❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❈❛♠♣✐♥❛ ●r❛♥❞❡
Pr♦❣r❛♠❛ ❆ss♦❝✐❛❞♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛
❉♦✉t♦r❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛
❊q✉❛çõ❡s ❞❡ ❙❝❤rö❞✐♥❣❡r
q✉❛s❡❧✐♥❡❛r❡s ❝♦♠ ♣♦t❡♥❝✐❛✐s
s✐♥❣✉❧❛r❡s ♦✉ s❡ ❛♥✉❧❛♥❞♦ ♥♦ ✐♥✜♥✐t♦
♣♦r
●✐❧s♦♥ ▼❛♠❡❞❡ ❞❡ ❈❛r✈❛❧❤♦
❊q✉❛çõ❡s ❞❡ ❙❝❤rö❞✐♥❣❡r
q✉❛s❡❧✐♥❡❛r❡s ❝♦♠ ♣♦t❡♥❝✐❛✐s
s✐♥❣✉❧❛r❡s ♦✉ s❡ ❛♥✉❧❛♥❞♦ ♥♦ ✐♥✜♥✐t♦
♣♦r●✐❧s♦♥ ▼❛♠❡❞❡ ❞❡ ❈❛r✈❛❧❤♦
†s♦❜ ❛ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ❯❜❡r❧❛♥❞✐♦ ❇❛t✐st❛ ❙❡✈❡r♦
❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ Pr♦❣r❛♠❛ ❆ss♦❝✐❛❞♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ✲ ❯❋P❇✴❯❋❈●✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ▼❛t❡♠át✐❝❛✳
❏♦ã♦ P❡ss♦❛ ✲ P❇ ❏✉❧❤♦✴✷✵✶✻
†❊st❡ tr❛❜❛❧❤♦ ❝♦♥t♦✉ ❝♦♠ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❞❛ ❈❆P❊❙
C331e Carvalho, Gilson Mamede de.
Equações de Schrödinger quaselineares com potenciais singulares ou se anulando no infinito / Gilson Mamede de Carvalho.- João Pessoa, 2016.
113f.
Orientador: Uberlandio Batista Severo Tese (Doutorado) - UFPB-UFCG
1. Matemática. 2. Equações de Schrödinger. 3. Potenciais singulares ou se anulando no infinito. 4. Métodos variacionais. 5. Espaços de Orlicz. 6. Desigualdade de Trudinger-Moser.
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❈❛♠♣✐♥❛ ●r❛♥❞❡
Pr♦❣r❛♠❛ ❆ss♦❝✐❛❞♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❉♦✉t♦r❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ❆♥á❧✐s❡ ❆♣r♦✈❛❞❛ ❡♠✿ ✶✾✴✵✼✴✷✵✶✻
❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ Pr♦❣r❛♠❛ ❆ss♦❝✐❛❞♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ✲ ❯❋P❇✴❯❋❈●✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ▼❛t❡♠át✐❝❛✳
❏✉❧❤♦✴✷✵✶✻
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛♠♦s ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ❞♦ t✐♣♦ ♦♥❞❛ ❡st❛❝✐♦♥ár✐❛ ♣❛r❛ ✉♠❛ ❝❧❛ss❡ ❞❡ ❡q✉❛çõ❡s ❞❡ ❙❝❤rö❞✐♥❣❡r q✉❛s❡❧✐♥❡❛r❡s✱ ❡♥✈♦❧✈❡♥❞♦ ♣♦♥t❡♥❝✐❛s q✉❡ ♣♦❞❡♠ s❡r s✐♥❣✉❧❛r ♥❛ ♦r✐❣❡♠ ♦✉ q✉❡ ♣♦❞❡♠ s❡ ❛♥✉❧❛r ♥♦ ✐♥✜♥✐t♦✳ P❛r❛ ❞✐♠❡♥sõ❡s ♠❛✐♦r❡s q✉❡ ❞♦✐s✱ ❝♦♥s✐❞❡r❛♠♦s ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡s ❝♦♠ ❝r❡s❝✐♠❡♥t♦ s✉❜❝rít✐❝♦✳ ❊♠ ❞✐♠❡♥sã♦ ❞♦✐s✱ tr❛❜❛❧❤❛♠♦s ❝♦♠ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡s ♣♦ss✉✐♥❞♦ ❝r❡s❝✐♠❡♥t❡ ❝rít✐❝♦ ❡①♣♦♥❡♥❝✐❛❧✳ P❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ♥♦ss♦s r❡s✉❧t❛❞♦s✱ ✉s❛♠♦s té❝♥✐❝❛s ✈❛r✐❛❝✐♦♥❛✐s✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ✉♠❛ ✈❡rsã♦ ❞♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✱ ✉♠ r❡s✉❧t❛❞♦ ❞❡ r❡❣✉❧❛r✐❞❛❞❡ ❞♦ t✐♣♦ ❇ré③✐s✲ ❑❛t♦✱ ❛r❣✉♠❡♥t♦s ❞♦ t✐♣♦ ♣r✐♥❝í♣✐♦ ❞❛ ❝r✐t✐❝❛❧✐❞❛❞❡ s✐♠étr✐❝❛✱ ♠ét♦❞♦ ❞❡ ✐t❡r❛çã♦ ❞❡ ▼♦s❡r ❡ ✉♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞♦ t✐♣♦ ❚r✉❞✐♥❣❡r✲▼♦s❡r✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❊q✉❛çõ❡s ❞❡ ❙❝❤rö❞✐♥❣❡r❀ P♦t❡♥❝✐❛✐s s✐♥❣✉❧❛r❡s ♦✉ s❡ ❛♥✉❧❛♥❞♦ ♥♦ ✐♥✜♥✐t♦❀ ▼ét♦❞♦s ✈❛r✐❛❝✐♦♥❛✐s❀ ❊s♣❛ç♦s ❞❡ ❖r❧✐❝③❀ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❚r✉❞✐♥❣❡r✲▼♦s❡r
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦✱ ✇❡ st✉❞② ❡①✐st❡♥❝❡ ♦❢ st❛♥❞✐♥❣ ✇❛✈❡ s♦❧✉t✐♦♥ ❢♦r ❛ ❝❧❛ss ♦❢ q✉❛s✐❧✐♥❡❛r ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ ♣♦t❡♥t✐❛❧s t❤❛t ♠❛② ❜❡ s✐♥❣✉❧❛r ❛t t❤❡ ♦r✐❣✐♥ ♦r ✈❛♥✐s❤✐♥❣ ❛t ✐♥✜♥✐t②✳ ❋♦r ❞✐♠❡♥s✐♦♥s ❜✐❣❣❡r t❤❛♥ t✇♦✱ ✇❡ ❝♦♥s✐❞❡r ♥♦♥❧✐♥❡❛r✐t✐❡s ✇✐t❤ s✉❜❝r✐t✐❝❛❧ ❣r♦✇t❤✳ ■♥ ❞✐♠❡♥s✐♦♥ t✇♦✱ ✇❡ ✇♦r❦ ✇✐t❤ ♥♦♥❧✐♥❡❛r✐t✐❡s ❤❛✈✐♥❣ ❡①♣♦♥❡♥t✐❛❧ ❝r✐t✐❝❛❧ ❣r♦✇t❤✳ ❚♦ ♦❜t❛✐♥ ♦✉r r❡s✉❧ts✱ ✇❡ ❤❛✈❡ ✉s❡❞ ✈❛r✐❛t✐♦♥❛❧ t❡❝❤♥✐q✉❡s✱ ♠♦r❡ s♣❡❝✐✜❝❛❧❧②✱ ❛ ✈❡rs✐♦♥ ♦❢ t❤❡ ▼♦✉♥t❛✐♥ P❛ss ❚❤❡♦r❡♠✱ ❛ r❡❣✉❧❛r✐t② r❡s✉❧t ♦❢ ❇ré③✐s✲❑❛t♦ t②♣❡✱ ❛r❣✉♠❡♥ts ♦❢ s②♠♠❡tr✐❝❛❧ ❝r✐t✐❝❛❧✐t② ♣r✐♥❝✐♣❧❡ t②♣❡✱ ▼♦s❡r ✐t❡r❛t✐♦♥ ♠❡t❤♦❞ ❛♥❞ ❛ ❚r✉❞✐♥❣❡r✲▼♦s❡r t②♣❡ ✐♥❡q✉❛❧✐t②✳
❑❡②✇♦r❞s✿ ◗✉❛s✐❧✐♥❡❛r ❙❝❤rö❞✐♥❣❡r❀ ❙✐♥❣✉❧❛r ♣♦t❡♥t✐❛❧s ♦r ✈❛♥✐s❤✐♥❣ ❛t ✐♥✜♥✐t②❀ ❖r❧✐❝s s♣❛❝❡s❀ ❚r✉❞✐♥❣❡r✲▼♦s❡r ✐♥❡q✉❛❧✐t②✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❆♦ ❝♦♥❝❧✉✐r♠♦s ✉♠❛ ❡t❛♣❛✱ é ♥❡❝❡ssár✐♦ ❛❣r❛❞❡❝❡r àq✉❡❧❡s q✉❡ ❡st✐✈❡r❛♠ ❛♦ ♥♦ss♦ ❧❛❞♦ ❞✉r❛♥t❡ ❡st❛ ❢❛s❡✳
Pr✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s✱ q✉❡ tr♦✉①❡ ♣❛③ ❛♦s ♠❡✉s ♣❡♥s❛♠❡♥t♦s ♥♦s ✈ár✐♦s ♠♦♠❡♥t♦s t✉r❜✉❧❡♥t♦s q✉❡ ♣❛ss❡✐✱ ♠❡ ❛❥✉❞❛♥❞♦ ❛ ♣❡r♠❛♥❡❝❡r ♣❡rs❡✈❡r❛♥t❡ ❡ ❢♦❝❛❞♦ ♥♦s ❡st✉❞♦s✳ ❆ ♠✐♥❤❛ ♠ã❡✱ ▲ú❝✐❛✱ s❡♠ ❞ú✈✐❞❛✱ ❛ ♣r✐♥❝✐♣❛❧ r❡s♣♦♥sá✈❡❧ ♣❡❧♦ ❤♦♠❡♠ q✉❡ ♠❡ t♦r♥❡✐✳ ❖ ♥♦rt❡ ❞❛ ♠✐♥❤❛ ❜úss♦❧❛✱ s❡♠♣r❡ ❛ ♠❡ ✐♥❞✐❝❛r ♦ ♠❡❧❤♦r ❝❛♠✐♥❤♦ ❛ s❡r s❡❣✉✐❞♦✳ ❆ ♠❡✉ ♣❛✐✱ ●❡r❛❧❞♦ ✭❡♠ ♠❡♠ór✐❛✮✱ ♣❡❧♦s ✈❛❧♦r❡s ❡ ❡♥s✐♥❛♠❡♥t♦s ❞❡✐①❛❞♦s ♥❛ ♠✐♥❤❛ ✐♥❢â♥❝✐❛ ❡ q✉❡ ❝❛rr❡❣♦ ❝♦♠✐❣♦ ❛té ❤♦❥❡✳
❆ ♠❡✉s ✐r♠ã♦s✱ ●❡r❧â♥✐❛ ❡ ●✐❧♠❛r✱ ♣❡❧♦s ❢♦rt❡s ❧❛ç♦s ❞❡ ❛♠♦r q✉❡ ♥♦s ✉♥❡✱ ♠❡ ❡♥❝♦r❛❥❛♥❞♦ ❡ tr❛♥s♠✐t✐♥❞♦ ❝♦♥✜❛♥ç❛ ♥❛ ♠✐♥❤❛ ❝❛♣❛❝✐❞❛❞❡✳
❆ ♠❡✉ ♣❛❞❛str♦✱ ❊❞✉❛r❞♦✱ ♣❡❧❛ ❛♠✐③❛❞❡ ❝♦♥str✉í❞❛ ❡ ❛t❡♥çã♦ ♥♦s ♠♦♠❡♥t♦s q✉❡ ♣r❡❝✐s❡✐✳
❆ t♦❞♦s ♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s ❡ ❛♠✐❣♦s✱ ♣♦r ❝♦♠♣r❡❡♥❞❡r❡♠ ♠✐♥❤❛ ❛✉sê♥❝✐❛ ❡♠ ♠✉✐t♦s ♠♦♠❡♥t♦s ✐♠♣♦rt❛♥t❡s✳
❆ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❯❜❡r❧❛♥❞✐♦✱ ♣♦r t♦❞♦s ♦s ❡♥s✐♥❛♠❡♥t♦s ♠❛t❡♠át✐❝♦s ❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♣❡❧❛ r❡❧❛çã♦ ❞❡ r❡s♣❡✐t♦ ❡ ❛♠✐③❛❞❡ ❝♦♥str✉í❞❛ ♥♦s ú❧t✐♠♦s ❛♥♦s✳ ❆♦s ♣r♦❢❡ss♦r❡s ●✐♦✈❛♥② ❞❡ ❏❡s✉s ▼❛❧❝❤❡r ❞❡ ❋✐❣✉❡✐r❡❞♦✱ ❊❞❝❛r❧♦s ❉♦♠✐♥❣♦s ❞❛ ❙✐❧✈❛✱ ❏❡✛❡rs♦♥ ❆❜r❛♥t❡s ❞♦s ❙❛♥t♦s✱ ❏♦ã♦ ▼❛r❝♦s ❇❡③❡rr❛ ❞♦ Ó✱ ❋r❛♥❝✐s❝♦ ❙✐❜ér✐♦ ❇❡③❡rr❛ ❆❧❜✉q✉❡rq✉❡ ❡ ❊✈❡r❛❧❞♦ ❙♦✉t♦ ❞❡ ▼❡❞❡✐r♦s ♣♦r t❡r❡♠ ❛❝❡✐t❛❞♦ ❝♦♠♣♦r ❛ ❜❛♥❝❛ ♣❛r❛ ♠✐♥❤❛ ❞❡❢❡s❛ ❞❡ t❡s❡ ❡ ♣❡❧❛s ✈❛❧✐♦s❛s s✉❥❡stõ❡s ❞❛❞❛s ❛ ❡st❡ tr❛❜❛❧❤♦✳
❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛ ♣♦r t❡r❡♠ ❝♦♥tr✐❜✉í❞♦ ♣❛r❛ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❛❝❛❞ê♠✐❝❛✳ ❊♠ ❡s♣❡❝✐❛❧ ❛♦s ♣r♦❢❡ss♦r❡s ❯❜❡r❧❛♥❞✐♦ ❇❛t✐st❛ ❙❡✈❡r♦✱ ❊✈❡r❛❧❞♦ ❙♦✉t♦ ❞❡ ▼❡❞❡✐r♦s✱ ❋❧á✈✐❛ ❏❡rô♥✐♠♦ ❇❛r❜♦s❛✱ ❏♦ã♦ ▼❛r❝♦s ❇❡③❡rr❛ ❞♦ Ó✱ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛✱ ❋á❣♥❡r ❉✐❛s ❆r❛r✉♥❛✱ ▼❛r✐✈❛❧❞♦ P❡r❡✐r❛ ▼❛t♦s ❡ ❆♥tô♥✐♦ ❞❡ ❆♥❞r❛❞❡ ❡ ❙✐❧✈❛ q✉❡ ❝♦♥tr✐❜✉✐r❛♠ ❞✐r❡t❛♠❡♥t❡ ♣❛r❛ ♦ ♠❡✉ ❛♣r❡♥❞✐③❛❞♦ ❡ sã♦ ❡①❡♠♣❧♦s ❞❡ ❞♦❝❡♥t❡s ❞♦s q✉❛✐s q✉❡r♦ s❡r ❡s♣❡❧❤♦✳
❆ t♦❞♦s ♦s ❛♠✐❣♦s ❞♦ ♣r♦❥❡t♦ ♠✐❧ê♥✐♦✱ ✉♠❛ ✈❡r❞❛❞❡✐r❛ ❢❛♠í❧✐❛ ❢♦r❛ ❞❡ ❝❛s❛✳ ❊st❡ ❛❝♦❧❤✐♠❡♥t♦ ♥♦ ✐♥í❝✐♦ ❞❛ ❣r❛❞✉❛çã♦ ❢♦r♠♦✉ ✉♠❛ ❜❛s❡ ♣❛r❛ t✉❞♦ q✉❡ ❢♦✐ ❝♦♥str✉í❞♦ ❛té ❛q✉✐✳
❆ t♦❞♦s ♦s ❛♠✐❣♦s ❞❛ ♣ós ❣r❛❞✉❛çã♦ ❡♠ ♠❛t❡♠át✐❝❛ ❞❛ ❯❋P❇✱ ▼❡str❛❞♦ ❡ ❉♦✉t♦r❛❞♦✱ ♠❡✉ s✐♥❝❡r♦s ❛❣r❛❞❡❝✐♠❡♥t♦s ♣♦r ❝❛❞❛ ❡①♣❡r✐❡♥❝✐❛ ✈✐✈✐❞❛ ❡ ❝♦♠♣❛rt✐❧❤❛❞❛✳
❆♦s ♠❡✉s ❛♠✐❣♦s✱ ❨❛♥❡✱ ❘❡❣✐♥❛❧❞♦✱ ◆❛❝✐❜✱ ❑❛ré✱ ❘✐❝❛r❞♦ ❇✉r✐t②✱ ❘❡♥❛t❛✱ ❉❛♥✐❧♦✱ ❏✉❧✐❛♥❛✱ ❱❛❧❞❡❝✐r✱ ❆♠❛♥❞❛✱ ●✉st❛✈♦✱ P❛♠♠❡❧❧❛✱ ❘♦❞r✐❣♦✱ ❉✐❡❣♦✱ ❘❛✐♥❡❧❧②✱ ❘✐❝❛r❞♦ P✐♥❤❡✐r♦ ❡ ❊st❡❜❛♥ r❡❣✐str♦ ♠✐♥❤❛ ❣r❛t✐❞ã♦ ♣♦r ❝❛❞❛ s♦rr✐s♦ ❡ ♣❛❧❛✈r❛s ❞❡ ✐♥❝❡♥t✐✈♦✳ ❱♦❝ês t❡♠ ♣❛rt✐❝✐♣❛çã♦ ❝♦♥❝r❡t❛ ♥❛ ❡❢❡t✐✈❛çã♦ ❞❡st❛ ❝♦♥q✉✐st❛✳
❆♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❘✉r❛❧ ❞❡ P❡r♥❛♠❜✉❝♦✱ ■♥st✐t✉✐çã♦ ❛ q✉❛❧ s♦✉ ❞♦❝❡♥t❡✱ ♣❡❧♦ ❛♣♦✐♦ ❛♦s ❡st✉❞❛♥t❡s ❞❡ ❉♦✉t♦r❛❞♦✳
❆♦ Pr♦❣r❛♠❛ ❆ss♦❝✐❛❞♦ ❞❡ Pós✲❣r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ✭❯❋P❇✴❯❋❈●✮ ❡ ❛ ❈❆P❊❙✱ ♣❡❧❛ ❡str✉t✉r❛ ❡ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ q✉❡ t♦r♥♦✉ ♣♦ssí✈❡❧ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ s♦♥❤♦✳ P♦r ✜♠✱ ❞❡✐①♦ ❛q✉✐ r❡❣✐str❛❞♦ ♦ ♠❡✉ r❡❝♦♥❤❡❝✐♠❡♥t♦✱ ♠✐♥❤❛ ❣r❛t✐❞ã♦ ❡ ❛❞♠✐r❛çã♦ ❛ ♠✐♥❤❛ ❡s♣♦s❛✱ ❏❛♥❛✐♥❛✱ q✉❡ t♦♠♦✉ ♣❛r❛ s✐ ❡st❡ s♦♥❤♦✱ q✉❡ ❛ ♣r✐♥❝í♣✐♦ ❡r❛ só ♠❡✉✱ ❡ s❡ tr❛♥❢♦r♠♦✉ ❡♠ ♥♦ss♦✳ ❙❡♠♣r❡ ❝♦♠♣r❡❡♥s✐✈❛ ❝♦♠ ❛s ♠✐♥❤❛s ♥♦✐t❡s ❞❡ ❡st✉❞♦✱ ♥♦s ♠♦♠❡♥t♦s ❞❡ ❝❛♥s❛ç♦✱ ❡❧❛ ❢♦✐ ❛ ♠✐♥❤❛ ♣r✐♥❝✐♣❛❧ ❢♦♥t❡ ❞❡ ❡♥❡r❣✐❛✳
✏❖ ú♥✐❝♦ ❧✉❣❛r ♦♥❞❡ ♦ s✉❝❡ss♦ ✈❡♠ ❛♥t❡s ❞♦ tr❛❜❛❧❤♦ é ♥♦ ❞✐❝✐♦♥ár✐♦✳✑
❆❧❜❡rt ❊✐♥st❡✐♥
❉❡❞✐❝❛tór✐❛
❆ ♠❡✉s ♣❛✐s✱ ✐r♠ã♦s ❡ ❛ ♠✐♥❤❛ ❡s♣♦s❛✳✳✳
❙✉♠ár✐♦
◆♦t❛çã♦ ❡ t❡r♠✐♥♦❧♦❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶ ❊q✉❛çõ❡s q✉❛s❡❧✐♥❡❛r❡s ❝♦♠ ♣♦t❡♥❝✐❛✐s s✐♥❣✉❧❛r❡s ♦✉ s❡ ❛♥✉❧❛♥❞♦ ♥♦
✐♥✜♥✐t♦ ❡♠ ❞✐♠❡♥sã♦ N ≥3 ✶✵
✶✳✶ ❆ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ f ❡ ♦ ❡s♣❛ç♦ ❞❡ ❖r❧✐❝③ E1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✶✳✷ ❯♠ r❡s✉❧t❛❞♦ ❞❡ r❡❣✉❧❛r✐❞❛❞❡ ❞♦ t✐♣♦ ❇ré③✐s✲❑❛t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✸ Pr♦♣r✐❡❞❛❞❡s ❞♦ ❢✉♥❝✐♦♥❛❧ I ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✶✳✸✳✶ P♦♥t♦s ❝rít✐❝♦s ❞❡ I ❡ s♦❧✉çõ❡s ❢r❛❝❛s ❞❡ ✭P1✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✶✳✹ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✶✳✵✳✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷ ❊q✉❛çõ❡s q✉❛s❡❧✐♥❡❛r❡s ❝♦♠ ♣♦t❡♥❝✐❛✐s s✐♥❣✉❧❛r❡s ♦✉ s❡ ❛♥✉❧❛♥❞♦ ♥♦
✐♥✜♥✐t♦ ❡♠ ❞✐♠❡♥sã♦ ❞♦✐s ❝♦♠ ❝r❡s❝✐♠❡♥t♦ ❝rít✐❝♦ ❡①♣♦♥❡♥❝✐❛❧ ✸✾ ✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❞♦s ❡s♣❛ç♦s Yrad ❡ Lq(R2, Q) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✷✳✷ ❖ ❡s♣❛ç♦ ❞❡ ❖r❧✐❝③ E2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
✷✳✸ ❯♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞♦ t✐♣♦ ❚r✉❞✐♥❣❡r✲▼♦s❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✷✳✹ Pr♦♣r✐❡❞❛❞❡s ❞♦ ❢✉♥❝✐♦♥❛❧ I ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✷✳✺ P♦♥t♦s ❝rít✐❝♦s ❞♦ ❢✉♥❝✐♦♥❛❧ I ❡ s♦❧✉çõ❡s ❢r❛❝❛s ❞❡ ✭P2✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
✷✳✻ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✷✳✵✳✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✸ ❊q✉❛çõ❡s q✉❛s❡❧✐♥❡❛r❡s ❝♦♠ ♣♦t❡♥❝✐❛✐s s✐♥❣✉❧❛r❡s ♦✉ s❡ ❛♥✉❧❛♥❞♦ ♥♦
✐♥✜♥✐t♦ ❡♥✈♦❧✈❡♥❞♦ ✉♠ ♣❛râ♠❡tr♦ ♣♦s✐t✐✈♦ ✼✹
✸✳✶ ❖ ♣r♦❜❧❡♠❛ ❛✉①✐❧✐❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✸✳✷ P♦♥t♦s ❝rít✐❝♦s ❞❡ Lκ ❡ s♦❧✉çã♦ ❢r❛❝❛s ❞❡ ✭✸✳✸✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷
✸✳✹ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✸✳✵✳✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵
❘❡❢❡rê♥❝✐❛s ✾✼
◆♦t❛çã♦ ❡ t❡r♠✐♥♦❧♦❣✐❛
• C✱ C0✱C1, . . . ❞❡♥♦t❛♠ ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s ✭♣♦ss✐✈❡❧♠❡♥t❡ ❞✐❢❡r❡♥t❡s✮❀
• |A|❞❡♥♦t❛ ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❞❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ A ❡♠ RN, N ≥1❀
• ❆ ❡①♣r❡ssã♦ |x|>> c q✉❡r ❞✐③❡r|x| ♠✉✐t♦ ♠❛✐♦r q✉❡ c❀
• supp(f)❞❡♥♦t❛ ♦ s✉♣♦rt❡ ❞❛ ❢✉♥çã♦ f❀
• C∞
0 (RN)r❡♣r❡s❡♥t❛ ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ✐♥✜♥✐t❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡✐s ❝♦♠ s✉♣♦rt❡
❝♦♠♣❛❝t♦ ❡♠ RN ❡C∞
0,rad(RN) =
u∈C∞
0 (RN);u é r❛❞✐❛❧ ❀
• Lp(Ω) =
u: Ω→R ♠❡♥s✉rá✈❡❧; Z
Ω|
u|pdx <∞
✱ ❡♠ q✉❡ 1 ≤ p < ∞ ❡
Ω⊆RN é ✉♠ ❛❜❡rt♦ ❝♦♥❡①♦✱ ❝♦♠ ♥♦r♠❛ ❞❛❞❛ ♣♦r
kukp = Z
Ω|
u|pdx
1/p
;
• L∞(Ω) ❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s q✉❡ sã♦ ❧✐♠✐t❛❞❛s q✉❛s❡ s❡♠♣r❡
❡♠ Ω❝♦♠ ♥♦r♠❛ ❞❛❞❛ ♣♦r
kuk∞ = inf{C > 0;|u(x)| ≤C q✉❛s❡ s❡♠♣r❡ ❡♠Ω};
• P❛r❛ N ≥3✱D1,2(RN) é ♦ ❢❡❝❤♦ ❞❡ C∞
0 (RN) ❝♦♠ r❡s♣❡✐t♦ ❛ ♥♦r♠❛ ❞♦ ❣r❛❞✐❡♥t❡
k∇uk2 =
Z
RN|∇
u|2dx
1
2
• H1(R2) ❞❡♥♦t❛ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ W1,2(R2)✱ ♦♥❞❡
W1,2(R2) =
u∈L2(R2); ❡①✐st❡♠ g1, g2 ∈L2(R2)t❛✐s q✉❡✱ ♣❛r❛ i= 1,2; Z
R2
u∂ϕ ∂xi
dx=−
Z
R2
giϕdx, ♣❛r❛ t♦❞♦ ϕ ∈C0∞(R2)
♠✉♥✐❞♦ ❝♦♠ ❛ ♥♦r♠❛
kuk2
1,2 =k∇uk22+kuk22
❡ H1
rad(R2) ={u∈H1(R2);ué r❛❞✐❛❧}❀
• BR(x) ❞❡♥♦t❛ ❛ ❜♦❧❛ ❛❜❡rt❛ ❞❡ ❝❡♥tr♦ x ❡ r❛✐♦ R ❡ BR ❛ ❜♦❧❛ ❛❜❡rt❛ ❞❡ r❛✐♦ R
❝❡♥tr❛❞❛ ♥❛ ♦r✐❣❡♠❀
• ∇u=
∂u ∂x1
, . . . , ∂u ∂xN
❞❡♥♦t❛ ♦ ❣r❛❞✐❡♥t❡ ❞❛ ❢✉♥çã♦ u❀
• ∆u=
N X
i=1
∂2u
∂x2
i
❞❡♥♦t❛ ♦ ▲❛♣❧❛❝✐❛♥♦ ❞❛ ❢✉♥çã♦ u❀
• P❛r❛ N ≥3✱2∗ = 2N
N −2 ❞❡♥♦t❛ ♦ ❡①♣♦❡♥t❡ ❝rít✐❝♦ ❞❡ ❙♦❜♦❧❡✈❀
• ❉✐③❡♠♦s q✉❡ ψ(s) = o(ω(s))q✉❛♥❞♦ s→0 s❡ lim
s→0
ψ(s)
ω(s) = 0;
• P❛r❛ N ≥2✱O(N)❞❡♥♦t❛ ♦ ❣r✉♣♦ ❞❛s tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s ♦rt♦❣♦♥❛✐s ❞❡ RN
■♥tr♦❞✉çã♦
❖ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ ❞❡ t❡s❡ ❝♦♥s✐st❡ ♥❛ ♦❜t❡♥çã♦ ❞❡ r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ ❡q✉❛çõ❡s ❞❡ ❙❝❤rö❞✐♥❣❡r q✉❛s❡❧✐♥❡❛r❡s ❞❛ ❢♦r♠❛
i∂ψ
∂t =−∆ψ+W(x)ψ−η(x,|ψ|
2)ψ
−ǫ∆ρ(|ψ|2)ρ′(|ψ|2)ψ, ✭✶✮
♦♥❞❡ ψ : RN × R → C é ❛ ❢✉♥çã♦ ✐♥❝ó❣♥✐t❛✱ ǫ é ✉♠❛ ❝♦♥st❛♥t❡ r❡❛❧✱ W é ✉♠
♣♦t❡♥❝✐❛❧ ❞❛❞♦ ❡ η : RN ×R+ → R✱ ρ : R+ → R sã♦ ❢✉♥çõ❡s ❛♣r♦♣r✐❛❞❛s✳ ❊q✉❛çõ❡s
q✉❛s❡❧✐♥❡❛r❡s ❞♦ t✐♣♦ ✭✶✮ ❛♣❛r❡❝❡♠ ♥❛t✉r❛❧♠❡♥t❡ ♥❛ ❋ís✐❝❛✲▼❛t❡♠át✐❝❛ ❡ t❡♠ ♠♦❞❡❧❛❞♦ ❛❧❣✉♥s ❢❡♥ô♠❡♥♦s ❢ís✐❝♦s✱ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡ρ ❝♦♥s✐❞❡r❛❞❛✳ P♦r ❡①❡♠♣❧♦✱
q✉❛♥❞♦ρ(s) =s✱ ✭✶✮ ❢♦✐ ✉s❛❞❛ ♣❛r❛ ♠♦❞❡❧❛r ✉♠❛ ❡q✉❛çã♦ ❞❡ ♠❡♠❜r❛♥❛ ❞❡ s✉♣❡r✢✉✐❞♦
♥❛ ❢ís✐❝❛ ❞❡ ♣❧❛s♠❛s✱ ✈❡❥❛ ❑✉r✐❤❛r❛ ❡♠ ❬✸✸❪✳ ◗✉❛♥❞♦ ρ(s) = (1 +s)1/2✱ ❛ ❡q✉❛çã♦ ✭✶✮
♠♦❞❡❧♦✉ ❛ ❛✉t♦✲❝❛♥❛❧✐③❛çã♦ ❞❡ ✉♠ ❧❛s❡r ❞❡ ❛✉t❛ ♣♦tê♥❝✐❛ ✉❧tr❛ ❝✉rt♦ ♥❛ ♠❛tér✐❛✱ ✈❡❥❛ ❬✶✹✱ ✶✺❪✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡s ❡ ❛♣❧✐❝❛çõ❡s ❢ís✐❝❛s✱ ✈❡❥❛ t❛♠❜é♠ ❬✸✽✱ ✹✶❪ ❡ s✉❛s r❡❢❡rê♥❝✐❛s✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❝❛s♦ ρ(s) = s✱ ♥♦ss♦ ♠❛✐♦r ✐♥t❡r❡ss❡ é ❣❛r❛♥t✐r ❛
❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❞♦ t✐♣♦ ♦♥❞❛ ❡st❛❝✐♦♥ár✐❛ ♣❛r❛ ❛ ❡q✉❛çã♦ ✭✶✮✱ ✐st♦ é✱ s♦❧✉çõ❡s ❞❛ ❢♦r♠❛
ψ(x, t) = exp(−iEt)u(x),
❡♠ q✉❡ E ∈ R ❡u :RN → R é ✉♠❛ ❢✉♥çã♦ r❡❛❧ ❝♦♠ u(x) ≥0 ♣❛r❛ t♦❞♦ x∈RN✳ P♦r
♠❡✐♦ ❞❡ ❝á❧❝✉❧♦s s✐♠♣❧❡s✱ é ❢á❝✐❧ ♠♦str❛r q✉❡ψ s❛t✐s❢❛③ ✭✶✮ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ué s♦❧✉çã♦
❞❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦ ❡❧í♣t✐❝❛ q✉❛s❡❧✐♥❡❛r✿
−∆u+V(x)u−ǫ∆(u2)u=h(x, u), x∈RN, ✭✷✮
◆♦s ú❧t✐♠♦s ❛♥♦s✱ ♠♦t✐✈❛❞♦s ♣❡❧❛s ❛♣❧✐❝❛çõ❡s ♥❛ ❋ís✐❝❛✱ ♦ ❡st✉❞♦ ❞❛ ❡q✉❛çã♦ ✭✷✮ t❡♠ ❛tr❛í❞♦ ❛ ❛t❡♥çã♦ ❞❡ ♠✉✐t♦s ♣❡sq✉✐s❛❞♦r❡s ❞❛ ár❡❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s✳ ❖ ❝❛s♦ s❡♠✐❧✐♥❡❛r✱ ✐st♦ é✱ q✉❛♥❞♦ ǫ = 0✱ ❥á ❢♦✐ ❡st✉❞❛❞♦ ❡①t❡♥s✐✈❛♠❡♥t❡ ♣♦r
✈ár✐♦s ❛✉t♦r❡s ❝♦♠ ✉♠❛ ❣r❛♥❞❡ ✈❛r✐❡❞❛❞❡ ❞❡ ❤✐♣ót❡s❡s s♦❜r❡ ♦ ♣♦t❡♥❝✐❛❧V ❡ s♦❜r❡ ❛ ♥ã♦
❧✐♥❡❛r✐❞❛❞❡h✱ ✈❡❥❛ ♣♦r ❡①❡♠♣❧♦ ❬✹✱ ✶✶✱ ✶✷✱ ✶✸✱ ✶✽✱ ✹✷✱ ✹✺✱ ✹✽❪✳ ❖ ♣r♦❜❧❡♠❛ q✉❛s❡❧✐♥❡❛r
✭ǫ6= 0✮ t❛♠❜é♠ ❞❡s♣❡rt♦✉ ❛ ❛t❡♥çã♦ ❞❡ ✈ár✐♦s ♣❡sq✉✐s❛❞♦r❡s ❞❛ ár❡❛✱ ✈❡❥❛✱ ♣♦r ❡①❡♠♣❧♦✱
❬✸✱ ✶✾✱ ✷✶✱ ✷✸✱ ✷✹✱ ✷✻✱ ✷✼✱ ✹✻❪ ❡ s✉❛s r❡❢❡rê♥❝✐❛s✳ ◗✉❛♥❞♦ ❝♦♠♣❛r❛♠♦s ♦ ❡st✉❞♦ ❞♦ ❝❛s♦ s❡♠✐❧✐♥❡❛r ❝♦♠ ♦ ❝❛s♦ q✉❛s❡❧✐♥❡❛r✱ ♦❜s❡r✈❛✲s❡ q✉❡ ♦ q✉❛s❡❧✐♥❡❛r ❛♣r❡s❡♥t❛ ❞✐✜❝✉❧❞❛❞❡s ❡①tr❛s q✉❛♥❞♦ q✉❡r❡♠♦s ❛♣❧✐❝❛r té❝♥✐❝❛s ✈❛r✐❛❝✐♦♥❛✐s✱ ✐st♦ é✱ ❛ss♦❝✐❛r ✉♠ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛ ❛♦ ♣r♦❜❧❡♠❛ ✭✷✮ ❡ ❢❛③❡r ✉♠ ❡st✉❞♦ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ ♣♦♥t♦s ❝rít✐❝♦s ❞♦ ♠❡s♠♦✳ ❊s♣❡❝✐✜❝❛♠❡♥t❡✱ ❞❡✈✐❞♦ ❛ ♣r❡s❡♥ç❛ ❞♦ t❡r♠♦ q✉❛s❡❧✐♥❡❛r ❡ ♥ã♦ ❝♦♥✈❡①♦ ∆(u2)u✱ ♥ã♦
♣♦❞❡♠♦s ❣❛r❛♥t✐r q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛ ✭✷✮ ❡st❡❥❛ ❜❡♠ ❞❡✜♥✐❞♦ s♦❜r❡ ♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ✉s✉❛✐s✱ ♣♦r ❝❛✉s❛ ❞♦ t❡r♠♦ ✐♥t❡❣r❛❧ RRNu2|∇u|2dx✱ ❡①❝❡t♦ q✉❛♥❞♦ N = 1
✭✈❡❥❛ ❬✹✶❪✮✳
❆q✉✐✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ✐♥✈❡st✐❣❛♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♥ã♦ ♥✉❧❛ ❡ ♥ã♦ ♥❡❣❛t✐✈❛ ♣❛r❛ ❛ s❡❣✉✐♥t❡ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s q✉❛s❡❧✐♥❡❛r❡s ❞♦ t✐♣♦ ✭✷✮✿
−∆u+V(|x|)u−ǫ[∆(u2)]u=Q(|x|)h(u), x∈RN,
u(x)→0 q✉❛♥❞♦ |x| → ∞,
✭P✮
❡♠ q✉❡ N ≥ 2✱ V, Q : (0,∞) → R sã♦ ♣♦t❡♥❝✐❛✐s ❝♦♥tí♥✉♦s ❡ ♥ã♦✲♥❡❣❛t✐✈♦s q✉❡
s❛t✐s❢❛③❡♠ ❤✐♣ót❡s❡s ❝♦♥✈❡♥✐❡♥t❡s ♥❛ ♦r✐❣❡♠ ❡ ♥♦ ✐♥✜♥✐t♦✱ ❡ h : R → R é ✉♠❛
❢✉♥çã♦ ❝♦♥tí♥✉❛ s❛t✐s❢❛③❡♥❞♦ ❝♦♥❞✐çõ❡s ❛❞❡q✉❛❞❛s ♣❛r❛ tr❛t❛r ❡st❛ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s✳ ❙❛❧✐❡♥t❛♠♦s q✉❡ ♦s tr❛❜❛❧❤♦s ❝✐t❛❞♦s ❛❝✐♠❛ ❛ss✉♠✐r❛♠ q✉❡ ♦ ♣♦t❡♥❝✐❛❧✱ ♥♦s ♣r♦❜❧❡♠❛s q✉❛s❡❧✐♥❡❛r❡s✱ s❛t✐s❢❛③✐❛ ❛s ❤✐♣ót❡s❡s
lim sup
|x|→0
V(x)<∞ ♦✉ lim inf
|x|→∞ V(x)>0.
❆té ♦ ♣r❡s❡♥t❡ ♠♦♠❡♥t♦✱ ♥ã♦ ❝♦♥❤❡❝❡♠♦s tr❛❜❛❧❤♦s q✉❡ ❧✐❞❛♠ ❝♦♠ ❛ ❡q✉❛çã♦ ✭✷✮ ❝✉❥♦ ♣♦t❡♥❝✐❛❧V ✈❡r✐✜❝❛ ❛♦ ♠❡s♠♦ t❡♠♣♦ ❛s ❝♦♥❞✐çõ❡s✿
✐✮ lim sup
|x|→0
V(x) = +∞ ✭s✐♥❣✉❧❛r ♥❛ ♦r✐❣❡♠✮❀
✐✐✮ lim inf
|x|→∞ V(x) = 0 ✭❛♥✉❧❛♥❞♦✲s❡ ♥♦ ✐♥✜♥✐t♦✮✳
❆s ♣r✐♥❝✐♣❛✐s ❞✐✜❝✉❧❞❛❞❡s ❡♥❝♦♥tr❛❞❛s ♥♦ ❡st✉❞♦ ❞♦ ♣r♦❜❧❡♠❛ ✭P✮ sã♦ ❛ ♣♦ssí✈❡❧ ♣❡r❞❛
❞❡ ❝♦♠♣❛❝✐❞❛❞❡✱ ✉♠❛ ✈❡③ q✉❡ ❡st❛♠♦s tr❛❜❛❧❤❛♥❞♦ ❡♠ t♦❞♦ ♦ RN ❡ ♣❡❧❛ ♣r❡s❡♥ç❛ ❞♦
t❡r♠♦ q✉❛s❡❧✐♥❡❛r ❡ ♥ã♦ ❝♦♥✈❡①♦ ∆(u2)u✱ ♦ q✉❛❧ ♥♦s ✐♠♣❡❞❡ ❞❡ ❛♣❧✐❝❛r ❞✐r❡t❛♠❡♥t❡
♠ét♦❞♦s ✈❛r✐❛❝✐♦♥❛✐s✱ ♣♦✐s ♦ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛ ❛ss♦❝✐❛❞♦ ❛ ✭P✮✱ ❣❡r❛❧♠❡♥t❡✱ ♥ã♦ ❡stá
❜❡♠ ❞❡✜♥✐❞♦ s♦❜r❡ ♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ✉s✉❛✐s✳ ❆❧é♠ ❞✐ss♦✱ t❡♠ ❛s ❝❛r❛❝t❡ríst✐❝❛s ♣ró♣r✐❛s ❞♦s ♣♦t❡♥❝✐❛✐s V ❡ Q q✉❡✱ ❝♦♠♦ ✈❡r❡♠♦s✱ ♣♦❞❡♠ s❡r s✐♥❣✉❧❛r❡s ♥❛ ♦r✐❣❡♠✱
❝♦❡r❝✐✈♦s ❡ ♣♦❞❡♠ s❡ ❛♥✉❧❛r ♥♦ ✐♥✜♥✐t♦✳ ❖ ❢❛t♦ ❞❡V ♥ã♦ s❡r ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❧✐♠✐t❛❞♦
✐♥❢❡r✐♦r♠❡♥t❡ ♣♦r ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ ❡ ♣♦❞❡r s❡r s✐♥❣✉❧❛r ♥❛ ♦r✐❣❡♠✱ ❢❛③ ❝♦♠ q✉❡ s✉r❥❛♠ ❞✐✜❝✉❧❞❛❞❡s ❡①tr❛s ♥❛s ❡st✐♠❛t✐✈❛s ♣❛r❛ s❡ ♦❜t❡r q✉❡ s♦❧✉çõ❡s ❞❡ ✭P✮ ❡st❡❥❛♠
❡♠ L∞
loc(RN)✳
❊st❡ tr❛❜❛❧❤♦ ❞❡ t❡s❡ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ três ❝❛♣ít✉❧♦s✳
◆♦ ❈❛♣ít✉❧♦ ✶✱ ❡st✉❞❛♠♦s ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦ ♥✉❧❛ ❡ ♥ã♦ ♥❡❣❛t✐✈❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭P✮ ❝♦♠N ≥3 ❡ǫ= 1✳ ◆❡st❡ ❝❛♣ít✉❧♦✱ ❛ss✉♠✐♠♦s q✉❡ ♦s ♣♦t❡♥❝✐❛✐s
V ❡Q s❛t✐s❢❛③❡♠ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿
(V1) V : (0,∞)→Ré ❝♦♥tí♥✉♦✱V(r)≥0♣❛r❛ t♦❞♦r >0❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡sa∈R
❡ a0 ≥ −2 t❛✐s q✉❡
0<lim inf
r→0+
V(r)
ra0 ≤lim sup
r→0+
V(r)
ra0 <∞ ❡ 0<lim infr→+∞
V(r)
ra ;
(Q1) Q : (0,∞) → R é ❝♦♥tí♥✉♦✱ Q(r) > 0 ♣❛r❛ t♦❞♦ r > 0 ❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡s
b, b0 ∈R ❝♦♠
✭✐✮ b0 > b q✉❛♥❞♦ b≥ −2, a≤ −2❀
✭✐✐✮ b0 ≥b, b0 >−2 q✉❛♥❞♦ b≤max{a,−2}✱
s❛t✐s❢❛③❡♥❞♦
0<lim inf
r→0+
Q(r)
rb0 ≤lim sup
r→0+
Q(r)
rb0 <∞ ❡ lim supr→+∞
Q(r)
rb <∞.
P❛r❛ ❡st❛❜❡❧❡❝❡r♠♦s ❛s ❤✐♣ót❡s❡s s♦❜r❡ ❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ h(s)✱ ✐♥tr♦❞✉③✐♠♦s ♦s
s❡❣✉✐♥t❡s ♥ú♠❡r♦s✿
α:=
2(N +b)
N −2 , s❡ b≥ −2 ❡ a≤ −2; 2, s❡ b≤max{a,−2};
❡
β := 2(N +b0)
N −2 .
❊st❡s ♥ú♠❡r♦s ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ ✐♠❡rsã♦ ❝♦♥tí♥✉❛ ❡ ❝♦♠♣❛❝t❛✱ ❝♦♠♦ ✈❡r❡♠♦s ♥♦ ❈❛♣ít✉❧♦ ✶✳ ❖❜s❡r✈❡ q✉❡✱ ♣❡❧❛ ❝♦♥❞✐çã♦ (V1)✱ 2 ≤ α ❡
♣❡❧❛ ❝♦♥❞✐çã♦(Q1)✱ α < β✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❡❞✐♠♦s ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s s♦❜r❡ h✿
(h1) h :R→R é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡ h(s) =o(|s|α−1) q✉❛♥❞♦ s→0❀
(h2) ❡①✐st❡♠ p∈(max{4, α+ 1},2β) ❡ C1 >0 t❛✐s q✉❡
|h(s)| ≤C1(1 +|s|p−1), ♣❛r❛ t♦❞♦ s∈R;
(h3) ❡①✐st❡ µ >2 t❛❧ q✉❡
0≤2µH(s) := 2µ
Z s
0
h(t)dt ≤sh(s), ♣❛r❛ t♦❞♦s ≥0.
❖ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡st❡ ❝❛♣ít✉❧♦ é ♦ s❡❣✉✐♥t❡✿
❚❡♦r❡♠❛ ✵✳✵✳✶ ❙✉♣♦♥❤❛ q✉❡ ❛s ❝♦♥❞✐çõ❡s (V1)✱ (Q1)❡ (h1)−(h3)sã♦ s❛t✐s❢❡✐t❛s ❝♦♠
b > −N+2
N ❡♠ (Q1)i ❡ b 6= −2 ❡♠ (Q1)ii✳ ❊♥tã♦✱ ♦ ♣r♦❜❧❡♠❛ ✭P✮ t❡♠ ✉♠❛ s♦❧✉çã♦
❢r❛❝❛ ♥ã♦ ♥✉❧❛ ❡ ♥ã♦ ♥❡❣❛t✐✈❛✳
P❛r❛ ❞❡♠♦♥str❛r ❡st❡ r❡s✉❧t❛❞♦✱ ✉s❛♠♦s ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ ✐♥tr♦❞✉③✐❞❛ ♥♦s tr❛❜❛❧❤♦s ❞❡ ❏✳ ▲✐✉✱ ❨✳ ❲❛♥❣ ❡ ❩✳✲◗✳ ❲❛♥❣ ❬✸✼❪ ❡ ▼✳ ❈♦❧✐♥ ❡ ▲✳ ❏❡❛♥❥❡❛♥ ❬✷✶❪✱ ❡ tr❛♥s❢♦r♠❛♠♦s ♦ ♣r♦❜❧❡♠❛ q✉❛s❡❧✐♥❡❛r ❡♠ ✉♠ ♣r♦❜❧❡♠❛ s❡♠✐❧✐♥❡❛r✱ ♦ q✉❛❧ t❡♠ ✉♠ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛ ❛ss♦❝✐❛❞♦ ❜❡♠ ❞❡✜♥✐❞♦ ❡ ●❛t❡❛✉①✲❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❖r❧✐❝③ ❝♦♠ ♣❡s♦✳ ❊st❛❜❡❧❡❝❡♠♦s ✉♠ r❡s✉❧t❛❞♦ q✉❡ r❡❧❛❝✐♦♥❛ ♦s ♣♦♥t♦s ❝rít✐❝♦s ❞❡ss❡ ❢✉♥❝✐♦♥❛❧ ❝♦♠ ❛s s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛ q✉❛s❡❧✐♥❡❛r✳ ❉❛í✱ ♠♦str❛♠♦s q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛ s❛t✐s❢❛③ ❛s ❤✐♣ót❡s❡s ❣❡♦♠étr✐❝❛s ❞❡ ✉♠❛ ✈❡rsã♦ ❞♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❡ ❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ❞❡ P❛❧❛✐s✲❙♠❛❧❡✳ ❆ss✐♠✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ♣♦♥t♦ ❝rít✐❝♦ ❞♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ é✱ ❞❡ ❢❛t♦✱ ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦ ♥✉❧❛ ❡ ♥ã♦ ♥❡❣❛t✐✈❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭P✮✳ P❛r❛ ✐st♦✱ ❢♦✐ ♥❡❝❡ssár✐♦ ♦❜t❡r♠♦s ✉♠❛ ✈❡rsã♦ ❞♦
r❡s✉❧t❛❞♦ ❞❡ r❡❣✉❧❛r✐❞❛❞❡ ❞♦ t✐♣♦ ❇ré③✐s✲❑❛t♦ ❡ ✉♠ r❡s✉❧t❛❞♦ ❞♦ t✐♣♦ ♣r✐♥❝í♣✐♦ ❞❡ ❝r✐t✐❝❛❧✐❞❛❞❡ s✐♠étr✐❝❛ ♣❛r❛ ♦ ♥♦ss♦ ❝♦♥t❡①t♦✳
◆♦ ❈❛♣ít✉❧♦ ✷✱ ❡st✉❞❛♠♦s ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦ ♥✉❧❛ ❡ ♥ã♦ ♥❡❣❛t✐✈❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭P✮✱ ❡♠ ❞✐♠❡♥sã♦ N = 2❡ ❝♦♠ ǫ= 1✳ ◆❡st❡ ❝❛♣ít✉❧♦✱ ❛ss✉♠✐♠♦s ❛s
s❡❣✉✐♥t❡s ❤✐♣ót❡s❡s s♦❜r❡ ♦s ♣♦t❡♥❝✐❛✐sV ❡ Q✿
(V2) V : (0,∞) → R é ❝♦♥tí♥✉♦✱ V(r) > 0 ♣❛r❛ t♦❞♦ r > 0 ❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡s
a0 >−2 ❡ a >−2 t❛✐s q✉❡
0<lim inf
r→0
V(r)
ra0 ≤lim sup
r→0
V(r)
ra0 <∞
❡
0<lim inf
r→+∞
V(r)
ra ≤lim sup
r→+∞
V(r)
ra <∞.
(Q2) Q : (0,∞) → R é ❝♦♥tí♥✉♦✱ Q(r) > 0 ♣❛r❛ t♦❞♦ r > 0 ❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡s
−2< b < a ❡ −2< b0 ≤0 s❛t✐s❢❛③❡♥❞♦
0<lim inf
r→0
Q(r)
rb0 ≤lim supr→0
Q(r)
rb0 <∞ ❡ lim supr→+∞
Q(r)
rb <∞.
❉❡st❛❝❛♠♦s q✉❡✱ ♥♦ r❡❢❡r✐❞♦ ❝❛♣ít✉❧♦✱ ❛ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡ h ♣♦ss✉✐ ❝r❡s❝✐♠❡♥t♦ ❝rít✐❝♦
❡①♣♦♥❡♥❝✐❛❧ ♣❛r❛ ❡st❛ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s q✉❛s❡❧✐♥❡❛r❡s✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ h
s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿
(h1) ✭❈r❡s❝✐♠❡♥t♦ ❝rít✐❝♦ ❡①♣♦♥❡♥❝✐❛❧✮ ❊①✐st❡ λ0 >0t❛❧ q✉❡
lim
s→∞
h(s)
eλs4 =
0, ♣❛r❛ t♦❞♦λ > λ0
+∞, ♣❛r❛ t♦❞♦λ < λ0;
(h2) lim
s→0h(s)/s= 0;
(h3) ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ µ >2t❛❧ q✉❡ ♣❛r❛ t♦❞♦ s >0✈❛❧❡
0≤2µH(s) := 2µ
Z s
0
h(t)dt≤sh(s);
(h4) ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ξ >0t❛❧ q✉❡
H(t)≥ ξ 4t
4, ♣❛r❛ t♦❞♦ t
≥0.
❖ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡st❡ ❝❛♣ít✉❧♦ é ❡♥✉♥❝✐❛❞♦ ❝♦♠♦ s❡❣✉❡✿
❚❡♦r❡♠❛ ✵✳✵✳✷ ❙✉♣♦♥❤❛ q✉❡ ❛s ❝♦♥❞✐çõ❡s (V2)✱ (Q2) ❡ (h1)−(h4) s❡❥❛♠ s❛t✐s❢❡✐t❛s ❡
q✉❡✱ ❡♠(h4)✱
ξ > λ0µ(3π+kVkL1(B2))
2
4πkQkL1(B1)[1 + (b0/2)](µ−2)
+ 12π
kQkL1(B1)
.
❊♥tã♦✱ ♦ ♣r♦❜❧❡♠❛ ✭P✮ t❡♠ ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦ ♥✉❧❛ ❡ ♥ã♦ ♥❡❣❛t✐✈❛✳
P❛r❛ ❞❡♠♦♥str❛r ❡st❡ r❡s✉❧t❛❞♦✱ ❛ss✐♠ ❝♦♠♦ ♥♦ ❈❛♣ít✉❧♦ ✶✱ ✉s❛♠♦s ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ ✐♥tr♦❞✉③✐❞❛ ❡♠ ❬✷✶❪ ❡ ❬✸✼❪ ♣❛r❛ tr❛♥s❢♦r♠❛r ♦ ♣r♦❜❧❡♠❛ q✉❛s❡❧✐♥❡❛r ❡♠ ♦✉tr♦ s❡♠✐❧✐♥❡❛r✱ ♦ q✉❛❧ ♠♦str❛♠♦s q✉❡ ♣♦ss✉✐ ✉♠ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛ ❛ss♦❝✐❛❞♦ ❜❡♠ ❞❡✜♥✐❞♦ ❡ ●❛t❡❛✉① ❞✐❢❡r❡♥❝✐á✈❡❧ s♦❜r❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❖r❧✐❝③ ❝♦♠ ♣❡s♦✳ P❛r❛ ✐st♦✱ ❢♦✐ ❡ss❡♥❝✐❛❧ ♦❜t❡r♠♦s ✉♠❛ ✈❡rsã♦ ❞❛ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❚r✉❞✐♥❣❡r✲▼♦s❡r ♣❛r❛ ❡st❡ ❡s♣❛ç♦ ❞❡ ❖r❧✐❝③ ✭✈❡❥❛ ♠❛✐s ❞❡t❛❧❤❡s ♥♦ ❞❡❝♦rr❡r ❞♦ ❈❛♣ít✉❧♦ ✷✮✳ ❉❡ss❛ ❢♦r♠❛✱ ❢♦✐ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ s❡♠✐❧✐♥❡❛r s❛t✐s❢❛③✐❛ ❛s ❝♦♥❞✐çõ❡s ❣❡♦♠étr✐❝❛s ❞❡ ✉♠❛ ✈❡rsã♦ ❞♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✳ ❊♠ s❡❣✉✐❞❛✱ ♠♦str❛♠♦s q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ♣❛r❛ ♥í✈❡✐s ❛❜❛✐①♦ ❞❡ ✉♠ ✈❛❧♦r ❡s♣❡❝í✜❝♦✳ ❯s❛♥❞♦ ❛ ❤✐♣ót❡s❡ (h4)✱ ♣r♦✈❛♠♦s q✉❡ ♦ ♥í✈❡❧ ❞♦ ♣❛ss♦ ❞❛ ♠♦♥t❛♥❤❛ ❡stá
❡♥tr❡ ♦s ♥í✈❡✐s ♦♥❞❡ s❡ t❡♠ ❝♦♠♣❛❝✐❞❛❞❡ ❡✱ ♣♦rt❛♥t♦✱ ♦❜t❡♠♦s ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ♥ã♦ ♥✉❧♦ ♥♦ ♥í✈❡❧ ❞♦ ♣❛ss♦ ❞❛ ♠♦♥t❛♥❤❛✳ ❉❛í✱ ❝♦♠ ❛ ♦❜t❡♥çã♦ ❞❡ ✉♠ r❡s✉❧t❛❞♦ ❞♦ t✐♣♦ ♣r✐♥❝í♣✐♦ ❞❛ ❝r✐t✐❝❛❧✐❞❛❞❡ s✐♠étr✐❝❛✱ ❝♦♥s❡❣✉✐♠♦s ♠♦str❛r q✉❡ ❡st❡ ♣♦♥t♦ ❝rít✐❝♦ é ✉♠❛ s♦❧✉çã♦ ♥ã♦ ♥✉❧❛ ❡ ♥ã♦ ♥❡❣❛t✐✈❛ ♣❛r❛ ✭P✮✳
Pr♦❜❧❡♠❛s ❡❧í♣t✐❝♦s ❝♦♠ ❝r❡s❝✐♠❡♥t♦ ❝rít✐❝♦ ❡①♣♦♥❡♥❝✐❛❧ ❡♠ R2✱ ❡♥✈♦❧✈❡♥❞♦ ❛
❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❚r✉❞✐♥❣❡r✲▼♦s❡r✱ t❡♠ r❡❝❡❜✐❞♦ ❛ ❛t❡♥çã♦ ❞❡ ♠✉✐t♦s ♣❡sq✉✐s❛❞♦r❡s ❛♣ós ♦s tr❛❜❛❧❤♦s ♣✐♦♥❡✐r♦s ❞❡ ◆✳ ❚r✉❞✐♥❣❡r ❬✸✶❪ ❡ ❏✳ ▼♦s❡r ❬✹✵❪✳ P❛r❛ ♣r♦❜❧❡♠❛s s❡♠✐❧✐♥❡❛r❡s ❡♠ R2 ❝♦♠ ❝r❡s❝✐♠❡♥t♦ ❝rít✐❝♦✱ ✈❡❥❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❬✺✱ ✶✼✱ ✷✷✱ ✷✾❪ ❡ s✉❛s
r❡❢❡rê♥❝✐❛s✳ ❈♦♠ r❡s♣❡✐t♦ ❛ ♣r♦❜❧❡♠❛s q✉❛s❡❧✐♥❡❛r❡s ❡♥✈♦❧✈❡♥❞♦ ❝r❡s❝✐♠❡♥t♦ ❝rít✐❝♦ ❡①♣♦♥❡♥❝✐❛❧✱ ♣♦❞❡♠♦s ❝✐t❛r ❬✷✹✱ ✷✺✱ ✷✼✱ ✷✽✱ ✸✾✱ ✺✶❪✳ ❱❛❧❡ s❛❧✐❡♥t❛r q✉❡ ❡st❡s tr❛❜❛❧❤♦s ❝♦♥s✐❞❡r❛♠ ❛♣❡♥❛s ♣♦t❡♥❝✐❛✐s V ❧✐♠✐t❛❞♦s ✐♥❢❡r✐♦r♠❡♥t❡ ♣♦r ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛
❡ q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠ ♥♦ ✐♥✜♥✐t♦✳ P♦rt❛♥t♦✱ ♥♦ss♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ♥♦ ❈❛♣ít✉❧♦ ✷ ♠❡❧❤♦r❛ ❡ ❝♦♠♣❧❡♠❡♥t❛ ♦s r❡❢❡r✐❞♦s tr❛❜❛❧❤♦s✱ ✉♠❛ ✈❡③ q✉❡ ♦❜t❡♠♦s s♦❧✉çã♦ ♥♦ ♥í✈❡❧ ❞♦ ♣❛ss♦ ❞❛ ♠♦♥t❛♥❤❛✳
◆♦ ❈❛♣ít✉❧♦ ✸✱ ✐♥✈❡st✐❣❛♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦ ♥✉❧❛ ❡ ♥ã♦ ♥❡❣❛t✐✈❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭P✮✱ ❝✉❥❛ ❞✐♠❡♥sã♦ N ≥ 3 ❡ ǫ = −κ/2 ❝♦♠ κ > 0✳ ❉❡ss❛ ❢♦r♠❛✱ ♦
♣r♦❜❧❡♠❛ ✭P✮ t♦r♥❛✲s❡
−∆u+V(|x|)u+ κ 2[∆(u
2)]u=Q(
|x|)h(u), x∈RN,
u(x)→0 q✉❛♥❞♦ |x| → ∞.
✭✸✮
◆❡st❡ ❝❛♣ít✉❧♦✱ ❛ss✉♠✐♠♦s q✉❡ ♦s ♣♦t❡♥❝✐❛✐sV ❡ Qs❛t✐s❢❛③❡♠ ❛s ❤✐♣ót❡s❡s✿
(V3) V : (0,∞) → R é ❝♦♥tí♥✉♦✱ V(r) ≥ 0 ♣❛r❛ t♦❞♦ r > 0 ❡ ❡①✐t❡♠ ❝♦♥st❛♥t❡s
a >−2(N −1)❡ a0 ≥ −2 t❛✐s q✉❡
0<lim inf
r→0
V(r)
ra0 ≤lim supr→0
V(r)
ra0 <∞
❡
0<lim inf
r→+∞
V(r)
ra ≤lim sup
r→+∞
V(r)
ra <∞.
(Q3) Q : (0,∞) → R é ✉♠ ♣♦t❡♥❝✐❛❧ ❝♦♥tí♥✉♦✱ Q(r) > 0 ♣❛r❛ t♦❞♦ r > 0 ❡ ❡①✐st❡♠
❝♦♥st❛♥t❡s b0 >−2 ❡b ∈R ❝♦♠
✐✮ b =a < b0 s❡ a <−2❀
✐✐✮ b ≤a ❡b0 > b s❡ a≥ −2
s❛t✐s❢❛③❡♥❞♦ ♦s ❧✐♠✐t❡s
0<lim inf
r→0
Q(r)
rb0 ≤lim supr→0
Q(r)
rb0 <∞ ❡ lim supr→+∞
Q(r)
rb <∞.
❊q✉❛çõ❡s ❞❡ ❙❝❤rö❞✐♥❣❡r ❞♦ t✐♣♦ ❛❝✐♠❛✱ ❝♦♠ ♦ ♣❛râ♠❡tr♦ κ > 0✱ ❞❡s❡♠♣❡♥❤❛
✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ❡♠ ✈ár✐♦s ❞♦♠í♥✐♦s ❞❛ ❋ís✐❝❛✱ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ót✐❝❛ ♥ã♦ ❧✐♥❡❛r ❡ ❢ís✐❝❛ ❞♦s ♣❧❛s♠❛s ✭✈❡❥❛ ♦s tr❛❜❛❧❤♦s ❬✶✻✱ ✸✵✱ ✸✺❪✮✳ ❘❡❝❡♥t❡♠❡♥t❡✱ ✈ár✐♦s ♣❡sq✉✐s❛❞♦r❡s ❛❜♦r❞❛r❛♠ ♣r♦❜❧❡♠❛s ❞♦ t✐♣♦ ✭✸✮✱ ♣♦❞❡♠♦s ❝✐t❛r ❬✷✱ ✻✱ ✽✱ ✺✵❪ ❡ s✉❛s r❡❢❡rê♥❝✐❛s✳ ❙❛❧✐❡♥t❛♠♦s✱ q✉❡ ❡st❡s tr❛❜❛❧❤♦s✱ ❝♦♠ ❡①❝❡çã♦ ❞❡ ❬✷❪✱ ❛❜♦r❞❛r❛♠ s♦♠❡♥t❡ ♣♦t❡♥❝✐❛✐s ❧✐♠✐t❛❞♦s ✐♥❢❡r✐♦r♠❡♥t❡ ♣♦r ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ ❡ ❛ss✐♥tót✐❝♦s ❛ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ ♥♦ ✐♥✜♥✐t♦✳ ▼❛s✱ ❆✐r❡s ❡♠ ❬✷❪ ♥ã♦ ❧✐❞❛ ❝♦♠ ♣♦t❡♥❝✐❛✐s s✐♥❣✉❧❛r❡s✳
❆❣♦r❛✱ ❞❡✜♥✐♥❞♦ ♦ ♥ú♠❡r♦β= 2(N+b0)/(N−2)✱ s❡❣✉❡ q✉❡β >2♣♦✐sb0 >−2✳
❆ss✐♠✱ ❝♦♠ r❡❧❛çã♦ ❛ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡h✱ ♣❡❞✐♠♦s ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿
(h1) h :R→R é ❝♦♥tí♥✉❛ ❡ lims→0h(s)/s= 0❀
(h2) ❊①✐st❡♠ p∈(2, β)❡ C1 >0 t❛✐s q✉❡
|h(s)| ≤C1(1 +|s|p−1) ∀ s∈R;
(h3) ❊①✐st❡ µ >2 t❛❧ q✉❡ ♣❛r❛ t♦❞♦s≥0✱
0≤µH(s) :=µ
Z s
0
h(t)dt ≤sh(s) ∀ s≥0.
❚❡♦r❡♠❛ ✵✳✵✳✸ ❙✉♣♦♥❤❛ q✉❡ ❛s ❝♦♥❞✐çõ❡s (V3)✱ (Q3) ❡ (h1)− (h3) sã♦ s❛t✐s❢❡✐t❛s✳
❊♥tã♦✱ ❡①✐st❡ κ0 > 0 t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ 0 < κ < κ0✱ ♦ ♣r♦❜❧❡♠❛ ✭✸✮ t❡♠ ✉♠❛ s♦❧✉çã♦
❢r❛❝❛ uκ ♥ã♦ ♥✉❧❛ ❡ ♥ã♦ ♥❡❣❛t✐✈❛✳
◆❛ ♣r♦✈❛ ❞❡st❡ t❡♦r❡♠❛✱ s✉r❣❡ ❛ ♠❡s♠❛ ❞✐✜❝✉❧❞❛❞❡ ❝♦♠♦ ♥♦s ❝❛s♦s ❛♥t❡r✐♦r❡s✳ ❖ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛ ❛ss♦❝✐❛❞♦ ♥ã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ♥♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ✉s✉❛✐s ♣♦r ❝❛✉s❛ ❞♦ t❡r♠♦ RRNu2|∇u|2dx✳ ▼❛s✱ ❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ ✉s❛❞❛ ♥♦s ❝❛♣ít✉❧♦s
❛♥t❡r✐♦r❡s ♥ã♦ ❢✉♥❝✐♦♥❛ ♥❡st❡ ❝❛s♦✱ ♣♦✐s ♦ t❡r♠♦1−κu2 ♣♦❞❡ ♠✉❞❛r ❞❡ s✐♥❛❧✳ ❙❡❣✉✐♥❞♦
✐❞é✐❛s ✐♥tr♦❞✉③✐❞❛s ♣♦r ❨✳ ❙❤❡♥ ❡ ❨✳ ❲❛♥❣ ❡♠ ❬✹✻❪ ❡ ❈✳ ❆❧✈❡s✱ ❨✳ ❙❤❡♥ ❡ ❨✳ ❲❛♥❣ ❡♠ ❬✻❪✱ ❝♦♥s✐❞❡r❛♠♦s ✉♠ ♣r♦❜❧❡♠❛ ❛✉①✐❧✐❛r ❡ ✉♠❛ ♦✉tr❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧✳ ❖ ♥♦✈♦ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛✱ ♦ q✉❛❧ ❡stá ❛ss♦❝✐❛❞♦ ❛ ✉♠ ♣r♦❜❧❡♠❛ s❡♠✐❧✐♥❡❛r✱ ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ❡ é ❞❡ ❝❧❛ss❡ C1 ♥✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❛❞❡q✉❛❞♦ ❡ ♠♦str❛♠♦s q✉❡ ♣♦♥t♦s ❝rít✐❝♦s
❞❡st❡ ❢✉♥❝✐♦♥❛❧✱ ❝♦♠ ♥♦r♠❛ L∞ ♠❡♥♦r q✉❡ 1/√3κ✱ sã♦ s♦❧✉çõ❡s ❢r❛❝❛s ❞❡ ✭✸✮✳ P❛r❛
✐st♦✱ ✉s❛♠♦s ♦ ♠ét♦❞♦ ❞❡ ✐t❡r❛çã♦ ❞❡ ▼♦s❡r ♣❛r❛ ❡st✐♠❛r ❛ ♥♦r♠❛ L∞ ❞❛ s♦❧✉çã♦
❡✱ ♥♦✈❛♠❡♥t❡✱ é ❡ss❡♥❝✐❛❧ ♦❜t❡r♠♦s ✉♠ r❡s✉❧t❛❞♦ ❞♦ t✐♣♦ ♣r✐♥❝í♣✐♦ ❞❡ ❝r✐t✐❝❛❧✐❞❛❞❡ s✐♠étr✐❝❛ ♣❛r❛ ❡st❡ ❝❛s♦✳
❈♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ♥ã♦ ✜❝❛r♠♦s r❡❝♦rr❡♥❞♦ à ■♥tr♦❞✉çã♦ ❡ ❞❡ t♦r♥❛r ♦s ❝❛♣ít✉❧♦s ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❡♥✉♥❝✐❛r❡♠♦s ♥♦✈❛♠❡♥t❡✱ ❡♠ ❝❛❞❛ ❝❛♣ít✉❧♦✱ ♦s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s✱ ❜❡♠ ❝♦♠♦✱ ❛s ❤✐♣ót❡s❡s s♦❜r❡ ♦s ♣♦t❡♥❝✐❛✐s V ❡ Q ❡ s♦❜r❡ ❛ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡ h✳ ◆♦
❞❡❝♦rr❡r ❞❡ t♦❞♦ ❡st❡ tr❛❜❛❧❤♦✱ ❛ ♠❡♥♦s q✉❡ s❡❥❛ ❡①♣❧✐❝✐t❛❞♦✱ ❛s ❝♦♥✈❡r❣ê♥❝✐❛s s❡rã♦ ❡♠ n∈N✱ q✉❛♥❞♦ n→ ∞✳
❈❛♣ít✉❧♦ ✶
❊q✉❛çõ❡s q✉❛s❡❧✐♥❡❛r❡s ❝♦♠ ♣♦t❡♥❝✐❛✐s
s✐♥❣✉❧❛r❡s ♦✉ s❡ ❛♥✉❧❛♥❞♦ ♥♦ ✐♥✜♥✐t♦
❡♠ ❞✐♠❡♥sã♦
N
≥
3
❖ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❡st❡ ❝❛♣ít✉❧♦ ❝♦♥s✐st❡ ❡♠ ❡st❛❜❡❧❡❝❡r ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦ ♥✉❧❛ ❡ ♥ã♦ ♥❡❣❛t✐✈❛ ♣❛r❛ ✭P✮ ❝♦♠ǫ = 1❡ ❡♠ ❞✐♠❡♥sã♦
N ≥3✳ ❖✉ s❡❥❛✱ ❡st✉❞❛♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ❢r❛❝❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛
−∆u+V(|x|)u−[∆(u2)]u=Q(|x|)h(u), x∈RN,
u(x)→0, q✉❛♥❞♦ |x| → ∞.
✭P1✮
❆q✉✐✱ ❛ss✉♠✐♠♦s q✉❡ ♦s ♣♦t❡♥❝✐❛✐s V ❡ Q s❛t✐s❢❛③❡♠ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿
(V1) V : (0,∞)→Ré ❝♦♥tí♥✉♦✱V(r)≥0♣❛r❛ t♦❞♦r >0❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡sa∈R
❡ a0 ≥ −2 t❛✐s q✉❡
0<lim inf
r→0+
V(r)
ra0 ≤lim sup
r→0+
V(r)
ra0 <∞ ❡ 0<lim infr→+∞
V(r)
ra ;
(Q1) Q : (0,∞) → R é ❝♦♥tí♥✉♦✱ Q(r) > 0 ♣❛r❛ t♦❞♦ r > 0 ❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡s
b, b0 ∈R ❝♦♠
✭✐✮ b0 > b q✉❛♥❞♦ b≥ −2, a≤ −2❀
s❛t✐s❢❛③❡♥❞♦
0<lim inf
r→0+
Q(r)
rb0 ≤lim sup
r→0+
Q(r)
rb0 <∞ ❡ lim supr→+∞
Q(r)
rb <∞.
P❛r❛ ✉♠ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦ ❞❛ ✈❛r✐❛çã♦ ❞❛s ❝♦♥st❛♥t❡s b ❡ b0✱ ✈❡❥❛ ♦s ❣rá✜❝♦s
❛❜❛✐①♦
❖❜s❡r✈❡ q✉❡ s♦❜ ❛s ❝♦♥❞✐çõ❡s(V1)❡(Q1)✱ ♦s ♣♦t❡♥❝✐❛✐sV ❡Q♣♦❞❡♠ s❡r s✐♥❣✉❧❛r
♥❛ ♦r✐❣❡♠ ♦✉ ♣♦❞❡♠ s❡ ❛♥✉❧❛r ♥♦ ✐♥✜t✐t♦✳
❊①❡♠♣❧♦ ✶✳✵✳✹ ❙❡❥❛♠ V, Q: (0,∞) → R ❞❡✜♥✐❞♦s ♣♦r V(r) = 4r−1 ❡ Q(r) = 3r−32✳ ❋❛❝✐❧♠❡♥t❡✱ s❡ ❝♦♥st❛t❛ q✉❡V ❡Qs❛t✐s❢❛③❡♠ ❛s ❝♦♥❞✐çõ❡s(V1)❡(Q1)✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❖❜s❡r✈❛çã♦ ✶✳✵✳✺ ❈♦♠♦ r❡ss❛❧t❛❞♦ ♥❛ ■♥tr♦❞✉çã♦✱ ♣r♦❜❧❡♠❛s ❞♦ t✐♣♦ ✭P1✮
❞❡s♣❡rt❛r❛♠ ❛ ❛t❡♥çã♦ ❞❡ ✈ár✐♦s ♣❡sq✉✐s❛❞♦r❡s ❞❛ ár❡❛✱ ♣♦❞❡♠♦s ❝✐t❛r✱ ♣♦r ❡①❡♠♣❧♦✱ ❬✸✱ ✶✾✱ ✷✶✱ ✷✹✱ ✷✻✱ ✷✼✱ ✹✻❪ ❡ s✉❛s r❡❢❡rê♥❝✐❛s✳ ▼❛s✱ ❛té ♦♥❞❡ s❛❜❡♠♦s✱ ♥ã♦ ❤á ♣r♦❜❧❡♠❛s q✉❛s❡❧✐♥❡❛r❡s ❞♦ t✐♣♦ ❛❝✐♠❛ ❛❜♦r❞❛♥❞♦ ♣♦t❡♥❝✐❛✐s s✐♥❣✉❧❛r❡s ❡ q✉❡ s❡ ❛♥✉❧❛♠ ♥♦ ✐♥✜♥✐t♦✳ ❆té ♥♦ ❝❛s♦Q≡1✱ ♦ ♥♦ss♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ ❝❛♣ít✉❧♦ ❝♦♠♣❧❡♠❡♥t❛ ♦s tr❛❜❛❧❤♦s
❝✐t❛❞♦s✳
❆ ♣❛rt✐r ❞❛s ❝♦♥st❛♥t❡s a, a0, b ❡ b0✱ ✐♥tr♦❞✉③✐♠♦s ♦s s❡❣✉✐♥t❡s ♥ú♠❡r♦s✿
α:=
2(N +b)
N −2 , s❡ b≥ −2 ❡ a≤ −2; 2, s❡ b≤max{a,−2};
❡
β := 2(N +b0)
N −2 .
❖❜s❡r✈❛çã♦ ✶✳✵✳✻ ❙❡ b0 ≥0 ❡♥tã♦ β ≥2∗✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ b0 <0 ❡♥tã♦β <2∗✳
❉❛s ❝♦♥❞✐çõ❡s (V1) ❡ (Q1)✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r ❢❛❝✐❧♠❡♥t❡ q✉❡ α < β✳ ❆q✉✐✱
❛ss✉♠✐♠♦s h s❛t✐s❢❛③❡♥❞♦ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿
(h1) h :R→R é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡ h(s) =o(|s|α−1) q✉❛♥❞♦ s→0❀
(h2) ❡①✐st❡♠ p∈(α,2β) ❡C1 >0 t❛✐s q✉❡
|h(s)| ≤C1(1 +|s|p−1), ♣❛r❛ t♦❞♦ s∈R;
(h3) ❡①✐st❡ µ >2 t❛❧ q✉❡
0≤2µH(s) := 2µ
Z s
0
h(t)dt ≤sh(s), ♣❛r❛ t♦❞♦s ≥0.
◆❡st❡ ❝❛♣ít✉❧♦✱ ✉♠❛ ❢✉♥çã♦ u : RN → R é ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✭P
1✮ s❡
u∈D1,2(RN)∩L∞
loc(RN) ❡ s❛t✐s❢❛③ ❛ ✐❣✉❛❧❞❛❞❡ Z
RN
(1 + 2u2)∇u∇φ dx+
Z
RN
[2u|∇u|2+V(|x|)u]φ dx=
Z
RN
Q(|x|)h(u)φ dx, ✭✶✳✶✮
♣❛r❛ t♦❞♦φ∈C∞
0 (RN)✳
❖❜s❡r✈❡ q✉❡ ❛ ❢✉♥çã♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛ é ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ♣❛r❛ ✭P1✮✳ ▼❛s✱
♦ ♥♦ss♦ ✐♥t❡r❡ss❡ ❝♦♥s✐st❡ ♥❛ ♦❜t❡♥çã♦ ❞❡ s♦❧✉çã♦ ♥ã♦ ♥✉❧❛ ❡ ♥ã♦ ♥❡❣❛t✐✈❛✳ P❛r❛ ❛♣r❡s❡♥t❛r♠♦s ♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡st❡ ❝❛♣ít✉❧♦✱ ✐♥tr♦❞✉③✐♠♦s ❛❧❣✉♥s ❡s♣❛ç♦s ♥♦r♠❛❞♦s q✉❡ ❞❡s❡♠♣❡♥❤❛♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ❡♠ ♥♦ss♦s ❡st✉❞♦s✳ P❛r❛ t♦❞♦ 1 ≤
q <∞✱ ❞❡✜♥✐♠♦s ♦ ❡s♣❛ç♦ ❞❡ ▲❡❜❡s❣✉❡ ❝♦♠ ♦ ♣❡s♦Q ♣♦r Lq(RN;Q) =
u:RN →R; u é ♠❡♥s✉rá✈❡❧ ❡ Z
RN
Q(|x|)|u|q dx <∞
,
♠✉♥✐❞♦ ❝♦♠ ❛ ♥♦r♠❛
kukq,Q= Z
RN
Q(|x|)|u|q dx
1/q
.
❚❛♠❜é♠ ❞❡✜♥✐♠♦s ♦s ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt
X =
u∈D1,2(RN); Z
RN
V(|x|)u2dx <∞
❡
Xrad ={u∈X :u(x) = u(gx) ∀g ∈O(N)}
=
u∈D1rad,2(RN); Z
RN
V(|x|)u2dx <∞
,
♠✉♥✐❞♦s ❝♦♠ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ hu, vi=
Z
RN
(∇u∇v+V(|x|)uv)dx
❡ ❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ♥♦r♠❛
kukX = Z
RN
[|∇u|2+V(|x|)u2]dx
1
2
.
◆♦ q✉❡ s❡❣✉❡✱ ❡♥✉♥❝✐❛♠♦s ♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ ❝❛♣ít✉❧♦✳
❚❡♦r❡♠❛ ✶✳✵✳✼ ❙✉♣♦♥❤❛ q✉❡ ❛s ❝♦♥❞✐çõ❡s (V1)✱ (Q1)❡ (h1)−(h3)sã♦ s❛t✐s❢❡✐t❛s ❝♦♠
b >−NN+2 ❡♠ (Q1)i✱ b 6= −2 ❡♠ (Q1)ii ❡ p > max{4, α+ 1}✳ ❊♥tã♦✱ ♦ ♣r♦❜❧❡♠❛ ✭P1✮
t❡♠ ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦ ♥✉❧❛ ❡ ♥ã♦ ♥❡❣❛t✐✈❛ u∈Xrad✳
❖❜s❡r✈❛çã♦ ✶✳✵✳✽ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❚❡♦r❡♠❛ ✶✳✵✳✼✱ ♦❜t❡♠♦s ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦ ♥✉❧❛ ❡ ♥ã♦ ♥❡❣❛t✐✈❛ ♣❛r❛ ❛ s❡❣✉✐♥t❡ ❝❧❛ss❡s ❞❡ ♣r♦❜❧❡♠❛s q✉❛s❡❧✐♥❡❛r❡s
−∆u+ 1
|x|γu−[∆(u
2)]u=Q(
|x|)|u|p−2u, x∈RN \ {0},
u(x)→0 q✉❛♥❞♦ |x| → ∞,
❝♦♠ 0 < γ ≤ 2✱ max{4, α + 1} < p < 2β ❡ Q s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ (Q1)✳
◗✉❛♥❞♦ γ = 2✱ ♦ ♣♦t❡♥❝✐❛❧ V(|x|) = 1/|x|2 é ❝♦♥❤❡❝✐❞♦ ♥❛ ❧✐t❡r❛t✉r❛ ❝♦♠♦ P♦t❡♥❝✐❛❧
❞❡ ❍❛r❞②✳ ❊st✉❞♦s ❡♥✈♦❧✈❡♥❞♦ ♣♦t❡♥❝✐❛✐s V✱ ❝♦♠♦ ♥❛ ♦❜s❡r✈❛çã♦ ❛❝✐♠❛✱ ♥♦ ❝❛s♦
s❡♠✐❧✐♥❡❛r (κ = 0)✱ ❢♦r❛♠ ❛❜♦r❞❛❞♦s ❡♠ ❞✐✈❡rs♦s tr❛❜❛❧❤♦s✱ ✈❡❥❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❬✼✱ ✶✵❪
❡ s✉❛s r❡❢❡rê♥❝✐❛s✳
❯♠❛ ❞❛s ❞✐✜❝✉❧❞❛❞❡s ❡♥❝♦♥tr❛❞❛ ♥♦ ❡st✉❞♦ ❞♦ ♣r♦❜❧❡♠❛ ✭P1✮ é ❛ ♣♦ssí✈❡❧ ♣❡r❞❛
❞❡ ❝♦♠♣❛❝✐❞❛❞❡✱ ✉♠❛ ✈❡③ q✉❡ ❡st❛♠♦s tr❛❜❛❧❤❛♥❞♦ ❡♠ t♦❞♦ ♦ RN✳ ❆❧é♠ ❞✐ss♦✱
♥ã♦ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♠ét♦❞♦s ✈❛r✐❛❝✐♦♥❛✐s ❞✐r❡t❛♠❡♥t❡ ❛♦ ♣r♦❜❧❡♠❛ ❡st✉❞❛❞♦✱ ♣♦✐s ♦ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ✭P1✮✱ ❞❛❞♦ ♣♦r
J(u) = 1 2
Z
RN
(1 + 2u2)|∇u|2dx+1 2
Z
RN
V(|x|)u2dx−
Z
RN
Q(|x|)H(u)dx, ✭✶✳✷✮
♥ã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ♥♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ✉s✉❛✐s✱ ❞❡✈✐❞♦ ❛ ♣r❡s❡♥ç❛ ❞♦ t❡r♠♦
R
RNu2|∇u|2dx✳ ❖❜s❡r✈❡ q✉❡ s♦❧✉çõ❡s ❢r❛❝❛s ❞❡ ✭P1✮ ♣♦❞❡♠ s❡r ✈✐st❛s ❝♦♠♦ ♣♦♥t♦s
❝rít✐❝♦s ❞♦ ❢✉♥❝✐♦♥❛❧J ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❢✉♥çõ❡s ❛♣r♦♣r✐❛❞♦✱ ♣♦✐s ❛ ❞❡r✐✈❛❞❛ ❢♦r♠❛❧ ❞❡
●❛t❡❛✉① ❞❡ J é ❞❛❞♦ ♣♦r J′(u).φ=
Z
RN
(1 + 2u2)∇u∇φ dx+ 2
Z
RN
u|∇u|2φ dx+
Z
RN
V(|x|)uφ dx
−
Z
RN
Q(|x|)h(u)φ dx.
❈♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❝♦♥t♦r♥❛r ❛s ❞✐✜❝✉❧❞❛❞❡s ❞❡st❛❝❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡✱ ✉s❛♠♦s ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ ❛ ✜♠ ❞❡ tr❛♥s❢♦r♠❛r ♦ ♣r♦❜❧❡♠❛ q✉❛s❡❧✐♥❡❛r ❡♠ ✉♠ s❡♠✐❧✐♥❡❛r✳ ❉❡♣♦✐s✱ ♦❜t❡♠♦s ✉♠ r❡s✉❧t❛❞♦ q✉❡ r❡❧❛❝✐♦♥❛ ❛s s♦❧✉çõ❡s ❢r❛❝❛s ❞❡st❡ ♣r♦❜❧❡♠❛ s❡♠✐❧✐♥❡❛r ❝♦♠ ❛s s♦❧✉çõ❡s ❢r❛❝❛s ❞❡ ✭P1✮✳ ❉❛í✱ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❡ ✉♠ r❡s✉❧t❛❞♦
❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ❡ ✉s❛♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❝r❡s❝✐♠❡♥t♦ s♦❜r❡ ❛ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡ h✱
✈❡r✐✜❝❛♠♦s q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ s❡♠✐❧✐♥❡❛r ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ❡ é ●❛t❡❛✉① ❞✐❢❡r❡♥❝✐á✈❡❧ s♦❜r❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❖r❧✐❝③ ❝♦♠ ♣❡s♦ E1 ✭q✉❡ s❡rá ❞❡✜♥✐❞♦
♠❛✐s ❛❞✐❛♥t❡✮✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♥st❛t❛♠♦s q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ s❛t✐s❢❛③ ❛s ❤✐♣ót❡s❡s ❞❡ ✉♠❛ ✈❡rsã♦ ❞♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✱ ❝♦♠♦ t❛♠❜é♠✱ ❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ❞❡ P❛❧❛✐s✲❙♠❛❧❡✳ ❈♦♠ ✐st♦ ❡ ✉s❛♥❞♦ ✉♠ r❡s✉❧t❛❞♦ ❞♦ t✐♣♦ Pr✐♥❝í♣✐♦ ❞❛ ❈r✐t✐❝❛❧✐❞❛❞❡ ❙✐♠étr✐❝❛✱ ❣❛r❛♥t✐♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦ ♥✉❧❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ s❡♠✐❧✐♥❡❛r✳ ❊♠ s❡❣✉✐❞❛✱ ✉s❛♥❞♦ ✉♠ r❡s✉❧t❛❞♦ ❞❡ r❡❣✉❧❛r✐❞❛❞❡ ❞♦ t✐♣♦ ❇ré③✐s✲❑❛t♦ ❡ ❞❡ ✉♠ ▲❡♠❛ ❘❛❞✐❛❧✱ ✈❡r✐✜❝❛♠♦s q✉❡ ❛ s♦❧✉çã♦ ♣❡rt❡♥❝❡ ❛ L∞(RN) ❡ q✉❡ é ♥ã♦ ♥❡❣❛t✐✈❛✳
❋✐♥❛❧♠❡♥t❡✱ ✉s❛♥❞♦ ♦ r❡s✉❧t❛❞♦ q✉❡ r❡❧❛❝✐♦♥❛ ❛s s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛ s❡♠✐❧✐♥❡❛r ❝♦♠ ❛s s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛ q✉❛s❡❧✐♥❡❛r✱ ❝♦♥❝❧✉í♠♦s q✉❡ ✭P1✮ ♣♦ss✉✐ ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦
♥✉❧❛ ❡ ♥ã♦ ♥❡❣❛t✐✈❛✳
✶✳✶ ❆ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧
f
❡ ♦ ❡s♣❛ç♦ ❞❡ ❖r❧✐❝③
E
1❈♦♠♦ ❥á ❢♦✐ ♦❜s❡r✈❛❞♦ ❛❝✐♠❛✱ ♥ã♦ ♣♦❞❡♠♦s ❛♣❧✐❝❛r té❝♥✐❝❛s ✈❛r✐❛❝✐♦♥❛✐s ❞✐r❡t❛♠❡♥t❡ ♣❛r❛ ❡st✉❞❛r ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭P1✮✱ ♣♦r
❝❛✉s❛ ❞❛ ♣r❡s❡♥ç❛ ❞♦ t❡r♠♦RRNu2|∇u|2dx ♥❛ ❞❡✜♥✐çã♦ ❞♦ ❢✉♥❝✐♦♥❛❧J✳ P❛r❛ tr❛♥s♣♦r
❡st❛ ❞✐✜❝✉❧❞❛❞❡✱ ✉s❛♠♦s ❛ ✐❞❡✐❛ ❞❡s❡♥✈♦❧✈✐❞❛ ♣♦r ▲✐✉✱ ❲❛♥❣ ❡ ❲❛♥❣ ❡♠ ❬✸✼❪ ✭✈❡❥❛ t❛♠❜é♠ ❬✷✶❪✮✱ ✐st♦ é✱ ✉t✐❧✐③❛♠♦s ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧v =f−1(u)✱ ♦♥❞❡f é ❞❡✜♥✐❞❛
♣♦r
f′(t) = 1
(1 + 2f2(t))1/2 ❡♠ [0,+∞),
f(t) = −f(−t) ❡♠ (−∞,0].
❈♦♠ ❡st❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧✱ ❛ ♣❛rt✐r ❞♦ ❢✉♥❝✐♦♥❛❧ J✱ ♦❜t❡♠♦s ✉♠ ♥♦✈♦
❢✉♥❝✐♦♥❛❧ ❞❡✜♥✐❞♦ ♣♦r
I(v) = J(f(v)) = 1 2
Z
RN
[|∇v|2+V(|x|)f2(v)]dx−
Z
RN
Q(|x|)H(f(v))dx. ✭✶✳✸✮
◆♦ ❧❡♠❛ ❛❜❛✐①♦✱ ❧✐st❛♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❢✉♥çã♦ f(t)✱ ❝✉❥❛ ♣r♦✈❛ ♣♦❞❡
s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷✶✱ ✷✻❪✳
▲❡♠❛ ✶✳✶✳✶ ❆ ❢✉♥çã♦ f(t) s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣✐❡❞❛❞❡s✿
✭✶✮ f ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ é ✉♠❛ ❢✉♥çã♦ ❞❡ ❝❧❛ss❡ C∞ ❡ é ✉♠❛ ❢✉♥çã♦ ✐♥✈❡rtí✈❡❧❀
✭✷✮ |f′(t)| ≤1 ♣❛r❛ t♦❞♦ t∈R❀
✭✸✮ |f(t)| ≤ |t| ♣❛r❛ t♦❞♦ t ∈R❀
✭✹✮ f(t)/t→1 q✉❛♥❞♦ t→0❀
✭✺✮ f(t)/√t→21/4 q✉❛♥❞♦ t →+∞❀
✭✻✮ f(t)/2≤tf′(t)≤f(t) ♣❛r❛ t♦❞♦ t ≥0❀
✭✼✮ |f(t)| ≤21/4|t|1/2 ♣❛r❛ t♦❞♦ t∈R❀
✭✽✮ f2(t) é ✉♠❛ ❢✉♥çã♦ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛❀
✭✾✮ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ C t❛❧ q✉❡
|f(t)| ≥
C|t|, |t| ≤1
C|t|1/2, |t| ≥1;
✭✶✵✮ ❡①✐st❡♠ ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s C1 ❡ C2 t❛✐s q✉❡
|t| ≤C1|f(t)|+C2|f(t)|2 para todo t∈R;
✭✶✶✮ |f(t)f′(t)| ≤1/√2 ♣❛r❛ t♦❞♦ t∈R❀
✭✶✷✮ ❡①✐st❡ C > 0 t❛❧ q✉❡ f2(2t)≤Cf2(t) ♣❛r❛ t♦❞♦ t ∈R❀
✭✶✸✮ ❙❡ ̺ >1✱ ❡♥tã♦ |f(̺t)|2 ≤̺2|f(t)|2✱ ♣❛r❛ t♦❞♦ t∈R❀
✭✶✹✮ ❙❡ 0≤̺≤1✱ ❡♥tã♦ |f(̺t)|2 ≤̺|f(t)|2✱ ♣❛r❛ t♦❞♦ t∈R✳
❆❣♦r❛✱ ❞❡✜♥✐♠♦s ✉♠ s✉❜❡s♣❛ç♦✱ ❞♦ t✐♣♦ ❖r❧✐❝③✱ ❞❡ Drad1,2(RN) ❞❛❞♦ ♣♦r
E1 =
v ∈D1rad,2(RN); Z
RN
V(|x|)f2(v)dx <∞
♠✉♥✐❞♦ ❝♦♠ ❛ ♥♦r♠❛
kvk=k∇vk2+ inf
λ>0
1
λ
1 +
Z
RN
V(|x|)f2(λv)dx
. ✭✶✳✹✮
(E1,k · k) é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ✭✈❡❥❛ ♣♦r ❡①❡♠♣❧♦ ❬✷✻❪✮✳ ❖ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡st❛
s❡çã♦ ❝♦♥s✐st❡ ❡♠ ♦❜t❡r ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ❡♥✈♦❧✈❡♥❞♦ ♦ ❡s♣❛ç♦ E1✳ P❛r❛
