(x, v) +c|v|p−₂

v+tφ+h

✭✶✳✼✮

♣❛r❛ t♦❞❛ v ∈C1_(Ω) _❝♦♠ _v+₌_R

1✱ ∀τ ∈[0,1]✱ ∀t ≤T✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ ♣♦r ❛❜s✉r❞♦ q✉❡ ❡①✐st❛♠ s❡q✉ê♥❝✐❛s {vn} ⊂C1(Ω)❝♦♠ v+n=

R1✱ {τn} ⊂[0,1] ❡{tn} ⊂R✱ tn→ −∞✱ t❛❧ q✉❡

vn=H τn f(x, vn) +c|vn|p

−2_v

n+tnφ+h

✭✶✳✽✮

✉s❛♥❞♦ ❛ ❤✐♣ót❡s❡ ✭✶✳✹✮ ♦❜t❡♠♦s✱

f(x, s) +c|s|p−2_s _≤_c|s|p−2_s₊_|f₍_{x, s}₎_|

✭✶✳✾✮

≤c|s|p−2_s₊_c₊_c|s|p−1

✭✶✳✶✵✮

=c

✭✶✳✶✶✮

≤f(x,0) + 2c

✭✶✳✶✷✮

♣❛r❛s ≤0✳ ❆ss✐♠✱ ♣❛r❛ ❛ s❡q✉ê♥❝✐❛ vn t❡♠♦s

τn f(x, vn) +c|vn|p

−₂

vn+tnφ+h

≤τn f(x, vn+) +c|vn+| p−₂

vn++ 2c+tnφ+h

≤τn(M +tnφ+h)≤M +tnφ+h.

♦♥❞❡✱M = max

x∈Ω,vn+∈[0,R1]

f(x, vn+) +c|vn+| p−₂

vn++ 2c

✳ ❙❡❣✉❡ q✉❡

vn ≤ H(M +tnφ+h)

❙❡❥❛wn=H((M+tnφ+h))✱ ♦✉ s❡❥❛wn s❛t✐s❢❛③

−∆pwn+c|wn|p

−2_w

(25)

❙❊➬➹❖ ✶✳✷ ∗ Pr✐♠❡✐r❛ ❙♦❧✉çã♦ ✶✵

❉❡✜♥❛ sn t❛❧ q✉❡ tn=|sn|p−2sn✱ ❛ss✐♠

−∆p(

wn

sn

) +c|wn sn

|p−2₍wn

sn

) = (M

tn

+φ+ h

tn

)

▲♦❣♦✱ (M_t_n +φ+ _th_n)→ φ✱ q✉❛♥❞♦ tn → −∞✳ P❡❧♦ ▲❡♠❛ ✵✳✶✵ t❡♠♦s q✉❡ H é ❢♦rt❡♠❡♥t❡

❝r❡s❝❡♥t❡ ❡ ❝♦♥tí♥✉♦ q✉❛♥❞♦ r❡str✐t♦ ❛ ✉♠ ❝♦♥❥✉♥t♦ S ⊂ L∞

(Ω) ✱ L∞

(Ω)✲❧✐♠✐t❛❞♦✱ s❡❣✉❡

q✉❡

wn

sn

→ H(φ)>0.

❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡wn<0✳ ▲♦❣♦ vn+= 0✱ ❝♦♥tr❛❞✐③❡♥❞♦ ♦ ❢❛t♦ ❞❡ q✉❡ v+n=R1✳

▲❡♠❛ ✶✳✹✳ ❙❡❥❛t ∈_R ✜①♦✳ ❊♥tã♦ ❡①✐st❡ R2 >0 t❛❧ q✉❡

v =H τ f(x, v) +c|v|p−2_v₊_tφ₊_h

✭✶✳✶✸✮

♣❛r❛ t♦❞❛ v ∈C1_(Ω) _❝♦♠ _v−

=R2✱ ∀τ ∈[0,1]✳

❉❡♠♦♥str❛çã♦✳ ❆ ✐❞é✐❛ é ♦❜t❡r♠♦s ❡st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐ ♣❛r❛ ❡✈❡♥t✉❛✐s s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦

vτ =H τ f(x, vτ) +c|vτ|p

−2_v

τ +tφ+h

τ ∈[0,1].

s❡❣✉❡ ❞❛s ❤✐♣ót❡s❡s ✭✶✳✶✮❡ ✭✶✳✷✮q✉❡ ❡①✐st❡♠ ǫ >0 ❡c1 ∈R t❛❧ q✉❡

f(x, s)≥ −ǫ|s|p−2_s₊_c

1 ∀(x, s)∈Ω×R.

❧♦❣♦✱

−∆pvτ+c|vτ|p

−2_v

τ =τ f(x, vτ) +c|vτ|p

−2_v

τ +tφ+h

≥τ c|vτ|p

−2_v

τ−ǫ|vτ|p

−2_v

τ +c1+tφ+h

=τ (c−ǫ)|vτ|p

−2_v

τ +c1+tφ+h

P♦r ♦✉tr♦ ❧❛❞♦✱ s❡❥❛ wτ ❛ ú♥✐❝❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛

⎧ ⎨ ⎩

−∆pwτ+ (c−τ(c−ǫ))|wτ|p−2wτ =τ(c1 +tφ+h) ;x∈Ω

|∇wτ|p−2

∂wτ

∂ν = 0 ;x∈∂Ω

◆♦t❡ q✉❡ τ(c−ǫ)< c✱ ❧♦❣♦ ♦ ❝♦♥❥✉♥t♦(wτ)_τ∈[0,1] é ❧✐♠✐t❛❞♦ ❡♠C1(Ω)✱ ✐st♦ é✱ ❡①✐st❡ c2 t❛❧

q✉❡ wτC1_(Ω) ≤ c2✳ ❙❡❣✉❡ ❞♦ ♣r✐♥❝í♣✐♦ ❞❡ ❝♦♠♣❛r❛çã♦ ❢r❛❝♦ ▲❡♠❛ ✵✳✶✱ q✉❡ vτ ≥ wτ ≥

(26)

❙❊➬➹❖ ✶✳✸ ∗ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✶✳✶ ✶✶

▼♦str❛r❡♠♦s ❛❣♦r❛ q✉❡ ❛ ❡q✉❛çã♦ ✭✶✳✻✮ t❡♠ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ❛❧❣✉♠ t ✉s❛♥❞♦ ♦ ❣r❛✉

❞❡ ▲❡r❛②✲❙❤❛✉❞❡r✳ ❉❛❞♦ R1 > 0✱ ✜①❡♠♦s t ≤ T(R1) ❝♦♠ T(R1) ❞❛❞♦ ♣❡❧♦ ▲❡♠❛ ✶✳✸ ❡

❝♦♥s✐❞❡r❡ R2 >0❣❛r❛♥t✐❞♦ ♥♦ ▲❡♠❛ ✶✳✹✳ ❈♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦

Λ =v ∈C1(Ω) :v+< R1,

v−< R2

◆♦t❡ q✉❡ Λ é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❡♠ C1_(Ω) _{❝♦♥t❡♥❞♦} ₀_{✳ P❡❧♦s ▲❡♠❛s ✶✳✸ ❡ ✶✳✹ t❡♠♦s q✉❡}

v =H τ f(x, v) +c|v|p−₂

v+tφ+h ∀v ∈∂Λ,

▲♦❣♦✱ ♣❡❧❛ ✐♥✈❛r✐â♥❝✐❛ ❤♦♠♦tó♣✐❝❛ ❞♦ ❣r❛✉ ❞❡ ▲❡r❛②✲❙❝❤❛✉❞❡r s❡❣✉❡ q✉❡

deg I − H f(x, v) +c|v|p−2_v₊_tφ₊_h_,_Λ_,₀ _{= deg (}_I,_Λ_,_{0) = 1}_.

❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ ❡①✐st❡ v ∈Λ t❛❧ q✉❡

v =H f(x, v) +c|v|p−2_v₊_tφ₊_h

❡ ♣♦rt❛♥t♦✱ ♦ ♣r♦❜❧❡♠❛ (Pt) ♣♦ss✉✐ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ t≤T(R1)✳

✶✳✸ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✶✳✶

◆❡st❛ s❡çã♦ s❡❣✉✐r❡♠♦s ❛s ✐❞é✐❛s ❞❡ ❉❡ P❛✐✈❛ ❡ ▼♦♥t❡♥❡❣r♦ ❡♠ [✶✻]♣❛r❛ ❣❛r❛♥t✐r♠♦s ❛

❡①✐stê♥❝✐❛ ❞❡ s✉❜s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛(Pt)✳ ❊♠ s❡❣✉✐❞❛✱ ♠♦str❛r❡♠♦s ♦ ❜❡♠ ❝♦♥❤❡❝✐❞♦

♠ét♦❞♦ ❞❡ s✉❜✲s✉♣❡rs♦❧✉çã♦✳ ❯s❛♥❞♦ ❡st❡s r❡s✉❧t❛❞♦s✱ ♠♦str❛r❡♠♦s q✉❡ s❡ ♦ ♣r♦❜❧❡♠❛

(Pt) t❡♠ s♦❧✉çã♦ ♣❛r❛ ❛❧❣✉♠ t ❡♥tã♦ ♣❛r❛ t♦❞♦ s ≤ t t❛♠❜é♠ t❡♠ s♦❧✉çã♦ ❡ ❡♠ s❡❣✉✐❞❛

❢❛r❡♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✶✳✶✳ ❈♦♠❡ç❛r❡♠♦s ❝♦♠ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿

▲❡♠❛ ✶✳✺✳ ❖ ♣r♦❜❧❡♠❛ (Pt) ♣♦ss✉✐ s✉❜s♦❧✉çã♦ ♣❛r❛ t♦❞♦ t∈R✳

❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ⎧

⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

−∆pz ≤f(x, z) +tφ(x) +h(x) ;x∈Ω

z ≤0 ;x∈Ω

|∇z|p−₂∂z

∂ν = 0 ;x∈∂Ω

▼♦str❛r❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ zt ♥❡❣❛t✐✈❛ s❛t✐s❢❛③❡♥❞♦ ♦ ♣r♦❜❧❡♠❛ ❛❝✐♠❛✳ ❉❡

❢❛t♦✱ ✉s❛♥❞♦ ❛ ❤✐♣ót❡s❡ ✭✶✳✶✮ s❡❣✉❡ q✉❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡s ǫ > 0 ❡ C > 0 t❛✐s q✉❡

(27)

❙❊➬➹❖ ✶✳✸ ∗ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✶✳✶ ✶✷

zt =−

_{|t| φ}

∞+h∞+C

ǫ

1 (p−1) ✭✶✳✶✹✮

s❡❣✉❡ q✉❡ ♣❛r❛ t♦❞♦t ∈R✱

f(x, zt) +tφ+h≥ −ǫ|zt|p

−2_z

t−C+tφ+h=ǫ

_{|t| φ}

∞+h∞+C

ǫ

−C+tφ+h ≥ |t| φ∞+tφ+h∞+h

≥0.

❉❡ ♦♥❞❡ s❡❣✉❡ q✉❡zté s✉❜s♦❧✉çã♦ ❞❡(Pt)✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦f(x, k)+tφ+h≥ −ǫ|k|p−2k−

C+tφ+h≥ −ǫ|zt|p−2zt−C+tφ+h é ❢á❝✐❧ ✈❡r q✉❡ t♦❞❛ ❝♦♥st❛♥t❡ k < zt é s✉❜s♦❧✉çã♦

❡str✐t❛ ❞❡ (Pt)✳

❖ ♣ró①✐♠♦ ❚❡♦r❡♠❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ▼ét♦❞♦ ❞❡ s✉❜ ❡ s✉♣❡rs♦❧✉çã♦✳

❚❡♦r❡♠❛ ✶✳✻✳ ❙❡❥❛♠ zt✱w ∈ C1(Ω) s✉❜ ❡ s✉♣❡rs♦❧✉çã♦ ❞❡ (Pt) r❡s♣❡❝t✐✈❛♠❡♥t❡ t❛✐s q✉❡

zt ≤ w ❡♠ Ω✳ ❊♥tã♦ ❡①✐st❡ u ∈ C1(Ω) s♦❧✉çã♦ ❞❡ (Pt) t❛❧ q✉❡ zt ≤ u ≤ w ❡♠ Ω ❡

|∇u|p−₂∂u

∂ν = 0 s♦❜r❡ ∂Ω✳

❉❡♠♦♥str❛çã♦✳ ❉❡✜♥❛ ♦s ♦♣❡r❛❞♦r❡s Nt:C1(Ω)→L∞(Ω) ♣♦r

Nt(v) =f(x, v) +λ|v|p

−₂

v+tφ+h ✱ v ∈C1(Ω)

❡T :L∞

(Ω)→C1_(Ω) _♣♦r _T₍_v_{) =}_w _{s❡✱ ❡ s♦♠❡♥t❡ s❡✱} _w _{é s♦❧✉çã♦ ❞❡}

⎧ ⎨ ⎩

−∆pw+λ|w|p−2w=v ;x∈Ω

|∇w|p−2∂w

∂ν = 0 ;x∈∂Ω

❉❡✜♥❛ t❛♠❜é♠ Kt:C1(Ω)→C1(Ω) ♣♦r Kt =T ◦Nt✳

◆♦t❡ q✉❡ u é ✉♠ ♣♦♥t♦ ✜①♦ ❞❡ Kt s❡✱ ❡ s♦♠❡♥t❡ s❡✱ u é ✉♠❛ s♦❧✉çã♦ ❞❡ (Pt)✳ ▼♦str❡♠♦s

q✉❡Kt é ❝♦♠♣❛❝t♦✳ ❈♦♠ ❡❢❡✐t♦✱ ❝♦♠♦ Kt=T ◦Nt ❡ Nt é ❝♦♥tí♥✉❛✱ ❜❛st❛ ♠♦str❛r q✉❡ T

é ❝♦♠♣❛❝t♦✳ ❉❡ ❢❛t♦✱ s❡❥❛ {un}✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❡♠ L∞(Ω) ❡ wn =T(un)✱ ❧♦❣♦

−∆pwn+λ|wn|p−2wn=un.

P❡❧♦ ▲❡♠❛ ✵✳✾ wn ∈ C1(Ω) ❝♦♠ wnC1_(Ω) < C✳ ❖❜t❡♠♦s ♣❡❧❛ ✐♠❡rsã♦ ❝♦♠♣❛❝t❛

C1,α_(Ω) _֒_→ _C1,β_(Ω) _{q✉❡✱ ❛ ♠❡♥♦s ❞❡ s✉❜s❡q✉ê♥❝✐❛✱} _w

(28)

❙❊➬➹❖ ✶✳✸ ∗ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✶✳✶ ✶✸

é ❝♦♠♣❛❝t♦✳ ❆❣♦r❛✱ ❞❡✜♥❛

A =u∈C1(Ω) :zt(x)≤u(x)≤w

.

◆♦t❡ q✉❡Aé ❢❡❝❤❛❞♦ ❡ ❝♦♥✈❡①♦✳ P❡❧❛ ❤✐♣ót❡s❡ ✭✶✳✸✮❡ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❡ ❝♦♠♣❛r❛çã♦ t❡♠♦s

q✉❡ Kt(A)⊂ A✱ ❡ ♣❡❧♦ ▲❡♠❛ ✵✳✾ Kt(A) é ❧✐♠✐t❛❞♦✳ ❆ss✐♠✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦

❞❡ ❙❝❤❛✉❞❡r ❡①✐st❡ u∈C1_(Ω) _{♣♦♥t♦ ✜①♦ ❞❡} _K

t✱ ♦✉ s❡❥❛✱ ❡①✐st❡ u∈C1(Ω) s♦❧✉çã♦ ❞❡(Pt)

t❛❧ q✉❡zt(x)≤u(x)≤w ❡♠ Ω✳

❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ ♥♦s ❞✐③ q✉❡✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s t ♣❛r❛ ♦s q✉❛✐s ♦ ♣r♦❜❧❡♠❛ (Pt)t❡♠

s♦❧✉çã♦ é ✉♠ ✐♥t❡r✈❛❧♦ ♥ã♦ ❞❡❣❡♥❡r❛❞♦✳

▲❡♠❛ ✶✳✼✳ ❙❡ ♦ ♣r♦❜❧❡♠❛ (Pt) t❡♠ s♦❧✉çã♦ ♣❛r❛ ❛❧❣✉♠ t ∈ R✱ ❡♥tã♦ (Pt) t❡♠ s♦❧✉çã♦

♣❛r❛ t♦❞♦ s ≤t✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ut ✉♠❛ s♦❧✉çã♦ ❞❡(Pt)✳ P❛r❛ t♦❞♦s≤t t❡♠♦s q✉❡ut é s✉♣❡rs♦❧✉çã♦

❞♦ ♣r♦❜❧❡♠❛ (Pt) ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ s✱ ♣♦✐s

−∆pu=f(x, u) +tφ(x) +h(x)≥f(x, u) +sφ(x) +h(x).

P♦r ♦✉tr♦ ❧❛❞♦✱ s❡❣✉❡ ❞♦ ▲❡♠❛ ✶✳✺ q✉❡ ❡①✐st❡ ✉♠❛ s✉❜s♦❧✉çã♦ zs< ut✳ ▲♦❣♦✱ ♣❡❧♦ ❚❡♦r❡♠❛

✶✳✻ ❡①✐st❡ ✉♠❛ s♦❧✉çã♦ us ❞❡ (Pt) ♣❛r❛ t♦❞♦ s≤t✳

✶✳✸✳✶ ◆ã♦ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛

t

s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡

❖ ❧❡♠❛ q✉❡ ❛♣r❡s❡♥t❛r❡♠♦s ❛❣♦r❛ ♠♦str❛ q✉❡ ♦ ♣r♦❜❧❡♠❛ (Pt)♥ã♦ ♣♦ss✉✐ s♦❧✉çã♦ ♣❛r❛

t ❣r❛♥❞❡ ♦ s✉✜❝✐❡♥t❡✳

▲❡♠❛ ✶✳✽✳ ❖ ♣r♦❜❧❡♠❛ (Pt) ♥ã♦ ♣♦ss✉✐ s♦❧✉çã♦ ♣❛r❛ t >0 s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛♠♦s ♣♦r q✉❡ (Pt) ♣♦ss✉✐ s♦❧✉çã♦ ut ♣❛r❛ ❛❧❣✉♠ t✳ ❙❡❣✉❡ ❞❛s

❤✐♣ót❡s❡s ✭✶✳✶✮❡ ✭✶✳✷✮q✉❡ ♣❛r❛ t♦❞♦ s∈R

f(x, s)≥ǫ|s|p−₁

−C.

(29)

❙❊➬➹❖ ✶✳✸ ∗ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✶✳✶ ✶✹

❯s❛♥❞♦ϕ ≡1 ❝♦♠♦ ❢✉♥çã♦ t❡st❡ ❡♠ (Pt) t❡♠♦s

0 =

Ω

|∇ut|p

−2_∇u

t· ∇ϕ dx=

Ω

f(x, ut)dx+

Ω

tφ dx+

Ω

h(x)dx ≥

Ω

(ǫ|ut|p

−₁

−C)dx+

Ω

tφ dx+

Ω

h(x)dx

=ǫ

Ω

|ut|p

−1_dx₋_C|_Ω_|₊_t Ω

φ dx+

Ω

h(x)dx

❧♦❣♦✱

ǫ

Ω

|ut|p

−₁

dx+t

Ω

φ dx+

Ω

h(x)dx≤C|Ω|.

P♦rt❛♥t♦

t

Ω

φ dx+

Ω

h(x)dx≤C|Ω|.

❆ss✐♠ t é ❧✐♠✐t❛❞♦✱ ♦ q✉❡ ♣r♦✈❛ ♦ ❧❡♠❛✳

✶✳✸✳✷ ❈♦♥❝❧✉sã♦ ❞❛ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✶✳✶

Pr♦✈❛ ❞❡ ✭✐✮

❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ❝♦♥❥✉♥t♦✱

S ={t : (Pt) t❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦}.

♣❡❧♦ q✉❡ ❢♦✐ ❢❡✐t♦ ♥❛ s❡çã♦ (1.2) t❡♠♦s q✉❡ S é ♥ã♦ ✈❛③✐♦✳ ❆❧é♠ ❞✐ss♦✱ s❡❣✉❡ ❞♦s ▲❡♠❛s

✶✳✼ ❡ ✶✳✽ q✉❡ S é ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡ ❧♦❣♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r t0 = sup

S

t✳ ❆❧é♠ ❞✐ss♦✱

♥♦t❡ q✉❡ s❡t∈ S ❡♥tã♦ (−∞, t]⊂ S✳ ▲♦❣♦✱ q✉❡ ♣❛r❛ ❝❛❞❛ t < t0✱ ♦ ♣r♦❜❧❡♠❛(Pt)♣♦ss✉✐

♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦✳

Pr♦✈❛ ❞❡ ✭✐✐✮

❙❡❣✉❡ ❞♦ ▲❡♠❛ ✶✳✽ ❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ s✉♣r❡♠♦ q✉❡ ♣❛r❛ t♦❞♦ t > t0✱ (Pt) ♥ã♦ t❡♠

(30)

❙❊➬➹❖ ✶✳✹ ∗ ❊st✐♠❛t✐✈❛ ❛ ♣r✐♦r✐ ✶✺

✶✳✹ ❊st✐♠❛t✐✈❛ ❛ ♣r✐♦r✐

P❛r❛ ♦❜t❡r♠♦s ❛ s❡❣✉♥❞❛ s♦❧✉çã♦ ✈✐❛ t❡♦r✐❛ ❞♦ ❣r❛✉✱ ♣r❡❝✐s❛♠♦s ❞❡ ❡st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐ ♣❛r❛ ❡✈❡♥t✉❛✐s s♦❧✉çõ❡s ❞❡ (Pt)✳ ◆❡st❛ s❡çã♦✱ ♦❜t❡♠♦s ❡st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐ ♣❛r❛ ❛s ♣❛rt❡s

♥❡❣❛t✐✈❛ ❡ ♣♦s✐t✐✈❛ ❞❡ ✉♠❛ ❡✈❡♥t✉❛❧ ❞❡ (Pt)✳ ❊♠ s❡❣✉✐❞❛✱ ♦❜t❡♠♦s ✉♠❛ ❡st✐♠❛t✐✈❛ ♥❛

♥♦r♠❛L∞ _{❡ ♣♦r ✜♠ ❡st✐♠❛t✐✈❛ ♥❛ ♥♦r♠❛}

C1_✳

▲❡♠❛ ✶✳✾✳ ❙❡❥❛ u ∈ W1,p_(Ω) _{✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡} ₍_P

t)✳ ❙❡ t ♣❡rt❡♥❝❡ ❛ ✉♠ ✐♥t❡r✈❛❧♦

❧✐♠✐t❛❞♦ ❡♥tã♦✱ ❡①✐st❡ M =M(t)>0 t❛❧ q✉❡ u−

∞≤M✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛u✉♠❛ s♦❧✉çã♦ ❞❡(Pt)✳ ❈♦♥s✐❞❡r❛♥❞♦ ❛ ❢✉♥çã♦ϕ=max(u−−k,0)∈

W1,p_(Ω)_{✱ ❝♦♠} _{k >}₀ _{✜①♦ ❝♦♠♦ ❢✉♥çã♦ t❡st❡ ♦❜t❡♠♦s}

Ω

|∇u|p−2_∇u∇ϕ₌ Ω

(f(x, u) +tφ+h)ϕ

❉❡✜♥✐♥❞♦Ωk ={x∈Ω :u− > k}✱ ❝♦♠♦ ∇u−=∇(u−−k) = −∇u ❡♠ Ωk✱ t❡♠♦s q✉❡

Ωk

|∇(u−−k)|p

=−

Ωk

f(x,−u−

) +tφ+hϕ

❈♦♠♦ f s❛t✐s❢❛③ ❛ ❤✐♣ót❡s❡ ✭✶✳✶✮ t❡♠♦s q✉❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡s ǫ > 0 ❡ C1 > 0 t❛❧ q✉❡

f(x, u)≥ −ǫ|u|p−2_u₋_C

1✱ ♣❛r❛ t♦❞♦ u <0✳ ❊♥tã♦✱

Ωk

|∇(u−

−k)|p ₌₋

Ωk

f(x,−u−

) +tφ+hϕ

✭✶✳✶✻✮

≤

Ωk

ǫ(u−)p−1+C1+|t| φ∞+h∞

(u−−k)

✭✶✳✶✼✮

=

Ωk

ǫ(u−

)p−₁

(u−

−k) + (C1+|t| φ∞+h∞)

Ωk

(u−

−k)

✭✶✳✶✽✮

≤

Ωk

C2 (u−−k)p+kp−1(u−−k)

✭✶✳✶✾✮

+ (C1+|t| φ∞+h∞)

Ωk

(u−−k)

✭✶✳✷✵✮

=C2

Ωk

(u−

−k)p₊ _C

2kp

−₁

+C1+|t| φ∞+h∞

Ωk

(u−

−k).

✭✶✳✷✶✮

(31)

❙❊➬➹❖ ✶✳✹ ∗ ❊st✐♠❛t✐✈❛ ❛ ♣r✐♦r✐ ✶✻

Ωk

(u−

−k)p ≤ |Ωk|

p n

Ωk

(u−

−k)(nnp−p) (n−p)

n ✭✶✳✷✷✮

≤ |Ωk|

p nC₃

Ωk

|∇(u−

−k)|p₊

Ωk

(u−

−k)p

,

✭✶✳✷✸✮

❞❡ ✭✶✳✶✻✮ ❡ ✭✶✳✷✷✮ s❡❣✉❡ q✉❡

|Ωk|

−p

n −C

3 Ωk

(u−−k)p ≤C3

Ωk

(u−−k)p+C4 kp

−₁

+ 1 +|t| φ∞+h∞

Ωk

(u−−k)

✭✶✳✷✹✮

❧♦❣♦✱

|Ωk|

−p

n −C

3 Ωk

(u−

−k)p _≤_C

4 kp

−1 _{+ 1 +}_{|t| φ}

∞+h∞

Ωk

(u−

−k).

❯s❛♥❞♦ ♦ ♠❡s♠♦ r❛❝í♦❝✐♥✐♦ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✶✳✽✱ ♦❜t❡♠♦s q✉❡

|Ωk|=

u−_>k

≤

Ωk

(u−

)

kp−₁ ≤C5k

1−p_.

✭✶✳✷✺✮

❧♦❣♦✱ ❛ ♠❡❞✐❞❛ |Ωk| → 0 q✉❛♥❞♦ k → +∞✱ ❞❡ ♦♥❞❡ s❡❣✉❡ q✉❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ k0

♥ã♦ ❞❡♣❡♥❞❡♥t❡ ❞❡ u t❛❧ q✉❡ |Ωk|−

p

n −C₃ > 0✱ ♣❛r❛ t♦❞♦ k ≥ k₀✳ ❙❡❣✉❡ ❞❛ ❞❡s✐❣✉❧❞❛❞❡ ❞❡ ❍♦❧❞❡r ❡ ❞❡ ✭✶✳✷✹✮✱

Ωk

(u−

−k)≤ |Ωk|

(p−1) p

Ωk

(u−

−k)p

1 p

≤C6|Ωk|

(p−1) p

kp−1_{+ 1 +}_{|t| φ}

∞+h∞

|Ωk|−

p n −C₃

Ωk

(u−−k)

1 p

❧♦❣♦✱

Ωk

(u−

−k)(p

−₁₎

p

≤C6|Ωk|

(p−₁₎

p

kp−1_{+ 1 +}_{|t| φ}

∞+h∞

|Ωk|−

p n −C₃

1 p

❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛✱

Ωk

(u−

−k)≤C7|Ωk|

kp−₁

+ 1 +|t| φ∞+h∞

|Ωk|

−p

n −C₃

1 p−₁

=C7|Ωk|1+(

p n(p−1))

kp−1_{+ 1 +}_{|t| φ}

∞+h∞

1− |Ωk|

p nC₃