Open Existência de soluções para equações elípticas semilineares envolvendo não linearidades do tipo côncavoconvexas

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

❈✉rs♦ ❞❡ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛

❊①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ❡q✉❛çõ❡s ❡❧í♣t✐❝❛s

s❡♠✐❧✐♥❡❛r❡s ❡♥✈♦❧✈❡♥❞♦ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡s ❞♦

t✐♣♦ ❝ô♥❝❛✈♦✲❝♦♥✈❡①❛s

❘♦s✐♥â♥❣❡❧❛ ❈❛✈❛❧❝❛♥t✐ ❞❛ ❙✐❧✈❛

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

❈✉rs♦ ❞❡ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛

❊①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ❡q✉❛çõ❡s ❡❧í♣t✐❝❛s

s❡♠✐❧✐♥❡❛r❡s ❡♥✈♦❧✈❡♥❞♦ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡s ❞♦

t✐♣♦ ❝ô♥❝❛✈♦✲❝♦♥✈❡①❛s

♣♦r

❘♦s✐♥â♥❣❡❧❛ ❈❛✈❛❧❝❛♥t✐ ❞❛ ❙✐❧✈❛

s♦❜ ♦r✐❡♥t❛çã♦ ❞❡

❊❧✐s❛♥❞r❛ ❞❡ ❋át✐♠❛ ●❧♦ss ❞❡ ▼♦r❛❡s

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

❊①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ❡q✉❛çõ❡s ❡❧í♣t✐❝❛s

s❡♠✐❧✐♥❡❛r❡s ❡♥✈♦❧✈❡♥❞♦ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡s ❞♦

t✐♣♦ ❝ô♥❝❛✈♦✲❝♦♥✈❡①❛s

❘♦s✐♥â♥❣❡❧❛ ❈❛✈❛❧❝❛♥t✐ ❞❛ ❙✐❧✈❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ❆♥á❧✐s❡ ❆♣r♦✈❛❞♦ ♣❡❧❛ ❇❛♥❝❛✿

Pr♦❢a_{✳ ❉r❛✳ ❊❧✐s❛♥❞r❛ ❞❡ ❋át✐♠❛ ●❧♦ss ❞❡ ▼♦r❛❡s✭❖r✐❡♥t❛❞♦r❛✮}

Pr♦❢✳ ❉r✳ ❊✈❡r❛❧❞♦ ❙♦✉t♦ ❞❡ ▼❡❞❡✐r♦s

Pr♦❢a_{✳ ❉r❛✳ ❏❛♥❡t❡ ❙♦❛r❡s ❞❡ ●❛♠❜♦❛}

Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛✭❙✉♣❧❡♥t❡✮

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

❆♦s ♠❡✉s ♣❛✐s ❡ ♠✐♥❤❛s ✐r♠ãs✳

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛ ❉❡✉s ♣♦r t❡r ♠❡ ❞❛❞♦ ❙❛❜❡❞♦r✐❛ ♣❛r❛ ❝♦♥q✉✐st❛r ♠❡✉ s♦♥❤♦✳ ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♣❛✐s ♣♦r t❡r❡♠ ❛❝r❡❞✐t❛❞♦ ❡♠ ♠✐♠✱ ♣♦r t♦❞♦ ❛♣♦✐♦✱ ❝❛r✐♥❤♦ ❡ ♣❡❧❛s ♣❛❧❛✈r❛s ❞❡ ❢♦rç❛✳ ❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛s ✐r♠ãs ❊❞♥â♥❣❡❧❛ ❡ ▼❛r✐♥â♥❣❡❧❛ ♣♦r t❡r❡♠ ♠❡ ❛❣✉❡♥t❛❞♦ ❤♦r❛s ❛♦ t❡❧❡❢♦♥❡ ♣❛r❛ ♠❛t❛r ❛ s❛✉❞❛❞❡ ❡ ♣♦r ❡st❛r❡♠ ❛♦ ♠❡✉ ❧❛❞♦ s❡♠♣r❡✳ ❆♦ ♠❡✉ ❝✉♥❤❛❞♦ ❉✐♦❣♦ ♣♦r t✉❞♦ q✉❡ ❢❡③ ♣♦r ♠✐♠✳ ❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♥❛♠♦r❛❞♦ ❘✉❜❡♥s ♣❡❧♦ s❡✉ ❝❛r✐♥❤♦ ❡ ❞❡❞✐❝❛çã♦✱ ♣♦r t❡r ♠❡ ❞❛❞♦ ❢♦rç❛ ❡ ❝♦♥s❡❧❤♦s q✉❡ ♥✉♥❝❛ ✈♦✉ ❡sq✉❡❝❡r✳ ❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ❛♠✐❣❛ P❛♠♠❡❧❧❛ ♣♦r t❛♥t❛s ♥♦✐t❡s ❞❡ ❡st✉❞♦s ❥✉♥t❛s✱ ♣♦r s❡❝❛r ♠✐♥❤❛s ❧á❣r✐♠❛s q✉❛♥❞♦ ❛ s❛✉❞❛❞❡ ❡ ❛ ❛♥❣úst✐❛ ❝❤❡❣❛✈❛♠ ❡ ♣♦r t❛♥t♦s r✐s♦s✳ ❆❣r❛❞❡ç♦ ❛ ❢❛♠í❧✐❛ ◗✉❡✐r♦③ ❡♠ ♥♦♠❡ ❞❡ ❉♦♥❛ ❋át✐♠❛✱ ♣♦r t❡r ♠❡ ❛❝♦❧❤✐❞♦ ❝♦♠♦ ✜❧❤❛✳ ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❝♦❧❡❣❛s✱ ❛ t♦❞♦s ♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ❡ ❛ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ❡ ❛♠✐❣❛ ❊❧✐s❛♥❞r❛ ♣❡❧❛ s✉❛ ♣❛❝✐ê♥❝✐❛✱ ❞❡❞✐❝❛çã♦ ❡ ♣❡❧❛ ❢♦rç❛ q✉❡ ♠❡ ❞❡✉ ❞✉r❛♥t❡ t♦❞♦ ❡ss❡ t❡♠♣♦ ❞❡ ❡st✉❞♦s✳

❘❡s✉♠♦

❖ ♦❜❥❡t✐✈♦ ❞❛ ♥♦ss❛ ❞✐ss❡rt❛çã♦ é ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ✉♠❛ ❝❧❛ss❡ ❞❡ ❡q✉❛çõ❡s ❡❧í♣t✐❝❛s s❡♠✐❧✐♥❡❛r❡s ❡♠ ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦✱ ❡♥✈♦❧✈❡♥❞♦ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡s ❞♦ t✐♣♦ ❝ô♥❝❛✈♦✲❝♦♥✈❡①❛s✳ ▼♦str❛r❡♠♦s ❛❧❣✉♥s ❝❛s♦s ❞✐❢❡r❡♥t❡s ❡ ♠ét♦❞♦s ❞✐✈❡rs✐✜❝❛❞♦s ♣❛r❛ ❡♥❝♦♥tr❛r t❛✐s s♦❧✉çõ❡s✱ ✉s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✱ ♦ Pr✐♥❝í♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞✱ ❚❡♦r❡♠❛ ❞♦s ▼✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡✱ ❛ ❱❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ ❡ s✉❜ ❡ s✉♣❡rs♦❧✉çã♦✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❊q✉❛çõ❡s s❡♠✐❧✐♥❡❛r❡s✱ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡s ❝ô♥❝❛✈♦✲❝♦♥✈❡①❛s✱ ▼✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡✱ ❱❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐✱ s✉❜ ❡ s✉♣❡rs♦❧✉çã♦✳

❆❜str❛❝t

❚❤❡ ❣♦❛❧ ♦❢ ♦✉r ✇♦r❦ ✐s t♦ ♣r♦✈❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s t♦ ❛ ❝❧❛ss ♦❢ s❡♠✐❧✐♥❡❛r ❡❧❧✐♣t✐❝ ❡q✉❛t✐♦♥s ✐♥ ❛ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥✱ ✐♥✈♦❧✈✐♥❣ ❝♦♥❝❛✈❡✲❝♦♥✈❡① t②♣❡ ♥♦♥❧✐♥❡❛r✐t✐❡s✳ ❲❡ ✉s❡ ❛ ✈❛r✐❡t② ♦❢ ♠❡t❤♦❞s t♦ ✜♥❞ t❤❡s❡ s♦❧✉t✐♦♥s✱ s✉❝❤ ❛s ▼♦✉♥t❛✐♥ P❛ss ❚❤❡♦r❡♠✱ ❊❦❡❧❛♥❞✬s ❱❛r✐❛t✐♦♥❛❧ Pr✐♥❝✐♣❧❡✱ ▲❛❣r❛♥❣❡ ▼✉❧t✐♣❧✐❡rs ❚❤❡♦r❡♠✱ ◆❡❤❛r✐ ▼❛♥✐❢♦❧❞ ❛♥❞ s✉❜ ❛♥❞ s✉♣❡rs♦❧✉t✐♦♥ ♠❡t❤♦❞✳

❑❡②✇♦r❞s✿ ❙❡♠✐❧✐♥❡❛r ❊q✉❛t✐♦♥s✱ ♥♦♥✲❧✐♥❡❛r✐t✐❡s ♦❢ t❤❡ ❝♦♥❝❛✈❡✲❝♦♥✈❡① t②♣❡✱ ▲❛❣r❛♥❣❡ ▼✉❧t✐✲ ♣❧✐❡rs ❚❤❡♦r❡♠✱ ◆❡❤❛r✐ ▼❛♥✐❢♦❧❞✱ s✉❜ ❛♥❞ s✉♣❡rs♦❧✉t✐♦♥✳

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

◆♦t❛çõ❡s ✸

✶ ❊①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ ❡q✉❛çõ❡s ❡❧í♣t✐❝❛s s❡♠✐❧✐♥❡❛r❡s ✈✐❛ P❛ss♦ ❞❛

▼♦♥t❛♥❤❛ ✺

✶✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ❖ ❝❛s♦ ❣❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ❖ ❝❛s♦ ♣❛rt✐❝✉❧❛r ♥❛ ❱❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷ ❯♠❛ ❡q✉❛çã♦ ❡❧í♣t✐❝❛ s❡♠✐❧✐♥❡❛r ❡♥✈♦❧✈❡♥❞♦ ✉♠❛ ❢✉♥çã♦ ♣❡s♦ ❝♦♠ ♠✉✲

❞❛♥ç❛ ❞❡ s✐♥❛❧ ✶✺

✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ❖ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❡ ❛ ❱❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✸ ❊st✐♠❛t✐✈❛s ♣❛r❛ ♦ í♥✜♠♦ ❞❡Jλ ❡♠ Mλ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✷✳✹ ❊①✐stê♥❝✐❛ ❞❡ s❡q✉ê♥❝✐❛s ♠✐♥✐♠✐③❛♥t❡s ♣❛r❛ Jλ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✷✳✺ ❊①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✻ ❘❡s✉❧t❛❞♦ ❞❡ ♥ã♦ ❡①✐stê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸ ❊①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ❡q✉❛çõ❡s ❡❧í♣t✐❝❛s s❡♠✐❧✐♥❡❛r❡s ❝♦♠ ❝❛s♦

s✉♣❡r❝rít✐❝♦ ✹✷

✸✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✷ ❋♦r♠✉❧❛çã♦ ❱❛r✐❛❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✸ ❊①✐stê♥❝✐❛ ❞❛ ♣r✐♠❡✐r❛ s♦❧✉çã♦ ♣❛r❛ λ∈(0, λ0) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✸✳✹ ❊①✐stê♥❝✐❛ ❞❛ s❡❣✉♥❞❛ s♦❧✉çã♦ ♣❛r❛λ∈(0, λ0) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✸✳✺ ❘❡❣✉❧❛r✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✻ ❊①✐stê♥❝✐❛ ❞❛ ♣r✐♠❡✐r❛ s♦❧✉çã♦ ♣❛r❛ λ∈(0,Λ) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✼ ❊①✐stê♥❝✐❛ ❞❛ s❡❣✉♥❞❛ s♦❧✉çã♦ ♣❛r❛λ∈(0,Λ) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵

❆ ❘❡s✉❧t❛❞♦s ❞❡ ❘❡❣✉❧❛r✐❞❛❞❡ ✻✸

❇ ❘❡s✉❧t❛❞♦s ❜ás✐❝♦s ✼✹

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✼✽

■♥tr♦❞✉çã♦

◆❡st❡ tr❛❜❛❧❤♦ ✐r❡♠♦s ♠♦str❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♥ã♦ tr✐✈✐❛✐s ♣❛r❛ ❛❧❣✉♠❛s ❡q✉❛✲ çõ❡s ❡❧í♣t✐❝❛s s❡♠✐❧✐♥❡❛r❡s ❡♠ ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❞❡RN✱ ♦♥❞❡ _{N ≥}3✳ ❊st❛ ❞✐ss❡rt❛çã♦

❡stá ❞✐✈✐❞✐❞❛ ❡♠ três ❝❛♣ít✉❧♦s ❡ ❞♦✐s ❛♥❡①♦s ♦r❣❛♥✐③❛❞♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✳

◆♦ ❈❛♣ít✉❧♦ ✶✱ ❜✉s❝❛♠♦s s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ♣❛r❛ ✉♠❛ ❡q✉❛çã♦ ❡❧í♣t✐❝❛ ✉s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✳ ❉✐✈✐❞✐♠♦s ❡st❡ ❝❛♣ít✉❧♦ ❡♠ ❞✉❛s s❡çõ❡s✳ ◆❛ ♣r✐♠❡✐r❛ s❡çã♦ ❡st✉❞❛♠♦s ♦ ❝❛s♦ ❣❡r❛❧

(

−∆u = g(x, u) ❡♠ Ω

u = 0 s♦❜r❡ ∂Ω

♥♦ q✉❛❧ g(x, u) é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✱ s✉❜❝rít✐❝❛ q✉❡ s❛t✐s❢❛③ ❛ ❝♦♥❤❡❝✐❞❛ ❝♦♥❞✐çã♦ ❞❡ ❆♠❜r♦s❡tt✐✲❘❛❜✐♥♦✇✐t③✳ ◆❛ s❡çã♦ s❡❣✉✐♥t❡✱ ❡st✉❞❛♠♦s ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛rg(x, u) = |u|p−1_u✱

♣❛r❛ 1< p <2∗−1✱ ♣r♦✈❛♥❞♦ q✉❡ ♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❞♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦

❝♦✐♥❝✐❞❡ ❝♦♠ ♦ í♥✜♠♦ ❞❡st❡ ❢✉♥❝✐♦♥❛❧ ♥❛ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐✳ ◆❡st❡ ❝❛♣ít✉❧♦ ✉s❛♠♦s ❝♦♠♦ r❡❢❡rê♥❝✐❛ ♦s ❧✐✈r♦s ❞❡ ❘❛❜✐♥♦✇✐t③ ❬✶✶❪ ❡ ❞❡ ❉❛✈✐ ●✳ ❈♦st❛ ❬✸❪✳

◆♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s ❡st✉❞❛♠♦s ❡q✉❛çõ❡s ❡♥✈♦❧✈❡♥❞♦ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡s ❞♦ t✐♣♦ ❝ô♥❝❛✈♦✲ ❝♦♥✈❡①❛s✳ ▼✉✐t♦s ❡st✉❞♦s t❡♠ s✐❞♦ r❡❛❧✐③❛❞♦s s♦❜r❡ ♣r♦❜❧❡♠❛s ❞❡st❡ t✐♣♦✳ ❈✐t❛♠♦s ♦ ❢❛♠♦s♦ ❛rt✐❣♦ ❞❡ ❆♠❜r♦s❡tt✐✱ ❇r❡③✐s ❡ ❈❡r❛♠✐ ❬✶❪✱ ♥♦ q✉❛❧ ❡st✉❞❛r❛♠ ♦ ♣r♦❜❧❡♠❛

(

−∆u = λuq_+up _{❡♠ Ω}

u = 0 s♦❜r❡ ∂Ω ✭✵✳✶✮

♦♥❞❡ Ω é ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❞❡ RN ❝♦♠ ❢r♦♥t❡✐r❛ s✉❛✈❡ ❡ _{0 < q < 1 < p}✳ ❯s❛♥❞♦

♦ ♠ét♦❞♦ ❞❡ s✉❜ ❡ s✉♣❡rs♦❧✉çã♦ ♦s ❛✉t♦r❡s ♠♦str❛r❛♠ q✉❡ ❡①✐st❡ Λ > 0 t❛❧ q✉❡ ♣❛r❛

λ∈ (0,Λ] ♦ ♣r♦❜❧❡♠❛ ✭✵✳✶✮ ♣♦ss✉✐ ✉♠❛ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ❡ ♥ã♦ ❤á s♦❧✉çã♦ ♣❛r❛ λ > Λ✳

◆♦ ❝❛s♦ ❡♠ q✉❡ 1 < p < (N + 2)/(N − 2)✱ ✉s❛♥❞♦ ♠ét♦❞♦s ✈❛r✐❛❝✐♦♥❛✐s ♣r♦✈❛r❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s❡❣✉♥❞❛ s♦❧✉çã♦ s❡0< λ <Λ✳

◆♦ ❈❛♣ít✉❧♦ ✷✱ ✐♥✈❡st✐❣❛♠♦s ✉♠❛ ❡q✉❛çã♦ ❡❧í♣t✐❝❛ ❡♥✈♦❧✈❡♥❞♦ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ q✉❡ ♠✉❞❛ ❞❡ s✐♥❛❧✱ ❞♦ t✐♣♦

(

−∆u = λf(x)uq_+up _{❡♠ Ω} u = 0 s♦❜r❡ ∂Ω

❝♦♥s✐❞❡r❛♥❞♦ 0 < q < 1 < p < 2∗ _{−1✱ t❡♥❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ♦ ❛rt✐❣♦ ❞❡ ❚s✉♥❣✲❋❛♥❣} ❲✉ ❬✶✹❪✳ ▼♦str❛♠♦s ❛q✉✐✱ q✉❡ s❡ f(x) é q✉❛❧q✉❡r ❢✉♥çã♦ ❝♦♥tí♥✉❛ q✉❡ ♠✉❞❛ ❞❡ s✐♥❛❧ ❡♠ _Ω¯_{✱ ❡①✐st❡ λ0 > 0 t❛❧ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❛❞♠✐t❡ ❞✉❛s s♦❧✉çõ❡s ♣♦s✐t✐✈❛s s❡ λ ∈ (0, λ0)✳} ▼♦str❛r❡♠♦s ✐st♦ ♣♦r ♠❡✐♦ ❞❡ ♠✐♥✐♠✐③❛çã♦ ❞♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ♥❛ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐✱ ✉s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞♦s ▼✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡ ❡ ♦ Pr✐♥❝í♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞✳

◆♦ ❈❛♣ít✉❧♦ ✸✱ ♠♦str❛♠♦s q✉❡ ❡①✐st❡♠ s♦❧✉çõ❡s ♣❛r❛ ❛ ❡q✉❛çã♦ ❞♦ t✐♣♦

(

−∆u = λuq_+h(x)up _{❡♠ Ω} u = 0 s♦❜r❡ ∂Ω

♦♥❞❡0< q <1< p <2∗_{−1 +τ ❡ h(x) é ✉♠❛ ❢✉♥çã♦ ❍ö❧❞❡r ❝♦♥tí♥✉❛✱ s❛t✐s❢❛③❡♥❞♦ ❝♦♥✲} ❞✐çõ❡s ❡s♣❡❝✐❛✐s✳ ◆❡st❡ ❝❛s♦✱ ♦ ♣r♦❜❧❡♠❛ tr❛③ ✉♠❛ ❢✉♥çã♦ ❝♦♠ ❝r❡s❝✐♠❡♥t♦ s✉♣❡r❝rít✐❝♦✳ ❊♥tã♦ ♦s ❛r❣✉♠❡♥t♦s ✈❛r✐❛❝✐♦♥❛✐s ❣❡r❛✐s ♥ã♦ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞♦s ❞✐r❡t❛♠❡♥t❡✳ ❯s❛♠♦s ✉♠ ❚❡♦r❡♠❛ q✉❡ ♥♦s ❣❛r❛♥t❡ ❛ ✐♠❡rsã♦ ❝♦♠♣❛❝t❛ ❞❡ H1

0(Ω) ❡♠ Lp(Ω) ❡♠ ✉♠ ❞♦♠í♥✐♦

❝✐❧✐♥❞r✐❝❛♠❡♥t❡ s✐♠étr✐❝♦ ❝♦♠ p ♠❛✐♦r q✉❡ ♦ ❡①♣♦❡♥t❡ ❝rít✐❝♦ ❞❡ ❙♦❜♦❧❡✈✱ r❡s✉❧t❛❞♦ ❡st❡

♣r♦✈❛❞♦ ♣♦r ❲❡♥③❤✐ ❲❛♥❣ ❬✶✻❪✳ ❉❡ss❛ ❢♦r♠❛✱ ♣♦r ♠❡✐♦ ❞❡ ❛r❣✉♠❡♥t♦s ✈❛r✐❛❝✐♦♥❛✐s ♠♦s✲ tr❛♠♦s q✉❡ ❡①✐st❡Λ ∈(0,∞) t❛❧ q✉❡✱ ♣❛r❛ 0< λ <Λ✱ ♦ ♣r♦❜❧❡♠❛ ♣♦ss✉✐ ❛♦ ♠❡♥♦s ❞✉❛s

s♦❧✉çõ❡s ♣♦s✐t✐✈❛s✱ s❡♥❞♦ ✉♠❛ ♠✐♥✐♠✐③❛♥t❡ ❧♦❝❛❧ ❞♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❡ ❛ ♦✉tr❛ ♦❜t✐❞❛ ♣♦r ♠❡✐♦ ❞♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛❀ ♣❛r❛λ= Λ♦ ♣r♦❜❧❡♠❛ t❡♠ ❛♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦ ❡ s❡ λ >Λ ♦ ♣r♦❜❧❡♠❛ ❡♠ q✉❡stã♦ ♥ã♦ ♣♦ss✉✐ s♦❧✉çã♦✳ ❯s❛♠♦s ❝♦♠♦ ❜❛s❡ ♣❛r❛ ♦ ❡st✉❞♦ ♥❡st❡ ❝❛♣ít✉❧♦✱ ♦ ❛rt✐❣♦ ❞❡ ❏✳ ●❛♦✱ ❨✳ ❩❤❛♥❣ ❡ P❡✐❤❛♦ ❩❤❛♦ ❬✽❪✳

◆♦ ❆♣ê♥❞✐❝❡ ❆ ♠♦str❛♠♦s q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛♦s ♣r♦❜❧❡♠❛s ❡st✉❞❛❞♦s é ❞❡ ❝❧❛ss❡C1_{✳ ❚r❛③❡♠♦s ❛✐♥❞❛✱ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ❞❡ r❡❣✉❧❛r✐❞❛❞❡ r❡❧❛❝✐♦♥❛❞♦s ❛♦s}

♣r♦❜❧❡♠❛s ❡st✉❞❛❞♦s ♥♦s ❝❛♣ít✉❧♦s ❛♥t❡r✐♦r❡s✳ ◆♦ ❆♣ê♥❞✐❝❡ ❇✱ ❛♣r❡s❡♥t❛♠♦s ♦s ♣r✐♥❝✐♣❛✐s t❡♦r❡♠❛s ✉t✐❧✐③❛❞♦s ♥♦ ❞❡❝♦rr❡r ❞❛ ♥♦ss❛ ❞✐ss❡rt❛çã♦✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ r❡❢❡rê♥❝✐❛ ❞♦s ♠❡s♠♦s✳

◆♦t❛çã♦

◆♦ ❞❡❝♦rr❡r ❞❡st❛ ❞✐ss❡rt❛çã♦ ✉s❛r❡♠♦s ❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s✿

R+ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♥ã♦ ♥❡❣❛t✐✈♦s

Br0(x) ❜♦❧❛ ❛❜❡rt❛ ❞❡ ❝❡♥tr♦ x❡ r❛✐♦ r0 ❡♠ R

N

⇀,→ ❝♦♥✈❡r❣ê♥❝✐❛ ❢r❛❝❛ ❡ ❢♦rt❡✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ |Ω| ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ Ω

q.t.p. ❡♠ q✉❛s❡ t♦❞❛ ♣❛rt❡ ∂u

∂xi ❞❡r✐✈❛❞❛ ♣❛r❝✐❛❧ ❞❡u ❡♠ r❡❧❛çã♦ ❛xi

∇u=

∂u ∂x1

, ∂u ∂x2

, ..., ∂u ∂xn

❣r❛❞✐❡♥t❡ ❞❡ u

△u=PN_i=1 ∂

2_u

∂x2

i

❧❛♣❧❛❝✐❛♥♦ ❞❡ u ∂u

∂ν =∇uν ❞❡r✐✈❛❞❛ ♥♦r♠❛❧ ❡①t❡r✐♦r Lp_{(Ω) =}

u: Ω→R ♠❡♥s✉rá✈❡❧_;

Z

Ω

|u|p_{dx <∞}

✱ 1≤p <∞ ||u||Lp =

Z

Ω

|u|p

1

p

♥♦r♠❛ ♥♦ ❡s♣❛ç♦ ❞❡ ▲❡❜❡s❣✉❡ Lp_(Ω)

L∞(Ω) ={u: Ω→R ♠❡♥s✉rá✈❡❧;_{|u(x)| ≤C}q✳t✳♣✳ s♦❜r❡_Ω♣❛r❛ ❛❧❣✉♠_{C >0}}

Ck_{(Ω) ❢✉♥çõ❡sk ✈❡③❡s ❝♦♥t✐♥✉❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡✐s s♦❜r❡Ω✱ k ∈N}

W1,p_{(Ω) =}

  u∈L

p_(Ω)

∃ g1, g2, ..., gN ∈Lp(Ω) t❛✐s q✉❡ Z

Ω

u∂ϕ ∂xi =−

Z

Ω

giϕ,∀ϕ∈C_c∞(Ω),∀i= 1, ..., N

  

W01,p(Ω) ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡ Cc1(Ω) ❡♠ W1,p(Ω)

H1

0(Ω) =W 1,2 0 (Ω)

H−1_{(Ω) é ♦ ❡s♣❛ç♦ ❞✉❛❧ ❞❡ H}1 0(Ω)

p∗ ₌ N p

N −p ♣❛r❛ 1≤p < N ❡①♣♦❡♥t❡ ❝rít✐❝♦ ❞❡ ❙♦❜♦❧❡✈ f =o(g) q✉❛♥❞♦ x→x0 s❡ lim

x→x0

|f(x)|/|g(x)|= 0

❈❛♣ít✉❧♦ ✶

❊①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ ❡q✉❛çõ❡s

❡❧í♣t✐❝❛s s❡♠✐❧✐♥❡❛r❡s ✈✐❛ P❛ss♦ ❞❛

▼♦♥t❛♥❤❛

✶✳✶ ■♥tr♦❞✉çã♦

◆❡st❡ ❝❛♣ít✉❧♦ ♣r♦✈❛r❡♠♦s ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ♣❛r❛ ❡q✉❛çõ❡s s❡♠✐❧✐♥❡❛r❡s ❞♦ t✐♣♦

(

−∆u=g(x, u), ❡♠ Ω

u= 0, s♦❜r❡ ∂Ω ✭✶✳✶✮

♦♥❞❡ Ω⊂RN é ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❝♦♠ ❢r♦♥t❡✐r❛ s✉❛✈❡✱ _{N ≥ 3}✱ _∆u é ♦ ▲❛♣❧❛❝✐❛♥♦ ❞❡

u✱ u : ¯Ω → R é ❛ ❢✉♥çã♦ ✈❛r✐á✈❡❧ ❡ _{g : ¯Ω×}R _→ R é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ s❛t✐s❢❛③❡♥❞♦

❛❧❣✉♠❛s ❤✐♣ót❡s❡s ❛❞✐❝✐♦♥❛✐s q✉❡ s❡rã♦ ❞❡s❝r✐t❛s ❛ s❡❣✉✐r✳ ❆ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ❡st❡ ♣r♦❜❧❡♠❛ s❡rá ❣❛r❛♥t✐❞❛ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✳

▼♦str❛r❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♥ã♦ tr✐✈✐❛✐s ♣❛r❛ ✉♠ ❝❛s♦ ♠❛✐s ❣❡r❛❧ ❡ ❡♠ s❡❣✉✐❞❛✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r g(x, t) = |t|s−1_{t✱ ♠♦str❛r❡♠♦s q✉❡ ♦}

í♥✜♠♦ ❞♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮ s♦❜r❡ ❛ ❱❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❞❡st❡ ❢✉♥❝✐♦♥❛❧✳

✶✳✷ ❖ ❝❛s♦ ❣❡r❛❧

◆❡st❛ s❡çã♦✱ ❜❛s❡❛❞♦s ♥♦ ❧✐✈r♦ ❞❡ ❘❛❜✐♥♦✇✐t③ ❬✶✶❪✱ ❡st✉❞❛r❡♠♦s ♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮ ♥♦ ❝❛s♦ ❡♠ q✉❡ g : ¯Ω×R_→R é ✉♠❛ ❢✉♥çã♦ s❛t✐s❢❛③❡♥❞♦✿

(g1) g ∈C( ¯Ω×R,R)❀

(g2) ❡①✐st❡♠ ❝♦♥st❛♥t❡s a1, a2 >0 ❡0≤p <2∗−1 t❛✐s q✉❡

|g(x, ξ)| ≤a1+a2|ξ|p, ∀ x∈Ω¯ ❡ ξ ∈R;

(g3) g(x, ξ) =o(|ξ|) s❡ ξ→0✱ ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ x∈Ω❀¯

(g4) ❡①✐st❡♠ ❝♦♥st❛♥t❡s µ >2 ❡r ≥0t❛✐s q✉❡ ♣❛r❛ |ξ| ≥r,

0< µG(x, ξ)≤ξg(x, ξ) ♦♥❞❡ G(x, ξ) =

Z ξ

0

g(x, t)dt.

❊①❡♠♣❧♦ ✶✳✶ ❯♠ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❡ s❛t✐s❢❛③ ❡st❛s ❝♦♥❞✐çõ❡s ég(x, u) = |u|p−1_u

❛ q✉❛❧ ✐r❡♠♦s ❡st✉❞❛r ♥❛ ♣ró①✐♠❛ s❡çã♦✳

❖❜s❡r✈❛çã♦ ✶✳✷ ❆ ❤✐♣ót❡s❡ (g3) ✐♠♣❧✐❝❛ q✉❡ g(x,0) = 0 ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦ ♣r♦✲

❜❧❡♠❛ ✭✶✳✶✮ ❛❞♠✐t❡ u≡0 ❝♦♠♦ s♦❧✉çã♦✳ ❆ ❝♦♥❞✐çã♦ (g4) é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❈♦♥❞✐çã♦ ❞❡

❆♠❜r♦s❡tt✐✲❘❛❜✐♥♦✇✐t③ ❡ ♥♦s ❣❛r❛♥t❡ q✉❡ 0≤ µ

ξ ≤

g(x, ξ)

G(x, ξ). ■♥t❡❣r❛♥❞♦ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ♦❜t❡♠♦s

0≤

Z ξ

r0

µ tdt≤

Z ξ

r0

g(x, t)

G(x, t)dt. ❚❡♠♦s

Z ξ

r0

µ

tdt =µ(ln|ξ| −ln|r0|) = ln

|ξ|µ

|r0|µ

❡

Z ξ

r0

g(x, t)

G(x, t)dt=

Z ξ

r0

(lnG(x, t))′dt = lnG(x, ξ)−lnG(x, r0) = ln

G(x, ξ)

G(x, r0)

.

❆ss✐♠

0≤ln |ξ|

µ

|r0|µ

≤ln G(x, ξ)

G(x, r0)

♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡

|ξ|µ

|r0|µ

≤ G(x, ξ)

G(x, r0)

.

❈♦♠ ✐ss♦ ✈❡♠♦s q✉❡ G(x, ξ)≥ c1|ξ|µ ♦♥❞❡ c1 =G(x, r0)/|r0|µ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ t♦♠❛♥❞♦

c2 = sup (x,ξ)∈Ω×(−r,r)

|G(x, ξ)| t❡♠♦s q✉❡ G(x, ξ)≥ −c2✳ ▲♦❣♦

G(x, ξ)≥c1|ξ|µ−c2, ∀ x∈Ω¯, ξ∈R. ✭✶✳✷✮

◆♦ss♦ ♦❜❥❡t✐✈♦✱ ♥❡st❛ s❡çã♦✱ é ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦ tr✐✈✐❛❧ ♣❛r❛ ✭✶✳✶✮✱ ♦✉ s❡❥❛✱ ♠♦str❛r❡♠♦s q✉❡ ❡①✐st❡u∈H1

0(Ω)✱ u6= 0 ❡♠ Ω✱ t❛❧ q✉❡ Z

Ω

∇u∇vdx=

Z

Ω

g(x, u)vdx, ∀v ∈H₀1(Ω).

❖ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮ éI :H1

0(Ω) →R❞❛❞♦ ♣♦r

I(u) =

Z

Ω

1 2|∇u|

2_{−G(x, u)}

dx.

P❡❧❛ Pr♦♣♦s✐çã♦ ❆✳✸ s❛❜❡♠♦s q✉❡ s❡ ✭✐✮ ❡ ✭✐✐✮ sã♦ s❛t✐s❢❡✐t❛s ❡♥tã♦I ∈C1_(H1

0(Ω),R) ❡

I′(u)v =

Z

Ω

(∇u∇v−g(x, u)v)dx, ∀v ∈H₀1(Ω).

P♦rt❛♥t♦✱ ❡♥❝♦♥tr❛r s♦❧✉çã♦ ❢r❛❝❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❡♥❝♦♥tr❛r ♣♦♥t♦ ❝rít✐❝♦ ♣❛r❛ ♦ ❢✉♥❝✐♦♥❛❧ I✳ ❯s❛r❡♠♦s ♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ♣❛r❛ ❡♥❝♦♥tr❛r

t❛❧ ♣♦♥t♦ ❝rít✐❝♦✱ ❞❡✈❡♠♦s ❡♥tã♦ ✈❡r✐✜❝❛r ❛s s✉❛s ❤✐♣ót❡s❡s✳

❯♠❛ ✈❡③ q✉❡Ω é ❧✐♠✐t❛❞♦ ♣♦❞❡♠♦s t♦♠❛r ❝♦♠♦ ♥♦r♠❛ ❡♠H1 0(Ω)

||u||2 _≡Z Ω

|∇u|2_dx.

◆❛ ♣r♦♣♦s✐çã♦ q✉❡ s❡❣✉❡✱ ♠♦str❛r❡♠♦s q✉❡ ♦ ❢✉♥❝✐♦♥❛❧I ♣♦ss✉✐ ❛ ●❡♦♠❡tr✐❛ ❞♦ P❛ss♦

❞❛ ▼♦♥t❛♥❤❛✳

Pr♦♣♦s✐çã♦ ✶✳✸ ❙❡g s❛t✐s❢❛③ ✭✐✮ ✲ ✭✐✈✮✱ ❡♥tã♦ I s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s✿

✭■✮ I(0) = 0 ❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡s ρ, α >0 t❛✐s q✉❡ I|∂Bρ ≥α; ✭■■✮ ❡①✐st❡ e∈H1

0(Ω)\Bρ¯ t❛❧ q✉❡ I(e)≤0.

❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ✈❡r✐✜❝❛r ❛ ❝♦♥❞✐çã♦ ✭■✮✳ ❈♦♠♦G(x,0) = 0✱ t❡♠♦s q✉❡I(0) = 0✳

❉❡✜♥❛

J(u)≡

Z

Ω

G(x, u)dx.

P❡❧❛ ❤✐♣ót❡s❡ ✭✐✐✐✮ t❡♠♦sg(x, ξ) = o(|ξ|)✱ s❡ ξ →0✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ x∈Ω✳ ❆ss✐♠ ❞❛❞♦¯

ε >0✱ ❡①✐st❡ δ >0 t❛❧ q✉❡ 0<|ξ|< δ ✐♠♣❧✐❝❛ q✉❡ |g(x, ξ)|< ε|ξ|✳ ❉❛í✱ |G(x, ξ)| ≤ 1

2ε|ξ|

2_{, s❡ |ξ|< δ.}

P♦r ✭✐✐✮ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ A=A(δ)>0 t❛❧ q✉❡ |ξ| ≥δ ✐♠♣❧✐❝❛ q✉❡

|G(x, ξ)| ≤ |ξg(x, ξ)| ≤ |ξ|(a1 +a2|ξ|p) = a1|ξ|+a2|ξ|p+1 =A|ξ|p+1

♣❛r❛ t♦❞♦x∈Ω✳ ❈♦♠❜✐♥❛♥❞♦ ❡ss❛s ❞✉❛s ❡st✐♠❛t✐✈❛s✱ ♣❛r❛ t♦❞♦¯ ξ ∈R❡ _x∈Ω¯ t❡♠♦s

|G(x, ξ)| ≤ ε

2|ξ|

2_+A|ξ|s+1_.

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ✉s❛♥❞♦ ❛ ✐♠❡rsã♦ ❝♦♥tí♥✉❛ ❞❡H1

0(Ω)❡♠L2(Ω)❡ ❞❡H01(Ω)❡♠Ls+1(Ω)

t❡♠♦s

|J(u)| ≤ ε

2||u||

2

L2_(Ω)+A||u||s_L+1s+1_(Ω)

≤ cε

2||u||

2_+A||u||s+1

= c||u||2ε

2 +A||u||

s−1_.

❊s❝♦❧❤❡♥❞♦ ||u|| ≤( ε

2A)

1

s−₁ ♦❜t❡♠♦s

|J(u)| ≤εc||u||2_.

▲♦❣♦ J(u) =o(||u||2_{)q✉❛♥❞♦ u→0. ❆ss✐♠✱ ❡①✐st❡ρ >0 t❛❧ q✉❡}

|J(u)| ||u||2 <

1

4 s❡ ||u|| ≤ρ. ❈♦♠♦I(u) = 1

2||u||

2_{−J(u) t❡♠♦s q✉❡ s❡ ||u|| ≤ρ ❡♥tã♦}

I(u)> 1

4||u||

2_.

❚♦♠❛♥❞♦α = 1₄ρ2 _{t❡♠♦s q✉❡ I(u) ≥α✱ s❡ ||u|| =ρ✱ ♦✉ s❡❥❛✱ I|}

∂Bρ ≥α✳ ▲♦❣♦ ❛ ♣r✐♠❡✐r❛ ❝♦♥❞✐çã♦ é s❛t✐s❢❡✐t❛✳

❱❡r✐✜❝❛r❡♠♦s ❛❣♦r❛ ❛ ❝♦♥❞✐çã♦ ✭■■✮✳ P♦r ✭✐✈✮ ❡ ♣♦r ✭✶✳✷✮ t❡♠♦s q✉❡

J(u)≥c1 Z

Ω

|u|µ_dx−c

2|Ω| ✭✶✳✸✮

♣❛r❛ t♦❞♦ u∈H1

0(Ω)✱ ♦♥❞❡|Ω|❞❡♥♦t❛ ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❞❡ Ω✳ ❊s❝♦❧❤❡♥❞♦ q✉❛❧q✉❡r

u∈H1

0(Ω)\{0}♣♦r ✭✶✳✸✮✱ ✉♠❛ ✈❡③ q✉❡ µ >2 t❡♠♦s

I(tu) =

Z

Ω

(1

2|∇(tu)|

2_{−G(x, tu))dx}

≤ t

2

2||u||

2_−c 1

Z

Ω

|tu|µ

dx+c2|Ω|

= t

2

2||u||

2_−c 1tµ

Z

Ω

|u|µ_dx+c

2|Ω| → −∞

q✉❛♥❞♦t → ∞. ▲♦❣♦✱ ❡①✐st❡ t >0s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ t❛❧ q✉❡ e=tu s❛t✐s❢❛③ ||e||> ρ

❡I(e)≤0✱ ♦✉ s❡❥❛✱ ❛ s❡❣✉♥❞❛ ❝♦♥❞✐çã♦ é s❛t✐s❢❡✐t❛✳

P❛r❛ ✉t✐❧✐③❛r♠♦s ♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❇✳✸✱ ♣r❡❝✐s❛♠♦s ✈❡r✐✜❝❛r q✉❡ ♦ ❢✉♥❝✐♦♥❛❧I s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ P❛❧❛✐s✲❙♠❛❧❡ ✭P❙✮✳ ❯♠❛ s❡q✉ê♥❝✐❛ é ❞✐t❛ ✭P❙✮ s❡ I(um) é ❧✐♠✐t❛❞♦ ❡I′_{(um)→0q✉❛♥❞♦m →0✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❢✉♥❝✐♦♥❛❧ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ P❛✲} ❧❛✐s ❙♠❛❧❡ s❡ t♦❞❛ s❡q✉ê♥❝✐❛ ✭P❙✮ ♣♦ss✉✐ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡✳ ❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ ❣❛r❛♥t❡ q✉❡ ❡st❛ ❝♦♥❞✐çã♦ é ✈á❧✐❞❛ ♣❛r❛ s❡q✉ê♥❝✐❛s ❧✐♠✐t❛❞❛s✳

Pr♦♣♦s✐çã♦ ✶✳✹ ❙❡❥❛g s❛t✐s❢❛③❡♥❞♦ ✭✐✮ ❡ ✭✐✐✮ ❡I ❞❡✜♥✐❞♦ ❝♦♠♦ ❛♥t❡r✐♦r♠❡♥t❡✳ ❙❡{um}

é ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❡♠ H1

0(Ω) t❛❧ q✉❡ I′(um) → 0 s❡ m → ∞✱ ❡♥tã♦ {um} é

♣ré✲❝♦♠♣❛❝t❛ ❡♠ H1 0(Ω).

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ D : H1

0(Ω) → H−1(Ω) ❞❡♥♦t❛♥❞♦ ❛ ❛♣❧✐❝❛çã♦ ❞✉❛❧✐❞❛❞❡ ❡♥tr❡

H1

0(Ω) ❡ s❡✉ ❞✉❛❧✱ ❛ q✉❛❧ s❛❜❡♠♦s q✉❡ é ✉♠❛ ✐s♦♠❡tr✐❛ ❧✐♥❡❛r ❞❡✈✐❞♦ ❛♦ ❚❡♦r❡♠❛ ❞❡

❘❡♣r❡s❡♥t❛çã♦ ❞❡ ❘✐❡s③✳ ❊♥tã♦ ♣❛r❛u, ϕ ∈H1

0(Ω) t❡♠♦s

Du(ϕ) =

Z

Ω

∇u∇ϕdx=hu, ϕi.

❈♦♠♦ I′_{(u)ϕ = hu, ϕi −J}′_{(u)ϕ✱ t❡♠♦s Du = I}′_{(u) +J}′_{(u)✳ ❆✐♥❞❛✱ ♣❡❧❛ Pr♦♣♦s✐çã♦} ❆✳✸ t❡♠♦s q✉❡ J′ é ❝♦♠♣❛❝t♦✳ ❙❡♥❞♦ {um} ❧✐♠✐t❛❞❛✱ ❡♥tã♦ J′(um) ♣♦ss✉✐ s✉❜sq✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡✳ P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ D−1 _t❡♠♦s

umk =D

−1_I′_(um

k) +D

−1_J′_(um

k)→limD

−1_J′_(um k). ▲♦❣♦ {um}♣♦ss✉✐ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡✳

Pr♦♣♦s✐çã♦ ✶✳✺ I s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ P❛❧❛✐s✲❙♠❛❧❡✳

❉❡♠♦♥str❛çã♦✿ ❉❡✈❡♠♦s ♠♦str❛r q✉❡ t♦❞❛ s❡q✉ê♥❝✐❛(P S) ♣♦ss✉✐ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✲ ✈❡r❣❡♥t❡✳ ❙❡❥❛ {um} ✉♠❛ s❡q✉ê♥❝✐❛ (P S)✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✹✱ ❜❛st❛ ♠♦str❛r q✉❡ {um}

é ❧✐♠✐t❛❞❛✳

❈♦♠♦ I(um) é ❧✐♠✐t❛❞❛✱ ❡①✐st❡ M > 0 t❛❧ q✉❡ |I(um)| < M✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ I′_{(um)→0✱ ❡①✐st❡ m}

0 ∈Nt❛❧ q✉❡

||I′(um)||< µ, ∀ m > m0.

❆ss✐♠✱

I(um)− 1

µI

′_{(um)um ≤M +||um||, ∀ m > m}

0. ✭✶✳✹✮

P♦r ♦✉tr♦ ❧❛❞♦✱

I(um)− 1

µI

′_{(um)um =} 1 2||um||

2 ₋Z Ω

G(x, um)dx− 1

µ

||um||2−

Z

Ω

g(x, um)umdx

=

1 2 −

1

µ

||um||2+

Z

Ω

1

µg(x, um)um−G(x, um)

dx.

❚♦♠❛♥❞♦Tm = 1

µg(·, um)um−G(·, um) ♥❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r t❡♠♦s I(um)− 1

µI

′_{(um)um =}

1 2 −

1

µ

||um||2+

Z

{x∈Ω;|um(x)|≥r}

Tm(x)dx+

Z

{x∈Ω;|um(x)|<r}

Tm(x)dx.

❙❡❣✉❡ ❞❛ ❝♦♥❞✐çã♦ ✭✐✈✮ q✉❡

Z

{x∈Ω;|um(x)|≥r}

Tm(x)dx≥0.

❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ g ❡ G sã♦ ❝♦♥tí♥✉❛s ❡ ❧✐♠✐t❛❞❛s ❡♠Ω¯ ×[−r, r]✱ ❡♥tã♦Tm é ❝♦♥tí♥✉❛ ❡

❡①✐st❡ c >0 ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡m t❛❧ q✉❡

|Tm(x)| ≤c ∀ x∈ {y∈Ω;¯ |um(y)|< r}.

❉❛í _Z

{x∈Ω;|um(x)|<r}

Tm(x)dx≥ −c|Ω|✱ ∀m ∈N_.

❙❡❣✉❡ ❡♥tã♦ q✉❡

I(um)− 1

µI

′_{(um)um ≥}

1 2−

1

µ

||um||2 _{−c|Ω|. ✭✶✳✺✮}

❆ss✐♠✱ ♣♦r ✭✶✳✹✮ ❡ ✭✶✳✺✮ t❡♠♦s

1 2−

1

µ

||um||2_{−c|Ω| ≤I(um)−} 1

µI

′_{(um)um ≤M +} 1

µε||um||

♣❛r❛ t♦❞♦m > m0✳ ■st♦ ✐♠♣❧✐❝❛ q✉❡

1 2 −

1

µ

||um||2₋ ε

µ||um|| ≤M +c|Ω|✱ ∀m > m0.

❈♦♠♦µ >2❝♦♥❝❧✉✐♠♦s q✉❡{um}é ❧✐♠✐t❛❞❛✳ ❆ss✐♠✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✹ t❡♠♦s q✉❡{um}

♣♦ss✉✐ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡✳ P♦rt❛♥t♦✱ I s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✭P❙✮✳

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

❛ ❣❡♦♠❡tr✐❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❡ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✭P❙✮✳ P♦❞❡♠♦s ❡♥tã♦ ✉s❛r ♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ♣❛r❛ ♠♦str❛r q✉❡ ❡st❡ ♣r♦❜❧❡♠❛ ♣♦ss✉✐ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧✳

❚❡♦r❡♠❛ ✶✳✻ ❙❡ g s❛t✐s❢❛③ ✭✐✮ ✲ ✭✐✈✮✱ ♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮ ♣♦ss✉✐ ✉♠❛ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧✳

❉❡♠♦♥str❛çã♦✿ ❊♥❝♦♥tr❛r s♦❧✉çã♦ ❢r❛❝❛ ♣❛r❛ ❛ ❡q✉❛çã♦ ✭✶✳✶✮ é ❡q✉✐✈❛❧❡♥t❡ ❛ ♦❜t❡r ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ♣❛r❛ ♦ ❢✉♥❝✐♦♥❛❧I✱ ❞❛❞♦ ♣♦r

I(u) =

Z

Ω

1 2|∇u|

2_{−G(x, u)}

dx.

P❡❧❛s Pr♦♣♦s✐çõ❡s ✶✳✸✱ ✶✳✹✱ ✶✳✺✱ s❛❜❡♠♦s q✉❡ sã♦ s❛t✐s❢❡✐t❛s ❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✱ q✉❡ ♥♦s ❣❛r❛♥t❡ q✉❡I ♣♦ss✉✐ ✉♠ ✈❛❧♦r ❝rít✐❝♦ cP M ≥α❝❛r❛❝t❡r✐③❛❞♦

♣♦r

cP M = inf

γ∈Γ_u∈sup_γ([0,1])I(u)

♦♥❞❡ Γ = {γ ∈C([0,1], H1

0(Ω)); γ(0) = 0 ❡ I(γ(1))<0}✳ P♦rt❛♥t♦✱ ❡①✐st❡ u ∈ H01(Ω)

♣♦♥t♦ ❝rít✐❝♦ ❞❡ I t❛❧ q✉❡ I(u) =cP M✳ ❈♦♠♦ cP M >0 t❡♠♦s q✉❡u 6= 0❡ ♣♦r s✉❛ ✈❡③✱ u

é s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ♣❛r❛ ❛ ❡q✉❛çã♦ ✭✶✳✶✮✳

✶✳✸ ❖ ❝❛s♦ ♣❛rt✐❝✉❧❛r ♥❛ ❱❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐

❈♦♥s✐❞❡r❛♠♦s ❛ s❡❣✉✐r✱ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮ ❞❛ s❡çã♦ ❛♥t❡r✐♦r✱ ❡♠ q✉❡g(x, u) =|u|p−1_u

(

−∆u=|u|p−1_{u ❡♠ Ω}

u= 0 s♦❜r❡ ∂Ω ✭✶✳✻✮

♦♥❞❡ 1 < p < 2∗ _{−1 ❡ Ω é ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❡♠ R}N_{✳ ❆ss♦❝✐❛❞♦ à ❡q✉❛çã♦ ✭✶✳✻✮✱}

❝♦♥s✐❞❡r❡ ♦ ❢✉♥❝✐♦♥❛❧K :H1

0(Ω)→R ❞❛❞♦ ♣♦r

K(u) = 1 2

Z

Ω

|∇u|2dx− 1

p+ 1

Z

Ω

|u|p+1_dx.

❙❛❜❡♠♦s ♣❡❧❛ Pr♦♣♦s✐çã♦ ❆✳✸ q✉❡ K ∈C1_(H1

0(Ω),R) ❡

hK′(u), vi=

Z

Ω

∇u∇v−

Z

Ω

|u|p−1_uvdx.

Pr♦✈❛♠♦s ♥❛ s❡çã♦ ❛♥t❡r✐♦r q✉❡ ❛ ❡q✉❛çã♦ ✭✶✳✻✮ ♣♦ss✉✐ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧✱ ❛ q✉❛❧ é ♦❜t✐❞❛ ❝♦♠♦ ♣♦♥t♦ ❝rít✐❝♦ ❞❡K✱ ♥♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✳ ◆❡st❡ ❝❛s♦ ♣❛rt✐❝✉❧❛r✱ ✉s❛♥❞♦

Pr✐♥❝í♣✐♦ ❞♦ ▼á①✐♠♦✱ ✈❡♠♦s q✉❡ ❡①✐st❡ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭✶✳✻✮✳

❖s r❡s✉❧t❛❞♦s ❛ s❡❣✉✐r t❡♠ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ♦ ❧✐✈r♦ ❞❡ ❉❛✈✐ ❈♦st❛ ❬✸❪✳ ❉❡✜♥✐♠♦s ❛ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ ❛ss♦❝✐❛❞❛ ❛K✱

N ={u∈H₀1(Ω)\{0};hK′(u), ui= 0}.

❖❜s❡r✈❡ q✉❡✱ s❡u∈ N ❡♥tã♦

||u||2 _=||u||p+1

p+1. ✭✶✳✼✮

P♦r ♠❡✐♦ ❞♦s r❡s✉❧t❛❞♦s q✉❡ s❡❣✉❡♠✱ ♠♦str❛r❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ N é ❞❡ ❢❛t♦ ✉♠❛

✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛ ❡ ♥ã♦ ✈❛③✐❛✳ Pr♦✈❛r❡♠♦s ❛✐♥❞❛ q✉❡ ♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❞♦ ❢✉♥❝✐♦♥❛❧K ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ í♥✜♠♦ ❞❡ K s♦❜r❡ ❛ ✈❛r✐❡❞❛❞❡ N✳

Pr♦♣♦s✐çã♦ ✶✳✼ ❆ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ N é ♥ã♦ ✈❛③✐❛ ❡ N é ✉♠❛ C1 _{✲ s✉❜✈❛r✐❡❞❛❞❡ ❞❡}

H1 0(Ω)✳

❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ s❡❥❛φ :H1

0(Ω)→R ❞❡✜♥✐❞❛ ♣♦r

φ(u) =hK′(u), ui=||u||2−

Z

Ω

|u|p+1dx.

❙❛❜❡♠♦s ♣❡❧❛ Pr♦♣♦s✐çã♦ ❆✳✸ q✉❡ φ∈C1_(H1

0(Ω),R) ❡

φ′(u)v = 2

Z

Ω

∇u∇vdx−(p+ 1)

Z

Ω

|u|p_vdx.

❙❡❥❛ 0 6= v ∈ H1

0(Ω) ❡ ❝♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ 0 < t 7→ φ(tv)✳ ❊♥tã♦✱ ✉s❛♥❞♦ ❛ ✐♠❡rsã♦

❝♦♥tí♥✉❛ ❞❡H1

0(Ω) ❡♠ Lp+1(Ω)✱ ❝♦♠♦ p+ 1>2t❡♠♦s

φ(tv) =hK′(tv), tvi =

Z

Ω

|∇(tv)|2_dx−Z Ω

|tv|p+1_dx

= t2||v||2−tp+1_||v||p+1

Lp+1

≥ t2||v||2−tp+1_cp+1_||v||p+1

> 0 s❡ t >0 é ♣❡q✉❡♥♦.

❆❧é♠ ❞✐ss♦✱

lim

t→∞φ(tv) = limt→∞ t

2_||v||2_−tp+1_||v||p+1

Lp+1

=−∞.

❊♥tã♦✱ ❡①✐st❡ ✉♠¯_{t > 0 t❛❧ q✉❡ φ(¯tv) = 0✱ ♦✉ s❡❥❛✱} ¯_{tv ∈ N✳ ❉❡ss❛ ❢♦r♠❛✱ ❝♦♥❝❧✉✐♠♦s q✉❡}

N 6=∅✳

❱❡❥❛♠♦s ❛❣♦r❛ q✉❡ φ ♥ã♦ ♣♦ss✉✐ ♣♦♥t♦ ❝rít✐❝♦ ❡♠ N✳ P❛r❛u∈ N✱ ♣♦r ✭✶✳✼✮ t❡♠♦s

φ′(u)u = 2||u||2_{−(p+ 1)} Z

Ω

|u|p+1_dx

= 2||u||2_{−(p+ 1)||u||}2

= (1−p)||u||2 _{6= 0.}

▲♦❣♦ φ′_{(u) 6= 0✱ ♣❛r❛ t♦❞♦ u ∈ N✳ ❙❡❥❛ M = H}1

0(Ω)\{0}✳ ❉❡ss❛ ❢♦r♠❛✱ 0 é ♦ ú♥✐❝♦

♣♦♥t♦ ❝rít✐❝♦ ❡♠ φ−1_{(0) ❡ 0 6∈ M✳ ▲♦❣♦ 0 ∈ R é ✈❛❧♦r r❡❣✉❧❛r ❞❡ φ|}

M✳ P❡❧♦ ❚❡♦r❡♠❛

❞❛ ❙✉❜♠❡rsã♦✱ s❡❣✉❡ q✉❡φ−1_|

M(0) é ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ M✳ P♦rt❛♥t♦✱ N é ✉♠❛ C1 ✲

s✉❜✈❛r✐❡❞❛❞❡ ❞❡H1 0(Ω)✳

❖❜s❡r✈❛çã♦ ✶✳✽ ◆♦t❡ q✉❡ _t¯_{é ú♥✐❝♦ t❛❧ q✉❡} ¯_{tv ∈ N✱ ♣♦✐s s❡ v ∈H}1

0(Ω) ❡ tv ∈ N ❡♥tã♦

♣♦r ✭✶✳✼✮ t❡♠♦s q✉❡

0 =

Z

Ω

|∇(tv)|2_dx− Z

Ω

|tv|p+1_dx

= t2

Z

Ω

|∇v|2dx−tp+1Z Ω

|v|p+1_dx

= t2

Z

Ω

|v|p+1_dx−tp+1Z Ω

|v|p+1_dx

= t2_(1−tp−1₎ Z

Ω

|v|p+1_dx.

❈♦♠♦ t >0 ❡ v 6= 0✱ ❞❡✈❡♠♦s t❡r t= 1 ❡ ♣♦r s✉❛ ✈❡③ t é ú♥✐❝♦ t❛❧ q✉❡ tv∈ N✳

Pr♦♣♦s✐çã♦ ✶✳✾ ❆ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ ❛ss♦❝✐❛❞❛ ❛K(u)✱

N ={u∈H₀1(Ω)\{0};hK′(u), ui= 0}

é ❢❡❝❤❛❞❛ ❡♠H1 0(Ω)✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ un ∈ N t❛❧ q✉❡ un → u ❡♠ H1

0(Ω)✳ ❊♥tã♦ ♣❡❧❛s ✐♠❡rsõ❡s ❞❡

❙♦❜♦❧❡✈ t❡♠♦s q✉❡un →u ❡♠ Lp+1_{(Ω)✱ ♣♦✐s1< p+ 1 <2}∗_{✳ ❆ss✐♠} 0 =hK′(un), uni=||un||2−

Z

Ω

|un|p+1_{dx→ ||u||}2_{− ||u||}p+1

Lp+1_(Ω)=hK′(u), ui,

♦ q✉❡ ✐♠♣❧✐❝❛hK′_{(u), ui= 0✳ ❉❡ss❛ ❢♦r♠❛✱ ❜❛st❛ ♠♦str❛r q✉❡u6= 0✳ ❈♦♠♦ un ∈ N✱ ♣❡❧❛} ✐♠❡rsã♦ ❞❡H1

0(Ω) ❡♠ Lp+1(Ω) t❡♠♦s

||un||2 ₌ Z

Ω

|un|p+1_dx≤c||un||p+1_.

❆ss✐♠ s❡❣✉❡ q✉❡

||un|| ≥

1

c

1

p−₁

, ∀n∈N_.

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱u6= 0 ❡u∈ N✱ ♦ q✉❡ ♠♦str❛ q✉❡ N é ❢❡❝❤❛❞❛ ❡♠H1 0✳

◆♦t❡ q✉❡✱ s❡u∈ N ❡♥tã♦

K(u) = 1 2||u||

2₋ 1

p+ 1

Z

Ω

|u|p+1 ₌

1 2 −

1

p+ 1

||u||2 ≥0.

▲♦❣♦✱ K|N é ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡✳

❆❣♦r❛ ❡st❛♠♦s ♣r♦♥t♦s ♣❛r❛ ♠♦str❛r ♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡st❛ s❡çã♦✱ ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ♦ í♥✜♠♦ ❞♦ ❢✉♥❝✐♦♥❛❧ K ♥❛ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ ❡ ♦ ♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛

❞❡st❡ ❢✉♥❝✐♦♥❛❧✳

Pr♦♣♦s✐çã♦ ✶✳✶✵ ❙❡❥❛ β = inf{K(u) : u∈ N }✳ ❊♥tã♦ β > 0 ❡ β =cP M ♦♥❞❡ cP M é ♦

♥í✈❡❧ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❞❡ K✳

❉❡♠♦♥str❛çã♦✿ ❉❡♥♦t❛♥❞♦ Γ = {γ ∈ C([0,1], H1

0(Ω)) : γ(0) = 0, K(γ(1)) < 0}✱ ♣❡❧♦

❚❡♦r❡♠❛ ✶✳✻ s❛❜❡♠♦s q✉❡ ♦ ♣r♦❜❧❡♠❛ ✭✶✳✻✮ ♣♦ss✉✐ ✉♠❛ s♦❧✉çã♦ u ∈ H1

0(Ω)\{0} t❛❧ q✉❡

K(u) =cP M✱ ♦♥❞❡

cP M = inf

γ∈Γ₀sup_≤t≤1K(γ(t))>0.

▼♦str❛r❡♠♦s✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ q✉❡ cP M é ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ β✳ ❈♦♠♦ ✈✐♠♦s ♥❛ Pr♦♣♦s✐çã♦

✶✳✸✱ ❡①✐st❡δ >0 t❛❧ q✉❡ ||v||p_L+1p+1 ≤ 1₂||v||2 ❡ K(v)≥ 1₄||v||2 ♣❛r❛ t♦❞♦v ∈B¯(0, δ)✳ ❊♥tã♦✱

❞❛❞♦ γ ∈ Γ t❡♠♦s γ(0) = 0 ❡ K(γ(1)) < 0✱ ❞♦♥❞❡ γ(1) 6∈ B¯(0, δ)✳ ❈♦♠♦ γ é ❝♦♥tí♥✉❛✱

❡①✐st❡ t ∈ (0,1) t❛❧ q✉❡ γ(t) ∈ ∂B(0, δ)✱ ♦✉ s❡❥❛✱ ||γ(t)|| = δ✳ ❙❡❥❛ t0 = max{t ∈ (0,1) :

||γ(t)||=δ}✳ ❉❛í

φ(γ(t0)) =||γ(t0)||2− ||γ(t0)||p_L+1p+1 ≥

1

2||γ(t0)||

2 ₌ 1

2δ

2 _>0.

❈♦♠♦

0> K(γ(1)) = 1

2||γ(1)||

2₋ 1

p+ 1

Z

Ω

|γ(1)|p+1

t❡♠♦s

||γ(1)||2 _< 2

p+ 1

Z

Ω

|γ(1)|p+1_.

❆ss✐♠

φ(γ(1)) =||γ(1)||2₋ Z

Ω

|γ(1)|p+1 _<

2

p+ 1 −1

Z

Ω

|γ(1)|p+1 _<0.

❉❡ss❛ ❢♦r♠❛✱ ❡①✐st❡ ¯_{t ∈ (t0,1) t❛❧ q✉❡ γ(¯t) ♥ã♦ ♣❡rt❡♥❝❡ ❛ B}¯_{(0, δ) ❡ φ(γ(¯t)) = 0✳ ▲♦❣♦}

γ(¯t)∈N✳ ■st♦ ✐♠♣❧✐❝❛ q✉❡

max

t∈[0,1]K(γ(t))≥K(γ(¯t))≥β.

P♦rt❛♥t♦✱ cP M ≥ β✳ ▼♦str❛r❡♠♦s ❛❣♦r❛ q✉❡ β é ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ cP M✳ ❉❡ ❢❛t♦✱ ❞❛❞♦ u∈ N ♣♦r ✭✶✳✼✮ t❡♠♦s q✉❡

K(tu) = t

2

2||u||

2₋ tp+1

p+ 1

Z

Ω

|u|p+1_dx=

t2

2 −

tp+1

p+ 1

kuk2

❡max

t>0 K(tu) = K(u)✱ ♣♦✐s ♦❝♦rr❡ ❡♠t = 1✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ p+ 1>2s❡❣✉❡ q✉❡

lim

t→∞K(tu) = −∞.

❋✐①❛♥❞♦ t0 > 1 t❛❧ q✉❡ K(t0u) ≤ 0✱ ❞❡✜♥❛ γ(t) = t(t0u) ♣❛r❛ t ∈ [0,1]✳ ❆ss✐♠✱ t❡♠♦s

γ ∈Γ❡

cP M ≤ sup

0≤t≤1

K(γ(t)) = sup

0≤t≤1

K(t(t0u)) =K(u).

■st♦ ✐♠♣❧✐❝❛ q✉❡ K(u) ≥ cP M✱ ♣❛r❛ t♦❞♦ u ∈ N✱ ❡ ♣♦r s✉❛ ✈❡③ β ≥ cP M✳ ▲♦❣♦✱ β =

cP M >0✳

❈❛♣ít✉❧♦ ✷

❯♠❛ ❡q✉❛çã♦ ❡❧í♣t✐❝❛ s❡♠✐❧✐♥❡❛r

❡♥✈♦❧✈❡♥❞♦ ✉♠❛ ❢✉♥çã♦ ♣❡s♦ ❝♦♠

♠✉❞❛♥ç❛ ❞❡ s✐♥❛❧

✷✳✶ ■♥tr♦❞✉çã♦

◆❡st❡ ❝❛♣ít✉❧♦ ✐r❡♠♦s ❡st✉❞❛r ✉♠❛ ❝❧❛ss❡ ❞❡ ❡q✉❛çõ❡s ❡❧í♣t✐❝❛s s❡♠✐❧✐♥❡❛r❡s ❞♦ t✐♣♦

(

−∆u=|u|p−1_u+λf(x)|u|q−1_{u, ❡♠ Ω}

u= 0 s♦❜r❡ ∂Ω ✭✷✳✶✮

♦♥❞❡Ωé ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❡♠ RN_{✱0< q <1< p <2}∗_{−1✱λ >0 ❡f : ¯Ω→Ré ✉♠❛} ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❝♦♠ ♠✉❞❛♥ç❛ ❞❡ s✐♥❛❧ ❡♠_{Ω✳ ❊st❡ ❝❛♣ít✉❧♦ ❢♦✐ ❜❛s❡❛❞♦ ♥♦ ❛rt✐❣♦ ❞❡ ❚s✉♥❣✲}¯ ❋❛♥❣ ❲✉ ❬✶✹❪✱ ♦♥❞❡ ✉s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞♦s ▼✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡ ❇✳✷ ♠♦str❛r❡♠♦s q✉❡ ♣❛r❛λ ♣❡q✉❡♥♦✱ ♦ ♣r♦❜❧❡♠❛ ✭✷✳✶✮ ♣♦ss✉✐ ❛♦ ♠❡♥♦s ❞✉❛s s♦❧✉çõ❡s ♣♦s✐t✐✈❛s✳

❖ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡st❡ ❝❛♣ít✉❧♦ é ♦ s❡❣✉✐♥t❡✳

❚❡♦r❡♠❛ ✷✳✶ ❊①✐st❡ Λ0 > 0 t❛❧ q✉❡ ♣❛r❛ λ ∈ (0,Λ0)✱ ❛ ❡q✉❛çã♦ ✭✷✳✶✮ t❡♠ ♣❡❧♦ ♠❡♥♦s

❞✉❛s s♦❧✉çõ❡s ♣♦s✐t✐✈❛s✳

✷✳✷ ❖ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❡ ❛ ❱❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐

0(Ω) →R ❞❡✜♥✐❞♦ ♣♦r

Jλ(u) = 1 2

Z

Ω

|∇u|2dx− 1

p+ 1

Z

Ω

|u|p+1dx− λ

q+ 1

Z

Ω

f(x)|u|q+1dx.

❯♠❛ s♦❧✉çã♦ ✭❢r❛❝❛✮ ♣❛r❛ ❛ ❡q✉❛çã♦ ✭✷✳✶✮ é ✉♠❛ ❢✉♥çã♦ u∈H1

0(Ω) q✉❡ s❛t✐s❢❛③ Z

Ω

∇u∇vdx=

Z

Ω

|u|p−1_uvdx+λZ Ω

f(x)|u|q−1_{uvdx, ∀v ∈H}1 0(Ω).

❖ r❡s✉❧t❛❞♦ ❛ s❡❣✉✐r ♠♦str❛ ❛ r❡❣✉❧❛r✐❞❛❞❡ ❞♦ ❢✉♥❝✐♦♥❛❧Jλ ♣❛r❛ t♦❞♦ λ >0✳

▲❡♠❛ ✷✳✷ ❖ ❢✉♥❝✐♦♥❛❧ Jλ ♣❡rt❡♥❝❡ ❛ C1_(H1

0(Ω),R)✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ gλ : ¯Ω×R _→ R ❞❛❞❛ ♣♦r _{gλ(x, t) = |t|}p−1_t+λf(x)|t|q−1_{t ♣❛r❛}

❝❛❞❛ λ ✜①❛❞♦✳ ❈♦♠♦ f ∈ C( ¯Ω,R₎ ❡ _{p, q > 0} t❡♠♦s _{g ∈ C( ¯Ω ×}R_,R)✳ ❉❡♥♦t❛♥❞♦

Gλ(x, t) =

Z t

0

gλ(x, r)dr✱ t❡♠♦s

Gλ(x, t) =

Z t

0

|r|p−1_r+λf(x)|r|q−1_{rdr =} 1

p+ 1|t|

p+1₊ λ

q+ 1f(x)|t|

q+1_.

❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ f é ❧✐♠✐t❛❞❛ ❡♠ Ω¯ t❡♠♦s

|gλ(x, t)|=|t|p_{+|λ||f(x)||t|}q_{≤ |t|}p_+c

1|t|q✱ ∀(x, t)∈Ω¯ ×R.

❉❛í

|gλ(x, t)| ≤

(

|t|p_+c

1, s❡ |t| ≤1

(1 +c1)|t|p, s❡ |t|>1

♦ q✉❡ ✐♠♣❧✐❝❛ |gλ(x, t)| ≤ c1+c2|t|p ♣❛r❛ t♦❞♦ (x, t)∈Ω¯ ×R✱ ❝♦♠ 1< p <2∗−1✳ ❯♠❛

✈❡③ q✉❡

Jλ(u) =

Z

Ω

1 2|∇u|

2_{−Gλ(x, u)}

dx,

s❡❣✉❡ ❞❛ Pr♦♣♦s✐çã♦ ❆✳✸ q✉❡ Jλ é ❞❡ ❝❧❛ss❡ C1 _❡

hJλ′(u), vi =

Z

Ω

[∇u∇v−gλ(x, u)v]dx

=

Z

Ω

∇u∇vdx−

Z

Ω

|u|p−1_uvdx−λZ Ω

f(x)|u|q−1_uvdx. ✭✷✳✷✮

■st♦ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞♦ ❧❡♠❛✳ ❖❜s❡r✈❡ q✉❡ u ∈ H1

0(Ω) é s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✭✷✳✶✮ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ u é ♣♦♥t♦ ❝rít✐❝♦

❞❡ Jλ✳ ◆ã♦ ♣♦❞❡♠♦s ✉s❛r ❛q✉✐ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♥♦ ❈❛♣ít✉❧♦ ✶✱ ♣♦rq✉❡ gλ(·, t) ♥ã♦ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦(g3)❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ✈❡♠♦s q✉❡Jλ ♥ã♦ ♣♦ss✉✐ ❛ ❣❡♦♠❡tr✐❛ ❞♦ P❛ss♦

❞❛ ▼♦♥t❛♥❤❛✳ P❛r❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ✭✷✳✶✮ ✉s❛r❡♠♦s ♦✉tr❛s té❝♥✐❝❛s q✉❡ ❡♥✈♦❧✈❡♠ ❛ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ ❡ ♦ ❚❡♦r❡♠❛ ❞♦s ▼✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡ ❇✳✷✳

❆ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐ ❛ss♦❝✐❛❞❛ ❛♦ ❢✉♥❝✐♦♥❛❧ Jλ é ❞❛❞❛ ♣♦r

Mλ ={H01(Ω)\{0}:hJλ′(u), ui= 0}.

❆ss✐♠✱ s❡u∈ Mλ✱ s❡❣✉❡ ❞❡ ✭✷✳✷✮ q✉❡

||u||2−

Z

Ω

|u|p+1_dx−λZ Ω

f(x)|u|q+1_{dx= 0. ✭✷✳✸✮}

❆ ✜♠ ❞❡ ♠♦str❛r♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ Mλ é ❞❡ ❢❛t♦ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛ ❡ ♥ã♦ ✈❛③✐❛✱

❝♦♠❡ç❛♠♦s ❞❡✜♥✐♥❞♦ψ :H1

0(Ω) →R♣♦r

ψλ(u) = hJ_λ′(u), ui=||u||2−

Z

Ω

|u|p+1_dx−λZ Ω

f(x)|u|q+1_dx.

❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❛♦ ▲❡♠❛ ✷✳✷✱ ♠♦str❛♠♦s ♣❡❧❛ Pr♦♣♦s✐çã♦ ❆✳✸ q✉❡ψλ ∈C1_(H1

0(Ω),R)❡

hψλ′(u), vi= 2

Z

Ω

∇u∇v−(p+ 1)

Z

Ω

|u|p−1_{uv −(q+ 1)λ}Z Ω

f(x)|u|q−1_uvdx.

P❛r❛ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ψλ′ ♥❛ ✈❛r✐❡❞❛❞❡ ❞❡ ◆❡❤❛r✐✱ ❞✐✈✐❞✐✲

♠♦s Mλ ❡♠ três ♣❛rt❡s✿

M0_λ = {u∈ Mλ :hψλ′(u), ui= 0}

M+_λ = {u∈ Mλ :hψλ′(u), ui>0}

M_λ− = {u∈ Mλ :hψλ′(u), ui<0}.

P❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✷✳✶ ✉t✐❧✐③❛r❡♠♦s ❛❧❣✉♥s ❧❡♠❛s✳ ❊st❡ ♣r✐♠❡✐r♦ ❣❛r❛♥t❡ q✉❡ ♦ ❝♦♥❥✉♥t♦M0

λ é ✈❛③✐♦ ♣❛r❛ λ♣❡q✉❡♥♦ ❡ ❛ss✐♠✱ q✉❛❧q✉❡r u∈ Mλ é ♣♦♥t♦ r❡❣✉❧❛r ❞❡ ψ✳ ❯♠❛ ✈❡③ q✉❡ M=ψ|−_H11

0\{0}(0) t❡♠♦s q✉❡Mλ é ❞❡ ❢❛t♦ ✉♠❛ ✈❛r✐❡❞❛❞❡✳

▲❡♠❛ ✷✳✸ ❊①✐st❡Λ1 >0 t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ λ∈(0,Λ1) t❡♠♦s M0λ =∅✳

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛♠♦s ♣♦r ❝♦♥tr❛❞✐çã♦ q✉❡ M0

λ 6= ∅ ♣❛r❛ t♦❞♦ λ > 0 ❡ s❡❥❛ u∈ M0

λ✳ ❈♦♠♦ u∈ Mλ✱ t❡♠♦s

Z

Ω

|u|p+1_dx=||u||2_−λ Z

Ω

f(x)|u|q+1_{dx. ✭✷✳✹✮}

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱

hψ′(u), ui = 2||u||2−(p+ 1)

Z

Ω

|u|p+1_{dx−(q+ 1)λ}Z Ω

f(x)|u|q+1_dx

= 2||u||2_{−(p+ 1)}

||u||2_−λ Z

Ω

f(x)|u|q+1_dx

−(q+ 1)λ

Z

Ω

f(x)|u|q+1_dx

= (1−p)||u||2_{+ (p−q)λ}Z Ω

f(x)|u|q+1_dx.

❏á q✉❡ hψ′_{(u), ui= 0 s❡❣✉❡ q✉❡}

p−1

p−q

||u||2 =λ

Z

Ω

f(x)|u|q+1_dx.

❈♦♥s✐❞❡r❡σ = (p+ 1)/(p−q)✳ ❯♠❛ ✈❡③ q✉❡ 1/σ+ (q+ 1)/(p+ 1) = 1✱ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✱ ❚❡♦r❡♠❛ ❇✳✶✱ t❡♠♦s q✉❡

p−1

p−q

||u||2 _{= λ}Z Ω

f(x)|u|q+1_dx≤λ Z

Ω

|f(x)|σ

1

σ Z

Ω

(|u|q+1₎pq+1+1dx q+1

p+1

= λ||f||Lσ

"Z

Ω

|u|p+1_dx

1

p+1#

q+1

= λ||f||Lσ||u||q+1

Lp+1.

❉❡s❞❡ q✉❡ p+ 1 ∈ [1,2∗_{)✱ ❝♦♥s✐❞❡r❡ A ❛ ♠❡❧❤♦r ❝♦♥st❛♥t❡ ❞❛ ✐♠❡rsã♦ ❞❡ H}1

0(Ω) ❡♠

Lp+1_{(Ω)✳ ❊♥tã♦}

p−1

p−q

||u||2 ≤λ||f||Lσ||u||q+1

Lp+1 ≤λ||f||LσAq+1||u||q+1 ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡

||u|| ≤

λ

p−q p−1

||f||LσAq+1

1

1−q

. ✭✷✳✺✮

❙❡❥❛Iλ :Mλ →R ❞❡✜♥✐❞❛ ♣♦r Iλ(u) = C(p, q)

||u||2p

R

Ω|u|p+1dx

1

p−1

−λ

Z

Ω

f(x)|u|q+1_dx,

♦♥❞❡ C(p, q) =

1−q p−q

p

p−1 _p−₁

1−q

✳ ◆♦t❡ q✉❡✱ Iλ(u) = 0 ♣❛r❛ t♦❞♦ u ∈ M0

λ✳ ❉❡ ❢❛t♦✱

✉s❛♥❞♦ ♦ ❢❛t♦ ❞❡ u∈ M0

λ ❡ ✭✷✳✹✮ t❡♠♦s

0 = 2||u||2_{−(p+ 1)} Z

Ω

|u|p+1_{dx−(q+ 1)}

||u||2₋ Z

Ω

|u|p+1_dx

= (1−q)||u||2_−(p−q)Z Ω

|u|p+1_dx

♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡

||u||2 ₌ p−q

1−q

Z

Ω

|u|p+1_dx.

❆ss✐♠

Iλ(u) = C(p, q)

||u||2p

R

Ω|u|p+1dx

1

p−1

−λ

Z

Ω

f(x)|u|q+1_dx

=

1−q p−q

p

p−₁ p−1

1−q





p−q

1−q

p R

Ω|u|

p+1_dxp

R

Ω|u|p+1dx

 

1

p−1

−p−1

1−q

Z

Ω

|u|p+1_dx

=

1−q p−q

p

p−1 _p−₁

1−q

p−q

1−q

p

p−1 Z

Ω

|u|p+1_dx p−₁

p−1

−

p−1 1−q

Z

Ω

|u|p+1_dx

=

p−1 1−q

Z

Ω

|u|p+1_dx−

p−1 1−q

Z

Ω

|u|p+1_dx

= 0. ✭✷✳✻✮

P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✱ s❡♥❞♦σ = p_p+1_−q✱ ♣❛r❛ u∈ M0

λ t❡♠♦s Iλ(u) = C(p, q)

||u||2p

R

Ω|u|p+1dx

1

p−1

−λ

Z

Ω

f(x)|u|q+1_dx

≥ C(p, q)

||u||2p

R

Ω|u|p+1dx

1

p−1

−λ||f||Lσ||u||q+1

Lp+1.

❈♦♠♦ ♣❡❧❛ ✐♠❡rsã♦ ❞❡H1

0(Ω)❡♠Lp+1(Ω)✱ Pr♦♣♦s✐çã♦ ❆✳✷✱ t❡♠✲s❡||u||Lp+1 ≤A||u||✱ ❡♥tã♦

||u||(_Lqp+1)(+1 p−1) Z

Ω

|u|p+1_{dx = ||u||}(q+1)(p−1)

Lp+1 ||u||

p+1

Lp+1

≤ A(q+1)(p−1)||u||(q+1)(p−1) Ap+1||u||p+1

= Aq(p−1)+2p_||u||q(p−1)+2p_.

❆ss✐♠✱ ♣♦r ✭✷✳✺✮

Iλ(u) ≥ ||u||q_L+1p+1 C(p, q)

_||

u||2p

Aq(p−1)+2p_||u||q(p−1)+2p

1

p−1

−λ||f||Lσ

!

= ||u||q_L+1p+1 C(p, q)

1

Aq(p−1)+2p

1

p−1 ₁

||u||q −λ||f||Lσ

!

≥ ||u||q_L+1p+1 (

C(p, q)

1

Aq(p−1)+2p

1

p−1

λ

p−q p−1

||f||LσAq+1

−q

1−q

−λ||f||Lσ

)

= ||u||q_L+1p+1 (

C(p, q)

1

Aq(p−1)+2p

1

p−1

λ1−−qq

p−q p−1

||f||LσAq+1

−q

1−q

−λ||f||Lσ

)

.

■st♦ ✐♠♣❧✐❝❛ q✉❡ ♣❛r❛ λ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ t❡♠♦s Iλ(u) >0 ♣❛r❛ t♦❞♦ u ∈ M0

λ✱ ♦

q✉❡ ❝♦♥tr❛❞✐③ ✭✷✳✻✮✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ❡①✐st❡ Λ1 >0 t❛❧ q✉❡ ♣❛r❛λ ∈(0,Λ1)

t❡♠♦sM0

λ =∅✳

P❡❧♦ ▲❡♠❛ ✷✳✸ t❡♠♦s M0

λ = ∅ ♣❛r❛ λ ∈ (0,Λ1)✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

Mλ =M+_λ SM−_λ✳

❱❛♠♦s ❛❣♦r❛ ❞❡✜♥✐r ✉♠❛ ❢✉♥çã♦ ❝ô♥❝❛✈❛✱ ❛♥❛❧✐s❛r s❡✉ ❝♦♠♣♦rt❛♠❡♥t♦ ❡ ✐❞❡♥t✐✜❝❛r s❡✉ ♣♦♥t♦ ❞❡ ♠á①✐♠♦✱ ✐♥❢♦r♠❛çõ❡s q✉❡ ✉s❛r❡♠♦s ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛ ♣♦st❡r✐♦r✳

▲❡♠❛ ✷✳✹ P❛r❛ ❝❛❞❛ u∈H1

0(Ω)\{0} ❞❡✜♥✐♠♦s s:R→R ♣♦r

s(t) = t1−q||u||2_−tp−q

Z

Ω

|u|p+1_{dx ♣❛r❛ t≥0.}

❊♥tã♦st❡♠ ✉♠ ú♥✐❝♦ ♣♦♥t♦ ❝rít✐❝♦ q✉❡ étmax=

(1−q)||u||2 (p−q)R_Ω|u|p+1_dx

1

p−1

✉♠ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❣❧♦❜❛❧✳ ❆❧é♠ ❞✐ss♦✱

s(tmax)≥ ||u||q+1

1−q p−q

1−q

p−1 _p−₁

p−q

1

Ap+1 1−q

p−1

.

❉❡♠♦♥str❛çã♦✿ ◆♦t❡ q✉❡s(0) = 0✳ ❙❡♥❞♦ p−q >1−q t❡♠♦s q✉❡ s(t)>0♣❛r❛ t >0 ♣❡q✉❡♥♦ ❡

lim

t→∞s(t) = limt→∞t

1−q_||

u||2_−tp−q

Z

Ω

|u|p+1_dx=−∞.

❊♥tã♦✱ s❡♥❞♦s❝♦♥tí♥✉❛ ❛t✐♥❣❡ s❡✉ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❡♠ ❛❧❣✉♠t >0✳ ▼♦str❛r❡♠♦s ❛❣♦r❛ q✉❡s(t)❛t✐♥❣❡ ♦ ♠á①✐♠♦ ❡♠ tmax✳ ❙❡ t0 >0é ♣♦♥t♦ ❝rít✐❝♦ ❞❡ s(t)✱ ❡♥tã♦

0 = s′(t0) = (1−q)t0−q||u||2 −(p−q)t

p−q−1 0

Z

Ω

|u|p+1_dx