Open Sobre uma classe de problemas elípticos com não linearidades do tipo côncavoconvexa

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

❈✉rs♦ ❞❡ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛

❙♦❜r❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s ❡❧í♣t✐❝♦s ❝♦♠

♥ã♦ ❧✐♥❡❛r✐❞❛❞❡s ❞♦ t✐♣♦ ❝ô♥❝❛✈♦✲❝♦♥✈❡①❛

♣♦r

▼❛①✇❡❧❧ ❞❡ ❙♦✉s❛ P✐t❛

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ❊✈❡r❛❧❞♦ ❙♦✉t♦ ❞❡ ▼❡❞❡✐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✐❝❛✳

❧✐♥❡❛r✐❞❛❞❡s ❞♦ t✐♣♦ ❝ô♥❝❛✈♦✲❝♦♥✈❡①❛ ✴ ▼❛①✇❡❧❧ ❞❡ ❙♦✉s❛ P✐t❛✳✕❏♦ã♦ P❡ss♦❛✱ ✷✵✶✸✳

✼✹❢✳

❖r✐❡♥t❛❞♦r✿ ❊✈❡r❛❧❞♦ ❙♦✉t♦ ❞❡ ▼❡❞❡✐r♦s ❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦✮ ✲ ❯❋P❇✴❈❈❊◆

✶✳ ▼❛t❡♠át✐❝❛✳ ✷✳ Pr♦❜❧❡♠❛s ❡❧í♣t✐❝♦s s❡♠✐❧✐♥❡❛r❡s✳ ✸✳ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✳ ✹✳ Pr✐♥❝í♣✐♦ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞✳

❆ ❉❡✉s✱ ♣♦r ♥♦s ❞❛r ❛ ❡s♣❡r❛♥ç❛ ❞❡ ✉♠❛ ✈✐❞❛ ❡♠ ♣❛③✳

❆♦s ♠❡✉s ♣❛✐s✱ ♣❡❧♦ ❡♥♦r♠❡ ❛♣♦✐♦ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦s ♠❛✐s ❞✐❢í❝❡✐s✳

➚ ▼✐♥❤❛ ♥❛♠♦r❛❞❛ ❊❞♥❛✱ ♣♦r ❡st❛r s❡♠♣r❡ ♣r❡s❡♥t❡✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❊✈❡r❛❧❞♦✱ ♣♦r ❡st❛r s❡♠♣r❡ ❞✐s♣♦♥í✈❡❧ ❛ ❡s❝❧❛r❡❝❡r q✉❛✐sq✉❡r ❞ú✲ ✈✐❞❛s✱ ♣❡❧❛ ❛♠✐③❛❞❡✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ ♣♦r ❛❝r❡❞✐t❛r ❡♠ ♠✐♠✱ ❛♣❡s❛r ❞❡ t♦❞❛s ❛s ♠✐♥❤❛s ❞✐✜❝✉❧❞❛❞❡s✳

❆♦ ♣r♦❢❡ss♦r ❇r✉♥♦✱ ♣❡❧❛ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ ❡♠ ♠❡ ❛❥✉❞❛r✳

❆♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ❞❛ ❣r❛❞✉❛çã♦ ❡♠ ♠❛t❡♠át✐❝❛ ❞♦ ■❋❈❊✿ ▼ár✐♦ ❞❡ ❆ss✐s ❖❧✐✈❡✐r❛ ❡ ❏♦sé ❆❧✈❡s✱ ♣❡❧♦s ❡①❝❡❧❡♥t❡s ❡♥s✐♥❛♠❡♥t♦s✳

❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ♣ós✲❣r❛❞✉❛çã♦ ❡♠ ♠❛t❡♠át✐❝❛ ❞❛ ❯❋P❇✿ ❈❛r❧♦s ❇♦❝❦❡r✱ ▲✐③❛♥❞r♦✱ ◆❛♣♦❧❡♦♥ ❡ ❆❧❡①❛♥❞r❡✱ ♣❡❧❛s ❞✐s❝✐♣❧✐♥❛s ❧❡❝✐♦♥❛❞❛s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♠✉✐t♦ ♣❛r❛ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳

❆♦s ♠❡✉s ♣❛✐s✱ ▼❛r✐❛ ❙♦❝♦rr♦ ❞❡ ❙♦✉s❛ P✐t❛ ❡ ❏♦sé P✐t❛ ◆❡t♦✳

◆❡st❡ tr❛❜❛❧❤♦✱ ✈❛♠♦s ❡st❛❜❡❧❡❝❡r ✉♠❛ ✈❡rsã♦ ❞♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❞❡✈✐❞♦ ❛ ▼❛rt✐♥ ❙❝❤❡❝❤t❡r ❬✶✷❪✱ ❛ q✉❛❧ ✐rá ❢♦r♥❡❝❡r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ❡♠ ✉♠ ♥í✈❡❧ ♠❛①✲♠✐♥✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡st❡✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ Pr✐♥❝í♣✐♦ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞✱ ✈❛♠♦s ♦❜t❡r ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❡ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ s♦❧✉çã♦ ♣❛r❛ ✉♠❛ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s ❡❧í♣t✐❝♦s s❡♠✐❧✐♥❡❛r❡s ❡♥✈♦❧✈❡♥❞♦ ✉♠❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ ❞♦ t✐♣♦ ❝ô♥❝❛✈♦✲❝♦♥✈❡①❛✳

■♥ t❤✐s ✇♦r❦✱ ✇❡ ✇✐❧❧ ❡st❛❜❧✐s❤ ❛ ✈❡rs✐♦♥ ♦❢ t❤❡ ▼♦✉♥t❛✐♥ P❛ss ❚❤❡♦r❡♠ ❞✉❡ t♦ ▼❛rt✐♥ ❙❝❤❡❝❤t❡r ❬✶✷❪✱ ✇❤✐❝❤ ✇✐❧❧ ♣r♦✈✐❞❡ ❛ ❈❡r❛♠✐ s❡q✉❡♥❝❡ ❛t ❛ ♠❛①✲♠✐♥ ❧❡✈❡❧✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤✐s r❡s✉❧t✱ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❊❦❡❧❛♥❞ ✈❛r✐❛t✐♦♥❛❧ ♣r✐♥❝✐♣❧❡✱ ✇❡ ♦❜t❛✐♥ s♦♠❡ r❡s✉❧ts ♦❢ ❡①✐st❡♥❝❡ ❛♥❞ ♠✉❧t✐♣❧✐❝✐t② ♦❢ s♦❧✉t✐♦♥ ❢♦r ❛ ❝❧❛ss ♦❢ s❡♠✐❧✐♥❡❛r ❡❧❧✐♣t✐❝ ♣r♦❜❧❡♠s ✐♥✈♦❧✈✐♥❣ ❛ ♥♦♥❧✐♥❡❛r✐t② ♦❢ ❝♦♥❝❛✈❡✲❝♦♥✈❡① t②♣❡✳

■♥tr♦❞✉çã♦ ✽ ✶ ❯♠ t❡♦r❡♠❛ ❞♦ t✐♣♦ ♠❛①✲♠✐♥ ❡ ♦ ♣r✐♥❝í♣✐♦ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞ ✶✷ ✶✳✶ ❖ ❝❛♠♣♦ ♣s❡✉❞♦✲❣r❛❞✐❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷ ❖ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✸ ❖ ♣r✐♥❝í♣✐♦ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✷ ❯♠ ♣r♦❜❧❡♠❛ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❧✐♥❡❛r ✷✸

✷✳✶ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✷✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✷✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✸ ❯♠ ♣r♦❜❧❡♠❛ ❙✉♣❡r❧✐♥❡❛r ✹✷

✸✳✶ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✸✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✷ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✸✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

✹ ❖ ❝❛s♦ f(x, u)_≡λu ✺✼

✹✳✶ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✹✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✹✳✷ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✹✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

❆ ❘❡s✉❧t❛❞♦s ✉t✐❧✐③❛❞♦s ✻✷

❆✳✶ ❉✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ❞❡ ❢✉♥❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ❆✳✷ ❘❡❣✉❧❛r✐❞❛❞❡ ❞♦s ❢✉♥❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ❆✳✸ ❆✉t♦✈❛❧♦r❡s ❞♦ ▲❛♣❧❛❝✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ❆✳✹ ❘❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧ ❡ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✼✶

❱❛♠♦s ✉t✐❧✐③❛r ❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s✿

◆♦t❛çõ❡s ❣❡r❛✐s✿

• Br(x) ❞❡♥♦t❛ ❛ ❜♦❧❛ ❛❜❡rt❛ ❞❡ r❛✐♦r ❡ ❝❡♥tr♦ x✳

• ⇀ ❡_→ ❞❡♥♦t❛♠ ❝♦♥✈❡r❣ê♥❝✐❛ ❢r❛❝❛ ❡ ❢♦rt❡✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

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

• ∆u=Pn_i=1 ∂u

∂xi ❞❡♥♦t❛ ♦ ❧❛♣❧❛❝✐❛♥♦ ❞❛ ❢✉♥çã♦ ✉✳

• ∇u= (_∂x∂u₁, ...,_∂x∂u_n) ❞❡♥♦t❛ ♦ ❣r❛❞✐❡♥t❡ ❞❛ ❢✉♥çã♦ ✉✳

• q✳t✳♣ é ✉♠❛ ❛❜r❡✈✐❛çã♦ ❞❡ q✉❛s❡ t♦❞♦ ♣♦♥t♦✳ • u+ _{=♠❛①{0, u}✳}

• u− _{=♠❛①{0,−u}✳}

• I′ _{❞❡♥♦t❛ ❛ ❞❡r✐✈❛❞❛ ❛ ●ât❡❛✉① ❞♦ ❢✉♥❝✐♦♥❛❧ I✳}

• ∂Ω ❞❡♥♦t❛ ❛ ❢r♦♥t❡✐r❛ ❞♦ ❝♦♥❥✉♥t♦ Ω✳

❊s♣❛ç♦s ❞❡ ❢✉♥çõ❡s✿

• Lp_{(Ω)❂{u: Ω→R❀u é ♠❡♥s✉rá✈❡❧ ❡} R

Ω|u|p <∞}✳

• L∞_{(Ω)❂{u: Ω→R❀ ué ♠❡♥s✉rá✈❡❧ ❡ ∃C > 0t❛❧ q✉❡ |u(x)| ≤C q✳t✳♣ ❡♠ Ω}✳}

• 0 ❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ❞✉❛❧ ❞♦ ❊s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ 0 ✳

• C(Ω) ❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❡♠ Ω✳

• C1(Ω) ❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❝♦♥t✐♥✉❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡✐s ❡♠ Ω✳

• C∞

0 (Ω) ❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❞❡ ❝❧❛ss❡ C∞ ❡♠ Ω❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✳

◆♦r♠❛s✿

• ku_k= R_Ω|∇u_|2_dx12✱ ♣❛r❛ _u∈H1

0(Ω)✳

• kh_k∞ = inf{C >0;|u(x)| ≤C q✳t✳♣ ❡♠ Ω} ❞❡♥♦t❛ ❛ ♥♦r♠❛ ❡♠ L∞(Ω)✳

• kh_kp = (

R

Ω|u|p)

1

p ❞❡♥♦t❛ ❛ ♥♦r♠❛ ❡♠ Lp(Ω) ❝♦♠ 1≤p < ∞✳

❖ ♦❜❥❡t✐✈♦ ❞❡st❛ ❞✐ss❡rt❛çã♦ é ❡st✉❞❛r ❛ ❡①✐stê♥❝✐❛ ❡ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ s♦❧✉çã♦ ♥ã♦ ♥❡❣❛t✐✈❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛

  

−∆u=h(x)uq+f(x, u) ❡♠ Ω

u= 0 s♦❜r❡ ∂Ω,

✭✶✮

♦♥❞❡ Ω é ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❡ s✉❛✈❡ ❡♠ RN _{(N ≥ 1)}✱ _{0 < q < 1} ❡ ❛s ❢✉♥çõ❡s _h(x) ❡ f(x, s)s❛t✐s❢❛③❡♠ ❛❧❣✉♠❛s ❝♦♥❞✐çõ❡s ❞❡ ❝r❡s❝✐♠❡♥t♦✳

❊st❛ ❞✐ss❡rt❛çã♦ é ❜❛s❡❛❞❛ ♥♦s ❛rt✐❣♦s ❞❡ ▲✐✱ ❲✉ ❡ ❩❤♦✉ ❬✶✵❪ ❡ ❙❝❤❡❝❤t❡r ❬✶✷❪✳

❖ ♣r♦❜❧❡♠❛ ✭✶✮ ❢♦✐ ❛♠♣❧❛♠❡♥t❡ ❡st✉❞❛❞♦ s♦❜ ✈ár✐❛s ❤✐♣ót❡s❡s ❡♠ h(x) ❡ f(x, s)✳ ❊♠

❬✻❪ ❢♦✐ ❡st✉❞❛❞♦ ♦ ❝❛s♦ ♦♥❞❡ h(x) _≡ λ ❡ f(x, s) _≡ 0✳ ◆❡st❡ ❝❛s♦ ❢♦✐ ♣r♦✈❛❞♦ q✉❡ ❡①✐st❡

✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭✶✮✳ ❆❧é♠ ❞✐ss♦✱ ❲❛♥❣ ❡♠ ❬✶✸❪ ♣r♦✈♦✉ q✉❡✱ s❡ h(x) _≡λ é ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛✱ ♦ ♣r♦❜❧❡♠❛ ✭✶✮ ♣♦ss✉✐ ✐♥✜♥✐t❛s s♦❧✉çõ❡s un ∈H01(Ω)

t❛✐s q✉❡ |un|∞ → 0✱ I(un) < 0 ❡ I(un) → 0✳ P❛r❛ ♦ ❝❛s♦ ♦♥❞❡ f(x, s) é s✉♣❡r❧✐♥❡❛r

♣♦❞❡♠♦s ❝✐t❛r ♦ ❢❛♠♦s♦ ❛rt✐❣♦ ❞❡ ❆♠❜r♦s❡tt✐✱ ❇r❡③✐s ❡ ❈❡r❛♠✐✱ ❬✶❪✱ ♥♦ q✉❛❧ ❡st✉❞❛r❛♠ ♦ ❝❛s♦ ♦♥❞❡h(x) =λ ❡f(x, u) =up_{✱ ❝♦♠ 0< q <1< p ❡λ >0✳ ❆tr❛✈és ❞♦ ♠ét♦❞♦ ❞❡ s✉❜}

❡ s✉♣❡rs♦❧✉çã♦ ❡st❡s ❛✉t♦r❡s ♣r♦✈❛r❛♠ q✉❡ ❡①✐st❡ Λ >0 t❛❧ q✉❡ λ _∈ (0,Λ) s❡✱ ❡ s♦♠❡♥t❡

s❡✱ ♦ ♣r♦❜❧❡♠❛ ✭✶✮ ♣♦ss✉✐ ✉♠❛ s♦❧✉çã♦ ♥ã♦✲tr✐✈✐❛❧✳ ❆❧é♠ ❞✐ss♦✱ s❡p_∈(1,N_N+2₋₂)❢♦✐ ♣r♦✈❛❞❛

❛ ❡①✐stê♥❝✐❛ ❞❡ ♦✉tr❛ s♦❧✉çã♦✱ ❝❛s♦ t❡♥❤❛♠♦s 0< λ <Λ✳

❆s ❤✐♣ót❡s❡s ✉t✐❧✐③❛❞❛s ♥❡st❛ ❞✐ss❡rt❛çã♦✱ ❛s q✉❛✐s ✈❡r❡♠♦s ♥♦ ✐♥í❝✐♦ ❞❡ ❝❛❞❛ ❝❛♣ít✉❧♦✱ ❞✐❢❡r❡♠ ❞❛s ❤✐♣ót❡s❡s ✉s❛❞❛s ♥❡st❡s ♦✉tr♦s ❛rt✐❣♦s✱ ♦♥❞❡ ❞❡s❝❛rt❛♠♦s ✈ár✐❛s ❤✐♣ót❡s❡s ❞♦s ♠❡s♠♦s✳ ❖ ♠ét♦❞♦ ✉s❛❞♦ ♥❡st❡ ❛rt✐❣♦ é ❜❛s❡❛❞♦ ❡♠ ✉♠❛ s✐♠♣❧❡s ✈❛r✐❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✳ ❊st❡ ♠ét♦❞♦ t❛♠❜é♠ ✈❛❧❡ ♣❛r❛ ♦ ❝❛s♦ ♦♥❞❡ f(x, s) é s✉♣❡r❧✐♥❡❛r

❡♠ r❡❧❛çã♦ ❛ s ♥♦ ✐♥✜♥✐t♦ ♦✉ f(x, s)_≡λs♣❛r❛ ❛❧❣✉♠ λ >0✱ ✐st♦ é✱ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s

♥❡st❛ ❞✐ss❡rt❛çã♦ ❝♦❜r❡♠ t♦❞❛s ❛s ❝♦♥❞✐çõ❡s ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❡ f(x, s) ❡♠ s✱ ❛ s❛❜❡r✱

❛ss✐♠ s✉♣♦♠♦s f(x, s)_≥0 ♣❛r❛ s_≥0✳

❊st❛ ❞✐ss❡rt❛çã♦ ❡stá ♦r❣❛♥✐③❛❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

◆♦ ❈❛♣ít✉❧♦ ✶✱ ❡stã♦ ❡♥✉♥❝✐❛❞♦s ❡ ❞❡♠♦♥str❛❞♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❝❧áss✐❝♦s ❞❛ ❆♥á❧✐s❡ ♥ã♦✲❧✐♥❡❛r q✉❡ sã♦✿ ✉♠❛ ✈❡rsã♦ ♠❛✐s ❣❡r❛❧ ❞♦ q✉❡ ❝♦♥❤❡❝❡♠♦s ❝♦♠♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❡ ♦ Pr✐♥❝í♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞✳ ❆♠❜♦s t❡♠ ✐♠♣♦rtâ♥❝✐❛ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ q✉❡ ♣♦ss❛♠♦s ❡♥❝♦♥tr❛r s♦❧✉çõ❡s ❢r❛❝❛s ❞♦ ♣r♦❜❧❡♠❛ ✭✶✮✳

◆♦ ❈❛♣ít✉❧♦ ✷✱ s❡rá ❡st✉❞❛❞♦ ♦ ♣r♦❜❧❡♠❛ ✭✶✮ ♥♦ ❝❛s♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❧✐♥❡❛r✳ ❆ss✐♠✱ ❛ ❢✉♥çã♦ f(x, s)❞❡✈❡ s❛t✐s❢❛③❡r ❛ s❡❣✉✐♥t❡ ❤✐♣ót❡s❡✿

(f3) lims→+∞f(x,s_s ) =ℓ∈(λ1,+∞) ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ x∈Ω✳

❙❡rã♦ ❡♥✉♥❝✐❛❞♦s ❡ ❞❡♠♦♥str❛❞♦s ❞♦✐s t❡♦r❡♠❛s✳ ◆♦ ♣r✐♠❡✐r♦ ❞❡❧❡s✱ ♦ ❚❡♦r❡♠❛ ✷✳✶✱ ✈❛♠♦s ♦❜t❡r ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦✲♥❡❣❛t✐✈❛ ♣❛r❛ ❡st❡ ♣r♦❜❧❡♠❛✱ ❝♦♠ ❡♥❡r❣✐❛ ♣♦s✐t✐✈❛✳ ❆❧é♠ ❞✐ss♦✱ s❡ ❛ ❢✉♥çã♦ h ❢♦r ♥ã♦✲♥❡❣❛t✐✈❛ ❡st❛ s♦❧✉çã♦ ❞❡✈❡rá s❡r ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈❛✳ ❙✉❛ ❞❡♠♦♥str❛çã♦ ❝♦♥s✐st❡ ❞❡ ✈ár✐♦s ❧❡♠❛s✱ ♠❛s ❞♦✐s ❞❡ss❡s ❧❡♠❛s t❡♠ ♠❛✐♦r ✐♠♣♦rtâ♥❝✐❛✱ ♥✉♠ s❡♥t✐❞♦ q✉❡ ✜❝❛rá ♣r❡❝✐s♦ ♥♦ ❞❡❝♦rr❡r ❞❡st❛ ❞✐ss❡rt❛çã♦✱ ♦♥❞❡ ❞❛r❡♠♦s ❛❣♦r❛ ❛❧❣✉♠❛s ✐♥❢♦r♠❛çõ❡s q✉❡ ❥✉st✐✜❝❛♠ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡st❡s ❞♦✐s ❧❡♠❛s✳ ◆♦ ♣r✐♠❡✐r♦ ❞❡❧❡s✱ ♦ ▲❡♠❛ ✷✳✶✱ ✈❛♠♦s ♠♦str❛r q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛ I ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ✭✶✮ s❛t✐s❢❛③ ❛ ❣❡♦♠❡tr✐❛ ❞♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✳ ■st♦ ♥♦s ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ♥í✈❡❧ c > 0 q✉❡ s❡rá ❞❡✜♥✐❞♦ ♥♦ ❚❡♦r❡♠❛ ✶✳✷✳ ◆♦ ♦✉tr♦ ❧❡♠❛ ❝✐t❛❞♦✱ ♦ ▲❡♠❛

✷✳✹✱ ♠♦str❛♠♦s q✉❡ t♦❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♣❛r❛ ♦ ❢✉♥❝✐♦♥❛❧ I é ❧✐♠✐t❛❞❛✳ ❆ss✐♠✱ ❛ ❝♦♠♣❛❝✐❞❛❞❡ ❞❛ ✐♠❡rsã♦ H1

0(Ω) ֒→ Lp(Ω)✱ ❝♦♠ 1 ≤ p < 2∗✱ ❣❛r❛♥t✐rá ❛ ❡①✐stê♥❝✐❛ ❞❡

✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦✲♥❡❣❛t✐✈❛ ♣❛r❛ ❡st❡ ♣r♦❜❧❡♠❛✳ ◆♦ t❡♦r❡♠❛ s❡❣✉✐♥t❡✱ ♦ ❚❡♦r❡♠❛ ✷✳✷✱ s❡rá ❛❞✐❝✐♦♥❛❞❛ ✉♠❛ ♦✉tr❛ ❤✐♣ót❡s❡ s♦❜r❡ ❛ ❢✉♥çã♦ h✳ ❱❛♠♦s ❞❡♥♦t❛r ♣♦r c1 ♦ í♥✜♠♦

❞♦ ❢✉♥❝✐♦♥❛❧ I ❡♠ ✉♠❛ ❜♦❧❛ Bρ ⊂ H01(Ω) ❡ ♣r♦✈❛r❡♠♦s q✉❡ c1 < 0✳ ❯t✐❧✐③❛r❡♠♦s ♦

Pr✐♥❝í♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞ ♣❛r❛ ❝♦♥str✉✐r ✉♠❛ s❡q✉ê♥❝✐❛ (un) ⊂ H01(Ω) ❧✐♠✐t❛❞❛✳

❆ss✐♠✱ ♥♦✈❛♠❡♥t❡ ♣❡❧❛ ✐♠❡rsã♦ ❝♦♠♣❛❝t❛ H1

0(Ω) ֒→ Lp(Ω)✱ ❝♦♠ 1 ≤ p < 2∗✱ ❡ ♣❡❧♦

♠❡s♠♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞♦ ❚❡♦r❡♠❛ ✷✳✶ ♦❜t❡r❡♠♦s ✉♠❛ s♦❧✉çã♦ ♥ã♦✲♥❡❣❛t✐✈❛✱ ❝♦♠ ❡♥❡r❣✐❛ ♥❡❣❛t✐✈❛✳ ❖ Pr✐♥❝í♣✐♦ ❞♦ ▼á①✐♠♦ ❋♦rt❡ s❡rá út✐❧ ♣❛r❛ ♠♦str❛r q✉❡ s❡ h ❢♦r ♥ã♦✲♥❡❣❛t✐✈❛ ❡♥tã♦ ❛s s♦❧✉çõ❡s ♦❜t✐❞❛s ♥❡ss❡s ❞♦✐s ❞♦✐s t❡♦r❡♠❛s ❞❡✈❡♠ s❡r ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈❛s✳

❝♦♠♣♦st♦ ♣♦r ❞♦✐s t❡♦r❡♠❛s✳ ◆♦ ♣r✐♠❡✐r♦ ❞❡❧❡s✱ ♦ ❚❡♦r❡♠❛ ✸✳✶✱ ✈❛♠♦s ♦❜t❡r ❞✉❛s s♦❧✉✲ çõ❡s ❢r❛❝❛s ♥ã♦ ♥❡❣❛t✐✈❛s✱ ✉♠❛ ❝♦♠ ❡♥❡r❣✐❛ ♣♦s✐t✐✈❛ ❡ ♦✉tr❛ ❝♦♠ ❡♥❡r❣✐❛ ♥❡❣❛t✐✈❛✳ P❛r❛ ❞❡♠♦♥strá✲❧♦✱ ✈❛♠♦s ✉t✐❧✐③❛r ✉♠ ❧❡♠❛ té❝♥✐❝♦ ❡ ♠❛✐s ❞♦✐s ❧❡♠❛s ❛♥á❧♦❣♦s ❛♦s ❞♦ ❝❛♣ít✉❧♦ ✷✱ ♠❛s ❝♦♠ ❞❡♠♦♥str❛çã♦ ❞✐❢❡r❡♥t❡✱ ♣♦✐s ❛❣♦r❛ f(x, s) é s✉♣❡r❧✐♥❡❛r✳ ◆♦ ♣r✐♠❡✐r♦ ❞❡❧❡s✱

♦ ▲❡♠❛ ✸✳✶✱ ♥♦✈❛♠❡♥t❡ ✈❛♠♦s ♠♦str❛r q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ s❛t✐s❢❛③ ❛ ❣❡♦♠❡tr✐❛ ❞♦ ♣❛ss♦ ❞❛ ♠♦♥t❛♥❤❛✳ ■st♦ ♥♦s ❞á ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ✱ ❛ q✉❛❧ ♠♦str❛r❡♠♦s s❡r ❧✐♠✐t❛❞❛ ♥♦ ❧❡♠❛ s❡❣✉✐♥t❡✱ ♦ ▲❡♠❛ ✸✳✷✳ ❆ ♣❛rt✐r ❞❛í✱ ♣r♦❝❡❞❡♠♦s ❛♥❛❧♦✲ ❣❛♠❡♥t❡ ❛♦ ❚❡♦r❡♠❛ ✷✳✶ ❡ ♦❜t❡r❡♠♦s ✉♠❛ s♦❧✉çã♦ ♥ã♦✲♥❡❣❛t✐✈❛ ❝♦♠ ❡♥❡r❣✐❛ ♣♦s✐t✈❛✳ P❛r❛ ♦❜t❡r♠♦s ✉♠❛ ♦✉tr❛ s♦❧✉çã♦ ♥ã♦✲♥❡❣❛t✐✈❛✱ ♠❛s ❝♦♠ ❡♥❡r❣✐❛ ♥❡❣❛t✐✈❛✱ ♣r♦❝❡❞❡♠♦s ❞❡ ♠❛✲ ♥❡✐r❛ ❛♥á❧♦❣❛ ❛♦ ❚❡♦r❡♠❛ ✷✳✷✱ ✉t✐❧✐③❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞✳ ◆♦ t❡♦r❡♠❛ s❡❣✉✐♥t❡✱ ♦ ❚❡♦r❡♠❛ ✸✳✷✱ ✈❛♠♦s ♦❜t❡r ✉♠❛ s♦❧✉çã♦ ♥ã♦✲♥❡❣❛t✐✈❛ ❝♦♠ ❡♥❡r❣✐❛ ♣♦s✐t✐✈❛✱ ♠❛s ❛♥❛❧✐s❛♥❞♦ ♦ ❝❛s♦ ♦♥❞❡ ❛ ❢✉♥çã♦ h é ♥ã♦✲♥❡❣❛t✐✈❛✳ P❛r❛ t❛❧✱ ❞❡s❝❛rt❛♠♦s ❛❧❣✉♠❛s ❤✐♣ó✲ t❡s❡s ❡♥✈♦❧✈❡♥❞♦ h❡f(x, s)✉t✐❧✐③❛❞❛s ♥♦ ❚❡♦r❡♠❛ ✸✳✷ ❡ ❛❝r❡s❝❡♥t❛♠♦s ♠♦♥♦t♦♥✐❝✐❞❛❞❡ à

❢✉♥çã♦ f(x, s)✳ ❆ ❞❡♠♦♥str❛çã♦ é ❛♥á❧♦❣❛✳

◆♦ ❈❛♣ít✉❧♦ ✹✱ ❝♦♠♦ ❛♣❧✐❝❛çã♦✱ ❛♥❛❧✐s❛r❡♠♦s ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ♣r♦❜❧❡♠❛ ✭✶✮❀ ❛ s❛❜❡r✱ ♦ ❝❛s♦ ❡s♣❡❝✐❛❧ ♦♥❞❡ f(x, u) é ❣❧♦❜❛❧♠❡♥t❡ ❧✐♥❡❛r ❡♠ r❡❧❛çã♦ ❛ u✱ ✐st♦ é✱ f(x, u) =

λu✳ ❙❡rã♦ ❡♥✉♥❝✐❛❞♦s ❡ ❞❡♠♦♥str❛❞♦s ❞♦✐s t❡♦r❡♠❛s✳ ◆♦ ♣r✐♠❡✐r♦ ❞❡❧❡s✱ ♦ ❚❡♦r❡♠❛ ✹✳✶✱ s✉♣♦♥❞♦ q✉❡ ❛ ❢✉♥çã♦ h s❡❥❛ ♥ã♦✲♥❡❣❛t✐✈❛ ✈❛♠♦s ♣r♦✈❛r ❡①✐stê♥❝✐❛ ❡ ♥ã♦ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s✱ ❛♥❛❧✐s❛♥❞♦ s❡♣❛r❛❞❛♠❡♥t❡ ♦s ❝❛s♦s ♦♥❞❡ λ _≥ λ1 ❡ λ < λ1✳ ❱❛♠♦s

♠♦str❛r q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛J ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ s❛t✐s❢❛③ ❛ ❣❡♦♠❡tr✐❛ ❞♦ ♣❛ss♦ ❞❛ ♠♦♥t❛♥❤❛ ❡ ❛ss✐♠ ♦ r❡st❛♥t❡ ❞❛ ❞❡♠♦♥str❛çã♦ s❡rá ❛♥á❧♦❣❛ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✷✳✶✱ ♥♦ ❝❛s♦ λ < λ1✳ ◆♦ t❡♦r❡♠❛ s❡❣✉✐♥t❡✱ ♦ ❚❡♦r❡♠❛ ✹✳✷✱ s✉♣♦♥❞♦ q✉❡ h s❡❥❛ ♥ã♦✲

♣♦s✐t✐✈❛ ✈❛♠♦s ♥♦✈❛♠❡♥t❡ ♣r♦✈❛r ❡①✐stê♥❝✐❛ ❡ ♥ã♦✲❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ♣❛r❛ ❡st❡ ♣r♦❜❧❡♠❛✱ ❛♥❛❧✐s❛♥❞♦ s❡♣❛r❛❞❛♠❡♥t❡ ♦s ❝❛s♦s ♦♥❞❡λ > λ1 ❡λ≤λ1✳ ❆❧é♠ ❞✐ss♦✱ s❡rá

❞❛❞❛ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ♣❛r❛ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ❞❡st❡ ♣r♦❜❧❡♠❛✱ s❡ h ❢♦r ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛ ❡ λ =λ1✳

❋✐♥❛❧♠❡♥t❡✱ ♥♦ ❆♣ê♥❞✐❝❡ ✈❛♠♦s ❞❡♠♦♥str❛r ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ r❡❣✉❧❛r✐❞❛❞❡ ❞♦s ❢✉♥❝✐♦♥❛✐s ❡♥❡r❣✐❛ ✉t✐❧✐③❛❞♦s✱ s❡rã♦ ❡♥✉♥❝✐❛❞♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧ ❡ ♦✉tr♦s ❡♥✈♦❧✈❡♥❞♦ ♦s ❛✉t♦✈❛❧♦r❡s ❞♦ ▲❛♣❧❛❝✐❛♥♦✳ ❊st❡s r❡s✉❧t❛❞♦s s❡rã♦ ✉t✐❧✐③❛❞♦s ♥♦

❯♠ t❡♦r❡♠❛ ❞♦ t✐♣♦ ♠❛①✲♠✐♥ ❡ ♦

♣r✐♥❝í♣✐♦ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞

◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ ❚❡♦r❡♠❛ ❞♦ t✐♣♦ ♠❛①✲♠✐♥ ❞❡✈✐❞♦ ❛ ▼✳ ❙❝❤❡❝❤t❡r ❡♠ ❬✶✷❪ ❜❡♠ ❝♦♠♦ ♦ ❡♥✉♥❝✐❛❞♦ ❡ ❛ ♣r♦✈❛ ❞♦ ♣r✐♥❝í♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞✳ P❛r❛ ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ♠❛①✲♠✐♥✱ ♣r❡❝✐s❛♠♦s ❞❡✜♥✐r ❡ ❝♦♥str✉✐r ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ Ps❡✉❞♦✲●r❛❞✐❡♥t❡✱ ♦ q✉❛❧ ❡stá ✐♥❝❧✉s♦ ♥❛ ♣ró①✐♠❛ s❡çã♦✳

✶✳✶ ❖ ❝❛♠♣♦ ♣s❡✉❞♦✲❣r❛❞✐❡♥t❡

❉❡✜♥✐çã♦ ✶ ❙❡❥❛ I : E _→ R ✉♠ ❢✉♥❝✐♦♥❛❧ ❞❡ ❝❧❛ss❡ _C1 ❞❡✜♥✐❞♦ ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ E✳ ❯♠ ❝❛♠♣♦ ♣s❡✉❞♦✲❣r❛❞✐❡♥t❡ ♣❛r❛ I é ✉♠❛ ❛♣❧✐❝❛çã♦ ❧♦❝❛❧♠❡♥t❡ ❧✐♣s❝❤✐t③✐❛♥❛ γ :Ee_→E t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ α_∈(0,1)✱ t❡♠♦s✿

kγ(w)_{k ≤}1, α_kI′(w)_{k ≤}I′(w)γ(w), _∀w_∈E,e ♦♥❞❡ Ee=_{u_∈E;I′_{(u)6= 0}✳}

▲❡♠❛ ✶✳✶ ❊①✐st❡ ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ♣s❡✉❞♦✲❣r❛❞✐❡♥t❡ ♣❛r❛ ♦ ❢✉♥❝✐♦♥❛❧ I✳ ❉❡♠♦♥str❛çã♦✿ P❛r❛ ❝❛❞❛ u_∈Ee✱ ❝♦♠♦I′_{(u)é ❝♦♥tí♥✉❛✱ t❡♠♦s}

kI′(u)_k= sup

kwk=1

I′(u)w. ❙❡❥❛α1 ∈(α,1)✳ ❊①✐st❡ w(u)∈E t❛❧ q✉❡

α1kI′(u)k ≤I′(u)w(u)

❝♦♠

kw(u)_k= 1. P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ I′_{✱ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ V}

u ❞❡u t❛❧ q✉❡

α_kI′(v)_{k ≤}I′(v)w(u), _∀v _∈Vu.

❖❜s❡r✈❡ q✉❡ ❛ ❢❛♠í❧✐❛ {Vu;u ∈ Ee} ❢♦r♠❛ ✉♠❛ ❝♦❜❡rt✉r❛ ♣❛r❛ Ee✳ ❈♦♠♦ Ee ⊂ E

é ♠❡tr✐③á✈❡❧ ❡ ♣♦rt❛♥t♦ ♣❛r❛❝♦♠♣❛❝t♦✱ ❡①✐st❡ ✉♠❛ ❝♦❜❡rt✉r❛ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t❛ {Vi}i∈J

❡ ✉♠❛ ♣❛rt✐çã♦ ❞❛ ✉♥✐❞❛❞❡ {λi}i∈J ❧♦❝❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③ ❝♦♥tí♥✉❛ s✉❜♦r❞✐♥❛❞❛ à ❡st❛

❝♦❜❡rt✉r❛✱ ♦♥❞❡ ♣♦❞❡♠♦s s✉♣♦r q✉❡ J =_{1,2, ..., n0}✳ ❈♦♠ ✐ss♦✱ ♣❛r❛ ❝❛❞❛ i ∈ J t❡♠♦s

Vi ⊂Vui✳ ❆ss✐♠✱ ❝♦♥s✐❞❡r❡♠♦s ❛ ❢✉♥çã♦ γ :Ee →E ❞❡✜♥✐❞❛ ♣♦r

γ(v) =

n0

X

i=1

λi(v)w(ui).

▲♦❣♦✱ ❝♦♠♦ ❝❛❞❛ λi é ❧♦❝❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③ ❝♦♥tí♥✉❛✱ ❡ ✉♠❛ s♦♠❛ ✜♥✐t❛ ❞❡ ❛♣❧✐❝❛çõ❡s

❧♦❝❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③ t❛♠❜é♠ ♦ é✱ t❡♠♦s q✉❡ γ é ❧♦❝❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③ ❝♦♥tí♥✉❛✳ ❆ss✐♠✱

kγ(v)_{k ≤}

n0

X

i=1

λi(v) = 1.

❆❧é♠ ❞✐ss♦✱ t❡♠♦s

I′(v)γ(v)_≥α

n0

X

i=1

λi(v)kI′(v)k=αkI′(v)k,

♦ q✉❡ ♣r♦✈❛ ♦ r❡s✉❧t❛❞♦✳

✶✳✷ ❖ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛

■♥✐❝✐❛❧♠❡♥t❡✱ ❞❡♥♦t❡♠♦s ♦ s❡❣✉✐♥t❡ ❝♦♥❥✉♥t♦✿

Ψ =_{ψ : (0,+_∞)_→R_{, ψ} é ♥ã♦ ❝r❡s❝❡♥t❡ ❡

Z +∞

1

ψ(r)dr = +_∞}. ❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✳

❊①❡♠♣❧♦✿ ❆ ❢✉♥çã♦ ψ : (0,+_∞)_→R ❞❡✜♥✐❞❛ ♣♦r

ψ(s) = 1

1 +s, s >0 ❝❧❛r❛♠❡♥t❡ ♣❡rt❡♥❝❡ ❛ Ψ✳

❚❡♦r❡♠❛ ✶✳✶ ✭❙❝❤❡❝❤t❡r✱ ❬✶✷❪✮ ❙❡❥❛ I : E _→ R ✉♠ ❢✉♥❝✐♦♥❛❧ ❞❡✜♥✐❞♦ ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ E ❡ ✉♠ ❡❧❡♠❡♥t♦ e _∈ E ❝♦♠ e ₆= 0✳ ❙❡❥❛ Λ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❛❜❡rt♦s

❧✐♠✐t❛❞♦s N _⊂ E t❛✐s q✉❡ 0 _∈N ❝♦♠ e /_∈N✳ ❙✉♣♦♥❤❛ q✉❡ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s s❡❥❛♠ s❛t✐s❢❡✐t❛s✿

✭✐✮ I _∈C1_(E,R)

✭✐✐✮ ❊①✐st❡ N0 ∈Λ ❡ η∈R t❛❧ q✉❡ max{I(0), I(e)} ≤η ❡

I(u)_≥η, _∀u_∈∂N0. ✭✶✳✶✮

❙❡❥❛ b_≥η ❞❡✜♥✐❞♦ ♣♦r

b = sup

N∈Λ

inf

u∈∂NI(u), ✭✶✳✷✮

❡♥tã♦✱ ♣❛r❛ t♦❞♦ ψ _∈Ψ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ (un) ❡♠ E t❛❧ q✉❡

I(un)→b, q✉❛♥❞♦ n →+∞,

❡

kI′_(u

n)k

ψ(_kunk) →

0, q✉❛♥❞♦ n_→+_∞.

❉❡♠♦♥str❛çã♦✿ ❖❜s❡r✈❡ ♥❛ ❋✐❣✉r❛ ✶✳✶ ❛ s❡❣✉✐r ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ ✉♠ ❢✉♥❝✐♦♥❛❧ I s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ✭✐✮ ❡ ✭✐✐✮✿

❋✐❣✉r❛ ✶✳✶✿

■♥✐❝✐❛❧♠❡♥t❡✱ s✉♣♦♥❤❛♠♦s q✉❡ b > η✳ ❙✉♣♦♥❤❛ ♣♦r ❝♦♥tr❛❞✐çã♦ q✉❡ ❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ t❡♦r❡♠❛ ♥ã♦ s❡❥❛ ✈❡r❞❛❞❡✐r❛✳ ❆ss✐♠✱ ❡①✐st❡ ε >0❡ ψ _∈Ψt❛❧ q✉❡

kI′(u)_{k ≥}ψ(_ku_k), ✭✶✳✸✮ ♣❛r❛ t♦❞♦ u_∈E s❛t✐s❢❛③❡♥❞♦

|I(u)₋b_{| ≤}3ε. ✭✶✳✹✮

❙❡ ♥❡❝❡ssár✐♦✱ ♣♦❞❡♠♦s t♦♠❛r ε >0 s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ t❛❧ q✉❡

3ε < b₋η. ❉❡✜♥✐♠♦s ♦s s❡❣✉✐♥t❡s ❝♦♥❥✉♥t♦s✿

Q=_{u_∈E,_|I(u)₋b_{| ≤}2ε_}, Q1 ={u∈E;|I(u)−b| ≤ε},

Q2 =E\Q.

❙❡❥❛ g :E _→R❞❡✜♥✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛

g(u) = d(u, Q2)

d(u, Q1) +d(u, Q2)

.

❆✜r♠❛♠♦s q✉❡ g é ✉♠❛ ❛♣❧✐❝❛çã♦ ❧♦❝❛❧♠❡♥t❡ ❧✐♣s❝❤✐t③✐❛♥❛ ❡ s❛t✐s❢❛③

g(u) =

  

1, u_∈Q1,

0, u_∈Q₂

✭✶✳✺✮

❡

0< g(u)<1, ❝❛s♦ ❝♦♥trár✐♦. ❉❡ ❢❛t♦✱ ✈❛♠♦s ❞❡♥♦t❛r g1 ❡g2 ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛

g1(u) = d(u, Q1) ❡ g2(u) =d(u, Q2).

❉❛❞♦su2, u2 ∈E✱ t❡♠♦s

g(u1)−g(u2) =

g2(u1)

g1(u1) +g2(u1) −

g2(u2)

g1(u2) +g2(u2)

.

❉❛í✱

g(u1)−g(u2) =

g1(u2)g2(u1) +g2(u2)g2(u1)−g1(u1)g2(u2)−g2(u1)g2(u2)

[g1(u1) +g2(u1)][g1(u2) +g2(u2)]

.

❈♦♠ ✐ss♦✱ t❡♠♦s

g(u1)−g(u2) =

g1(u2)g2(u1)−g1(u1)g2(u2)

[g1(u1) +g2(u1)][g1(u2) +g2(u2)]

.

❆ss✐♠✱

g(u1)−g(u2) =

g1(u2)g2(u1)−g2(u1)g1(u1) +g2(u1)g1(u1)−g1(u1)g2(u2)

[g1(u1) +g2(u1)][g1(u2) +g2(u2)]

.

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱

|g(u1)−g(u2)| ≤

g2(u1)

[g1(u1) +g2(u1)][g1(u2) +g2(u2)]|

g1(u2)−g1(u1)|

+ g1(u1)

[g1(u1) +g2(u1)][g1(u2) +g2(u2)]|

g2(u1)−g2(u2)|.

❈♦♠♦ g1 ❡ g2 sã♦ ❢✉♥çõ❡s ❧✐♣s❝❤✐t③✐❛♥❛s✱ ❡①✐st❡♠ c1 >0 ❡ c2 >0 t❛✐s q✉❡

|g(u1)−g(u2)| ≤

g2(u1)

[g1(u1) +g2(u1)][g1(u2) +g2(u2)]

c1|u2−u1|

+ g1(u1)

[g1(u1) +g2(u1)][g1(u2) +g2(u2)]

c2|u1 −u2|.

❈♦♠♦ g1(u1) +g2(u1)>0✱ ❡①✐st❡♠a >0 ❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ W ❞❡u1 t❛❧ q✉❡

g1(u) +g2(u)> a >0 ∀u∈W.

❆❧é♠ ❞✐ss♦✱ ❝♦♠♦

g2(u1)

g1(u2) +g2(u2) ≤

1, ❡

g1(u1)

g1(u2) +g2(u2) ≤

1, t❡♠♦s q✉❡

|g(u)₋g(v)_{| ≤} c1+c2

a ku−vk ∀u, v ∈W, ❞♦♥❞❡ ♦❜t❡♠♦s q✉❡ g é ❧♦❝❛❧♠❡♥t❡ ❧✐♣s❝❤✐t③✐❛♥❛✳

❙❡❥❛γ ✉♠ ❝❛♠♣♦ ♣s❡✉❞♦✲❣r❛❞✐❡♥t❡ ♣❛r❛ I✱ ✐st♦ é✱ ✉♠❛ ❛♣❧✐❝❛çã♦γ :E _→E t❛❧ q✉❡

I′(u)γ(u)_≥α_kI′(u)_k, ❝♦♠ _kγ(u)_{k ≤}1, ✭✶✳✻✮ ♣❛r❛ ❛❧❣✉♠α >0✳ ❙❡❥❛ ϕ:E _→E ❛ ❛♣❧✐❝❛çã♦ ❞❡✜♥✐❞❛ ♣♦rϕ(u) = g(u)γ(u)✳ ❆ss✐♠✱ ϕ é ❧♦❝❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③ ❝♦♥tí♥✉❛ ❝♦♠

❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧✿

    

d

dtσ(t, u) =ϕ(σ(t, u)), σ(0, u) =u.

✭✶✳✽✮

❈♦♠♦ϕé ❧♦❝❛❧♠❡♥t❡ ❧✐♣s❝❤✐t③✐❛♥❛ ❡ ❧✐♠✐t❛❞❛✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ♣❛r❛ ✭✶✳✽✮ ❞❡✜♥✐❞❛ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ♠❛①✐♠❛❧(t−_{(u), t}+_{(u))✳ ❆✜r♠❛♠♦s q✉❡ t}−_{(u) = −∞❡t}+_{(u) = +∞✳ ❉❡}

❢❛t♦✱ s✉♣♦♥❤❛♠♦s ♣♦r ❝♦♥tr❛❞✐çã♦ q✉❡ t+_{(u) <∞✳ ❙❡❥❛ ✉♠❛ s❡q✉ê♥❝✐❛ (t}

n) ♥♦ ✐♥t❡r✈❛❧♦

(_−∞, t+_{(u))t❛❧ q✉❡ t}

n→t+(u)✳ ❆ss✐♠✱ t❡♠♦s

kσ(tm, u)−σ(tn, u)k=

Z tm

tn

d

dτ(σ(τ, u))dτ

=

Z tm

tn

ϕ(σ(τ, u))dτ

≤

Z tm

tn

kϕ(σ(τ, u))_kdτ

≤ ktm−tnk.

❆ss✐♠✱ ❝♦♠♦ (tn) é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✱ ♣♦✐s é ❝♦♥✈❡r❣❡♥t❡ ❡♠ R✱ t❡♠♦s q✉❡ ❛

s❡q✉ê♥❝✐❛ (σ(tn, u)) ⊂ E t❛♠❜é♠ é ❞❡ ❈❛✉❝❤②✳ ▲♦❣♦✱ (σ(tn, u)) ❝♦♥✈❡r❣❡ ♣❛r❛ ❛❧❣✉♠

♣♦♥t♦v _∈E✱ ❞❡s❞❡ q✉❡ tn →t+(u)✳ ❆❣♦r❛✱ ❝♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧

    

d

dtσ(t, u) =ϕ(σ(t, u)), σ(t+(u), u) =v.

■st♦ ♥♦s ❞á ✉♠❛ ❡①t❡♥sã♦ ❞❡ σ(t, u)♥♦ ✐♥t❡r✈❛❧♦ [t−_{(u)−δ, t}+_{(u) +δ]✱ ♣❛r❛ ❛❧❣✉♠ δ >0✳}

■st♦ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ♣♦✐s ♣♦r ❤✐♣ót❡s❡ t❡♠♦s q✉❡(t−_{(u), t}+_{(u))é ✉♠ ✐♥t❡r✈❛❧♦ ♠❛①✐♠❛❧✳}

❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ ♣r♦✈❛♠♦s q✉❡ t−_{(u) = −∞✳ P♦r ✭✶✳✼✮ t❡♠♦s}

kσ(t, u)₋u_{k ≤}t. ✭✶✳✾✮

❆❧é♠ ❞✐ss♦✱ ♣♦r ✭✶✳✻✮ ❡ ✭✶✳✽✮ t❡♠♦s d

dt(I(σ(t, u)) =I

′_{(σ(t, u))ϕ(σ(t, u))}

≥αg(σ(t, u))_kI′(σ(t, u))_k

>0.

✭✶✳✶✵✮

■st♦ ✐♠♣❧✐❝❛ q✉❡ I(σ(., u)) é ♥ã♦✲❞❡❝r❡s❝❡♥t❡✱ ♦✉ s❡❥❛✱

I(σ(t1, u))≤I(σ(t2, u)) 0< t1 < t2. ✭✶✳✶✶✮

P♦r ✭✶✳✷✮ ❡①✐st❡ N _∈Λ t❛❧ q✉❡

I(u)> b₋ε, u_∈∂N. ✭✶✳✶✷✮ ❙❡❥❛

M = sup

u∈∂Nk

u_k, ✭✶✳✶✸✮

❡ s❡❥❛ T t❛❧ q✉❡

2ε < α

Z T+M M

ψ(t)dt. ✭✶✳✶✹✮

P♦r ✭✶✳✶✶✮ ❡ ✭✶✳✶✷✮✱ t❡♠♦s

I(σ(t, u))> b₋ε, u_∈∂N, t_≥0. ✭✶✳✶✺✮ ❆ss✐♠

σ(t, u)₆= 0, σ(t, u)₆=e, u_∈∂N, t_≥0, ✭✶✳✶✻✮ ♣♦✐s t❡♠♦s η < b₋3ε✳ ❙❡ u_∈∂N✱ ❝♦♠ u /_∈Q1✱ t❡♠♦s

I(u)> b+ε, ✭✶✳✶✼✮ ♣♦✐s ♥ã♦ ♣♦❞❡♠♦s t❡r✱ ♣♦r ✭✶✳✶✷✮✱ ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡

I(u)< b₋ε. ✭✶✳✶✽✮ ❆ss✐♠✱ ♣♦r ✭✶✳✶✶✮ t❡♠♦s

I(σ(t, u))_≥I(u)> b+ε, u_∈∂N, u /_∈Q1,0≤t ≤T. ✭✶✳✶✾✮

P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ u_∈∂N _∩Q1✱ s❡❥❛t1 ♦ ♠❛✐♦r ♥ú♠❡r♦ ♣♦ssí✈❡❧ t❛❧ q✉❡ t1 ≤T ❡

σ(t, u)_∈Q1

♣❛r❛ 0_≤t_≤t1✳ ❙❡ t1 < T✱ ❡♥tã♦ ♣❛r❛ δ >0s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ t❡♠♦s

I(σ(t1+δ, u))≥I(σ(t1, u))≥b−ε.

❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ σ(t1+δ, u)∈/ Q1 t❡♠♦s

I(σ(t1+δ, u))> b+ε.

❈♦♥s❡q✉❡♥t❡♠❡♥t❡

I(σ(T, u))> b+ε. ✭✶✳✷✵✮ ❙❡ t1 =T✱ ❡♥tã♦ ♣♦r ✭✶✳✶✵✮✱ ✭✶✳✺✮✱ ✭✶✳✸✮✱ ✭✶✳✾✮✱ ✭✶✳✶✸✮ ❡ ✭✶✳✶✹✮✱ t❡♠♦s

I(σ(T, u))₋I(u)_≥α

Z T

0 k

I′(σ(t, u))_kdt

≥α

Z T

0

ψ(_k(σ(t, u))_k)dt

≥α

Z T

0

ψ(_ku_k+t)dt

≥α

Z T

0

ψ(M +t)dt

=α

Z T+M M

ψ(τ)dτ >2ε.

❆ss✐♠✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✷✵✮ s❡❣✉❡ ❞❡ ✭✶✳✶✷✮✳ P♦rt❛♥t♦✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✷✵✮ é ✈á❧✐❞❛ ♣❛r❛ t♦❞♦ u_∈∂N✳ ❉❡✜♥✐♠♦s ♦s s❡❣✉✐♥t❡s ❝♦♥❥✉♥t♦s

NT ={σ(T, u);u∈N},

❡

∂NT ={σ(T, u);u∈∂N}.

P♦r ✭✶✳✼✮ ❡ ♣❡❧❛ ❞❡♣❡♥❞ê♥❝✐❛ ❝♦♥tí♥✉❛ ❞❡σ(T, u)❡♠ut❡♠♦s q✉❡NT é ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦

❧✐♠✐t❛❞♦ ❡♠ E✳ ❈♦♠♦0, e_∈Q2 ❡ g ≡0❡♠Q2✱ ♣❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ❞❡ ✭✶✳✽✮ t❡♠♦s

q✉❡ σ(T,0) = 0✱ σ(T, e) = e✳ ❈♦♠♦ σ(T,0) _∈ NT ❡ σ(T, e) ∈/ NT✱ t❡♠♦s q✉❡ 0 ∈ NT ❡

e /_∈NT✳ ❆ss✐♠✱ NT ∈Λ ❡✱ ♣♦r ✭✶✳✷✵✮

I(u)> b+ε, u_∈∂NT. ✭✶✳✷✶✮

▼❛s ✐st♦ ❝♦♥tr❛❞✐③ ✭✶✳✷✮✳ ❆ss✐♠✱ ♦ t❡♦r❡♠❛ ❡stá ♣r♦✈❛❞♦ ♣❛r❛ ♦ ❝❛s♦ η < b✳ ❆❣♦r❛✱ s✉♣♦♥❤❛ q✉❡ η=b✳ ❊♥tã♦✱ ♣♦r ✭✶✳✶✮✱ t❡♠♦s

I(u)_≥b, u_∈∂N0. ✭✶✳✷✷✮

❙❡❥❛

T = 1

2min[d(0, ∂N0), d(e, ∂N0)]. ✭✶✳✷✸✮

❊s❝♦❧❤❛ ε >0 s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ t❛❧ q✉❡ ✭✶✳✶✹✮ s❡❥❛ ✈❡r❞❛❞❡✐r♦✳ ❊♥tã♦✱ ♣r♦❝❡❞❡♥❞♦

❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❝♦♠ N0 ❡♠ ✈❡③ ❞❡ N s✉❜st✐t✉í♠♦s ✭✶✳✶✷✮ ♣♦r ✭✶✳✷✷✮✳ P❛r❛ ♦❜t❡r♠♦s

✭✶✳✶✻✮✱ ♣♦r ✭✶✳✾✮ ❡ ✭✶✳✷✸✮ t❡♠♦s ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡

k(σ(t, u)₋u_{k ≤}t_≤T.

■st♦ ✐♠♣❧✐❝❛ q✉❡ ✭✶✳✶✻✮ é ✈❡r❞❛❞❡✐r♦ ♣❛r❛ t♦❞♦ u _∈ ∂N0 ❡ 0 ≤ t ≤ T✳ P❛r❛ ♦❜t❡r ✭✶✳✷✶✮

♣r♦❝❡❞❡♠♦s ❛♥❛❧♦❣❛♠❡♥t❡ ❝♦♥tr❛❞✐③❡♥❞♦ ♠❛✐s ✉♠❛ ✈❡③ ✭✶✳✷✮✳ ❆ss✐♠✱ ❡stá ❝♦♥❝❧✉í❞♦ ♦ t❡♦r❡♠❛✳

❆❣♦r❛✱ ✈❛♠♦s r❡❧❡♠❜r❛r ❛ ❝♦♥❞✐çã♦ ❞❡ ❈❡r❛♠✐✿

❉❡✜♥✐çã♦ ✷ ❙❡❥❛ E ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ r❡❛❧ ❡I _∈C1_{(E,R)✳ ❯♠❛ s❡q✉ê♥❝✐❛(u}

n)⊂E

é ❞✐t❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ♥í✈❡❧ c_∈R q✉❛♥❞♦

I(un)→c ❡ (1 +kunk)kI′(un)kE∗ →0. ✭✶✳✷✹✮

❈♦♠♦ ❛♣❧✐❝❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✶✳✶✱ t❡♠♦s ❛ s❡❣✉✐t❡ ✈❛r✐❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞♦ ♣❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✳

❚❡♦r❡♠❛ ✶✳✷ ❙❡❥❛ E ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ r❡❛❧ ❝♦♠ s❡✉ ❡s♣❛ç♦ ❞✉❛❧ E∗ _{❡ s✉♣♦♥❤❛ q✉❡}

I _∈C1_{(E,R) s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦}

max_{I(0), I(e)_{} ≤}0< η _≤ inf

kuk=ρI(u)

♣❛r❛ ❛❧❣✉♠ 0< η✱ ρ >0 ❡ e_∈E ❝♦♠ _ke_k> ρ✳ ❙❡ cé ❞❡✜♥✐❞♦ ♣♦r

c= sup

N∈Λ

inf

u∈∂N0

I(u),

♦♥❞❡ N0 = Bρ(0) ❡ Λ = {N ⊂ E;N é ❛❜❡rt♦ ❡ ❧✐♠✐t❛❞♦}✱ ❡♥tã♦ c > 0 ❡ ❡①✐st❡ ✉♠❛

s❡q✉ê♥❝✐❛ (un)⊂E q✉❡ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ ❈❡r❛♠✐✱ ♦✉ s❡❥❛✱

I(un)→c ❡ I′(un)(1 +kunk) = 0.

❉❡♠♦♥str❛çã♦✿ ❖❜s❡r✈❡ q✉❡ c > 0✳ ❉❡ ❢❛t♦✱

c= sup

N∈Λ

inf

u∈∂N0

I(u)_≥ inf

u∈∂N0

I(u)_≥η >0.

❉❡s❞❡ q✉❡ ψ(s) = ₁₊1_{s ∈}Ψ✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✶✱ ♦❜t❡♠♦s ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✳

✶✳✸ ❖ ♣r✐♥❝í♣✐♦ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞

❖ s❡❣✉✐♥t❡ ♣r✐♥❝í♣✐♦ ❞❡✈✐❞♦ ❛ ■✳ ❊❦❡❧❛♥❞ ❬✼❪ s❡rá ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦❜t❡r♠♦s s♦❧✉çõ❡s ❝♦♠ ❡♥❡r❣✐❛ ♥❡❣❛t✐✈❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❛ s❡r ❡st✉❞❛❞♦✳

❚❡♦r❡♠❛ ✶✳✸ ✭Pr✐♥❝í♣✐♦ ❱❛r✐❛❝✐♦♥❛❧ ❞❡ ❊❦❡❧❛♥❞✮ ❙❡❥❛ V ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠✲ ♣❧❡t♦ ❡ F :V _→R_{∪ {+∞}} ✉♠❛ ❢✉♥çã♦ s❡♠✐❝♦♥tí♥✉❛ ✐♥❢❡r✐♦r♠❡♥t❡ ❡ ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r✲ ♠❡♥t❡✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ ε >0 ❡①✐st❡ v _∈V t❛❧ q✉❡✿

F(v)_≤ inf

v∈V F(v) +ε e F(w)≥F(v) − εd(v, w) para todo w∈V.

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ u0 ∈V t❛❧ q✉❡

F(u0)≤ inf

v∈V F(v) +ε.

❉❡✜♥✐r❡♠♦s ❛ s❡❣✉✐r ✐♥❞✉t✐✈❛♠❡♥t❡ ✉♠❛ s❡q✉ê♥❝✐❛ (un) ❝✉❥♦ ♣r✐♠❡✐r♦ ❡❧❡♠❡♥t♦ s❡❥❛ u0✳

❊s❝♦❧❤❛♠♦s ✉♠ ❡❧❡♠❡♥t♦ un ∈V✳ ❊♥tã♦ ♣♦❞❡♠♦s t❡r ✉♠ ❞♦s s❡❣✉✐♥t❡s ❝❛s♦s✿

✭❛✮ F(w)> F(un)−εd(un, w), ∀w6=un❀

✭❜✮ ❡①✐st❡ w₆=un t❛❧ q✉❡ F(w)≤F(un)−εd(un, w)✳

❙✉♣♦♥❞♦ q✉❡ ✭❛✮ s❡❥❛ ✈❡r❞❛❞❡✐r♦✱ ❞❡✜♥✐♠♦s un+1 =un✳ ❈❛s♦ ❝♦♥trár✐♦✱ s❡ ✭❜✮ ❢♦r ✈❡r❞❛✲

❞❡✐r♦✱ ❡♥tã♦ ❞❡✜♥✐♠♦s un+1 ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

❙❡❥❛Sn ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s w∈V s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭❜✮✳ ❆ss✐♠✱ ❡s❝♦❧❤❛

un+1 ∈Sn t❛❧ q✉❡

F((un+1)− inf

u∈Sn

F(u)_≤ 1 2

F(un)− inf u∈Sn

F(u)

.

❆✜r♠❛♠♦s q✉❡ ❛ s❡q✉ê♥❝✐❛ (un) é ❞❡ ❈❛✉❝❤②✳ ❉❡ ❢❛t♦✱ s❡ ✭❛✮ ❢♦r ✈❡r❞❛❞❡✐r♦✱ ❡♥tã♦ ❛

s❡q✉ê♥❝✐❛ é ❝♦♥✈❡r❣❡♥t❡ ❡ ♣♦rt❛♥t♦ ❞❡ ❈❛✉❝❤②✳ ❈❛s♦ ❝♦♥trár✐♦✱ s❡ ♦ ✐t❡♠ ✭❜✮ ❢♦r ✈❡r❞❛❞❡✐r♦✱ t❡♠♦s s❛t✐s❢❡✐t❛ ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡✿

εd(un, un+1)≤F(un)−F(un+1), ∀n ∈N. ✭✶✳✷✺✮

❘❡♦r❞❡♥❛♥❞♦ ♦s t❡r♠♦s ❞❛ s❡q✉ê♥❝✐❛✱ t❡♠♦s

εd(un, up)≤F(un)−F(up), ∀n≤p. ✭✶✳✷✻✮

❆ss✐♠✱ F(un)é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡❝r❡s❝❡♥t❡ ❡ ❧✐♠✐t❛❞❛✳ ❉❡ ❢❛t♦✱

F(un)≥εd(un, un+1) +F(un+1)≥F(un+1).

❆ss✐♠✱ F(un) é ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡ ❡ ❞❡❝r❡s❝❡♥t❡✱ ❧♦❣♦ é ❧✐♠✐t❛❞❛✳ P♦rt❛♥t♦✱ é ❝♦♥✲

✈❡r❣❡♥t❡✳ ❆ss✐♠✱ ❢❛③❡♥❞♦ n, p _→ +_∞ ♦❜t❡♠♦s ❝❧❛r❛♠❡♥t❡ q✉❡ (un) é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡

❈❛✉❝❤②✳ ❈♦♠♦ V é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✱(un) ❝♦♥✈❡r❣❡ ♣❛r❛ ❛❧❣✉♠ ♣♦♥t♦ v ∈V✳

❯s❛♥❞♦ ♦ ❢❛t♦ ❞❡ q✉❡ F é s❡♠✐❝♦♥tí♥✉❛ ✐♥❢❡r✐♦r♠❡♥t❡✱ t❡♠♦s F(v)_≤lim

n F(un).

❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ F(un)é ❞❡❝r❡s❝❡♥t❡✱ ❞❛❞♦ q✉❛❧q✉❡r u∈V✱ t❡♠♦s

F(v)_≤lim

n F(un)≤F(u)≤vinf∈V F(v) +ε,

❞♦♥❞❡ ♦❜t❡♠♦s ❛ ♣r✐♠❡✐r❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞♦ t❡♦r❡♠❛✳

P❛r❛ ❞❡♠♦♥str❛r♠♦s ❛ ♦✉tr❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞♦ t❡♦r❡♠❛✱ s✉♣♦♥❤❛♠♦s q✉❡ ❡❧❛ ♥ã♦ s❡❥❛ ✈❡r❞❛❞❡✐r❛✱ ♦✉ s❡❥❛✱ s✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❡ w_∈V t❛❧ q✉❡

F(w)< F(v)₋εd(v, w). ✭✶✳✷✼✮ ❋❛③❡♥❞♦ p_→+_∞ ♥❛ ❡q✉❛çã♦ ✭✶✳✷✻✮ ♦❜t❡♠♦s

F(w)_≤limF(up)≤F(un)−εd(un, w).

P♦rt❛♥t♦✱ w_∈Sn, ∀ n ∈N✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ✭✶✳✷✺✮ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

2F(un+1)−F(un)≤ inf u∈Sn

F(u)_≤F(w). ✭✶✳✷✽✮ ❉❡s❞❡ q✉❡

F(un)→l,

t♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ ❡♠ ✭✶✳✷✽✮✱ t❡♠♦s

l _≤F(w). ❈♦♠♦ F é s❡♠✐❝♦♥tí♥✉❛ ✐♥❢❡r✐♦r♠❡♥t❡✱ t❡♠♦s

F(v)_≤l_≤F(w, ♦ q✉❡ ❝♦♥tr❛❞✐③ ✭✶✳✷✼✮✳ ▲♦❣♦✱ ❞❡✈❡♠♦s t❡r

F(w)_≥F(v)₋εd(v, w).

❯♠ ♣r♦❜❧❡♠❛ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❧✐♥❡❛r

◆❡st❡ ❝❛♣ít✉❧♦✱ ❜❛s❡❛❞♦ ♥♦ ❛rt✐❣♦ ❞❡ ▲✐✲❲✉✲❩❤♦✉ ❬✶✵❪✱ ✈❛♠♦s ❡st✉❞❛r ♦ ♣r♦❜❧❡♠❛

          

−∆u=h(x)uq+f(x, u) ❡♠ Ω

u_≥0 ❡♠ Ω

u= 0 s♦❜r❡ ∂Ω,

✭✷✳✶✮

♦♥❞❡ Ω é ✉♠ ❞♦♠í♥✐♦ s✉❛✈❡ ❡ ❧✐♠✐t❛❞♦ ❡♠ RN _{(N ≥1)}✱_{0< q <1}✳

❙❡❥❛λ1 >0 ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ✭✈❡❥❛ ❆♣ê♥❞✐❝❡✮ ❞❡−∆ ❡♠ H01(Ω)✱ ✐st♦ é✱

  

−∆u=λ1u ❡♠ Ω

u= 0 s♦❜r❡ ∂Ω. ❙❛❜❡♠♦s q✉❡λ1 é ❝❛r❛❝t❡r✐③❛❞♦ ♣♦r

λ1 = inf

R

Ω|∇u| 2_dx

R

Ωu2dx

:u_∈H₀1(Ω), u₆= 0

.

◆❡st❡ ❝❛♣ít✉❧♦ ✈❛♠♦s ❛ss✉♠✐r q✉❡ h(x) s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿

(h1) h(x)∈L∞(Ω) ❡ h(x)6= 0❀

(h2) ❡①✐st❡ v ∈H01(Ω) t❛❧ q✉❡

R

Ωh(x)(v

+₎q+1_{dx >0.}

❚❡♠♦s ❝♦♠♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❡♥❝♦♥tr❛r s♦❧✉çõ❡s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭✷✳✶✮ q✉❛♥❞♦f(x, s)

t❡♠ ❝r❡s❝✐♠❡♥t♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❧✐♥❡❛r ♥♦ ✐♥✜♥✐t♦✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✿

(f1) f(x, s)∈C( ¯Ω×R,R)❀ f(x,0)≡0;f(x, s)≥(6≡)0 ♣❛r❛ t♦❞♦s ≥0✱ x∈Ω.

s ∈ ∞ ∈

P❛r❛ ✈✐s✉❛❧✐③❛r ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ❛s ❝♦♥❞✐çõ❡s(f1)✱(f2) ❡(f3)✱ ♦❜s❡r✈❡ ❛ ❋✐❣✉r❛ ✷✳✶✿

❋✐❣✉r❛ ✷✳✶✿

❊①❡♠♣❧♦✿ ❙❡ α >0❡ ℓ > λ1✱ ❡♥tã♦ ❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r

f(x, s) :=

    

ℓsα

1 +sα, s❡ s≥0

0, s❡ s_≤0, s❛t✐s❢❛③ ❛s ❤✐♣ót❡s❡s (f1)✱ (f2) ❡ (f3)✳

❉❡✜♥✐çã♦ ✸ ❉✐③❡♠♦s q✉❡ u_∈H1

0(Ω) é ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭✷✳✶✮ s❡

Z

Ω∇

u_∇ϕdx=

Z

Ω

h(x)(u+₎q_ϕdx+

Z

Ω

f(x, u+_{)ϕdx, ♣❛r❛ t♦❞♦ϕ ∈H}1

0(Ω). ✭✷✳✷✮

◆♦t❡ q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ✭✷✳✶✮ é ❞❛❞♦ ♣♦r✭✈❡❥❛ ❆♣ê♥❞✐❝❡✮ I(u) = 1

2

Z

Ω|∇

u_|2dx₋ 1 q+ 1

Z

Ω

h(x)(u+)q+1dx₋

Z

Ω

F(x, u+)dx. ✭✷✳✸✮ P❡❧❛ Pr♦♣♦s✐çã♦ ❆✳✶ t❡♠♦s q✉❡ I _∈ C1_(H1

0(Ω),R)✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ t♦❞♦ ϕ ∈ H01(Ω)

t❡♠♦s

I′(u)ϕ =

Z

Ω∇

u_∇ϕdx₋

Z

Ω

h(x)(u+)qϕdx₋

Z

Ω

f(x, u+)ϕdx. ✭✷✳✹✮ ❙❡ u_∈H1

0(Ω) é ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ I✱ ❡s❝♦❧❤❡♥❞♦ϕ =u− ❡♠ ✭✷✳✹✮✱ ♦❜t❡♠♦s

−

Z

Ω|∇

u−_|2 =

Z

Ω

h(x)(u+)qu−dx+

Z

Ω

f(x, u+)u−dx= 0.

▲♦❣♦✱ u=u+ _{❡ ❛ss✐♠ ♣♦♥t♦s ❝rít✐❝♦s ❞❡ I sã♦ s♦❧✉çõ❡s ❢r❛❝❛s ♥ã♦✲♥❡❣❛t✐✈❛s ❞❡ ✭✷✳✶✮✳}

❖❜s❡r✈❛çã♦ ✷✳✶ ❙✉♣♦♥❤❛ q✉❡ h(x) _≥ 0✳ ❊♥tã♦✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞♦ ▼á①✐♠♦ ❋♦rt❡✱ ♦

q✉❛❧ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✽❪✱ ♦s ♣♦♥t♦s ❝rít✐❝♦s ♥ã♦✲♥✉❧♦s ❞❡ ✭✷✳✸✮ sã♦ s♦❧✉çõ❡s ❢r❛❝❛s ♣♦s✐t✐✈❛s ❞♦ ♣r♦❜❧❡♠❛ ✭✷✳✶✮✳

◆♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r✱ s♦❜ ❛s ❤✐♣ót❡s❡s ❛♥t❡r✐♦r❡s✱ ❝♦♠ ❡①❝❡çã♦ ❞❛ ❤✐♣ót❡s❡(h2)✱ ❡♥❝♦♥✲

tr❛r❡♠♦s ✉♠❛ s♦❧✉çã♦ ♥ã♦✲♥❡❣❛t✐✈❛ ❝♦♠ ❡♥❡r❣✐❛ ♣♦s✐t✐✈❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭✷✳✶✮ q✉❛♥❞♦ ❛ ♥♦r♠❛ kh_k∞ é ♣❡q✉❡♥❛✳

❚❡♦r❡♠❛ ✷✳✶ ❙✉♣♦♥❤❛ q✉❡ ❛s ❢✉♥çõ❡s h ❡f(x, s)s❛t✐s❢❛ç❛♠ ❛s ❤✐♣ót❡s❡s (h1)✱ (f1)✱ (f2)

❡ (f3)✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ m > 0 t❛❧ q✉❡ s❡ khk∞ < m✱ ♦ ♣r♦❜❧❡♠❛ ✭✷✳✶✮ t❡♠

✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ♥ã♦✲♥❡❣❛t✐✈❛✱ u1 ∈ H01(Ω)✱ ❝♦♠ I(u1) > 0✳ ❆❧é♠ ❞✐ss♦✱ s❡ h(x) ≥ 0

❡♥tã♦ u1 >0✳

❆ss✉♠✐♥❞♦ ✉♠❛ ❤✐♣ót❡s❡ ❛❞✐❝✐♦♥❛❧✱ ❛❧é♠ ❞❛ s♦❧✉çã♦ ♦❜t✐❞❛ ♥♦ ❚❡♦r❡♠❛ ✷✳✶✱ ♦❜t❡♠♦s ✉♠❛ ♦✉tr❛ s♦❧✉çã♦ ♥ã♦✲♥❡❣❛t✐✈❛ u2 ❝♦♠ ❡♥❡r❣✐❛ ♥❡❣❛t✐✈❛✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ t❡♠♦s

❚❡♦r❡♠❛ ✷✳✷ ❙✉♣♦♥❤❛ q✉❡ ❛s ❢✉♥çõ❡s h ❡ f(x, s) s❛t✐s❢❛♠ ❛s ❤✐♣ót❡s❡s (h1)✱ (h2)✱ (f1)✱

(f2)❡(f3)✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡m >0t❛❧ q✉❡ s❡khk∞ < m✱ ♦ ♣r♦❜❧❡♠❛ ✭✷✳✶✮ t❡♠

♣❡❧♦ ♠❡♥♦s ❞✉❛s s♦❧✉çõ❡s ❢r❛❝❛s ♥ã♦✲♥❡❣❛t✐✈❛s u1, u2 ∈H01(Ω) t❛✐s q✉❡I(u1)<0< I(u2)✳

❆❧é♠ ❞✐ss♦✱ s❡ h(x)_≥0 ❞❡✈❡♠♦s t❡r u1 >0 ❡ u2 >0✳

✷✳✶ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✷✳✶

P❛r❛ ❞❡♠♦♥str❛r ♦ ❚❡♦r❡♠❛ ✷✳✶✱ ✉s❛r❡♠♦s ♦ ❚❡♦r❡♠❛ ✶✳✷✳ ❆ s❡❣✉✐r✱ s❡rã♦ ❡♥✉♥❝✐❛❞♦s ❡ ❞❡♠♦♥str❛❞♦s ❛❧❣✉♥s ❧❡♠❛s ✐♠♣♦rt❛♥t❡s ✉t✐❧✐③❛❞♦s ♥❛ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛✳ ■♥✐✲ ❝✐❛♠♦s ❡st❛ s❡çã♦ ❝♦♠ ♦ ▲❡♠❛ ✷✳✶✱ ♦ q✉❛❧ tr❛t❛ ❛ ❣❡♦♠❡tr✐❛ ❞♦ ❚❡♦r❡♠❛ ✶✳✷✳ ❊st❡ ❧❡♠❛ é ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❣❛r❛♥t✐r✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ❚❡♦r❡♠❛ ✶✳✷✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ♥í✈❡❧ ♠❛①✲♠✐♥ c >0✳

▲❡♠❛ ✷✳✶ ❙✉♣♦♥❤❛ q✉❡ ❛s ❢✉♥çõ❡s h ❡ f(x, s)s❛❛t✐s❢❛ç❛♠ ❛s ❤✐♣ót❡s❡s (h1)✱ (f1)✱ (f2) ❡

(f3)✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ m >0 t❛❧ q✉❡ s❡ khk∞< m✱ t❡♠♦s✿

✐✮ ❊①✐st❡♠ ρ >0✱ η >0 t❛✐s q✉❡

✐✐✮ ❊①✐st❡ e_∈H1

0(Ω) ❝♦♠ kek> ρ t❛❧ q✉❡ I(e)<0✳

❉❡♠♦♥str❛çã♦✿ Pr♦✈❛ ❞♦ ✐t❡♠ ✐✮✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ✈❛♠♦s ♣r♦✈❛r ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿ ❆✜r♠❛çã♦ ✶ ❉❛❞♦ ε >0 ❡①✐st❡ Cε=Cε(ε, f,Ω, k) t❛❧ q✉❡✿

F(x, s)_≤ (µ+ε) 2 s

2_+C

εsk+1, ∀s≥0, x∈Ω, ✭✷✳✺✮

♦♥❞❡ k é ✉♠❛ ❝♦♥st❛♥t❡ t❛❧ q✉❡ s❡ N _≥ 3✱ ❡♥tã♦ 1 < k < N+2

N−2✳ ❙❡ N = 1,2 ❡♥tã♦

1< k <+_∞✳

P❡❧❛ ❤✐♣ót❡s❡ (f3) t❡♠♦s

lim

s→+∞

f(x, s)

sk = 0, ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ x∈Ω. ✭✷✳✻✮

❉❡ ❢❛t♦✱

lim

s→+∞

f(x, s)

sk =

lim

s→+∞

f(x, s)

s s→lim+∞

1

sk−1

=ℓ.0 = 0.

❆ss✐♠✱ ♣❡❧❛s ❤✐♣ót❡s❡s (f1),(f2) ❡ (f3) ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ Cε = C(ε, k, f,Ω) > 0 t❛❧

q✉❡

f(x, s)_≤(µ+ε)s+Cεsk, ♣❛r❛ t♦❞♦s≥0, x∈Ω. ✭✷✳✼✮

❉❡ ❢❛t♦✱ ♣❡❧❛ ❤✐♣ót❡s❡ (f2)❡①✐st❡ s0 >0t❛❧ q✉❡

f(x, s)_≤(µ+ε)s, s❡0< s_≤s0, x∈Ω. ✭✷✳✽✮

P♦r ✭✷✳✻✮✱ ❡①✐st❡♠ ❝♦♥st❛♥t❡s s1 >0❡ C1 >0 t❛✐s q✉❡

f(x, s)_≤C1sk, ♣❛r❛ t♦❞♦s≥s1, x∈Ω. ✭✷✳✾✮

P❡❧❛ ❤✐♣ót❡s❡ (f1)✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C2 >0 t❛❧ q✉❡

f(x, s)_≤C2 =C2

sk

sk ≤C2

sk

s0

=C3sk, s❡s∈[s0, s1], x∈Ω. ✭✷✳✶✵✮

❉❛s ❞❡s✐❣✉❛❧❞❛❞❡s ✭✷✳✽✮✲✭✷✳✾✮✲✭✷✳✶✵✮ ♦❜t❡♠♦s ♦❜t❡♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✼✮✳ ■♥t❡❣r❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✼✮✱ ♦❜t❡♠♦s✿

Z s

0

f(x, t)dt _≤

Z s

0

(µ+ε)tdt+

Z s

0

Cεtkdt,

♦ q✉❡ ♣r♦✈❛ ✭✷✳✺✮✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s

h(x)_{≤ |}h(x)_{| ≤ k}h_k∞, q✳t✳♣✳ ❡♠ Ω.

❆❣♦r❛ ✜①❡♠♦s ε0 > 0 t❛❧ q✉❡ µ+ε0 < λ1✳ P❡❧❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r

t❡♠♦s

(µ+ε0)

2

Z

Ω

u2 _≤ (µ+ε0)

2λ1

Z

Ω|∇

u_|2dx.

P❡❧❛ ❞❡✜♥✐çã♦ ❞❡I ❞❛❞❛ ❡♠ ✭✷✳✸✮✱ s❡❣✉❡ ❞❡ ✭✷✳✺✮ ❡ ❞❛ ✐♠❡rsã♦ ❞❡ ❙♦❜♦❧❡✈H1

0(Ω)֒→Lp(Ω)

q✉❡

I(u)_≥ 1 2

Z

Ω|∇

u_|2dx₋ 1 q+ 1

Z

Ωk

h_k∞(u+)q+1dx−

Z

Ω

[(µ+ε0) 2 u

2_+C

ε|u|k+1]dx

≥(1 2−

(µ+ε0)

2λ1

)_ku_k2₋khk∞ q+ 1

Z

Ω

(u+)q+1dx₋Cε

Z

Ω|

u_|k+1dx

≥ C1kuk2 −C2khk∞kukq+1−C3kukk+1.

❆ss✐♠✱

I(u)_≥[C1− khk∞C2kukq−1−C3kukk−1]kuk2. ✭✷✳✶✶✮

❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r✿

g(t) =C2khk∞tq−1+C3tk−1,

♦♥❞❡ t _≥0✱ q _∈(0,1) ❡k _∈[1,N_N+2₋₂)s❡ N _≥3✱ ♦✉k _∈ [1,+_∞) s❡1 _≤N <3✳ ❉❡r✐✈❛♥❞♦

g ❡♠ r❡❧❛çã♦ ❛ t✱ t❡♠♦s

g′(t) =C2(q−1)khk∞tq−2+C3(k−1)tk−2.

❙❡ t0 >0é t❛❧ q✉❡ g′(t0) = 0✱ ❡♥tã♦

C3(k−1)tk0−q+C2(q−1)khk∞= 0,

♦ q✉❡ ✐♠♣❧✐❝❛

t0 =

(1₋q)C2

C3(k−1)k

h_k∞

1/(k−q)

, 0< q <1< k. ❱❛♠♦s ❞❡♥♦t❛r C4 = _C(1₃−_(kq)₋C₁₎2✳ ❆ss✐♠✱

g(t0) =C2khk∞(C4khk∞)

q−1

k−q +C

3(C4khk∞)

k−1

k−q =C 5khk

k−1

k−q

∞ ,

♦♥❞❡C5 =C2C q−1

k−q

4 +C3C k−1

k−q

4 ✱ ❡ kk−−1q >0✱ ♣♦✐s0< q <1< k✳ ❆ss✐♠✱ ❡①✐st❡m= ( C1

C5)

k−q k−1 >

0✱ t❛❧ q✉❡g(t0)< C1 s❡khk∞ < m✳ ❊♥tã♦✱ s❡khk∞< m ❡ ρ=t0✱ ♣♦r ✭✷✳✶✶✮ ❡ ♣❛r❛ t♦❞♦

u_∈H1

0(Ω) t❛❧ q✉❡ kuk=ρ✱ t❡♠♦s

I(u)_≥(C1−g(t0))t20 =η >0,

❞♦♥❞❡ ♦❜t❡♠♦s ✐✮✳

Pr♦✈❛ ❞♦ ✐t❡♠ ✐✐✮✳ ❙❡❣✉❡ ❞❛ ❤✐♣ót❡s❡(f3)q✉❡

lim

s→+∞

F(x, s)

s2 =

ℓ

2, ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ x∈Ω. ✭✷✳✶✷✮

❉❡ ❢❛t♦✱ ❞❛❞♦ ε >0✱ ❡①✐st❡ M > 0 t❛❧ q✉❡

ℓ₋ε_≤ f(x, s)

s ≤ℓ+ε, ♣❛r❛ t♦❞♦s≥M. ❆ss✐♠✱ ✐♥t❡❣r❛♥❞♦ ❞❡ 0 ❛s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ t❡♠♦s

Z s

0

(ℓ₋ε)tdt_≤

Z s

0

f(x, t)dt _≤

Z s

0

(ℓ+ε)tdt, ♣❛r❛ t♦❞♦ s_≥M. ▲♦❣♦✱

(ℓ₋ε) 2 ≤

F(x, s)

s2 ≤

(ℓ+ε)

2 , ♣❛r❛ t♦❞♦s≥M.

P♦rt❛♥t♦✱

lim

s→+∞

F(x, s)

s2 =

ℓ

2, ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ x∈Ω.

❆❣♦r❛✱ ✉s❛♥❞♦ q✉❡ℓ > λ1✱ ❡①✐st❡ τ >0t❛❧ q✉❡✱ ♣❛r❛s >0s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ t❡♠♦s

F(x, s)

s2 ≥

ℓ₋τ

2 >

λ1

2 , ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ x∈Ω. ✭✷✳✶✸✮

❙❡❥❛ ϕ1 >0 ❛ ❛✉t♦❢✉♥çã♦ ❛ss♦❝✐❛❞❛ ❛♦ ❛✉t♦✈❛❧♦r λ1✳ ❙❛❜❡♠♦s q✉❡

kϕ1k2 =λ1kϕ1k2L2_(Ω).

❆ss✐♠ t❡♠♦s

I(tϕ1) =

t2

2kϕ1k

2

− t

q+1

q+ 1

Z

Ω

h(x)ϕq₁+1dx₋

Z

Ω

F(x, tϕ1)dx

≤ t

2

2kϕ1k

2

− t

q+1

q+ 1

Z

Ω

h(x)ϕq1+1dx−

t2

2

Z

Ω

(ℓ₋τ)ϕ21dx

= t

2

2

Z

Ω

(λ1−ℓ+τ)ϕ21dx−

tq+1

q+ 1

Z

Ω

h(x)ϕq₁+1dx

= t

2

2

Z

Ω

(λ1 −ℓ+τ)ϕ21dx−

tq−1

q+ 1

Z

Ω

h(x)ϕq1+1dx

.

❈♦♠♦

0< q <1 ❡ λ1−ℓ+τ <0,

t❡♠♦s

lim

t→+∞

tq−1

q+ 1

Z

Ω

h(x)ϕq₁+1dx= 0 ❡

Z

Ω

(λ1−ℓ+τ)ϕ21dx <0.

❊s❝♦❧❤❡♥❞♦ t0 s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ t❛❧ q✉❡

kt0ϕ1k> ρ,

t❡♠♦s

I(t0ϕ1)<0.

▲♦❣♦✱ ❡s❝♦❧❤❡♥❞♦ e=t0ϕ1✱ ♦❜t❡♠♦s ✐✐✮ ❡ ✐st♦ ❝♦♠♣❧❡t❛ ❛ ♣r♦✈❛ ❞♦ ❧❡♠❛✳

▲❡♠❛ ✷✳✷ ❙❡❥❛ Ω _⊂ Rn ✉♠ ❛❜❡rt♦ ❧✐♠✐t❛❞♦ ❡ _{(un) ⊂ H}1

0(Ω) ✉♠❛ s❡q✉ê♥❝✐❛ t❛❧ q✉❡

un ⇀ u ❡♠ H01(Ω)✳ ❊♥tã♦ u+n ⇀ u+ ❡ u−n ⇀ u−✳

❉❡♠♦♥str❛çã♦✿ P❡❧❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❢r❛❝❛ ❞❡(un)❡♠H01(Ω)✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡C > 0

t❛❧ q✉❡ kunk ≤C.❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ un =u+n +u−n✱ t❡♠♦s

kunk2 =

Z

Ω|∇

un|2dx

=

Z

Ω|∇

(u+n +u−n)|2dx

=

Z

Ω|∇

u+_n|2dx+ 2

Z

Ω∇

u+_n∇u−_ndx+

Z

Ω|∇

u−_n|2dx

=

Z

Ω|∇

u+_n|2dx+

Z

Ω|∇

u−_n|2dx

=_ku+_nk2+_ku−_nk2. ▲♦❣♦ (u+

n) é ❧✐♠✐t❛❞❛ ❡♠ H01(Ω)✳ ❆ss✐♠✱ ❡①✐st❡ v ∈H01(Ω) t❛❧ q✉❡

          

u+_n ⇀ v ❡♠ H₀1(Ω),

u+_{n →}v ❡♠ Lp(Ω) 1_≤p <2∗, u+_n(x)_→v(x) q✳t✳♣ ❡♠ Ω.

✭✷✳✶✹✮

❉❡s❞❡ q✉❡ ❛ ✐♠❡rsã♦ H1

0(Ω)֒→Lp(Ω) é ❝♦♠♣❛❝t❛ t❡♠♦s

  

un→u, ❡♠ Lp(Ω) 1≤p < 2∗,

un(x)→u(x) q✳t✳♣ ❡♠Ω.

✭✷✳✶✺✮

P♦r ♦✉tr♦ ❧❛❞♦✱ s❛❜❡♠♦s q✉❡

u+

n =

un+|un|

2 ❡ u

+ ₌ u+|u|

2 .

❆ss✐♠✱ s❡❣✉❡ ❞❡ ✭✷✳✶✺✮ q✉❡

▲♦❣♦✱ ♣❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ❧✐♠✐t❡✱ s❡❣✉❡ ❞❡ ✭✷✳✶✹✮ q✉❡ v =u+_{✳ P♦rt❛♥t♦✱}

u+_n ⇀ u+.

❆♥❛❧♦❣❛♠❡♥t❡✱ ❝♦♥❝❧✉✐✲s❡ q✉❡ u−

n ⇀ u−,❡ ✐st♦ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞♦ ❧❡♠❛✳

▲❡♠❛ ✷✳✸ ❙❡❥❛ Ω_⊂Rn ✉♠ ❛❜❡rt♦ ❧✐♠✐t❛❞♦ ❡_(fn)⊂L2_(Ω) ✉♠❛ s❡q✉ê♥❝✐❛ t❛❧ q✉❡_{fn ⇀ f} ❡♠ L2_{(Ω) ❡ f}

n≥0✱ ♣❛r❛ t♦❞♦ n✳ ❊♥tã♦ t❡♠♦s q✉❡ f ≥0 q✳t✳♣ ❡♠ Ω✳

❉❡♠♦♥str❛çã♦✿ P❡❧♦ ❚❡♦r❡♠❛ ❞❛ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ❘✐❡s③ ❡♠ L2_{(Ω)✱ t❡♠♦s}

lim

n→+∞

Z

Ω

fnϕdx=

Z

Ω

f ϕdx ♣❛r❛ t♦❞♦ ϕ_∈L2(Ω). ❋❛③❡♥❞♦ ❛ s✉❜st✐t✉✐çã♦ ϕ =f−_{✱ t❡♠♦s}

0_≤ lim

n→+∞

Z

Ω

fnf−dx

=

Z

Ω

f f−dx

=

Z

Ω

(f+₋f−)f−dx

=₋

Z

Ω

(f−₎2_dx.

▲♦❣♦✱

f−= 0 q✳t✳♣ ❡♠ Ω. P♦rt❛♥t♦✱

f _≥0 q✳t✳♣ ❡♠ Ω, ❡ ✐st♦ ❝♦♥❝❧✉✐ ♦ ❧❡♠❛✳

❖ ❧❡♠❛ ❛ s❡❣✉✐r✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ✐♠❡rsã♦ ❝♦♠♣❛❝t❛ H1

0(Ω) ֒→ Lp(Ω)✱ s❡rá ✐♠♣♦r✲

t❛♥t❡ ♣❛r❛ ♦❜t❡r♠♦s ❞❡ ♠❛♥❡✐r❛ ❞✐r❡t❛ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭✷✳✶✮✳ ❊st❡ ❧❡♠❛ ♥♦s ❞✐③ q✉❡✱ s♦❜ ❛s ❝♦♥❞✐çõ❡s ❞❛❞❛s ♥♦ ✐♥í❝✐♦ ❞♦ ❝❛♣ít✉❧♦ s♦❜r❡ ❛s ❢✉♥çõ❡sf(x, s)❡h✱ q✉❛❧q✉❡r

s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ é ❧✐♠✐t❛❞❛✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ t❡♠♦s

▲❡♠❛ ✷✳✹ ❙✉♣♦♥❤❛ q✉❡ ❛s ❢✉♥çõ❡s f(x, s) ❡ h(x) s❛t✐s❢❛③❡♠ ❛s ❝♦♥❞✐çõ❡s (h1),(f1),(f2)

❡ (f3)✳ ❊♥tã♦ t♦❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ (un) ♣❛r❛ ♦ ❢✉♥❝✐♦♥❛❧ I é ❧✐♠✐t❛❞❛✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ (un)⊂H01(Ω) ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❡r❛♠✐ ♥♦ ♥í✈❡❧ c >0✱ ♦✉ s❡❥❛✱

I(un) =

1 2kunk

2

− 1

q+ 1

Z

Ω

h(x)(u+

n)q+1dx−

Z

Ω

F(x, u+

n)dx=c+o(1) ✭✷✳✶✻✮