❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆
Pr♦❜❧❡♠❛s ❡❧í♣t✐❝♦s s✉♣❡r❧✐♥❡❛r❡s ❝♦♠
♥ã♦ ❧✐♥❡❛r✐❞❛❞❡s ❛ss✐♠étr✐❝❛s
❲❛❧❧✐s♦♠ ❞❛ ❙✐❧✈❛ ❘♦s❛
❙ã♦ ❈❛r❧♦s ✲ ❙P
▼❛rç♦ ❞❡ ✷✵✶✺
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆
❲❛❧❧✐s♦♠ ❞❛ ❙✐❧✈❛ ❘♦s❛
❇♦❧s✐st❛ ❈❆P❊❙ ✭Pr♦❞♦✉t♦r❛❧ ❡ ❘❡✉♥✐✮
❖r✐❡♥t❛❞♦r✿ Pr♦❢ ❉r✳ ❋r❛♥❝✐s❝♦ ❖❞❛✐r ❱✐❡✐r❛ ❞❡ P❛✐✈❛
Pr♦❜❧❡♠❛s ❡❧í♣t✐❝♦s s✉♣❡r❧✐♥❡❛r❡s ❝♦♠
♥ã♦ ❧✐♥❡❛r✐❞❛❞❡s ❛ss✐♠étr✐❝❛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
Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária UFSCar Processamento Técnico
com os dados fornecidos pelo(a) autor(a)
R788p
Rosa, Wallisom da Silva
Problemas elípticos superlineares com não
linearidades assimétricas / Wallisom da Silva Rosa. -- São Carlos : UFSCar, 2015.
95 p.
Tese (Doutorado) -- Universidade Federal de São Carlos, 2015.
1. Problemas elípticos. 2. Métodos variacionais. 3. Métodos topológicos. 4. Linking. 5. Grau
❆●❘❆❉❊❈■▼❊◆❚❖❙
➚ ❊✈❛✱ ♠✐♥❤❛ ♠✉s❛ ✐♥s♣✐r❛❞♦r❛✱ ❝♦♠ q✉❡♠ ❞✐✈✐❞♦ s♦♥❤♦s ❡ r❡❛❧✐③❛çõ❡s ❞❡s❞❡ ✷✵✵✹✳ ➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ q✉❡ s✉♣♦rt♦✉ ♦s ♠♦♠❡♥t♦s ❞❡ ❛✉sê♥❝✐❛ ❝♦♠ ❛ ❝♦♠♣r❡❡♥sã♦ ♥❡❝❡ssá✲ r✐❛✳
➚ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❯❜❡r❧â♥❞✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ♣r♦❢❡ss♦r❡s ❞♦ ❈✉rs♦ ❞❡ ▼❛t❡✲ ♠át✐❝❛ ❞❛ ❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ■♥t❡❣r❛❞❛s ❞♦ P♦♥t❛❧✱ ♣❡❧❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞♦ ❛❢❛st❛♠❡♥t♦ ♣❛r❛ ❝✉♠♣r✐r ❡ss❛ ❡t❛♣❛ ♣r♦✜ss✐♦♥❛❧✳
❆ t♦❞♦s ♦s ❛♠✐❣♦s ❞♦ ❞♦✉t♦r❛❞♦ q✉❡ ❞✐✈✐❞✐r❛♠ ❝❛❞❛ ♠♦♠❡♥t♦ ❞❡ss❛ ❥♦r♥❛❞❛✱ ❡♠ ❡s♣❡❝✐❛❧ ♦s ❛♠✐❣♦s q✉❡ ✐♥❣r❡ss❛r❛♠ ❝♦♠✐❣♦✿ ❆❧❧❛♥✱ ❈❛r♦❧✱ ❊✈❛✱ ❏❛♣❛ ❡ ▼❛r❝♦s✱ ❡ ♦s ❝♦❧❡❣❛s ❞❡ ♦r✐❡♥t❛çã♦✿ ❋❛❜✐❛♥❛ ❋❡rr❡✐r❛ ❡ ▼♦✐sés ◆❛s❝✐♠❡♥t♦✳
❆♦s ♥♦✈♦s ❛♠✐❣♦s sã♦ ❝❛r❧❡♥s❡s q✉❡ ✜③ ♥❡ss❡ ♣❡rí♦❞♦ ❞❡ ♠♦r❛❞❛ ♣r♦✈✐sór✐❛ ❡ ❛♦s ❛♠✐❣♦s ❞❡ ■t✉✐✉t❛❜❛✲▼●✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♦s ❛✜❧❤❛❞♦s ❆♥❞ré ❡ ❏✉ss❛r❛ q✉❡ s❡♠♣r❡ ♥♦s ❛❝♦❧❤❡r❛♠ ❡♠ s✉❛ r❡s✐❞ê♥❝✐❛ q✉❛♥❞♦ ♣r❡❝✐s❛♠♦s✳
❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❜❛♥❝❛✱ q✉❡ ❞✐s♣♦♥✐❜✐❧✐③❛r❛♠ t❡♠♣♦✱ ❝♦♥❤❡❝✐♠❡♥t♦ ❡ ♣❛❝✐ê♥❝✐❛ ♣❛r❛ ❝♦♥tr✐❜✉✐r ❝♦♠ ♦ tr❛❜❛❧❤♦✳
❘❊❙❯▼❖
◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ✉♠❛ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s ❡❧í♣t✐❝♦s ♥ã♦ ❧✐♥❡❛r❡s ❛ss✐♠étr✐❝♦s✳ ❆ ❛ss✐♠❡tr✐❛ q✉❡ ❝♦♥s✐❞❡r❛♠♦s ❛q✉✐ t❡♠ ❝♦♠♣♦rt❛♠❡♥t♦ ❧✐♥❡❛r ❡♠−∞❡ s✉♣❡r❧✐♥❡❛r ❡♠+∞✳ P❛r❛ ♦❜t❡r t❛✐s r❡s✉❧t❛❞♦s ❛♣❧✐❝❛♠♦s ♠ét♦❞♦s ✈❛r✐❛❝✐♦♥❛✐s ❝♦♠♦ t❡♦r❡♠❛s ❞❡ ❧✐♥❦✐♥❣ ❡ ♠ét♦❞♦s t♦♣♦❧ó❣✐❝♦s ❝♦♠♦ ❛ t❡♦r✐❛ ❞♦ ❣r❛✉ t♦♣♦❧ó❣✐❝♦✳
❆❇❙❚❘❆❈❚
❚❤❡ ❛✐♠ ♦❢ t❤✐s ✇♦r❦ ✐s t♦ ♣r❡s❡♥t r❡s✉❧ts ♦❢ ❡①✐st❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s ❢♦r ❛ ❝❧❛ss ♦❢ ♥♦♥❧✐♥❡❛r ❛s②♠♠❡tr②❝ ❡❧❧✐♣t✐❝ ♣r♦❜❧❡♠s✳ ❚❤❡ ❛s②♠♠❡tr② t❤❛t ✇❡ ❝♦♥s✐❞❡r ❤❡r❡ ❤❛s ❧✐♥❡❛r ❜❡❤❛✈✐♦r ♦♥−∞❛♥❞ s✉♣❡r❧✐♥❡❛r ♦♥ +∞✳ ❚♦ ♦❜t❛✐♥ t❤❡s❡ r❡s✉❧ts ✇❡ ❛♣♣❧② ✈❛r✐❛t✐♦♥❛❧ ♠❡t❤♦❞s ❛s ❧✐♥❦✐♥❣ t❤❡♦r❡♠s ❛♥❞ t♦♣♦❧♦❣✐❝❛❧ ♠❡t❤♦❞s ❧✐❦❡ t♦♣♦❧♦❣✐❝❛❧ ❞❡❣r❡❡ t❤❡♦r②✳
❙❯▼➪❘■❖
❆❣r❛❞❡❝✐♠❡♥t♦s ✐
❘❡s✉♠♦ ✐✐
❆❜str❛❝t ✐✐✐
■♥tr♦❞✉çã♦ ✶
✶ ❯♠ ♣r♦❜❧❡♠❛ ❞❡ ◆❡✉♠❛♥♥ ❝♦♠ ❡①♣♦❡♥t❡ ❝rít✐❝♦ ✼ ✶✳✶ ❘❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷ Pr❡❧✐♠✐♥❛r❡s ❡ ♥♦t❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸ ❈♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✹ ❈♦♥❞✐çõ❡s ❣❡♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✺ Pr♦✈❛ ❞♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ◆♦t❛s ❞♦ ❝❛♣ít✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✷ ❯♠ s✐st❡♠❛ ❍❛♠✐❧t♦♥✐❛♥♦ ❝♦♠ ❝♦♥❞✐çõ❡s ❞❡ ◆❡✉♠❛♥♥ ♥❛ ❢r♦♥t❡✐r❛ ✸✸ ✷✳✶ ❍✐♣ót❡s❡s ❡ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✷ ❚❡♦r❡♠❛ ❞❡ ♠✐♥✐♠❛① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
❙❯▼➪❘■❖ ✈
✷✳✸ ❋♦r♠✉❧❛çã♦ ✈❛r✐❛❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✹ ❈♦♥❞✐çõ❡s ❣❡♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✺ ❖ ❝❛s♦ b= 0 ✭♦✉ c= 0✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✷✳✻ ❈♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✼ Pr♦✈❛ ❞♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ◆♦t❛s ❞♦ ❝❛♣ít✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✸ ❆ r❡ss♦♥â♥❝✐❛ ♥♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ✺✸
✸✳✶ ❖ ❝❛s♦ ❡s❝❛❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✷ ❊st✐♠❛t✐✈❛ ❛ ♣r✐♦r✐ ♣❛r❛ ♦ ❝❛s♦ ❡s❝❛❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✸ ❊①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ♦ ❝❛s♦ ❡s❝❛❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✸✳✹ ❙✐st❡♠❛ r❡ss♦♥❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✸✳✺ ❊st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐ ♣❛r❛ ❛s s♦❧✉çõ❡s ❞♦ s✐st❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✸✳✻ ❊①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ♦ s✐st❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ◆♦t❛s ❞♦ ❝❛♣ít✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷
❆ ❙í♠❜♦❧♦s ❡ ♥♦t❛çõ❡s ✼✹
❇ ❉❡s✐❣✉❛❧❞❛❞❡s ❡❧❡♠❡♥t❛r❡s ✼✽
❈ ❘❡s✉❧t❛❞♦s ❡❧❡♠❡♥t❛r❡s ✼✾
❉ ❖ ❣r❛✉ t♦♣♦❧ó❣✐❝♦ ✽✸
■◆❚❘❖❉❯➬➹❖
❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ✉♠❛ ❝❧❛ss❡ ❞❡ ❡q✉❛çõ❡s ❡ s✐st❡♠❛s ❡❧í♣t✐❝♦s ❝♦♠ ♣❛rt❡ ♥ã♦ ❧✐♥❡❛r ❛ss✐♠étr✐❝❛✳ ❊♠ ❝❛❞❛ ❝❛♣ít✉❧♦ ❝♦♥s✐❞❡r❛r❡♠♦s ✉♠ ♣r♦❜❧❡♠❛ q✉❡ é s✉♣❡r❧✐♥❡❛r ❡♠+∞❡ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❧✐♥❡❛r ❡♠−∞✳
◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❡st✉❞❛♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ◆❡✉♠❛♥♥
✭✶✮
−∆u=λu+g(x, u) + (u+)2∗−1
, x∈Ω, ∂u
∂ν = 0, x∈∂Ω,
♦♥❞❡Ω⊂RN✱N ≥3✱ é ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❝♦♠ ❢r♦♥t❡✐r❛ s✉❛✈❡∂Ω✱λ >0❡2∗ = 2N
N −2 é ♦ ❡①♣♦❡♥t❡ ❝rít✐❝♦ ♣❛r❛ ❛ ✐♠❡rsã♦ ❞❡ ❙♦❜♦❧❡✈✳
❈♦♠ ❛s ❤✐♣ót❡s❡s q✉❡ ✉t✐❧✐③❛♠♦s s♦❜r❡ ❛ ❢✉♥çã♦ g : Ω×R→ R✱ ♦ ♣r♦❜❧❡♠❛ ✭✶✮ é ❞♦
t✐♣♦ s✉♣❡r❧✐♥❡❛r ❡♠ +∞ ❡ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❧✐♥❡❛r ❡♠ −∞✳ P❛r❛ ♠♦str❛r ♦ r❡s✉❧t❛❞♦ ❞❡
❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ♣❛r❛ ❡ss❡ ♣r♦❜❧❡♠❛ ✉t✐❧✐③❛♠♦s ♦ ❚❡♦r❡♠❛ ❞♦ ✏▲✐♥❦✐♥❣✑✱ ❞❡✈✐❞♦ ❛ ❘❛❜✐♥♦✇✐t③ ❬❘❛❜✐♥♦✇✐t③❪✳ ❆ ♣r❡s❡♥ç❛ ❞♦ t❡r♠♦ ❝rít✐❝♦(u+)2∗−1
❞✐✜❝✉❧t❛ ❛ ♣r♦✈❛ ❞♦s r❡s✉❧t❛❞♦s ❞❡ ❝♦♠♣❛❝✐❞❛❞❡✱ ❣❡r❛❧♠❡♥t❡ ❝♦♠✉♥s ♥❛s ❤✐♣ót❡s❡s ❞♦s t❡♦r❡♠❛s ❞❡ ♣♦♥t♦s ❝rít✐❝♦s✳ ❊♠ ❣❡r❛❧✱ ❢✉♥❝✐♦♥❛✐s r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❡q✉❛çõ❡s q✉❡ ❛♣r❡s❡♥t❛♠ ❡ss❡ t❡r♠♦ s❛✲ t✐s❢❛③❡♠ ❛ ❝♦♥❞✐çã♦ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ✭❝♦♥❞✐çã♦ P❙✮ ♣♦ss✐✈❡❧♠❡♥t❡ ❛♣❡♥❛s ♣❛r❛ ✈❛❧♦r❡s ❞♦ ♥í✈❡❧c❞❛ s❡q✉ê♥❝✐❛ ♥✉♠ ✐♥t❡r✈❛❧♦ ❧✐♠✐t❛❞♦ ❞❛ r❡t❛✳ ■ss♦ ❢♦rç❛ ✉♠ ♣❛ss♦ ❛ ♠❛✐s✿ ♠♦str❛r
q✉❡ ♦ ♥í✈❡❧ ♠✐♥✲♠❛① ♣❡rt❡♥❝❡ ❛♦ ✐♥t❡r✈❛❧♦ ♦♥❞❡ ♦ ❢✉♥❝✐♦♥❛❧ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ P❙✳ ❆ ♠♦t✐✈❛çã♦ ♣❛r❛ ♦ ❡st✉❞♦ ❞❛ ❡q✉❛çã♦(1) é ♦ ♣r♦❜❧❡♠❛ s✉❜❝rít✐❝♦ ❞❡ ❆r❝♦②❛✲❱✐❧❧❡❣❛s
■♥tr♦❞✉çã♦ ✷
❬❆r❝♦②❛✲❱✐❧❧❡❣❛s❪✳ ❖ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❡♠ q✉❡g(x, s) = (s+)p,❝♦♠ p∈(1,2∗−1)❢♦r♥❡❝❡ ❛
❡q✉❛çã♦✿
−∆u=λu+ (u+)p+ (u+)2∗−1.
Pr♦❜❧❡♠❛s ❡❧í♣t✐❝♦s ❡♥✈♦❧✈❡♥❞♦ ♦ ❡①♣♦❡♥t❡ ❝rít✐❝♦ ❞❡ ❙♦❜♦❧❡✈ tê♠ s✐❞♦ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ❞❡ ❞✐✈❡rs♦s ♣❡sq✉✐s❛❞♦r❡s✳ ❖ ♣r✐♠❡✐r♦ tr❛❜❛❧❤♦ r❡❧❡✈❛♥t❡ ♥❡ss❡ t❡♠❛✱ ✉t✐❧✐③❛♥❞♦ ❛ t❡♦r✐❛ ❞❡ ♠✐♥✲♠❛①✱ é ♦ ❝❧áss✐❝♦ ❛rt✐❣♦ ❬❇r❡③✐s✲◆✐r❡♥❜❡r❣❪ q✉❡ ♠♦str♦✉ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛
−∆u=λu+u|u|2∗−2
, x∈Ω, u= 0, x∈∂Ω.
❆ ♣r✐♥❝✐♣❛❧ ❤✐♣ót❡s❡ ✉t✐❧✐③❛❞❛ ♥❡ss❡ tr❛❜❛❧❤♦ ❢♦✐λ < λ1 ✭♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ❞♦ ♦♣❡r❛❞♦r
−∆✱ ♦ ♦♣❡r❛❞♦r ▲❛♣❧❛❝✐❛♥♦✱ ❝♦♠ ❝♦♥❞✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t ♥❛ ❢r♦♥t❡✐r❛✮✳ P♦st❡r✐♦r♠❡♥t❡
♦s ❛rt✐❣♦s ❬❈❛♣♦③③✐✲❡t ❛❧✳❪ ❡ ❬❈❡r❛♠✐✲❡t ❛❧✳❪ ❛♠♣❧✐❛r❛♠ ❛ ❤✐♣ót❡s❡ s♦❜r❡ λ ❞❡ ❢♦r♠❛ ❛
❝♦❜r✐r q✉❛❧q✉❡r ✈❛❧♦r r❡❛❧ ❡ ♦❜t✐✈❡r❛♠ ❛✐♥❞❛ r❡s✉❧t❛❞♦s ❞❡ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ s♦❜r❡ ❛s s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛✱ r❡❧❛❝✐♦♥❛♥❞♦ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ❝♦♠ ❛ ♣♦s✐çã♦ ❞❡ λ ❡♠ r❡❧❛çã♦ ❛♦
❡s♣❡❝tr♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦✳
■♥s♣✐r❛❞♦s ♣♦r ❡ss❡s tr❛❜❛❧❤♦s s✉r❣✐r❛♠ ♠✉✐t♦s ♦✉tr♦s r❡s✉❧t❛❞♦s tr♦❝❛♥❞♦ λu ♣♦r
❢✉♥çõ❡s ♠❛✐s ❣❡r❛✐s g(x, u) q✉❡ ✐♠✐t❛♠ t❛❧ ❝♦♠♣♦rt❛♠❡♥t♦ ❧✐♥❡❛r ♣ró①✐♠♦ ❞❛ ♦r✐❣❡♠ ❡ s❡♠♣r❡ r❡❧❛❝✐♦♥❛♥❞♦λ ❝♦♠ ♦s ❛✉t♦✈❛❧♦r❡s ❞♦ ▲❛♣❧❛❝✐❛♥♦✱ ♣♦r ❡①❡♠♣❧♦ ❬❈❤❛❜r♦✇s❦✐✲❘✉❢❪
✭❝❛s♦ ◆❡✉♠❛♥♥✮ ❡ ❬●❛③③♦❧❛✲❘✉❢❪ ✭❝❛s♦ ❉✐r✐❝❤❧❡t✮✳
❘❡❝❡♥t❡♠❡♥t❡✱ ♠✉✐t♦s tr❛❜❛❧❤♦s tê♠ s✐❞♦ ❞❡s❡♥✈♦❧✈✐❞♦s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❛ss✐♠étr✐❝♦✱ ❡♠ q✉❡ ❛ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡ s❡ ❝♦♠♣♦rt❛ ❞❡ ♠❛♥❡✐r❛ ❞✐❢❡r❡♥t❡ ❡♠+∞ ❡−∞✳
❖s ❛rt✐❣♦s ❬❆r❝♦②❛✲❱✐❧❧❡❣❛s❪ ❡ ❬P❛♣❛❣❡♦r❣✐♦✉✲❙♠②r❧✐s❪ ❡st✉❞❛r❛♠ ✉♠❛ ❡q✉❛çã♦ ❝♦♠ ♣❛rt❡ ♥ã♦ ❧✐♥❡❛r ❛ss✐♠étr✐❝❛ ❡ ❝♦♥❞✐çõ❡s ❞❡ ◆❡✉♠❛♥♥ ♥❛ ❢r♦♥t❡✐r❛✱ ♣♦ré♠ s✉❜❝rít✐❝♦✳ ◆♦ss♦ r❡❢❡r❡♥❝✐❛❧ ♣❛r❛ ❛s ❤✐♣ót❡s❡s s♦❜r❡ λu+g(x, u) ❢♦r❛♠ ❛s ❤✐♣ót❡s❡s s♦❜r❡ ❛ ❢✉♥çã♦
f ❞♦ ❛rt✐❣♦ ❬❆r❝♦②❛✲❱✐❧❧❡❣❛s❪ ❡ ❛❝r❡s❝❡♥t❛♠♦s ♦ t❡r♠♦ ❝rít✐❝♦ ♠♦t✐✈❛❞♦s ♣❡❧♦ tr❛❜❛❧❤♦
❬❈❛❧❛♥❝❤✐✲❘✉❢❪✳
◆♦ ❛rt✐❣♦ ❬❈❛❧❛♥❝❤✐✲❘✉❢❪ ♦s ❛✉t♦r❡s ❡st✉❞❛r❛♠ ♦ ♣r♦❜❧❡♠❛
−∆u=λu+g(x, u+) + (u+)2∗−1
+f(x), x∈Ω,
■♥tr♦❞✉çã♦ ✸
❝♦♠ λ > λ1 ❡ f(x) = h +tϕ1✱ ♦♥❞❡ ϕ1 é ❛ ♣r✐♠❡✐r❛ ❛✉t♦❢✉♥çã♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦✱ ❝♦♠
❝♦♥❞✐çõ❡s ❞❡ ❉✐r✐❝❤❧❡t✱ ❡ h ∈ Lr(Ω) ♣❛r❛ ❛❧❣✉♠ r > N✳ ❯♠ ♣r♦❜❧❡♠❛ q✉❡ ♣♦ss✉✐ ♦
t❡r♠♦ f(x) ❝♦♠ ❛s ❝❛r❛❝t❡ríst✐❝❛s ❛❝✐♠❛ é ❞✐t♦ ❞♦ t✐♣♦ ❆♠❜r♦s❡tt✐✲Pr♦❞✐✳ ❆s té❝♥✐❝❛s ✈❛r✐❛❝✐♦♥❛✐s ✉t✐❧✐③❛❞❛s ♣♦r ❡ss❡s ❛✉t♦r❡s s❡r✈✐r❛♠ ❞❡ ❢♦♥t❡ ❞❡ ✐♥s♣✐r❛çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❡st✉❞❛❞♦ ♥❡st❡ ❝❛♣ít✉❧♦✳
◆♦ ❈❛♣ít✉❧♦ ✷ ❡st✉❞❛♠♦s ✉♠ s✐st❡♠❛ ❍❛♠✐❧t♦♥✐❛♥♦ ❝✉❥❛s ❡q✉❛çõ❡s s❡❣✉❡♠ q✉❛s❡ ❛s ♠❡s♠❛s ❝❛r❛❝t❡ríst✐❝❛s ❞♦ ♣r♦❜❧❡♠❛ ✭✶✮ q✉❛♥t♦ à ❛ss✐♠❡tr✐❛✱ ♣♦ré♠ ❝♦♠ ♦s ❡①♣♦❡♥t❡s s✉❜❝rít✐❝♦s✳
❯♠ s✐st❡♠❛ ❞♦ t✐♣♦
✭✷✮
−∆u+u=f(x, u, v)
−∆v +v =g(x, u, v)
é ❝❤❛♠❛❞♦ ❍❛♠✐❧t♦♥✐❛♥♦ s❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ H(x, u, v) t❛❧ q✉❡
✭✸✮ f = ∂H
∂v ❡ g = ∂H
∂u.
❊st✉❞❛♠♦s ♥♦ ❈❛♣ít✉❧♦ ✷ ♦ s✐st❡♠❛ ❍❛♠✐❧t♦♥✐❛♥♦ ❝♦♠ ❝♦♥❞✐çõ❡s ❞❡ ◆❡✉♠❛♥♥ ♥❛ ❢r♦♥t❡✐r❛✿
✭✹✮
−∆u+u=au+bv+ (v+)p, x∈Ω,
−∆v +v =cu+av+ (u+)q, x∈Ω, ∂u
∂ν = ∂v
∂ν = 0, x∈∂Ω,
♦♥❞❡Ω⊂RN✱N ≥3✱ é ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❝♦♠ ❢r♦♥t❡✐r❛ s✉❛✈❡∂Ω✱a, b, csã♦ ❝♦♥st❛♥t❡s
r❡❛✐s ❡p ❡ q sã♦ t❛✐s q✉❡
1< p, q < 2∗−1 = N + 2
N −2.
❖❜t✐✈❡♠♦s ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♥ã♦ tr✐✈✐❛✐s ♣❛r❛ ♦ s✐st❡♠❛ ✭✹✮ ✉t✐❧✐③❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞♦ ▲✐♥❦✐♥❣✱ ❞❡ ❋❡❧♠❡r ❬❋❡❧♠❡r❪✱ ❝♦♠ ❛r❣✉♠❡♥t♦s s✐♠✐❧❛r❡s ❛♦s ❞♦ ❛rt✐❣♦ ❬▼❛ss❛❪✳
■♥tr♦❞✉çã♦ ✹
❖ ❈❛♣ít✉❧♦ ✸ ❢♦✐ ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦ ❞♦s ♣r♦❜❧❡♠❛s ❝♦♠ r❡ss♦♥â♥❝✐❛ ♥♦ ♣r✐♠❡✐r♦ ❛✉✲ t♦✈❛❧♦r✳ ❊♥t❡♥❞❡♠♦s ❝♦♠♦ ♣r♦❜❧❡♠❛s ✏r❡ss♦♥❛♥t❡s✑ ♦s ♣r♦❜❧❡♠❛s ❝✉❥❛ ❡q✉❛çã♦ é ❞♦ t✐♣♦
−∆u=λu+f(x, u)❝♦♠λ ✐❣✉❛❧ ❛ ✉♠ ❛✉t♦✈❛❧♦r ✭❝♦♠ ❛ ❞❡✈✐❞❛ ❝♦♥❞✐çã♦ ♥❛ ❢r♦♥t❡✐r❛✮✳ ❆
r❡ss♦♥â♥❝✐❛ ♥♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ✭λ = λ1✮ é ❡s♣❡❝✐❛❧♠❡♥t❡ ♣❡❝✉❧✐❛r ♣♦rq✉❡ ♦ ❢✉♥❝✐♦♥❛❧
❛ss♦❝✐❛❞♦ ♥ã♦ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ❞❡ P❛❧❛✐s✲❙♠❛❧❡✳
❆ ♠♦t✐✈❛çã♦ ♣❛r❛ ♦ ❡st✉❞♦ ❞❡st❡ ❝❛♣ít✉❧♦ ❢♦✐ ♦ ❛rt✐❣♦ ❬❈✉❡st❛✲❡t ❛❧✳❪✱ ♥♦ q✉❛❧ ♦s ❛✉t♦r❡s tr❛❜❛❧❤❛r❛♠ ❝♦♠ ♦s s❡❣✉✐♥t❡s ♣r♦❜❧❡♠❛s ❞❡ ❉✐r✐❝❤❧❡t r❡ss♦♥❛♥t❡s✿
−∆u=λ1u+ (u+)p+f(x), x∈Ω,
u= 0, x∈∂Ω,
❡
−∆u=λ1u+ (v+)p+f(x), ❡♠ Ω,
−∆v =λ1v+ (u+)q+g(x), ❡♠ Ω,
u=v = 0, ❡♠ ∂Ω.
❙✉♣♦♥❞♦f ∈Lr(Ω)✱ r > N✱1< p < N + 1
N −1 ❡ ✭✺✮
Z
Ω
f(x)ϕ1(x)dx <0,
♦♥❞❡ ϕ1 é ❛ ♣r✐♠❡✐r❛ ❛✉t♦❢✉♥çã♦ ❞♦ ▲❛♣❧❛❝✐❛♥♦ ❝♦♠ ❝♦♥❞✐çõ❡s ❞❡ ❉✐r✐❝❤❧❡t ♥❛ ❢r♦♥t❡✐r❛
❡ ♥♦r♠❛❧✐③❛❞❛ ❡♠ L2(Ω)✱ ♦s ❛✉t♦r❡s ♠♦str❛r❛♠ q✉❡ ❛ ❡q✉❛çã♦ ❡s❝❛❧❛r ♣♦ss✉✐ ❛♦ ♠❡♥♦s
✉♠❛ s♦❧✉çã♦ ❡♠W2,r(Ω)∩H1
0(Ω)✳ P❛r❛ ♦ s✐st❡♠❛✱ ❛❧é♠ ❞❡ f ❡ g s❛t✐s❢❛③❡r❡♠ ❛ ❤✐♣ót❡s❡
(5)✱ ♦s ❡①♣♦❡♥t❡sp ❡ q ❞❡✈❡r✐❛♠ s❛t✐s❢❛③❡r
1
p+ 1 +
N −1
N + 1 1
q+ 1 >
N −1
N + 1 ❡
1
q+ 1 +
N −1
N + 1 1
p+ 1 >
N −1
N + 1.
❊♠ ❛♠❜♦s ♦s ❝❛s♦s✱ ♦s ❛✉t♦r❡s ✉t✐❧✐③❛r❛♠ ❛ t❡♦r✐❛ ❞♦ ❣r❛✉ t♦♣♦❧ó❣✐❝♦ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ í♥❞✐❝❡ ❞❡ ▼♦rs❡✳
❊st✉❞❛♠♦s ♥❡st❡ ❝❛♣ít✉❧♦ ❛ ❡q✉❛çã♦ ❡s❝❛❧❛r ❞❡ ◆❡✉♠❛♥♥✿
✭✻✮
−∆u= (u+)p+f(x), x∈Ω,
∂u
■♥tr♦❞✉çã♦ ✺
❡ ♦ s✐st❡♠❛ ❍❛♠✐❧t♦♥✐❛♥♦✿
✭✼✮
−∆u= (v+)p+f(x), ❡♠ Ω,
−∆v = (u+)q+g(x), ❡♠ Ω,
∂u ∂ν =
∂v
∂ν = 0, ❡♠ ∂Ω.
❆q✉✐ t❛♠❜é♠ s✉♣♦♠♦s q✉❡f ❡g s❛t✐s❢❛③❡♠ ✉♠❛ ❤✐♣ót❡s❡ t❛❧ q✉❛❧ ✭✺✮✳ ◆♦t❡ q✉❡✱ ♥♦ ❝❛s♦
◆❡✉♠❛♥♥✱ϕ1 é ❝♦♥st❛♥t❡ ✭❧♦❣♦✱ t❡♠ s✐♥❛❧ ❞❡✜♥✐❞♦✮ ❡ ♣♦r ✐ss♦ t✐✈❡♠♦s q✉❡ s✉♣♦r q✉❡ f ❡
g tê♠ ✐♥t❡❣r❛✐s ❡str✐t❛♠❡♥t❡ ♥❡❣❛t✐✈❛s ❡♠ Ω✳
❆ss✐♠ ❝♦♠♦ ♥♦ ❛rt✐❣♦ ❬❈✉❡st❛✲❡t ❛❧✳❪✱ ✉t✐❧✐③❛♠♦s ♠ét♦❞♦s t♦♣♦❧ó❣✐❝♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦s r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♦❜t✐❞♦s ♥❡ss❡ ❝❛♣ít✉❧♦✳ ❆ t❡♦r✐❛ ❞♦ ●r❛✉ ❞❡ ▲❡r❛②✲ ❙❝❤❛✉❞❡r ❢♦✐ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❢✉♥❞❛♠❡♥t❛❧ ♥❡st❡ ♣r♦❝❡ss♦✳ ◆♦ ❆♣ê♥❞✐❝❡ ❉ ✐♥❝❧✉í♠♦s ✉♠ r❡s✉♠♦ ❞❛ t❡♦r✐❛ ✉t✐❧✐③❛❞❛ ♣❛r❛ ♣r♦✈❛r ♦s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s✳
P❛r❛ ♦❜t❡r♠♦s ❛s ❡ss❡♥❝✐❛✐s ❡st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐ s♦❜r❡ ❛s s♦❧✉çõ❡s ❞❡ ✭✻✮✱ ❞✐❢❡r❡♥t❡✲ ♠❡♥t❡ ❞❛s ❤✐♣ót❡s❡s ❞♦ ❛rt✐❣♦ ❬❈✉❡st❛✲❡t ❛❧✳❪✱ s✉♣♦♠♦s1< p < N
N −2 ❡f ∈L
r(Ω)✱ ♣❛r❛
❛❧❣✉♠r > N/2✳ P❛r❛ ♦ s✐st❡♠❛ ✭✼✮ s✉♣♦♠♦s
1< p, q < N N −2
❡ f, g ∈ Lr(Ω)✱ ♣❛r❛ ❛❧❣✉♠ r > N/2✳ ❊ss❛ ♣❛rt❡ ❞♦ tr❛❜❛❧❤♦ ❢♦✐ ✐♥s♣✐r❛❞❛ ♣❡❧♦ ❛rt✐❣♦
[❑❛♥♥❛♥✲❖rt❡❣❛] ♥♦ q✉❛❧ ♦s ❛✉t♦r❡s ❡st✉❞❛♠ ✉♠❛ ❡q✉❛çã♦ ❞♦ t✐♣♦
−∆u=g(u) +f(x), x∈Ω, ∂u
∂ν = 0, x∈∂Ω,
♦♥❞❡g :R→R s❛t✐s❢❛③✿
✭G1✮ g é ❧♦❝❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③✐❛♥❛❀
✭G2✮ lim
s→−∞g(s) =∞❀
✭G3✮ |g(s)| ≤M ♣❛r❛ s≥0❀
✭G4✮ ❡①✐st❡♠ ❝♦♥st❛♥t❡s α, β ❡ ✉♠ ❡①♣♦❡♥t❡ p t❛✐s q✉❡
■♥tr♦❞✉çã♦ ✻
♣❛r❛ t♦❞♦s∈R ♦♥❞❡
1≤p < N
N −2, N ≥3. ❖❜s❡r✈❡ q✉❡ g(s) = (s+)p ♥ã♦ s❛t✐s❢❛③ ❛s ❤✐♣ót❡s❡s (G
2)❡ (G3)✳
❈❆P❮❚❯▲❖ ✶
❯▼ P❘❖❇▲❊▼❆ ❉❊ ◆❊❯▼❆◆◆
❈❖▼ ❊❳P❖❊◆❚❊ ❈❘❮❚■❈❖
◆❡ss❡ ❝❛♣ít✉❧♦ ❜✉s❝❛♠♦s ✉♠❛ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛
✭✶✳✶✮
−∆u=λu+g(x, u) + (u+)2∗−1
, x∈Ω, ∂u
∂ν = 0, x∈∂Ω,
♦♥❞❡Ω⊂RN✱N ≥3✱ é ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❝♦♠ ❢r♦♥t❡✐r❛ s✉❛✈❡∂Ω✱λ >0✱2∗ = 2N
N −2 é ♦ ❡①♣♦❡♥t❡ ❝rít✐❝♦ ♣❛r❛ ❛ ✐♠❡rsã♦ ❞❡ ❙♦❜♦❧❡✈
H1(Ω)֒→Lq(Ω), u+= max(u,0) ❡u− = max(−u,0)✳ ❆ss✐♠✱ u=u+−u−.
❆ ❢✉♥çã♦ g : Ω×R→R é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ s❛t✐s❢❛③❡♥❞♦✿
✭g1✮ g(x, s) = 0 s❡s≤0 ❡ g(x, s)>0 s❡s >0❀
✭g2✮ ❡①✐st❡♠σ ∈(1,2∗−1) ❡ ✉♠❛ ❝♦♥st❛♥t❡ K >0t❛✐s q✉❡✿ |g(x, s)| ≤K|s|σ, ∀x∈Ω, ∀s ∈R;
❙❊➬➹❖ ✶✳✶ ∗ ❘❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ✽
✭g3✮ ∂g
∂s(x,0) = 0❀
✭g4✮ ❡①✐st❡ θ∈
0,1
2
t❛❧ q✉❡
0< G(x, s)≤θsg(x, s), ∀x∈Ω, ∀s >0,
♦♥❞❡ G(x, s) =
Z s
0
g(x, t)dt é ✉♠❛ ♣r✐♠✐t✐✈❛ ❞❡ g✳
✶✳✶ ❘❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧
❙✉♣♦♥❤❛u✉♠❛ s♦❧✉çã♦ ❞❡ ✭✶✳✶✮ ❝♦♠ u+(x) = 0 ♣❛r❛ q✉❛s❡ t♦❞♦x∈Ω✳ ❊♥tã♦✱ ❞❛❞❛s
❛s ❤✐♣ót❡s❡s s♦❜r❡ ❛ ❢✉♥çã♦g✱ ♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮ s❡ r❡s✉♠❡ ❛ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❛✉t♦✈❛❧♦r ❞♦
t✐♣♦✿
−∆u=λu, ❡♠ Ω, u <0, ❡♠ Ω, ∂u
∂ν = 0, ❡♠ ∂Ω,
♦ q✉❛❧ só t❡rá s♦❧✉çã♦ s❡ λ = λ1 = 0✱ ♣♦✐s ❛ ú♥✐❝❛ ❛✉t♦❢✉♥çã♦ ❝♦♠ s✐♥❛❧ ❞❡✜♥✐❞♦ é ❛
♣r✐♠❡✐r❛ ✭♥❡ss❡ ❝❛s♦✱ q✉❛❧q✉❡r ❢✉♥çã♦ ❝♦♥st❛♥t❡ ♥❡❣❛t✐✈❛ s❡r✐❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛✮✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ λ= 0 ❡♥tã♦ ✉t✐❧✐③❛♥❞♦ ❛ ♣r✐♠❡✐r❛ ❛✉t♦❢✉♥çã♦ ϕ1 = 1 ❝♦♠♦ ❢✉♥çã♦✲
t❡st❡ ❡ ✐♥t❡❣r❛♥❞♦ ✭✶✳✶✮✱ ♦❜t❡♠♦s✿
Z
Ω
g(x, u)dx=−
Z
Ω
(u+)2∗−1dx≤0.
❈♦♠♦g é ❝♦♥tí♥✉❛ ❡ s❛t✐s❢❛③ (g1)✱ ❞❡✈❡♠♦s t❡r✿
Z
Ω
g(x, u)dx= 0
♦ q✉❡ ♦❜r✐❣❛ u+ = 0 q✳t✳♣✳ ❡♠ Ω✱ ❡ ❞❛í✱ ♣❡❧♦ ✈✐st♦ ❛❝✐♠❛✱ ❛s s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮
sã♦ ❝♦♥st❛♥t❡s ♥❡❣❛t✐✈❛s✳ P♦rt❛♥t♦✱ s❡λ= 0 ❛s ú♥✐❝❛s s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮ sã♦ ❛s ❛✉t♦❢✉♥çõ❡s ♥❡❣❛t✐✈❛s ❞♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ✭❝♦♥st❛♥t❡s<0✮✳
◗✉❛♥❞♦ λ <0✱ s❛❜❡✲s❡ q✉❡
kuk⋆ =
Z
Ω
[|∇u|2−λu2]dx
❙❊➬➹❖ ✶✳✷ ∗ Pr❡❧✐♠✐♥❛r❡s ❡ ♥♦t❛çõ❡s ✾
❞❡✜♥❡ ✉♠❛ ♥♦r♠❛ ❡q✉✐✈❛❧❡♥t❡ à ♥♦r♠❛ ♣❛❞rã♦ ❡♠H1(Ω)✳ ❈♦♠ ❡ss❡ ❢❛t♦✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠
❛ ❤✐♣ót❡s❡ (g2)✱ é ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮ ♣♦❞❡ s❡r
s❡♣❛r❛❞♦ ❡♠ ✉♠❛ ♣❛rt❡ ❛ss♦❝✐❛❞❛ ❛ ❡ss❛ ♥♦r♠❛ ❝✐t❛❞❛ ❡ ♦✉tr❛ ♣❛rt❡ q✉❡ é s✉♣❡rq✉❛❞rát✐❝❛✱ s❛t✐s❢❛③❡♥❞♦ ❛s ❤✐♣ót❡s❡s ❞♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✳
❊st❡ ❝❛♣ít✉❧♦ s❡ ♣r♦♣õ❡ ❛ ♠♦str❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ❞❡ ✭✶✳✶✮ ♣❛r❛
λ >0 q✉❛❧q✉❡r✳ ▼♦str❛r❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
❚❡♦r❡♠❛ ✶✳✶✳ ❙✉♣♦♥❤❛λ >0❡ ❛s ❤✐♣ót❡s❡s (g1)−(g4)✳ ❊♥tã♦ ♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮ ❛❞♠✐t❡ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧✳
✶✳✷ Pr❡❧✐♠✐♥❛r❡s ❡ ♥♦t❛çõ❡s
❊st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ s♦❧✉çõ❡su∈H1(Ω)✱ q✉❡ é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ♠✉♥✐❞♦ ❝♦♠
❛ ♥♦r♠❛✶
kuk= |u|22+|∇u|2212
,
♦♥❞❡| · |p ❞❡♥♦t❛rá s❡♠♣r❡ ❛ ♥♦r♠❛ ♥♦ ❡s♣❛ç♦ Lp(Ω)✱ p≥1✳
❙❡❥❛λ1♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ❞♦ ♦♣❡r❛❞♦r−∆❝♦♠ ❝♦♥❞✐çõ❡s ❞❡ ◆❡✉♠❛♥♥ ♥❛ ❢r♦♥t❡✐r❛✿
−∆φ=λφ, x∈Ω, ∂φ
∂ν = 0, x∈∂Ω.
❙❛❜❡♠♦s q✉❡λ1 = 0 ❡ ❛s ❛✉t♦❢✉♥çõ❡s ❝♦rr❡s♣♦♥❞❡♥t❡s sã♦ ❛s ❢✉♥çõ❡s ❝♦♥st❛♥t❡s✳
❉❡♥♦t❡ ♣♦r λ1 = 0 < λ2 ≤ λ3 ≤... ♦s ❛✉t♦✈❛❧♦r❡s ❞❡ −∆ ❝♦♠ ❝♦♥❞✐çã♦ ❞❡ ◆❡✉♠❛♥♥
♥❛ ❢r♦♥t❡✐r❛ ❡ ϕ1, ϕ2, ϕ3, ... ❛s ❛✉t♦❢✉♥çõ❡s ❛ss♦❝✐❛❞❛s ❝♦♠ ♥♦r♠❛1 ❡♠ H1(Ω)✳
❖ ❢✉♥❝✐♦♥❛❧ J :H1(Ω) →R✱ ❞❡ ❝❧❛ss❡ C1✱ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ (1.1)é ❞❛❞♦ ♣♦r✿
J(u) = 1 2
Z
Ω|∇
u|2dx− λ
2
Z
Ω
u2dx−
Z
Ω
G(x, u)dx− 1
2∗ Z
Ω
(u+)2∗dx.
❆ ❞❡r✐✈❛❞❛ ❞❡ J é ❞❛❞❛ ♣♦r
hJ′(u), ϕi=
Z
Ω
∇u∇ϕdx−λ
Z
Ω
uϕdx−
Z
Ω
g(x, u)ϕdx−
Z
Ω
(u+)2∗−1ϕdx,
✶❊♠ ✈ár✐♦s ♠♦♠❡♥t♦s ♥♦ t❡①t♦✱ ♣❛r❛ s✐♠♣❧✐✜❝❛r ❛s ♥♦t❛çõ❡s✱ ✉t✐❧✐③❛r❡♠♦s s✐♠♣❧❡s♠❡♥t❡ H1 ♣❛r❛ ♥♦s
❙❊➬➹❖ ✶✳✸ ∗ ❈♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ✶✵
♣❛r❛ t♦❞❛ϕ ∈H1(Ω). ❖s ♣♦♥t♦s ❝rít✐❝♦s ❞❡ J sã♦ s♦❧✉çõ❡s ❢r❛❝❛s ❞❡ (1.1)✳
▲❡♠❜r❡ q✉❡
S = inf
u∈D1,2(RN)\{0} R
RN |∇u|2dx R
RN|u|2
∗
dx22∗
,
♦♥❞❡
D1,2(RN) ={u : ∇u∈L2(RN), u∈L2∗(RN)}.
❆ ❝♦♥st❛♥t❡S é ❛t✐♥❣✐❞❛ ♣❡❧❛ ❢✉♥çã♦
✭✶✳✷✮ U(x) = cN
(N(N −2) +|x|2)N−22
,
♦♥❞❡ cN > 0 é ✉♠❛ ❝♦♥st❛♥t❡ q✉❡ ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞❡ N✳ ❆ ❢✉♥çã♦ ❡♠ ✭✶✳✷✮ s❛t✐s❢❛③ ❛
❡q✉❛çã♦
−∆U =U2∗−1 ❡♠ RN.
❆❞❡♠❛✐s✱
Z
RN|∇
U|2dx=
Z
RN
U2∗dx=SN2. ❯s❛r❡♠♦s ❛❞✐❛♥t❡ ❛ s❡❣✉✐♥t❡ ♥♦t❛çã♦✿
✭✶✳✸✮ u⋆ǫ(x) = ǫ−N−22U
x
ǫ
, ǫ >0.
✶✳✸ ❈♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡
▲❡♠❛ ✶✳✷✳ ✭❇ré③✐s✲▲✐❡❜✮ ❙❡❥❛♠ Ω ✉♠ ❛❜❡rt♦ ❞❡ RN ❡ {un}n∈N ⊂ Lp(Ω), 1≤ p < ∞✳
❙❡
✭❛✮{un} é ❧✐♠✐t❛❞❛ ❡♠ Lp(Ω)
✭❜✮ ❡ un→u q✳t✳♣✳ ❡♠Ω✱ q✉❛♥❞♦ n → ∞
❡♥tã♦✿
lim
n→∞(|un|
p
p − |un−u|pp) =|u|pp.
❉❡♠♦♥str❛çã♦✳ ❱❡r ❬❲✐❧❧❡♠❪✱ ♣✳ ✷✶✳
Pr♦♣♦s✐çã♦ ✶✳✸✳ ❉❛❞♦ s0 >0 ❡ θ ❝♦♠♦ ♥❛ ❤✐♣ót❡s❡ (g4)✱ ❡①✐st❡ K =K(s0)>0 t❛❧ q✉❡
✭✶✳✹✮ g(x, s)≥Ks1θ−1, ∀s≥s
❙❊➬➹❖ ✶✳✸ ∗ ❈♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ✶✶
❉❡♠♦♥str❛çã♦✳ ❈♦♠ ❡❢❡✐t♦✱ ❞❛ ❤✐♣ót❡s❡(g4)s❡❣✉❡ q✉❡✿ 1
θs ≤
g(x, s)
G(x, s), ∀s≥s0. ■♥t❡❣r❛♥❞♦ ❞❡ s0 ❛té s✱ ♦❜t❡♠♦s✿
1
θ(lns−lns0) ≤
Z s
s0
g(x, t)
G(x, t)dt
= ln(G(x, s))−ln(G(x, s0)),
❧♦❣♦✱ ♣❛r❛ t♦❞♦s≥s0✱
G(x, s)≥G(x, s0)s
−1
θ
0 s
1
θ ⇒ sθg(x, s)≥G(x, s
0)s
−1
θ
0 s
1
θ
⇒ g(x, s)≥ 1
θG(x, s0)s
−1θ
0 s
1
θ−1 =Ks
1
θ−1,
❞♦♥❞❡ s❡❣✉❡ ✭✶✳✹✮✳ ❈♦r♦❧ár✐♦ ✶✳✹✳ ❚❡♠✲s❡✿
✭✶✳✺✮ lim
s→+∞
g(x, s)
s = +∞.
❉❡♠♦♥str❛çã♦✳ ❇❛st❛ ♥♦t❛r q✉❡ ❞❛❞♦ s0 >0 ❛r❜✐trár✐♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✸ ✈❛❧❡ ❛ ❞❡s✐✲
❣✉❛❧❞❛❞❡ ✭✶✳✹✮ ❝♦♠ θ <1/2✳ ▲♦❣♦✱ ♣❛r❛ t♦❞♦ s ≥s0 t❡♠♦s✿
g(x, s)
s ≥Ks
1
θ−2 →+∞, s❡ s→+∞,
✈✐st♦ q✉❡ (1/θ)−2>0.
▲❡♠❛ ✶✳✺✳ ❙✉♣♦♥❤❛ λ >0✱ (g1)−(g4) ❡ s❡❥❛ {un} ⊂ H1(Ω) ✉♠❛ s❡q✉ê♥❝✐❛ (P S) ♣❛r❛
J✱ ✐st♦ é✱ ✉♠❛ s❡q✉ê♥❝✐❛ t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ n✿
|J(un)|=
1 2 Z Ω|∇
un|2dx−
λ
2
Z
Ω
u2ndx−
Z
Ω
G(x, un)dx−
1 2∗
Z
Ω
(u+n)2∗dx
≤ c, ✭✶✳✻✮
| hJ′(un), ϕi |=
Z Ω∇
un∇ϕdx−λ
Z
Ω
unϕdx−
Z
Ω
g(x, un)ϕdx−
Z
Ω
(u+n)2∗−1ϕdx
≤
ǫnkϕk,
✭✶✳✼✮
♣❛r❛ t♦❞❛ϕ ∈H1(Ω)✱ ♦♥❞❡ c >0 é ❝♦♥st❛♥t❡ ❡ lim
❙❊➬➹❖ ✶✳✸ ∗ ❈♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ✶✷
❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡{un}t❡♠ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ✐❧✐♠✐t❛❞❛ ✭❛ q✉❛❧
❝♦♥t✐♥✉❛r❡♠♦s ❛ ❝❤❛♠❛r ❞❡ {un}✮✱ ♦✉ s❡❥❛✱
lim
n→∞kunk= +∞.
❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡kunk ≥1, ∀n. ❉❡✜♥❛
zn =
un
kunk
.
❖❜✈✐❛♠❡♥t❡✱ kznk = 1, ∀n✳ ❊♥tã♦✱ ✉t✐❧✐③❛♥❞♦ ♦s t❡♦r❡♠❛s ❈✳✻ ❡ ❈✳✼ ✭✈❡r ❆♣ê♥❞✐❝❡ ❈✮✱
❡①✐st❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ✭q✉❡ ❝♦♥t✐♥✉❛r❡♠♦s ❛ ❝❤❛♠❛r {zn}✮ t❛❧ q✉❡✿
zn ⇀ z0, ❡♠ H1 :=H1(Ω),
✭✶✳✽✮
zn →z0, ❡♠ L2(Ω),
✭✶✳✾✮
zn(x)→z0(x), q✳t✳♣✳ ❡♠ Ω,
✭✶✳✶✵✮
|zn(x)| ≤q(x)q✳t✳♣✳ ❡♠ Ω,
✭✶✳✶✶✮
♦♥❞❡z0 ∈H1 ❡q ∈L2(Ω).
❉❡✜♥❛✿
f(x, s) =
λs+g(x, s) +s2∗−1
, s❡ s >0
λs, s❡ s≤0 ❉✐✈✐❞✐♥❞♦ (1.7) ♣♦rkunk✱ ♦❜t❡♠♦s✿
Z Ω∇
zn∇ϕdx−λ
Z
Ω
znϕdx−
Z
Ω
g(x, un)
kunk
ϕdx−
Z
Ω
(u+
n)2
∗−1
kunk
ϕdx
≤
ǫn k
ϕk kunk
,
♣❛r❛ t♦❞❛ϕ ∈H1. ❖✉ s✐♠♣❧❡s♠❡♥t❡✿
Z Ω∇
zn∇ϕdx−
Z
Ω
f(x, un)
kunk
ϕdx
≤
ǫn k
ϕk
kunk
, ∀ϕ∈H1.
P❛ss❛♥❞♦ ♦ ❧✐♠✐t❡ q✉❛♥❞♦n→ ∞✱ ❞❡❞✉③✐♠♦s ❞❡ (1.8)q✉❡✿
✭✶✳✶✷✮ ∃ lim
n→∞ Z
Ω
f(x, un)
kunk
ϕdx=
Z
Ω∇
z0∇ϕdx,
♣❛r❛ t♦❞❛ϕ ∈H1.
▼♦str❛r❡♠♦s q✉❡ z0 = 0 ❡ ❞❡♣♦✐s ❝❤❡❣❛r❡♠♦s ❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❋❛r❡♠♦s ✐ss♦ ❡♠
❙❊➬➹❖ ✶✳✸ ∗ ❈♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ✶✸
P❛ss♦ ✶✳ z0(x)≤0✱ ♣❛r❛ q✉❛s❡ t♦❞♦ x∈Ω
P❛ss♦ ✷✳
Z
Ω
z0(x)dx = 0.
❖s ♣❛ss♦s ✶ ❡ ✷ ✐♠♣❧✐❝❛♠ q✉❡ z0 ≡0.
P❛ss♦ ✶✳ z0(x)≤0✱ ♣❛r❛ q✉❛s❡ t♦❞♦ x∈Ω
❊s❝♦❧❤❛ ϕ=z0+ ❡♠ (1.12)✳ ❉❛í✿
✭✶✳✶✸✮ lim
n→∞ Z
Ω+
f(x, un)
kunk
z0(x)dx=
Z
Ω+|∇
z0|2dx <∞,
❝♦♠ Ω+ ={x : z
0(x)>0}✳
▼❛s✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ✉s❛♥❞♦ ♦ ❢❛t♦ q✉❡
lim
n→∞un(x) = +∞, q✳t✳♣✳ ❡♠Ω
+,
❡ ♦ ❈♦r♦❧ár✐♦ ✶✳✹✱ ♦❜t❡♠♦s✿
lim
n→∞
f(x, un)
kunk
z0(x)dx= lim
n→∞
λun+g(x, un) + (u+n)2
∗−1
kunk
z0(x)
un
un
= lim
n→∞
λun+g(x, un) + (u+n)2
∗−1
un
zn(x)z0(x)
= lim
n→∞
λ+g(x, un)
un
+ (u+n)2∗−2
zn(x)z0(x) = +∞,
♣❛r❛ q✉❛s❡ t♦❞♦x∈Ω+✳ P♦rt❛♥t♦✱ s❡|Ω+|>0✱ ♦❜t❡♠♦s✱ ♣❡❧♦ ▲❡♠❛ ❞❡ ❋❛t♦✉✱ q✉❡✿
+∞=
Z
Ω+ lim
n→∞
f(x, un)
kunk
z0dx=
Z
Ω+
lim inf
n→∞
f(x, un)
kunk
z0dx
≤lim inf
n→∞ Z
Ω+
f(x, un)
kunk
z0dx
= lim
n→∞ Z
Ω+
f(x, un)
kunk
z0dx,
❧♦❣♦✱
lim
n→∞ Z
Ω+
f(x, un)
kunk
z0(x)dx= +∞,
♦ q✉❡ ❝♦♥tr❛❞✐③(1.13)✳ P♦rt❛♥t♦✱ |Ω+|= 0 ❡z
0(x)≤0✱ ♣❛r❛ q✉❛s❡ t♦❞♦ x∈Ω✳
P❛ss♦ ✷✳ Z
Ω
❙❊➬➹❖ ✶✳✸ ∗ ❈♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ✶✹
▲❡♠❜r❡ q✉❡ ❛❣♦r❛ ♥♦ss♦ ♣r♦❜❧❡♠❛ ❡stá ♣♦st♦ ❛ss✐♠✿
−∆u=f(x, u) = λu+g(x, u+) + (u+)2∗−1
, x∈Ω, ∂u
∂ν = 0, x∈∂Ω,
❚♦♠❡ ϕ=un ❡♠ (1.7)✱ ♠✉❧t✐♣❧✐q✉❡ ♣♦r 12 ❡ s✉❜tr❛✐❛ ❞❡ (1.6) ♣❛r❛ ♦❜t❡r✿
Z Ω
f(x, un)un
2 −F(x, un)
dx ≤
C+ ǫnkunk
2 , ∀n∈N,
♦♥❞❡F(x, u) =Ru
0 f(x, s)ds.
❉✐✈✐❞✐♥❞♦ ♣♦r kunk✱ t❡♠♦s✿
Z Ω
f(x,un)un
2 −F(x, un)
kunk
dx ≤ C
kunk
+ǫn
2, ∀n ∈N, ❧♦❣♦✱
✭✶✳✶✹✮ lim
n→∞ Z
Ω
f(x,un)un
2 −F(x, un)
kunk
dx= 0.
P♦r ♦✉tr♦ ❧❛❞♦✱ ✜①❛♥❞♦s∗ >0 ❡ ✉s❛♥❞♦ ❛ ❝♦♥t✐♥✉✐t❛❞❡ ❞❡f✱ ❡①✐st❡ K
s∗✱ t❛❧ q✉❡✿
f(x, u)u
2 −F(x, u)
≤
Ks∗,∀u∈(−∞, s∗].
✭◆♦t❡ q✉❡ ♣❛r❛u≤0 ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ é ♥✉❧♦✦✮ ❉❛í✱ Z
un(x)≤s∗
f(x,un)un
2 −F(x, un)
kunk
dx
≤
Ks∗|Ω|
kunk
♣❛r❛ t♦❞♦n∈N✳ ❚♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ q✉❛♥❞♦ n→ ∞✱ t❡♠♦s✿
0≤lim sup
n→∞ Z
un(x)≤s∗
f(x,un)un
2 −F(x, un)
kunk
dx ≤ 0. ▲♦❣♦✱ ✭✶✳✶✺✮ lim n→∞ Z
un(x)≤s∗
f(x,un)un
2 −F(x, un)
kunk
dx
= 0.
❆✐♥❞❛ ❝♦♠ s∗ >0 ✜①❛❞♦ ❡ ❝❤❛♠❛♥❞♦
I =
Z
un(x)>s∗
f(x,un)un
2 −F(x, un)
kunk
❙❊➬➹❖ ✶✳✸ ∗ ❈♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ✶✺
t❡♠♦s ♣♦r(g4)❛s s❡❣✉✐♥t❡s ❞❡s✐❣✉❛❧❞❛❞❡s✿
I ≥
1 2−θ
Z
un(x)>s∗
f(x, un)un
kunk
dx≥
1 2−θ
s∗
Z
un(x)>s∗
f(x, un)
kunk
dx
=
1 2 −θ
s∗
Z
un(x)>s∗
[f(x, un)−λun+λun]
kunk
dx
=
1 2 −θ
s∗
Z
Ω
f(x, un)
kunk
dx−
1 2 −θ
s∗
Z
un(x)≤s∗
[f(x, un)−λun+λun]
kunk
dx
=
1 2 −θ
s∗
Z
Ω
f(x, un)
kunk
dx−
1 2 −θ
s∗
Z
un(x)≤s∗
[f(x, un)−λun]
kunk
dx
−
1 2−θ
s∗λ
Z
un(x)≤s∗
zn(x)dx
≥
1 2−θ
s∗
Z
Ω
f(x, un)
kunk
dx−
1 2 −θ
s∗ K2
kunk
−
1 2−θ
s∗λ
Z
un(x)≤s∗
zn(x)dx,
♦♥❞❡ K2 > 0 é ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ ✭q✉❡ ❞❡♣❡♥❞❡ ❞❛ ❧✐♠✐t❛çã♦ ❞❡ |f(x, u)−λu| ❡♠
(−∞, s∗])✳ ❙❡ ❝♦♥s✐❞❡r❛r♠♦s ✭✶✳✶✷✮✱ ❝♦♠ ϕ ≡ 1✱ ✭✶✳✶✹✮ ❡ (1.15) ❡ t♦♠❛r♠♦s ♦ ❧✐♠✐t❡
q✉❛♥❞♦n t❡♥❞❡ ❛ ✐♥✜♥✐t♦ ♥❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ ♦❜t❡♠♦s✿
0≥ −
1 2 −θ
s∗λ
Z
Ω
z0dx≥0
❝♦♠ s∗ >0✜①❛❞♦✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ Z
Ω
z0(x)dx= 0,
❡ ✐ss♦ ❝♦♥❝❧✉✐ ♦ ♣❛ss♦ ✷✳
❉♦s ♣❛ss♦s ✶ ❡ ✷✱ ❝♦♥❝❧✉í♠♦s q✉❡z0 ≡0✳ ▼♦str❡♠♦s q✉❡ ✐ss♦ ♥♦s ❧❡✈❛ ❛ ✉♠❛ ❝♦♥tr❛✲
❞✐çã♦✳
❆✜r♠❛çã♦ ✶✳✻✳ ❚❡♠✲s❡✿
lim
n→∞ Z
Ω
f(x, un)
kunk
zndx= 0.
❈♦♠ ❡❢❡✐t♦✱ ❢❛③❡♥❞♦ ϕ = un ❡♠ ✭✶✳✼✮✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r 1/2 ❡ s✉❜tr❛✐♥❞♦ ❞❡ ✭✶✳✻✮✱
♦❜t❡♠♦s✿ Z Ω
F(x, un)dx−
1 2
Z
Ω
f(x, un)undx
≤
❙❊➬➹❖ ✶✳✸ ∗ ❈♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ✶✻
P♦r (g4)✱ ♦❜t❡♠♦s✿
1 2−θ
Z
Ω
f(x, un)undx=
1 2
Z
Ω
f(x, un)undx−θ
Z
Ω
f(x, un)undx
≤ 1
2
Z
Ω
unf(x, un)dx−
Z
Ω
F(x, un)dx
≤c+ǫnkunk.
❉❛í✱
0≤lim sup
n→+∞ Z
Ω
f(x, un)
kunk
zndx≤lim sup n→+∞
C
1
kunk2
+ ǫn
kunk
= 0,
❞♦♥❞❡ s❡❣✉❡ ❛ ❛✜r♠❛çã♦✳
P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ✭✶✳✼✮✱ ❝♦♠ ϕ = zn ❡ ❞✐✈✐❞✐♥❞♦ ♣♦r kunk✱ ♦❜t❡♠♦s ❞❡
kznk= 1,♣❛r❛ t♦❞♦ n∈N✱ q✉❡✿
Z Ω|∇
zn|2dx−
Z
Ω
f(x, un)
kunk
zndx
= 1− Z Ω
zn2dx−
Z
Ω
f(x, un)
kunk
zndx
≤
ǫn
kunk
.
❚♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ q✉❛♥❞♦n → ∞✱ t❡♠♦s✿
lim
n→∞|zn|
2 2 = 1,
♦ q✉❡ ❝♦♥tr❛❞✐③ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❡♠(1.9)❝♦♠ z0 ≡0✳
P♦rt❛♥t♦✱ ❛ s❡q✉ê♥❝✐❛{un} é ❧✐♠✐t❛❞❛ ❡♠ H1(Ω)✳
▲❡♠❛ ✶✳✼✳ ❙✉♣♦♥❤❛λ >0 ❡ (g1)−(g4)✳ ❙❡❥❛ {un} ⊂H1(Ω) t❛❧ q✉❡✿
J(un) =
1 2
Z
Ω|∇
un|2dx−
λ
2
Z
Ω
u2ndx−
Z
Ω
G(x, un)dx−
1 2∗
Z
Ω
(u+n)2
∗
dx→c,
✭✶✳✶✻✮
| hJ′(un), ϕi |=
Z Ω∇
un∇ϕdx−λ
Z
Ω
unϕdx−
Z
Ω
g(x, un)ϕdx−
Z
Ω
(u+n)2∗−1ϕdx
≤
ǫnkϕk,
✭✶✳✶✼✮
♣❛r❛ t♦❞❛ϕ ∈H1(Ω)✱ ♦♥❞❡ ǫ
n →0 q✉❛♥❞♦ n→+∞. ❙❡
✭✶✳✶✽✮ c < S
N
2
N
❙❊➬➹❖ ✶✳✸ ∗ ❈♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ✶✼
❉❡♠♦♥str❛çã♦✳ ❙❛❜❡♠♦s✱ ♣❡❧♦ ▲❡♠❛ ✶✳✺✱ q✉❡ {un} é ❧✐♠✐t❛❞❛ ❡♠ H1(Ω)✳ ❉❛í✱ ♣❛ss❛♥❞♦
♣❛r❛ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱ s❡ ♥❡❝❡ssár✐♦✱ t❡♠♦s✿
un⇀ u, ❡♠ H1(Ω),
✭✶✳✶✾✮
un→u, ❡♠ L2(Ω),
✭✶✳✷✵✮
un(x)→u(x), q✳t✳♣✳ ❡♠ Ω.
✭✶✳✷✶✮
❈♦♠♦{u+
n}é ❧✐♠✐t❛❞❛ ❡♠H1(Ω)✱ s❡❣✉❡ q✉❡{u+n}é ❧✐♠✐t❛❞❛ ❡♠L2
∗
(Ω)✳ ❖❜s❡r✈❡ q✉❡✱ ❢❛③❡♥❞♦q = 2N
N + 2✱ ♦❜t❡♠♦s✿
|(u+n)2∗−1|qq =
Z
Ω
[(u+n)NN+2−2] 2N N+2dx=
Z
Ω
(u+n)N2N−2dx =|u+
n|2
∗
2∗,
❧♦❣♦✱ {(u+
n)2
∗−1
} é ❧✐♠✐t❛❞❛ ❡♠ LN2N+2(Ω)✳ P❛ss❛♥❞♦ ♥♦✈❛♠❡♥t❡ ❛ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱ s❡ ♥❡❝❡ssár✐♦✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡✿
(u+n)2∗−1 ⇀(u+)2∗−1, ❡♠ LN2N+2(Ω). ❯s❛♥❞♦ ❛ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ❡ ✭✶✳✷✵✮✱ t❡♠♦s✿
Z Ω
(unϕ−uϕ)dx
= Z Ω
(un−u)ϕdx
≤ |
un−u|2|ϕ|2 →0, ∀ϕ ∈H1,
q✉❛♥❞♦n →+∞✳ ▲♦❣♦✱ lim
n→+∞ Z
Ω
unϕdx=
Z
Ω
uϕdx, ∀ϕ ∈H1.
❙❡❣✉❡✱ ♣♦r ✭✶✳✶✾✮ ❡ ♦ ❚❡♦r❡♠❛ ❞❛s ■♠❡rsõ❡s ❞❡ ❙♦❜♦❧❡✈✱ q✉❡ ❡①✐st❡ q˜ ∈ Lσ(Ω)✱ 1 <
σ <2∗−1✱ t❛❧ q✉❡ |u
n(x)| ≤q˜(x)q✳t✳♣✳ ❡♠Ω✳ P❡❧❛ ❤✐♣ót❡s❡(g2)✱ t❡♠♦s ♣❛r❛ q✳t✳ x∈Ω✿
|g(x, un)| ≤K|un|σ ≤Kq˜(x)σ ∈L1(Ω).
▲♦❣♦✱ ❝♦♠♦ un(x) → u(x) q✳t✳♣✳ ❡♠ Ω✱ ♣♦❞❡♠♦s ✉s❛r ♦s ❢❛t♦s ❛❝✐♠❛ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦
❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ❉♦♠✐♥❛❞❛ ♣❛r❛ ❝♦♥❝❧✉✐r q✉❡
hJ′(un), ϕi=
Z
Ω∇
un∇ϕdx−λ
Z
Ω
unϕdx−
Z
Ω
g(x, un)ϕdx
Z
Ω
(u+n)2∗−1ϕdx
→
Z
Ω∇
u∇ϕdx−λ
Z
Ω
uϕdx−
Z
Ω
g(x, u)ϕdx−
Z
Ω
(u+)2∗−1ϕdx
❙❊➬➹❖ ✶✳✸ ∗ ❈♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ✶✽
♣❛r❛ t♦❞❛ϕ ∈H1(Ω)✳ P♦rt❛♥t♦✱ u r❡s♦❧✈❡ ❢r❛❝❛♠❡♥t❡ ❛ ❡q✉❛çã♦
−∆u=λu+g(x, u) + (u+)2∗−1.
❆❞❡♠❛✐s✱ ❡s❝♦❧❤❡♥❞♦ ϕ=u ❡ s✉❜st✐t✉✐♥❞♦ ♥❛ ❢ór♠✉❧❛ ❞❛ ❞❡r✐✈❛❞❛ ❞❡J ♦❜t❡♠♦s✿
0 =hJ′(u), ui=
Z
Ω|∇
u|2dx−λ
Z
Ω
u2dx−
Z
Ω
g(x, u)udx−
Z
Ω
(u+)2∗dx,
❡ ♣♦rt❛♥t♦✱
Z
Ω|∇
u|2dx=λ
Z
Ω
u2dx+
Z
Ω
g(x, u)udx+
Z
Ω
(u+)2∗dx.
❙✉❜st✐t✉✐♥❞♦ ♥❛ ❡①♣r❡ssã♦ ❞❡ J(u)✱ ♦❜t❡♠♦s✱ ✉s❛♥❞♦(g4)✿
J(u) = 1 2
Z
Ω|∇
u|2dx− λ
2
Z
Ω
u2dx−
Z
Ω
G(x, u)dx− 1
2∗ Z
Ω
(u+)2∗dx
= 1 2
Z
Ω
g(x, u)udx+1 2
Z
Ω
(u+)2∗dx−
Z
Ω
G(x, u)dx− 1
2∗ Z
Ω
(u+)2∗dx
= 1 2 − 1 2∗
|u+|2∗
2∗+
1 2
Z
Ω
g(x, u)udx−
Z
Ω
G(x, u)dx
= 1
N|u
+|2∗
2∗ +
Z
Ω
1
2g(x, u)udx−G(x, u)
dx
≥ N1|u+|22∗∗+
Z
Ω
[θg(x, u)u−G(x, u)]dx≥0,
✐st♦ é✱
✭✶✳✷✷✮ J(u)≥0.
❊s❝r❡✈❛ vn=un−u✳ P❡❧♦ ▲❡♠❛ ✶✳✷✱ t❡♠♦s✿
1 2∗
Z
Ω
(u+n)2∗dx= 1 2∗
Z
Ω
(u+)2∗dx+ 1 2∗
Z
Ω
(vn+)2∗dx+o(1).
❯s❛♥❞♦ ♦ ❢❛t♦ q✉❡un(x)→u(x)q✳t✳♣✳ ❡♠Ω❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛
❉♦♠✐♥❛❞❛ ❡ ♦ r❡s✉❧t❛❞♦ ❛❝✐♠❛ ✭✈✐❛ ▲❡♠❛ ✶✳✷✮✱ ♦❜t❡♠♦s✿ lim
n→∞[J(u) +J(vn)] = limn→∞
1 2
Z
Ω|∇
u|2dx−λ
2
Z
Ω
u2dx−
−
Z
Ω
G(x, u)dx− 1
2∗ Z
Ω
(u+)2∗dx
+ + 1 2 Z Ω|∇
(un−u)|2dx−
λ
2
Z
Ω
(un−u)2dx−
−
Z
Ω
G(x, un−u)dx−
1 2∗
Z
Ω
[(un−u)+]2
∗
dx
❙❊➬➹❖ ✶✳✸ ∗ ❈♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ✶✾ = lim n→∞ 1 2 Z Ω|∇
u|2dx+1
2
Z
Ω|∇
un|2dx−
Z
Ω∇
un∇udx+
+1 2
Z
Ω|∇
u|2dx− 1
2∗ Z
Ω
(u+)2∗− 1 2∗
Z
Ω
(v+n)2∗dx−
−λ
2
Z
Ω
u2ndx−
Z
Ω
G(x, un)dx
= lim n→∞ 1 2 Z Ω|∇
un|2dx−
λ
2
Z
Ω
u2ndx−
Z
Ω
G(x, un)dx−
1 2∗
Z
Ω
(u+n)2∗dx
= lim
n→∞J(un) =c.
P♦r ♦✉tr♦ ❧❛❞♦✱ ✉s❛♥❞♦ ♥♦✈❛♠❡♥t❡ ♦ ▲❡♠❛ ✶✳✷ ❡ ♦ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ❉♦♠✐♥❛❞❛✱ ❡ ❞❡♥♦t❛♥❞♦
L= lim
n→∞ Z
Ω|∇
vn|2dx−
Z
Ω
(vn+)2
∗
dx
,
t❡♠♦s✿
L= lim
n→∞ Z
Ω|∇
un|2dx+
Z
Ω|∇
u|2dx−2
Z
Ω∇
un∇udx−
−
Z
Ω
(u+n)2∗dx+
Z
Ω
(u+)2∗dx
= lim
n→∞ Z
Ω|∇
un|2dx−λ
Z
Ω
u2ndx−
Z
Ω
g(x, un)undx−
−
Z
Ω
(u+n)2∗dx−
Z
Ω|∇
u|2dx+λ
Z
Ω
u2dx+ +
Z
Ω
g(x, u)udx+
Z
Ω
(u+)2∗dx+ +2
Z
Ω|∇
u|2dx−2
Z
Ω∇
un∇udx
= lim
n→∞[hJ ′(u
n), uni − hJ′(u), ui+
+2
Z
Ω|∇
u|2dx−2
Z
Ω∇
un∇udx
= 0.
P♦❞❡♠♦s s✉♣♦r ❡♥tã♦ q✉❡ ❡①✐st❡ d >0t❛❧ q✉❡✿
|∇vn|22 →d ❡ |vn+|2
∗
❙❊➬➹❖ ✶✳✹ ∗ ❈♦♥❞✐çõ❡s ❣❡♦♠étr✐❝❛s ✷✵
P❡❧❛ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❙♦❜♦❧❡✈✱
Z
Ω|∇
vn|2dx=
Z
Ω|∇
(vn++v−n)|2dx=
Z
Ω|∇
v+n|2dx+
Z
Ω|∇
vn−|2dx≥
Z
Ω|∇
vn+|2dx≥S|v+n|22∗,
♦✉ s❡❥❛✱
|∇vn|22 ≥S|vn+|22∗.
❊♥tã♦✿
Z
Ω|∇
vn|2dx≥S|vn+|22∗ =S(|v+n|2 ∗
2∗)
2
2∗ ⇒d ≥Sd22∗.
❙❡d= 0 ❡♥tã♦✱ ❝♦♠♦ ❥á s❛❜❡♠♦s q✉❡ vn →0❡♠ L2(Ω)✱ s❡❣✉❡ q✉❡✿
kvnk2 =
Z
Ω|∇
vn|2dx+
Z
Ω
vn2dx→0, q✉❛♥❞♦ n→+∞,
❧♦❣♦✱un →u ❡♠ H1(Ω) ❡ ❛ ♣r♦✈❛ ❡stá ❝♦♠♣❧❡t❛✳
❙✉♣♦♥❤❛ d >0. ❊♥tã♦
d≥SN2.
P♦rt❛♥t♦✱ ❝♦♠♦ vn →0 ❡♠ L2(Ω) ❡ q✳t✳♣✳ ❡♠ Ω✱ t❡♠♦s✿
SN2
N = 1 2 − 1 2∗
SN2 ≤
1 2 − 1 2∗
d= 1 2d−
1 2∗d
= 1 2n→lim+∞
Z
Ω|∇
vn|2dx−
λ
2nlim→∞ Z
Ω
v2
ndx− lim n→∞
Z
Ω
G(x, vn)dx−
1
2∗ n→lim+∞|v
+
n|2
∗
2∗
≤J(u) + lim
n→+∞
1 2
Z
Ω|∇
vn|2dx−
λ
2
Z
Ω
v2ndx−
Z
Ω
G(x, vn)dx−
1 2∗|v
+
n|2
∗
2∗
= lim
n→+∞[J(u) +J(vn)] =c <
SN2
N ,
♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✦
▲♦❣♦✱ d= 0 ❡ {un} ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ ❡♠H1(Ω)✳
✶✳✹ ❈♦♥❞✐çõ❡s ❣❡♦♠étr✐❝❛s
❙❡❥❛♠ λ1 = 0< λ2 ≤λ3 ≤...♦s ❛✉t♦✈❛❧♦r❡s ❞❡ −∆ ❡ϕ1, ϕ2, ϕ3, ... ❛s ❝♦rr❡s♣♦♥❞❡♥t❡s
❙❊➬➹❖ ✶✳✹ ∗ ❈♦♥❞✐çõ❡s ❣❡♦♠étr✐❝❛s ✷✶
❣❡♥❡r❛❧✐❞❛❞❡✱ q✉❡ 0 ∈ Ω✳ P❛r❛ ❝❛❞❛ m ∈ N✱ s❡❥❛ ζm : Ω→ R ✉♠❛ ❢✉♥çã♦ s✉❛✈❡ t❛❧ q✉❡
0≤ζm ≤1✱|∇ζm|∞≤4m ❡
ζm(x) =
0, s❡ x∈B2/m,
1, s❡ x∈Ω\B3/m.
❉❡✜♥✐♠♦s ❛s ✏❛✉t♦❢✉♥çõ❡s ❛♣r♦①✐♠❛❞❛s✑ ♣♦rϕm
i =ζmϕi.❊♥tã♦✱ ❛ s❡❣✉✐♥t❡ ❡st✐♠❛t✐✈❛
♦❝♦rr❡ ✭✈❡r ❬❈❤❛❜r♦✇s❦✐✲❘✉❢❪✮✿
▲❡♠❛ ✶✳✽✳ ◗✉❛♥❞♦ m → ∞✱ t❡♠♦s ϕm
i →ϕi ❡♠ H1(Ω)✳ ❆❧é♠ ❞✐ss♦✱ ♥♦ ❡s♣❛ç♦ Hj,m− =
ϕm
1 , ..., ϕmj
✱ t❡♠♦s✿
max{|∇u|22 : u∈Hj,m− ,|u|22 = 1} ≤λj +cjm2−N
❡
Z
Ω∇
ϕmi ∇ϕmj dx=δij+O(m2−N),
♦♥❞❡ cj ✐♥❞❡♣❡♥❞❡ ❞❡ m✳
❙❡❥❛ξ ∈C1
0(B1/m)✉♠❛ ❢✉♥çã♦ ❝♦rt❡ t❛❧ q✉❡ ξ(x) = 1 s❡x∈B1/2m✱ 0≤ξ(x)≤1❡♠
B1/m ❡ |∇ξ|∞≤4m. ❈♦♥s✐❞❡r❡ ❛❣♦r❛ ❛ ❢❛♠í❧✐❛ ❞❡ ❢✉♥çõ❡s ❞❛❞❛s ♣♦r
uǫ(x) =ξ(x)u⋆ǫ(x)∈H, ǫ > 0.
▲❡♠❜r❡ q✉❡ ❛ ❢✉♥çã♦ u⋆
ǫ(x) ❢♦✐ ❞❡✜♥✐❞❛ ❡♠ ✭✶✳✸✮✳ ❖ ♣ró①✐♠♦ ❧❡♠❛ t❛♠❜é♠ ♣♦❞❡ s❡r
❡♥❝♦♥tr❛❞♦ ♥♦ ❛rt✐❣♦ ❬❈❤❛❜r♦✇s❦✐✲❘✉❢❪✳
▲❡♠❛ ✶✳✾✳ P❛r❛ ǫ→0 ❡ ♣❛r❛ m ✜①❛❞♦✱ t❡♠✲s❡✿
✭❛✮ |∇uǫ|22 =S
N
2 +O(ǫN−2) ✭❜✮ |uǫ|2
∗
2∗ =S
N
2 +O(ǫN)
✭❝✮ |uǫ|22 ≥K1ǫ2 +O(ǫN−2), ♦♥❞❡ K1 >0 é ❝♦♥st❛♥t❡✳
❙❊➬➹❖ ✶✳✹ ∗ ❈♦♥❞✐çõ❡s ❣❡♦♠étr✐❝❛s ✷✷
✭❞✮ |∇uǫ|22 =S
N
2 +O((ǫm)N−2) ✭❡✮ |uǫ|2
∗
2∗ =S
N
2 +O((ǫm)N)✳
❡♥q✉❛♥t♦ (c) ♦❝♦rr❡ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞❡ m✳
❖❜s❡r✈❛çã♦ ✶✳✶✵✳ ▲❡♠❜r❡ q✉❡✿
f(x) =o[g(x)] q✉❛♥❞♦ x→x0 s❡ lim
x→x0
f(x)
g(x) = 0 ❡ f(x) = O[g(x)] q✉❛♥❞♦ x→x0 s❡ lim sup
x→x0
|f(x)|
|g(x)| <+∞.
◆♦ss♦ ♦❜❥❡t✐✈♦ ❛q✉✐ é ❛♣❧✐❝❛r ♦ ❚❡♦r❡♠❛ ❞♦ ▲✐♥❦✐♥❣✱ ❞❡ ❘❛❜✐♥♦✇✐t③ ❬❘❛❜✐♥♦✇✐t③❪✱ ❛♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮✳
❉❡s❞❡ q✉❡ λ > 0✱ ❡①✐st❡ k ∈ N t❛❧ q✉❡ λ ∈ [λk, λk+1)✳ ❈♦♥s✐❞❡r❡ H+ =hϕ1, ..., ϕki⊥✳
❙❡❥❛♠ Sr =∂Br∩H+✱ Hm− =hϕm1 , ..., ϕmki ❡Qǫm = (BR∩Hm−)⊕[0, R]{uǫ}✱ ♣❛r❛ m ∈N
✜①❛❞♦✷✳ ❉❡✜♥❛ ❛ ❢❛♠í❧✐❛ ❞❡ ❛♣❧✐❝❛çõ❡s
H={h:Qǫm →H ❝♦♥tí♥✉❛ : h|∂Qǫ
m =Id},
❡ s❡❥❛
✭✶✳✷✸✮ c= inf
h∈Hu∈suph(Qǫ m)
J(u) = inf
h∈Hvsup∈Qǫ m
J(h(v)).
❖ ❚❡♦r❡♠❛ ❞♦ ▲✐♥❦✐♥❣✱ ❞❡ ❘❛❜✐♥♦✇✐t③ ❬❘❛❜✐♥♦✇✐t③❪✱ ❡st❛❜❡❧❡❝❡ q✉❡ s❡✿ ✭✶✮ J :H →R s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✭P❙✮c
✭✷✮ ❡①✐st❡♠ ♥ú♠❡r♦s r❡❛✐s 0< r < R ❡ β1 > β0 t❛✐s q✉❡
J(v)≥β1, ♣❛r❛ t♦❞♦v ∈Sr,
✭✶✳✷✹✮
J(v)≤β0, ♣❛r❛ t♦❞♦v ∈∂Qǫm,
✭✶✳✷✺✮
❡♥tã♦✸ ♦ ✈❛❧♦r c✱ ❞❡✜♥✐❞♦ ♣♦r ✭✶✳✷✸✮ s❛t✐s❢❛③c≥β
1✱ ❡ é ✉♠ ✈❛❧♦r ❝rít✐❝♦ ♣❛r❛ J✳
✷❊♥t❡♥❞❡♠♦s ♣♦r[0, R]{u
ǫ} ♦ ❝♦♥❥✉♥t♦{suǫ : 0≤s≤R}✳
✸❖❜s❡r✈❡ q✉❡ ♦s ❝♦♥❥✉♥t♦s ❛❝✐♠❛ ❞❡♣❡♥❞❡♠ ❞❡m❡ ❞❡ǫ✳ ❊s❝♦❧❤❡r❡♠♦sm s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ❡ǫ
❙❊➬➹❖ ✶✳✹ ∗ ❈♦♥❞✐çõ❡s ❣❡♦♠étr✐❝❛s ✷✸
◆♦t❡ ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡ ♣❛r❛ v ∈ H−
m⊕R{uǫ}✱ v = w+suǫ✱ t❡♠♦s✱ ♣♦r ❝♦♥str✉çã♦✱
s✉♣♣(uǫ)∩s✉♣♣(w) = ∅.■ss♦ ✐♠♣❧✐❝❛ q✉❡
J(v) =J(w+suǫ) =J(w) +J(suǫ).
❈♦♠❡❝❡♠♦s ♠♦str❛♥❞♦ q✉❡ ♦ ❢✉♥❝✐♦♥❛❧J s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✭✶✳✷✹✮✳
▲❡♠❛ ✶✳✶✶✳ ❊①✐st❡♠ r >0 ❡ β1 >0 t❛✐s q✉❡
J(v)≥β1, ♣❛r❛ t♦❞♦ v ∈Sr=∂Br∩H+.
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛v ∈H+✳
❆✜r♠❛çã♦ ✶✳✶✷✳ ❉❡✜♥❛ k · k∗ :H+ →R✱ ❞❛❞❛ ♣♦r kvk∗ =
Z
Ω|∇
v|2
12
=|∇v|2, v∈H+.
❊♥tã♦✱ k · k∗ ❞❡✜♥❡ ✉♠❛ ♥♦r♠❛ ❡♠ H+ q✉❡ é ❡q✉✐✈❛❧❡♥t❡ à ♥♦r♠❛ ✉s✉❛❧ k · k ❞❡ H1(Ω)✱
r❡str✐t❛ ❛ H+✳
❉❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❞❡✜♥✐çã♦ ❞❡ ♥♦r♠❛✱ ❛♣❡♥❛s ✉♠❛ ♥ã♦ é ✐♠❡❞✐❛t❛✿
kvk= 0⇔v = 0.
▲❡♠❜r❡ q✉❡ ❛s ❢✉♥çõ❡s ❝♦♥st❛♥t❡s ♥ã♦ ♥✉❧❛s sã♦ ❛✉t♦❢✉♥çõ❡s ❛ss♦❝✐❛❞❛s ❛♦ ❛✉t♦✈❛❧♦r
λ1 = 0✱ ❧♦❣♦✱ ♥ã♦ ♣❡rt❡♥❝❡♠ ❛ H+✳ P♦rt❛♥t♦✱ ❞❛❞♦ v ∈H+✱ t❡♠♦s✿
kvk∗ = 0⇔ ∇v ≡0⇔v ≡constante⇒v ≡0.
P❛r❛ ♠♦str❛r ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❛s ♥♦r♠❛s✱ ♥♦t❡ ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡ ✉♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ é ó❜✈✐❛✱ ♣♦✐s✿
kvk2∗ =|∇v|22 ≤ |v|22+|∇v|22 =kvk2, ∀v.
P♦r ♦✉tr♦ ❧❛❞♦✱ ✉s❛♥❞♦ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ✈❛r✐❛❝✐♦♥❛❧ ❞❡λk+1✱ ♦❜t❡♠♦s✿
kvk2 =|v|2
2+kvk2∗ ≤
1
λk+1k
vk2
∗+kvk2∗ =
1
λk+1
+ 1
kvk2
∗, ∀v ∈H+,
