UNIVERSIDADE FEDERAL DE SÃO CARLOS
CENTRO DE CIÊNCIAS EXATAS E DE TECNOLOGIA
PROGRAMA DE PÓS GRADUAÇÃO EM MATEMÁTICA
❋❛❜✐❛♥❛ ▼❛r✐❛ ❋❡rr❡✐r❛
Pr♦❜❧❡♠❛s ❊❧í♣t✐❝♦s ❙✉♣❡r❧✐♥❡❛r❡s
❝♦♠ ❘❡ss♦♥â♥❝✐❛
❙ã♦ ❈❛r❧♦s ✲ ❙P
❆❣♦st♦ ❞❡ ✷✵✶✺
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆
❋❛❜✐❛♥❛ ▼❛r✐❛ ❋❡rr❡✐r❛
❖r✐❡♥t❛❞♦r✿Pr♦❢ ❉r✳ ❋r❛♥❝✐s❝♦ ❖❞❛✐r ❱✐❡✐r❛ ❞❡ P❛✐✈❛
Pr♦❜❧❡♠❛s ❊❧í♣t✐❝♦s ❙✉♣❡r❧✐♥❡❛r❡s
❝♦♠ ❘❡ss♦♥â♥❝✐❛
❚❡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)
F383pe Ferreira, Fabiana Maria Problemas elípticos superlineares com ressonância / Fabiana Maria Ferreira. -- São Carlos : UFSCar, 2016. 79 p.
Tese (Doutorado) -- Universidade Federal de São Carlos, 2015.
❆❣r❛❞❡❝✐♠❡♥t♦s
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♥ã♦ tr✐✈✐❛✐s ♣❛r❛ ❝❧❛ss❡s ❞❡ s✐st❡✲ ♠❛s ❡❧í♣t✐❝♦s r❡ss♦♥❛♥t❡s ❡ s✉♣❡r❧✐♥❡❛r❡s✳ ❚❛✐s s✐st❡♠❛s sã♦ tr❛t❛❞♦s ✈✐❛ ♠ét♦❞♦s t♦♣♦❧ó❣✐✲ ❝♦s✳ ❊♥❝♦♥tr❛♠♦s ❡st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐ ♣❛r❛ ♣♦ssí✈❡✐s s♦❧✉çõ❡s ❞❡st❡s s✐st❡♠❛s ❡ ✉t✐❧✐③❛♠♦s ❡st❛s ❡st✐♠❛t✐✈❛s ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ t❡♦r✐❛ ❞♦ ❣r❛✉ t♦♣♦❧ó❣✐❝♦ ♣❛r❛ ❣❛r❛♥t✐r ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s✳
❆❜str❛❝t
❚❤❡ ❛✐♠ ♦❢ t❤✐s ✇♦r❦ ✐s t♦ ♣r❡s❡♥t r❡s✉❧ts ❛❜♦✉t t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♥♦♥✲tr✐✈✐❛❧ s♦❧✉t✐♦♥s ❢♦r s♦♠❡ ❝❧❛ss❡s ♦❢ r❡s♦♥❛♥t ❛♥❞ s✉♣❡r❧✐♥❡❛r ❡❧✐♣t✐❝ s②st❡♠s ❡♠♣❧♦②✐♥❣ t♦♣♦❧♦❣✐❝❛❧ ♠❡t❤♦❞s✳ ▼♦r❡ s♣❡❝✐✜❝❛❧❧②✱ ✇❡ ✉s❡ ❛✲♣r✐♦r✐ ❜♦✉♥❞s ♦♥ t❤❡ ❡✈❡♥t✉❛❧ s♦❧✉t✐♦♥s ♦❢ t❤✐s ♣r♦❜❧❡♠s ❛♥❞ t♦♣♦❧♦❣✐❝❛❧ ❞❡❣r❡❡ t❤❡♦r②✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ ❙✐st❡♠❛ ●r❛❞✐❡♥t❡ ✺
✶✳✶ ❊st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✶✳✷ ❘❡s✉❧t❛❞♦ ❞❡ ❊①✐stê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✶✳✸ ❘❡❣✉❧❛r✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷ ❙✐st❡♠❛ ❍❛♠✐❧t♦♥✐❛♥♦ ✷✶
✷✳✶ ❊st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷✳✷ ❘❡s✉❧t❛❞♦ ❞❡ ❊①✐stê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✷✳✸ ❘❡❣✉❧❛r✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✸ Pr♦❜❧❡♠❛ ❇✐✲❍❛r♠ô♥✐❝♦ ✸✽
✸✳✶ ❆❧❣✉♠❛s ♦❜s❡r✈❛çõ❡s s♦❜r❡ ♦ ♦♣❡r❛❞♦r ❜✐✲❤❛r♠ô♥✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✸✳✷ ❊st✐♠❛t✐✈❛ ❛ ♣r✐♦r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✸✳✸ ❘❡s✉❧t❛❞♦ ❞❡ ❊①✐stê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✸✳✹ ❘❡❣✉❧❛r✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
❆♣ê♥❞✐❝❡ ✺✹
❆ ◆♦t❛çõ❡s ✺✺
❆✳✶ ❆❧❣✉♠❛s ♥♦t❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
❆✳✷ ❊s♣❛ç♦s ❞❡ ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
❇ ❘❡s✉❧t❛❞♦s ❛✉①✐❧✐❛r❡s ✺✾
❈ ❖ Pr♦❜❧❡♠❛ ▲✐♥❡❛r ✻✹
❙✉♠ár✐♦
❈✳✷ Pr♦❜❧❡♠❛ ❞❡ ❛✉t♦✈❛❧♦r ❝♦♠ ♣❡s♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
❈✳✸ ❖❜s❡r✈❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽
❉ ●r❛✉ ❚♦♣♦❧ó❣✐❝♦ ✼✵
❉✳✶ ●r❛✉ ❞❡ ❇r♦✉✇❡r ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✵
❉✳✷ ●r❛✉ ❞❡ ▲❡r❛②✲❙❝❤❛✉❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✹
■♥tr♦❞✉çã♦
◆❡st❡ tr❛❜❛❧❤♦✱ t❡♠♦s ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r♦✈❛r r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ❛❧❣✉♥s s✐st❡♠❛s ❡❧í♣t✐❝♦s s✉♣❡r❧✐♥❡❛r❡s ❝♦♠ r❡ss♦♥â♥❝✐❛✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❝♦♥s✐❞❡r❛♠♦s s✐st❡♠❛s ❞♦ t✐♣♦ ❣r❛❞✐❡♥t❡ ❡ ❞♦ t✐♣♦ ❤❛♠✐❧t♦♥✐❛♥♦ ❡ t❛♠❜é♠ ✉♠ ♣r♦❜❧❡♠❛ ❡♥✈♦❧✈❡♥❞♦ ♦ ♦♣❡r❛❞♦r ❜✐✲❤❛r♠ô♥✐❝♦✳ ❚❛✐s ♣r♦❜❧❡♠❛s s❡rã♦ tr❛t❛❞♦s ✈✐❛ ♠ét♦❞♦s t♦♣♦❧ó❣✐❝♦s✱ ❡ ❛ ❡str❛✲ té❣✐❛ ✉s❛❞❛ ❝♦♥s✐st❡ ❡♠ ♦❜t❡r ❡st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐ ♣❛r❛ ♣♦ssí✈❡✐s s♦❧✉çõ❡s ❞♦s ♣r♦❜❧❡♠❛s✳ ❊ ❛ ♣❛rt✐r ❞❛í ✉t✐❧✐③❛r ❛ t❡♦r✐❛ ❞♦ ❣r❛✉ t♦♣♦❧ó❣✐❝♦ ♣❛r❛ ❣❛r❛♥t✐r♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s✳
❉❡ ✉♠❛ ♠❛♥❡✐r❛ ❣❡r❛❧✱ ❞❛❞♦ ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❡❧í♣t✐❝❛s
⎧ ⎪ ⎨ ⎪ ⎩
−∆u=h(x, u, v) x∈Ω
−∆v =k(x, u, v) x∈Ω,
✭✶✮
❡♠ q✉❡ Ω ∈ RN é ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ s✉❛✈❡ ❝♦♠ N ≥ 3✳ ❉✐③❡♠♦s q✉❡ ♦ s✐st❡♠❛ ❛❝✐♠❛ é
❣r❛❞✐❡♥t❡ s❡ ❡①✐st❡ K : Ω×R2 →R❞❡ ❝❧❛ss❡ C1✱ t❛❧ q✉❡
∂K
∂u =h ❡ ∂K
∂v =k;
❡ é ❞✐t♦ ✉♠ s✐st❡♠❛ ❤❛♠✐❧t♦♥✐❛♥♦ s❡ ❡①✐st❡ H : Ω×R2 →R✱ ❞❡ ❝❧❛ss❡ C1✱ t❛❧ q✉❡
∂H
∂u =k ❡ ∂H
∂v =h.
❖s ♣r♦❜❧❡♠❛s ❝♦♥s✐❞❡r❛❞♦s ♥❡st❛ t❡s❡ sã♦ ♠♦t✐✈❛❞♦s ♣♦r r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣♦r ▼✳ ❈✉❡st❛✱ ❉❡ ❋✐❣✉❡✐r❡❞♦ ❡ ❙r✐❦❛♥t❤✱ ❡♠ ❬✶✵❪✱ ♣❛r❛ ❛ s❡❣✉✐♥t❡ ❝❧❛ss❡ ❞❡ s✐st❡♠❛s ❤❛♠✐❧t♦♥✐❛♥♦s
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−∆u=λ1u+up++f(x) x∈Ω −∆v =λ1v+v+q +g(x) x∈Ω
u= 0 x∈∂Ω,
✭✷✮
❡♠ q✉❡ λ1 ❞❡♥♦t❛ ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ❞❡ (−∆, H01(Ω))✳ ❊♠ ❬✶✵❪✱ ❢♦✐ ♣r♦✈❛❞♦ ❡①✐stê♥❝✐❛
❞❡ s♦❧✉çã♦ ♣❛r❛ ✭✷✮ s✉♣♦♥❞♦ q✉❡ f, g ∈Lr(Ω)✱ r > N✱ s❛t✐s❢❛③❡♥❞♦
Ω
f φ1 <0 ❡
Ω
gφ1 <0, ✭✸✮
■♥tr♦❞✉çã♦ ✷
❡♠ q✉❡φ1 ❞❡♥♦t❛ ❛ ❛✉t♦❢✉♥çã♦ ❛ss♦❝✐❛❞❛ ❛♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ❡ p, q >1 s❛t✐s❢❛③❡♠
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 ❤✐♣ér❜♦❧❡s ❛❝✐♠❛ ❢♦r❛♠ ✐♥tr♦❞✉③✐❞❛s ♣♦r ❈❧é♠❡♥t✲❞❡ ❋✐❣✉❡✐r❡❞♦✲▼✐t✐❞✐❡r✐✱ ❡♠ ❬✼❪✱ ♣❛r❛ ♦❜t❡r ❡st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐ ♣❛r❛ s✐st❡♠❛s ❡❧í♣t✐❝♦s s✉♣❡r❧✐♥❡❛r❡s ✈✐❛ té❝♥✐❝❛ ❞❡ ❇r❡③✐s✲ ❚✉r♥❡r ✭❬✹❪✮✳ ◆♦t❡ q✉❡ s❡ p=q ❡♥tã♦ ✭✹✮ s❡ r❡❞✉③ ❛ ❝♦♥❞✐çã♦ ❞❡ ❇ré③✐s✲❚✉r♥❡r✱ p < N+1
N−1✳
❆s ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡s ❝♦♥s✐❞❡r❛❞❛s ❛q✉✐ ♣♦❞❡♠ s❡r ❝❛r❛❝t❡r✐③❛❞❛s ❝♦♠♦ ❛ss✐♠étr✐❝❛s✿ s✉♣❡r❧✐♥❡❛r❡s ❡♠ +∞ ❡ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❧✐♥❡❛r ❡♠ −∞✳ ❆❧é♠ ❞✐ss♦✱ ♥♦ss♦s ♣r♦❜❧❡♠❛s
sã♦ r❡ss♦♥❛♥t❡s ♥♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ❡♠−∞✳ Pr♦❜❧❡♠❛s ❞❡st❡ t✐♣♦ ❢♦r❛♠ ♣r✐♠❡✐r❛♠❡♥t❡
❝♦♥s✐❞❡r❛❞♦s ♣♦r ❲❛r❞ ❡♠ ❬✷✽❪✳ ◆❡st❡ ❛rt✐❣♦✱ ♣r♦✈♦✉✲s❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ s✉♣❡r❧✐♥❡❛r ❝♦♠ ❝♦♥❞✐çã♦ ❞❡ ◆❡✉♠❛♥♥
⎧ ⎪ ⎨ ⎪ ⎩
−∆u=up++f(x) x∈Ω
∂u
∂ν = 0 x∈∂Ω,
✭✺✮
♦♥❞❡ 1 < p < N
N−2 ❡ f ∈ L
1(Ω) s❛t✐s❢❛③❡♥❞♦
Ωf < 0✳ ❖ ❛✉t♦r ♦❜s❡r✈❛ q✉❡ ♦ ♠ét♦❞♦
✉t✐❧✐③❛❞♦ ♥ã♦ s❡ ❡st❡♥❞❡ ❛♦ ♣r♦❜❧❡♠❛ ❝♦♠ ❢r♦♥t❡✐r❛ ❞❡ ❉✐r✐❝❤❧❡t✳ ❖ ♣r♦❜❧❡♠❛ s✐♠✐❧✐❛r ❝♦♠ ❢r♦♥t❡✐r❛ ❞❡ ❉✐r✐❝❤❧❡t
⎧ ⎪ ⎨ ⎪ ⎩
−∆u=λ1u+up++f(x) x∈Ω
u= 0 x∈∂Ω,
✭✻✮
❢♦✐ ❝♦♥s✐❞❡r❛❞♦ ♣♦st❡r✐♦r♠❡♥t❡ ♣♦r ❑❛♥♥❛♥✲❖rt❡❣❛ ❡♠ ❬✷✵❪✱ ❡♠ q✉❡ 1 ≤ p < N N−1 ❡
f ∈C(Ω) s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ Ωf φ1 <0 ✳
❆ ❝♦♥❞✐çã♦ ✭✸✮ é ❝❤❛♠❛❞❛ ♥❛ ❧✐t❡r❛t✉r❛ ❞❡ ✧♦♥❡✲s✐❞❡❞ ▲❛♥❞❡s♠❛♥✲▲❛③❡r ❝♦♥❞✐t✐♦♥✧✳ ◆♦t❡ q✉❡✱ q✉❛♥❞♦ ♥ã♦ ❡①✐st❡ r❡ss♦♥â♥❝✐❛ ♥♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ♦ ♣r♦❜❧❡♠❛ é ❞♦ t✐♣♦ ❆♠❜r♦s❡tt✐✲Pr♦❞✐✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ s❡ λ = λ1 ♥♦ ♣r♦❜❧❡♠❛ ✭✻✮ ❡ f = tφ1 +h✱ ♦♥❞❡
Ωφ1h = 0✱ ❛ ❡①✐stê♥❝✐❛✱ ❛ ♥ã♦✲❡①✐stê♥❝✐❛ ❡ ❛ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡♣❡♥❞❡♠ ❞♦ ♣❛râ♠❡tr♦ t✳
■♥tr♦❞✉çã♦ ✸
◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ✈❛♠♦s ❡st✉❞❛r ♦ s✐st❡♠❛ ❣r❛❞✐❡♥t❡
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−∆u=au+bv+up++f(x) x∈Ω
−∆v =bu+cv+v+q +g(x) x∈Ω
u=v = 0 x∈∂Ω,
✭✼✮
❡♠ q✉❡ Ω∈ RN é ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ s✉❛✈❡✱ ❝♦♠ N ≥3✱1< p, q < N+1
N−1✱ ❛s ❢✉♥çõ❡s f✱
g ∈Lr(Ω)✱r > N✱ s❛t✐s❢❛③❡♠ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦
Ω
f φ1+
λ1−a
b
Ω
gφ1 <0. ✭✽✮
❆❧é♠ ❞✐ss♦✱ ♦s ♣❛râ♠❡tr♦s a, b, c ∈ R sã♦ t❛✐s q✉❡ max{a, c} > 0 ❡ b > 0✳ ❙✉♣♦✲
♠♦s t❛♠❜é♠ q✉❡ λ1 é ✉♠ ❛✉t♦✈❛❧♦r ❞❛ ♠❛tr✐③ A =
⎛ ⎝ a b
b c
⎞
⎠✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡
λ1 = a+2c+
(a−c)2
4 +b2✳ ❙♦❜ ❡ss❛s ❝♦♥❞✐çõ❡s ✈❛♠♦s ♠♦str❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛
♦ ♣r♦❜❧❡♠❛ ✭✼✮✳
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❡st✉❞❛♠♦s ❛ r❡s♦❧✉❜✐❧✐❞❛❞❡ ❞♦ s✐st❡♠❛ ❤❛♠✐❧t♦♥✐❛♥♦
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−∆u=au+bv+vp++f(x) x∈Ω −∆v =cu+av+uq++g(x) x∈Ω
u=v = 0 x∈∂Ω,
✭✾✮
❡♠ q✉❡ Ω∈RN é ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ s✉❛✈❡✱ ❝♦♠N ≥3✱1< p, q < N+1
N−1 ❡ ❛s ❢✉♥çõ❡sf✱
g ∈Lr(Ω)✱r > N✱ s❛t✐s❢❛③❡♠ ❛ ❝♦♥❞✐çã♦
Ω
φ1f +
Ω
φ1g <0. ✭✶✵✮
❖s ♣❛râ♠❡tr♦sa, b, c∈Rsã♦ t❛✐s q✉❡b, c >0✳ ❙✉♣♦♠♦s t❛♠❜é♠ q✉❡ ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r
❞♦ ▲❛♣❧❛❝✐❛♥♦λ1 s❡❥❛ t❛♠❜é♠ ❛✉t♦✈❛❧♦r ❞❛ ♠❛tr✐③ M =
⎛ ⎝ a b
b a
⎞
⎠✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡ s✉♣♦♠♦sλ1 =a+b✳ ◆♦t❡ q✉❡✱ ❛ ❝♦♥❞✐çã♦ ♥❛t✉r❛❧ s♦❜r❡ p ❡q s❡r✐❛ ✭✹✮✳ P♦r ❞✐✜❝✉❧❞❛❞❡s
■♥tr♦❞✉çã♦ ✹
◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ tr❛t❛♠♦s ♦ ♣r♦❜❧❡♠❛ r❡❧❛❝✐♦♥❛❞♦ ❛♦ ♦♣❡r❛❞♦r ❜✐✲❤❛r♠ô♥✐❝♦
⎧ ⎪ ⎨ ⎪ ⎩
(−∆)2u=λ2 1u+u
p
++f(x) x∈Ω
u= ∆u= 0 x∈∂Ω,
✭✶✶✮
❡♠ q✉❡ Ω é ✉♠ ❞♦♠í♥✐♦ s✉❛✈❡ ❧✐♠✐t❛❞♦ ❞❡ RN✱ ❝♦♠ N > 5✳ ❆ ❢✉♥çã♦ f ∈ Lr(Ω)✱ ❝♦♠
r > N/3❡ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✭✸✮✳ ❊ ♦ ❡①♣♦❡♥t❡ ps❛t✐s❢❛③
max
1, 4 N −4
< p < N + 1 N −3.
❙♦❜ ❡ss❛s ❝♦♥❞✐çõ❡s ✈❛♠♦s ♠♦str❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭✶✶✮✳
❆♦ ✜♥❛❧ ❞❡ ❝❛❞❛ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s s♦❜r❡ r❡❣✉❧❛r✐❞❛❞❡ ❞❛s s♦❧✉çõ❡s✳
❈❛♣ít✉❧♦ ✶
❙✐st❡♠❛ ●r❛❞✐❡♥t❡
❖ ♦❜❥❡t✐✈♦ ❞❡st❡ ❝❛♣ít✉❧♦ é ❡st✉❞❛r ♦ ♣r♦❜❧❡♠❛ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❞♦ s❡❣✉✐♥t❡ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❡❧í♣t✐❝❛s
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−∆u=au+bv+up++f(x) x∈Ω −∆v =bu+cv+v+q +g(x) x∈Ω
u=v = 0 x∈∂Ω,
✭✶✳✶✮
❡♠ q✉❡ Ω ∈ RN é ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ s✉❛✈❡✱ ❝♦♠ N ≥ 3 ❡ 1 < p, q < N+1
N−1✳ ❖s
♣❛râ♠❡tr♦s a, b, c∈ R sã♦ t❛✐s q✉❡ max{a, c} >0 ❡ b >0✳ ❉❡♥♦t❛♠♦s w+ = max{w,0}✳
❊ ❛s ❢✉♥çõ❡sf, g sã♦ t❛✐s q✉❡✱
f, g ∈Lr(Ω) ♣❛r❛ r > N. ✭✶✳✷✮
P❛r❛ tr❛t❛r ❡ss❡ ♣r♦❜❧❡♠❛ ✈❛♠♦s ✉t✐❧✐③❛r ♠ét♦❞♦s t♦♣♦❧ó❣✐❝♦s✳ ❆ ❡str❛té❣✐❛ é ❡♥❝♦♥tr❛r ❡st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐ ♣❛r❛ ♣♦ssí✈❡✐s s♦❧✉çõ❡s ❞❡ s✐st❡♠❛ ✭✶✳✶✮ ❡ ✉t✐❧✐③❛r ❛ ❚❡♦r✐❛ ❞♦ ●r❛✉ ❚♦♣♦❧ó❣✐❝♦ ♣❛r❛ ❣❛r❛♥t✐r ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s✳
➱ ❝♦♥✈❡♥✐❡♥t❡ ❡s❝r❡✈❡r♠♦s ♦ s✐st❡♠❛ ✭✶✳✶✮ ♥❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧✿
⎧ ⎪ ⎨ ⎪ ⎩
−∆U =AU +G(U) +F(x) x∈Ω
U = 0 x∈∂Ω,
✭✶✳✸✮
❡♠ q✉❡
U =
⎛ ⎝ u
v
⎞
⎠, A=
⎛ ⎝ a b
b c
⎞
⎠∈M2x2(R), G(U) =
⎛ ⎝ u
p
+
v+q
⎞
⎠ e F(x) =
⎛ ⎝ f(x)
g(x)
⎞ ⎠.
❈♦♥s✐❞❡r❛♠♦s λ1 < λ2 ≤ λ3 ≤ ... ≤ λn ≤ ... ♦s ❛✉t♦✈❛❧♦r❡s ❞❡ (−∆, H01(Ω)) ❡
φ1, φ2, ..., φn, ... ❛s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛✉t♦❢✉♥çõ❡s ❡ t❛❧ q✉❡ ❛ ♥♦r♠❛ L2 ❞❡ φ1 s❡❥❛ ♥♦r♠❛❧✐✲
③❛❞❛ ✐❣✉❛❧ ❛ 1✳ ❉❡♥♦t❛♠♦s H ♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H1
0(Ω)×H01(Ω) ♠✉♥✐❞♦ ❞❛ ♥♦r♠❛ U2 =u2+v2 =
Ω|∇
u|2+
Ω|∇
v|2
❙✐st❡♠❛ ●r❛❞✐❡♥t❡ ✻
♣❛r❛ (u, v) ∈ H✳ P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❙♦❜♦❧❡✈✱ ♣❛r❛ 1 ≤ σ ≤ 2∗ ✜①❛❞♦✱ ❛ ✐♠❡rsã♦ H ֒→ Lσ(Ω)×Lσ(Ω) é ❝♦♥tí♥✉❛✳ ❆❧é♠ ❞✐ss♦✱ s❡ σ < 2∗✱ ❡♥tã♦ ❛ ✐♠❡rsã♦ é ❝♦♠♣❛❝t❛✳
❈♦♠ r❡❧❛çã♦ ❛ ♥♦t❛çã♦ ✉s❛❞❛ ❞✉r❛♥t❡ ♦ ❝❛♣ít✉❧♦✱ q✉❛♥❞♦ tr❛t❛r♠♦s s♦❜r❡ ♦ ♣r♦❞✉t♦ ❞❡ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ A ×B✱ ❛❞♦t❛r❡♠♦s✱ ❛ ♠❡♥♦s q✉❡ s❡❥❛ ❡s♣❡❝✐✜❝❛❞♦✱ ❛ ♥♦r♠❛ ❞❛
s♦♠❛✳ P♦ss✐✈❡❧♠❡♥t❡✱ q✉❛♥❞♦ tr❛t❛r♠♦s ❞♦ ♣r♦❞✉t♦ ❞❡ ❡s♣❛ç♦s ✐❣✉❛✐s A×A❞❡♥♦t❛r❡♠♦s
❛ ♥♦r♠❛ ❞❡ ✉♠ ✈❡t♦r (a, b)∈A×A s✐♠♣❧❡s♠❡♥t❡ ♣♦r (a, b)A =aA+bA✳
✶✳✶✳ ❊st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐
◆❡st❛ s❡çã♦ ✈❛♠♦s ♠♦str❛r ✉♠❛ ❡st✐♠❛t✐✈❛ ❛ ♣r✐♦r✐ ♣❛r❛ ♣♦ssí✈❡✐s s♦❧✉çõ❡s ❞♦ s✐st❡♠❛ ✭✶✳✸✮✳
P❛r❛ ✐ss♦ ✈❛♠♦s s✉♣♦r q✉❡ ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r λ1 ❞❡ (−∆, H01(Ω))s❡❥❛ ✉♠ ❛✉t♦✈❛❧♦r
❞❛ ♠❛tr✐③ A✳ ❉❡st❛❝❛♠♦s q✉❡ ♦s ❛✉t♦✈❛❧♦r❡s ❞❛ ♠❛tr✐③ A sã♦
ξ= a+c 2 +
a−c
2
2
+b2 ❡ η= a+c
2 −
a−c
2
2
+b2.
▼❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡ ✈❛♠♦s s✉♣♦r q✉❡ λ1 = ξ✳ ❙❡♥❞♦ ❛ss✐♠ ♣❡❧♦s r❡s✉❧t❛❞♦s ❞❛❞♦s ♥♦
❆♣ê♥❞✐❝❡❈✱ ♦❜t❡♠♦s q✉❡ Φ = (αφ1, βφ1)✱ ❝♦♠ α= 1 ❡ β = λ1b−a✱ é s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛
⎧ ⎪ ⎨ ⎪ ⎩
−∆U =AU x∈Ω
U = 0 x∈∂Ω.
✭✶✳✹✮
◆♦t❡ q✉❡✱λ1 =ξ ✐♠♣❧✐❝❛ q✉❡ λ1 > a ❡ ❛ss✐♠ β >0✳
✶✳✶ ▲❡♠❛✳ ❈♦♥s✐❞❡r❡ 1 < p, q < NN+1−1 ❛s ❢✉♥çõ❡s f, g ∈ Lr(Ω)✱ r > N✱ s❛t✐s❢❛③❡♥❞♦ ❛
❝♦♥❞✐çã♦
Ω
f φ1+
λ1−a
b
Ω
gφ1 <0 ✭✶✳✺✮
❡ λ1 = ξ✳ ❙❡❥❛ U ∈ H ✉♠❛ ♣♦ssí✈❡❧ s♦❧✉çã♦ ❞❡ ✭✶✳✸✮✳ ❊♥tã♦ ❡①✐st❡ ❢✉♥çã♦ ❝r❡s❝❡♥t❡
ρ:R+ →R+✱ ❞❡♣❡♥❞❡♥❞♦ s♦♠❡♥t❡ ❞❡ p✱ q ❡ Ω✱ t❛❧ q✉❡ ρ(0) = 0 ❡
UC1
❙✐st❡♠❛ ●r❛❞✐❡♥t❡ ✼
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ U ∈ H ✉♠❛ ♣♦ssí✈❡❧ s♦❧✉çã♦ ❞❡ ✭✶✳✸✮✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ✭✶✳✸✮ ♣♦r Φ ❡
✐♥t❡❣r❛♥❞♦✱ ♦❜t❡♠♦s
Ω−
∆U ·Φ =
Ω
AU ·Φ +
Ω
G(U)·Φ +
Ω
F(x)·Φ,
❝♦♠♦ ❛ ♠❛tr✐③ A é ❛✉t♦❛❞❥✉♥t❛ ❡Φ é s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✭✶✳✹✮✱ t❡♠♦s q✉❡
Ω
G(U)·Φ = −
Ω
F(x)·Φ≤CF(x)r. ✭✶✳✼✮
P♦❞❡♠♦s ❞❡❝♦♠♣♦r U ❡♠ U = tΦ + U1 ❡♠ q✉❡ U1 = (u1, v1) é ♦rt♦❣♦♥❛❧ ❛ Φ ❡♠ H✱
✈❡❥❛ ❙❡çã♦ ✭❈✳✷✮✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ U1 é ♦rt♦❣♦♥❛❧ ❛ Φ t❛♠❜é♠ ❡♠ L2 ×L2✱ ✐st♦ é✱
ΩU1·Φ = 0✳ ▼✉❧t✐♣❧✐♣❧✐❝❛♥❞♦ t❛❧ ❞❡❝♦♠♣♦s✐çã♦ ♣♦r Φ❡ ✐♥t❡❣r❛♥❞♦✱ ♦❜t❡♠♦s
Ω
U ·Φ =t
Ω
Φ·Φ +
Ω
U1·Φ,
❧♦❣♦
t =C
Ω
U ·Φ =C
Ω
αuφ1+
Ω βvφ1 =C Ω
α(u+−u−)φ1+
Ω
β(v+−v−)φ1
≤C
Ω
αu+φ1+
Ω
βv+φ1.
❆♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✼✮✱ r❡s✉❧t❛ q✉❡
t≤C
Ω
up+φ1
1/p
+C
Ω
v+qφ1
1/q
≤C(F1r/p+F1r/q). ✭✶✳✽✮
▲❡♠❜r❡♠♦s q✉❡ ♥♦ss♦ ♦❜❥❡t✐✈♦ é ❧✐♠✐t❛r U✱ ♣❛r❛ ✐ss♦ ❞❡✈❡♠♦s ❧✐♠✐t❛r U1 ❡ |t|✳ ❙❡ t ≥
0✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✽✮ ♥♦s ❞á ✉♠❛ ❧✐♠✐t❛çã♦ ♣❛r❛ |t|✳ ❙❡♥❞♦ ❛ss✐♠✱ ✈❛♠♦s ❞✐✈✐❞✐r ❛
❞❡♠♦♥str❛çã♦ ❡♠ ❞✉❛s ♣❛rt❡s✿
❈❛s♦ ✶✿ t≥0✳ ❙♦❜ ❡ss❛ ❝♦♥❞✐çã♦ r❡st❛ ❡♥❝♦♥tr❛r♠♦s ✉♠❛ ❧✐♠✐t❛çã♦ ♣❛r❛ U1✳ ▼✉❧t✐✲
♣❧✐❝❛♥❞♦ ❛❣♦r❛ ❛ ❡q✉❛çã♦ ✭✶✳✸✮ ♣♦r U1 ❡ ✐♥t❡❣r❛♥❞♦✱ ♦❜t❡♠♦s
−
Ω
∆U ·U1 =
Ω
AU ·U1+
Ω
G(U)·U1+
Ω
❙✐st❡♠❛ ●r❛❞✐❡♥t❡ ✽
✉s❛♥❞♦ ❛ ❞❡❝♦♠♣♦s✐çã♦ U =tΦ +U1✱ t❡♠♦s U12 =
Ω
AU1·U1+
Ω
G(U)·U1+
Ω
F(x)·U1 ≤
Ω
AU1·U1+
Ω
G(U)·U1+
Ω
F(x)·U1
.
◆❡st❡ ♠♦♠❡♥t♦ ✈❛♠♦s ✉t✐❧✐③❛r ❛ t❡♦r✐❛ ❡s♣❡❝tr❛❧ ♣❛r❛ ♦♣❡r❛❞♦r❡s ❝♦♠♣❛❝t♦s✱ ✈❡❥❛ ❆♣ê♥✲ ❞✐❝❡ ❈✳ ❈♦♠♦ U1 é ♦rt♦❣♦♥❛❧ ❛ Φ ❡♠ H ❡ Φ = cΦA1 ❡♥tã♦ U1 t❛♠❜é♠ é ♦rt♦❣♦♥❛❧ ❛ ΦA1
❡♠ H✳ ▲♦❣♦ ♣♦❞❡♠♦s ✉s❛r ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭❈✳✹✮ ♣❛r❛ k = 1 ❡ ❛ss✐♠ ♦❜t❡♠♦s
U12 ≤ 1
λ2(A)
U12+
Ω
G(U)·U1 +
Ω|
F(x)·U1|,
✐st♦ é✱
1− 1
λ2(A)
U12 ≤
Ω
G(U)·U1+
Ω|
F(x)·U1|.
❈♦♠♦1 = λ1(A)< λ2(A)❡ ❛♣❧✐❝❛♥❞♦ ❛ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✱ ♦❜t❡♠♦s q✉❡ U12 ≤C
Ω
G(U)·U1+CF2U12.
❋✐♥❛❧♠❡♥t❡ ♣❡❧❛ ✐♠❡rsã♦ H1
0(Ω) ֒→L2(Ω)✱ ❝♦♥❝❧✉í♠♦s q✉❡ U12 ≤C
Ω
G(U).U1+CFrU1. ✭✶✳✾✮
❱❛♠♦s ❛❣♦r❛ ❡st✐♠❛r ❛ ♣r✐♠❡✐r❛ ✐♥t❡❣r❛❧ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✾✮✱
Ω
G(U)·U1 =
Ω
up+u1 +vq+v1
=
Ω
up+u+1 −
Ω
up+u−1 +
Ω
v+qv1+−
Ω
vq+v1−
≤
Ω
up+u+1 +
Ω
v+qv1+,
❝♦♠♦t ≥0t❡♠♦s q✉❡ u+1 ≤u+ ❡v1+≤v+✱ ❧♦❣♦
Ω
G(U)·U1 ≤
Ω
up++1+
Ω
❙✐st❡♠❛ ●r❛❞✐❡♥t❡ ✾
❆♣❧✐❝❛♥❞♦ ♦ ▲❡♠❛ ✭❇✳✾✮ ❡ ❡♠ s❡❣✉✐❞❛ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✼✮✱ ♦❜t❡♠♦s
Ω
G(U)·U1 ≤ Ω
up+φ1
α
Ω|∇
u+|2δ/2+ Ω
v+qφ1
α′
Ω|∇
v+|2δ
′/2
≤CFαr u+ δ
+Fαr′v+ δ′
≤CFαr uδ+Fαr′vδ′
≤CFαr Uδ+Fαr′Uδ′.
❯s❛♥❞♦ ❛ ❞❡❝♦♠♣♦s✐çã♦ U =tΦ +U1✱ t❡♠♦s
Ω
G(U)·U1 ≤CFαr (|t| Φ+U1) δ
+CFαr′(|t| Φ+U1)δ′,
❡ ♣❡❧❛ ❡st✐♠❛t✐✈❛ ❞❡ t=|t|✱ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✽✮✱ ❝♦♥❝❧✉í♠♦s
Ω
G(U)·U1 ≤CFαr
F1r/p+F1r/qδ+CFαr′Fr1/p+F1r/qδ
′
+CFαr U1δ+CFαr′U1δ′. ✭✶✳✶✵✮
❙✉❜st✐t✉✐♥❞♦ ❛ ❡st✐♠❛t✐✈❛ ✭✶✳✶✵✮ ❡♠ ✭✶✳✾✮✱ r❡s✉❧t❛
U12 ≤C
Fαr+δ/p+Fαr+δ/q+Frα′+δ′/p+Frα′+δ′/q
+CFαr U1 δ
+CFαr′U1 δ′
+CFrU1.
❈♦♠♦ p < N+1
N−1 s❡❣✉❡ ❞♦ ▲❡♠❛❇✳✾q✉❡ δ, δ
′ ∈(1,2)✳ ❙❡♥❞♦ ❛ss✐♠✱ ❛♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧✲ ❞❛❞❡ ❞❡ ❨♦✉♥❣✱ ♦❜t❡♠♦s q✉❡
U12 ≤C
Fαr+δ/p+Fαr+δ/q+Frα′+δ′/p+Frα′+δ′/q
+CFr2α/(2−δ)+CF
2α′/(2−δ′)
r +CF
2
r.
P♦rt❛♥t♦✱
U1 ≤CFαr+δ/p+F α+δ/q
r +F α′+δ′/p
r +F
α′+δ′/q
r
1/2
+CFα/r (2−δ)+CFrα′/(2−δ′)+CFr,
♥♦✈❛♠❡♥t❡ ✉s❛♥❞♦ ❛ ❞❡❝♦♠♣♦s✐çã♦ U =tΦ +U1✱ ❝♦♥❝❧✉í♠♦s q✉❡
U ≤CF1r/p+Fr1/q+CFαr+δ/p+Fαr+δ/q+Frα′+δ′/p+Frα′+δ′/q1/2
❙✐st❡♠❛ ●r❛❞✐❡♥t❡ ✶✵
❙✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s ❞❡ α✱ α′✱δ ❡δ′✱ ❞❛❞♦s ♣❡❧♦ ▲❡♠❛ ❇✳✾✱ ♣♦❞❡♠♦s ✈❡r q✉❡
U ≤CmaxF1r/p,F
1/q r ,F
α/(2−δ)
r ,F
α′/(2−δ′)
r
. ✭✶✳✶✶✮
❖❜s❡r✈❛✲s❡ q✉❡ ❡♥❝♦♥tr❛♠♦s ✉♠❛ ❧✐♠✐t❛çã♦ ♣❛r❛ U ❡♠ H✱ ♠❛s ♣r❡❝✐s❛♠♦s ❡♥❝♦♥tr❛r
✉♠❛ ❧✐♠✐t❛çã♦ ♣❛r❛ U ♥♦ ❡s♣❛ç♦ C1
0(Ω)2✳ ❖ ♣r✐♠❡✐r♦ ♣❛ss♦ ♣❛r❛ ❝❤❡❣❛r♠♦s ❛♦ ♥♦ss♦
♦❜❥❡t✐✈♦ é ❛♣❧✐❝❛r ✉♠ ❛r❣✉♠❡♥t♦ ❞❡ r❡❣✉❧❛r✐❞❛❞❡✱ ❝♦♥❤❡❝✐❞♦ ♣♦r ❛r❣✉♠❡♥t♦ ❞❡ ❜♦♦tstr❛♣✱ q✉❡ ❡♥❝♦♥tr❛✲s❡ ❡♠ ❞❡t❛❧❤❡s ♥❛ ❙❡çã♦ ✭✶✳✸✮✳ ❆♣❧✐❝❛♥❞♦ ♦ ❛r❣✉♠❡♥t♦ ❞♦ t✐♣♦ ❜♦♦tstr❛♣✱ ♦❜t❡♠♦s q✉❡ u, v ∈W2,r(Ω) ❡ ❡①✐st❡ C > 0t❛❧ q✉❡
U2,r ≤C(Fγr +Uη)
❝♦♠ η, γ ❝♦♥st❛♥t❡s✱ t❛✐s q✉❡ η, γ ≥1✳ ❊st❡ ❢❛t♦✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✶✶✮✱
✐♠♣❧✐❝❛ q✉❡
U2,r ≤ρ(Fr).
P♦r ✜♠✱ ❝♦♠♦ r > N✱ ✈❛❧❡ ❛ ✐♠❡rsã♦ W2,r(Ω)֒→C1(Ω)✱ ❡ ♣♦rt❛♥t♦ UC1
0( ¯Ω)≤ρ(Fr).
❈❛s♦ ✷✿ t <0✳ P❡❧♦ Pr✐♥❝í♣✐♦ ❞♦ ▼á①✐♠♦ ❞❡ ❍♦♣❢✱ ✭✈❡❥❛ ❚❡♦r❡♠❛ ❇✳✼✮✱ s❛❜❡♠♦s q✉❡
❛ ♣r✐♠❡✐r❛ ❛✉t♦❢✉♥çã♦ φ1 >0 s❡ ❡♥❝♦♥tr❛ ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠ ❝♦♥❡ ❞❡ ❢✉♥çõ❡s ♣♦s✐t✐✈❛s ♥♦
❡s♣❛ç♦C1
0(Ω)✳ ❊♥tã♦ ❡①✐st❡ ǫ >0t❛❧ q✉❡
w∈BC1
0( ¯Ω)(φ1, ǫ)⇒w >0 ❡♠ Ω ❡
∂w
∂η <0 ❡♠ ∂Ω,
❡♠ q✉❡ η ❞❡♥♦t❛ ♦ ✈❡t♦r ❡①t❡r✐♦r ♥♦r♠❛❧ ❛ ❢r♦♥t❡✐r❛ ❞❡ Ω✳ ❈♦♠♦ Φ∈ C1
0(Ω)×C01(Ω) é
t❛❧ q✉❡ Φ =
⎛ ⎝ αφ1
βφ1
⎞
⎠✱ ❝♦♠ α✱β >0✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ❡①✐st❡ ǫ >0✱ t❛❧ q✉❡ (w1, w2)∈BC1
0( ¯Ω)(φ1, ǫ)×BC01( ¯Ω)(φ1, ǫ)⇒w1, w2 >0 ❡♠ Ω ❡
∂w1
∂η , ∂w2
∂η <0 ❡♠ ∂Ω.
❉❡✜♥✐♠♦sǫ0 ♦ s✉♣r❡♠♦s ❞❡ t❛✐s ǫ✬s ❡ r❡❧❡♠❜r❛♠♦s q✉❡✱ ♣❡❧❛ ❙❡çã♦✶✳✸✱ U = (u, v)s♦❧✉çã♦
❞❡ ✭✶✳✸✮ ❜❡♠ ❝♦♠♦ U1 = (u1, v1) ♣❡rt❡♥❝❡♠ ❛ C01(Ω)×C01(Ω)✳ P♦❞❡♠♦s ❡s❝r❡✈❡r
u=tαφ1 +
u1
αt
❡ v =tβ
φ1+
v1
βt
❙✐st❡♠❛ ●r❛❞✐❡♥t❡ ✶✶
❆✜r♠❛♠♦s q✉❡ −u1/αt /∈ BC1
0( ¯Ω)(0, ǫ0) ❡ −v1/βt /∈ BC01( ¯Ω)(0, ǫ0)✳ P♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱ t❡rí❛♠♦s q✉❡
u
αt =φ1−(− u1
αt)∈BC10( ¯Ω)(φ1, ǫ0) ❡
v
βt =φ1−(− v1
βt)∈BC01( ¯Ω)(φ1, ǫ0) ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ❥❛ q✉❡ u+, v+≡0✱ α, β >0 ❡ t <0✳ ❙❡♥❞♦ ❛ss✐♠✱
|t| ≤ 1
αǫ0
u1C1
0( ¯Ω) ❡ |t| ≤
1
βǫ0
v1C1 0( ¯Ω).
P♦rt❛♥t♦✱
|t| ≤Cu1C1
0( ¯Ω)+Cv1C01( ¯Ω) =CU1C01( ¯Ω). ✭✶✳✶✷✮ ❘❡st❛ ❡♥❝♦♥tr❛r♠♦s ✉♠❛ ❧✐♠✐t❛çã♦ ❛ ♣r✐♦r✐ ♣❛r❛ U1C1
0(Ω)✳ ❖❜s❡r✈❛♠♦s q✉❡ ❛ ❞❡s✐❣✉❛❧✲ ❞❛❞❡ ✭✶✳✾✮ ❝♦♥t✐♥✉❛ ✈á❧✐❞❛ ♣❛r❛ t <0✱ ✐st♦ é✱
U12 ≤C
Ω
G(U)·U1+CFrU1. ✭✶✳✶✸✮
❆❣♦r❛ ✈❛♠♦s ❡st✐♠❛r ❛ ♣r✐♠❡✐r❛ ✐♥t❡❣r❛❧ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡ ✭✶✳✶✸✮✳ ◆♦t❡ q✉❡
Ω
G(U)·U1
≤ Ω
up+|u1|+
Ω
v+|q v1|,
❝♦♠♦ t < 0t❡♠♦s q✉❡ u+ <|u1| ❡ v+ < |v1|✱ s❡♥❞♦ ❛ss✐♠ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ ▲❡♠❛ ✭❇✳✾✮
❡ ♦❜t❡♠♦s q✉❡
Ω
G(U)·U1
≤ Ωup+φ1
α
Ω
|∇u1|δ/2+
Ω
vq+φ1
α′
Ω
|∇v1|δ
′/2
.
P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✼✮✱ r❡s✉❧t❛
Ω
G(U)·U1
≤CFαr u1 δ
+CFαr′v1δ′
≤CFαr U1δ+CFα
′
r U1 δ′
. ✭✶✳✶✹✮
❙✉❜st✐t✉✐♥❞♦ ✭✶✳✶✹✮ ❡♠ ✭✶✳✶✸✮✱ t❡♠♦s q✉❡
U12 ≤CFαr U1 δ
+CFαr′U1 δ′
+CFrU1.
◆♦✈❛♠❡♥t❡✱ ❝♦♠♦ δ, δ′ ∈(1,2)✱ ❛♣❧✐❝❛♠♦s ❛ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣ ❡ ❝♦♥❝❧✉í♠♦s q✉❡
U1 ≤CF
α
2−θ
r +CF
α′
2−θ′
❙✐st❡♠❛ ●r❛❞✐❡♥t❡ ✶✷
❈♦♠♦U1 é s♦❧✉çã♦ ❢r❛❝❛ ❞♦ ♣r♦❜❧❡♠❛
⎧ ⎪ ⎨ ⎪ ⎩
−∆U1 =AU1+G(U) +F(x) x∈Ω
U1 = 0 x∈∂Ω,
✭✶✳✶✻✮
♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ ❛r❣✉♠❡♥t♦ ❞♦ t✐♣♦ ❜♦♦tstr❛♣✱ ✭✈❡❥❛ ❙❡çã♦✶✳✸✮✱ ♣❛r❛ ♦ s✐st❡♠❛ ✭✶✳✶✻✮ ❡ ❝♦♥❝❧✉✐r q✉❡ U1 ∈W2,r×W2,r(Ω) ❡ t❛♠❜é♠ q✉❡ ❡①✐st❡ ❝♦♥st❛♥t❡ C >0✱ t❛❧ q✉❡
U12,r ≤C(Fγr +Uη)
❝♦♠ η, γ ❝♦♥st❛♥t❡s✱ t❛✐s q✉❡ η, γ ≥1✳ ❊st❡ ❢❛t♦✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✶✺✮✱
✐♠♣❧✐❝❛ q✉❡
U12,r ≤ρ(Fr).
❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ r > N✱ ✈❛❧❡ ❛ ✐♠❡rsã♦ W2,r(Ω) ֒→ C1(Ω)✳ ❙❡♥❞♦ ❛ss✐♠✱ ♣♦❞❡♠♦s
❝♦♥❝❧✉✐r q✉❡
U1C1
0(Ω) ≤ρ(Fr). ✭✶✳✶✼✮ ❈♦♠♦U =U1+tΦ❡ ✉s❛♥❞♦ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ✭✶✳✶✷✮ ❡ ✭✶✳✶✼✮✱ ♦❜t❡♠♦s q✉❡
UC1
0(Ω)≤ρ(Fr).
✶✳✷✳ ❘❡s✉❧t❛❞♦ ❞❡ ❊①✐stê♥❝✐❛
◆❡st❛ s❡çã♦ ✈❛♠♦s ♠♦str❛r ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮✳
✶✳✷ ❚❡♦r❡♠❛✳ ❆ss✉♠✐♥❞♦ ❛s ❤✐♣ót❡s❡s ❞♦ ▲❡♠❛ ✭✶✳✶✮ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦
U = (u, v)∈(W2,r(Ω)∩H1 0(Ω))
2 ❞♦ s✐st❡♠❛ ✭✶✳✸✮✳
◆❛ ❞❡♠♦♥str❛çã♦ ✈❛♠♦s ♣r♦❝❡❞❡r ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✱ ✈❛♠♦s ♠♦str❛r q✉❡ t♦❞❛ s♦❧✉çã♦
❙✐st❡♠❛ ●r❛❞✐❡♥t❡ ✶✸
♣❡q✉❡♥❛s ❡ s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭✶✳✺✮✳ ❈♦♥s✐❞❡r❡ ❛ ❧✐♥❡❛r✐③❛çã♦ ❞❡ ♣r♦❜❧❡♠❛ ✭✶✳✶✮✱ ♣❛r❛ (u0, v0) s♦❧✉çã♦ ❞❡ ✭✶✳✸✮✿
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−∆w=aw+bz+p(u+0)p−1w x∈Ω −∆z =bw+cz+q(v0+)q−1z x∈Ω
w=z = 0 x∈∂Ω.
✭✶✳✶✽✮
❆ ♥ã♦ ❞❡❣❡♥❡r❛❝✐❞❛❞❡ ❡ ♦ ❝á❧❝✉❧♦ ❞♦ í♥❞✐❝❡ sã♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦s s❡❣✉✐♥t❡s r❡s✉❧t❛❞♦s✿
✶✳✸ ▲❡♠❛✳ ❊①✐st❡ǫ >0t❛❧ q✉❡❀ ♣❛r❛ ❝❛❞❛ m, k ∈L∞✱ s∈R ❡t ∈[0,1]t❛✐s q✉❡ m, k ≥0 q✳t✳♣✳✱ m = 0✱ k = 0✱ m∞,k∞ < ǫ ❡ 0< s < ǫ❀ s✉♣♦♥❤❛ t❛♠❜é♠✱ q✉❡ ❡①✐st❛ Ω˜ ⊂ Ω ❝♦♠ ♠❡❞✐❞❛ ♣♦s✐t✐✈❛✱ t❛❧ q✉❡ ❛s ❢✉♥çõ❡s m ❡ k ♥ã♦ s❡ ❛♥✉❧❡♠ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ♣❛r❛ t♦❞♦ x∈Ω˜✳ ❊♥tã♦ ♦ s✐st❡♠❛
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−∆w=aw+bz+tm(x)w+ (1−t)sw x∈Ω
−∆z =bw+cz+tk(x)z+ (1−t)sz x∈Ω
w=z = 0 x∈∂Ω,
✭✶✳✶✾✮
♣♦ss✉✐ s♦♠❡♥t❡ s♦❧✉çã♦ tr✐✈✐❛❧ w=z = 0✳
❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ ❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧ ♣❛r❛ ♦ s✐st❡♠❛ ✭✶✳✶✾✮✿
⎧ ⎪ ⎨ ⎪ ⎩
−∆W = ˜AW x∈Ω
W = 0 x∈∂Ω,
❡♠ q✉❡
˜
A(x) =
⎛
⎝ a+tm(x) + (1−t)s b
b c+tk(x) + (1−t)s
⎞
⎠ e W =
⎛ ⎝ w
z
⎞ ⎠.
❱❛♠♦s ♠♦str❛r q✉❡ ❛s ♠❛tr✐③❡sA❡A˜(x)s❛t✐s❢❛③❡♠ ❛ r❡❧❛çã♦A≺A˜(x)✱ ✈❡❥❛ ❉❡✜♥✐çã♦
✭❈✳✷✮✳ ❉❛❞♦ (x, y)∈Ω×R2✱ ❡♠ q✉❡ y= (y1, y2)✱ t❡♠♦s
˜
❙✐st❡♠❛ ●r❛❞✐❡♥t❡ ✶✹
P❡❧❛s ❤✐♣ót❡s❡s ❞❛❞❛s s♦❜r❡m, k ❡♠ Ω❡Ω˜✱ ❡ s♦❜r❡t ❡s ✱ t❡♠♦s q✉❡ ❛ ❡①♣r❡ssã♦ ❛♥t❡r✐♦r
é ♥ã♦ ♥❡❣❛t✐✈❛ ❡♠Ω×R2 ❡ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈❛ ❡♠ Ω˜ ×R2✳ P♦rt❛♥t♦ A≺A˜(x)✳
❆❧é♠ ❞✐ss♦✱ s❛❜❡♠♦s q✉❡ ❛ ❛✉t♦❢✉♥çã♦ ΦA
1 ❛ss♦❝✐❛❞❛ ❛♦ ❛✉t♦✈❛❧♦r λ1(A) s❛t✐s❢❛③❛
Pr♦♣r✐❡❞❛❞❡ ❞❡ ❈♦♥t✐♥✉❛çã♦ Ú♥✐❝❛✱ ✈❡❥❛ ❞❡✜♥✐çã♦ ✭❈✳✸✮✳ ❙❡♥❞♦ ❛ss✐♠✱ ✉t✐❧✐③❛♥❞♦ ❛ Pr♦✲ ♣♦s✐çã♦ ✭❈✳✹✮✱ t❡♠♦s q✉❡ λ1( ˜A(x)) < λ1(A) = 1✳ ❚❡♠♦s t❛♠❜é♠ q✉❡ ❝❛❞❛ ❡♥tr❛❞❛ ❞❛
♠❛tr✐③ A˜ ❝♦♥✈❡r❣❡ ❡♠ L∞ ♣❛r❛ A✱ ✉s❛♥❞♦ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦s ❛✉t♦✈❛❧♦r❡s ❡♠ r❡❧❛çã♦ ❛s ♣❡s♦s A(x)✱ r❡s✉❧t❛ q✉❡ λj( ˜A(x)) → λj(A) ♣❛r❛ ❝❛❞❛ j = 1,2...✳ ❊♠ ♣❛rt✐❝✉❧❛r ♣❛r❛ j = 2 t❡♠♦s q✉❡ λ2( ˜A(x))→ λ2(A) > λ1(A) = 1✳ P♦rt❛♥t♦✱ λ1( ˜A(x))< 1< λ2( ˜A(x)) ❡
❝♦♥s❡q✉❡♥t❡♠❡♥t❡w=z = 0 é ❛ ú♥✐❝❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✾✮✳
✶✳✹ ▲❡♠❛✳ ❙❡❥❛ 0 < s < ǫ ✜①❛❞♦✱ ❝♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ❞❡ ❛✉t♦✈❛❧♦r ❝♦♠
♣❛râ♠❡tr♦ μ
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−∆w=λ(aw+bz+sw) x∈Ω
−∆z =λ(bw+cz+sz) x∈Ω
w=z = 0 x∈∂Ω.
✭✶✳✷✵✮
❊♥tã♦ ❡①✐st❡ s♦♠❡♥t❡ ✉♠ ❛✉t♦✈❛❧♦r λ ♥♦ ✐♥t❡r✈❛❧♦ [0,1]✳
❉❡♠♦♥str❛çã♦✳ Pr✐♠❡✐r❛♠❡♥t❡ ✈❛♠♦s ❡s❝r❡✈❡r ♦ ♣r♦❜❧❡♠❛ ✭✶✳✷✵✮ ♥❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧
⎧ ⎪ ⎨ ⎪ ⎩
−∆W =λAW˜ x∈Ω
W = 0 x∈∂Ω,
❡♠ q✉❡ A˜=
⎛
⎝ a+s b
b c+s
⎞
⎠ ❡ W =
⎛ ⎝ w
z
⎞ ⎠✳
❱❛♠♦s ♣r♦❝❡❞❡r ❝♦♠♦ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✭✶✳✸✮✳ ◆♦t❡ q✉❡✱ A ≺ A˜ ❡♥tã♦✱ ♣❡❧❛
Pr♦♣♦s✐çã♦ ✭❈✳✹✮✱ r❡s✉❧t❛ q✉❡ λ1( ˜A) < λ1(A) = 1✳ ❋❛③❡♥❞♦ ǫ s✉✜❝❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦
❙✐st❡♠❛ ●r❛❞✐❡♥t❡ ✶✺
❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✭✶✳✷✮✿
❈♦♥s✐❞❡r❡ ❛ ❛♣❧✐❝❛çã♦ TF :C01(Ω)2 −→C01(Ω)2 t❛❧ q✉❡
TF(U) = (−∆)−1(AU +G(U) +F(x))
=(−∆)−1(au+bv+up++f(x)),(−∆)−1(bu+cv+v+q +g(x))
.
◆♦t❡ q✉❡ TF é ✉♠ ♦♣❡r❛❞♦r ❝♦♠♣❛❝t♦ ❡ ❝♦♥tí♥✉♦ ❡ TF(u, v) = (u, v)s❡✱ ❡ s♦♠❡♥t❡ s❡✱ U = (u, v)é s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✭✶✳✶✮✳ ❈♦♥s✐❞❡r❡ F1 = (f1, g1) ❝♦♠
f1 =−(γαφ1)p ❡ g1 =−(γβφ1)q
❡♠ q✉❡ γ > 0✳ ❙❡♥❞♦ ❛ss✐♠✱ U0 = (u0, v0) = (γαφ1, γβφ1) é s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✭✶✳✸✮
♣❛r❛ F =F1✳ ❉❡ ❢❛t♦✱
AU0+G(U0) +F1(x) =AU0 =γAΦ =−γ∆Φ =−∆U0.
❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ❤♦♠♦t♦♣✐❛ H : [0,1]×C1
0(Ω)2 −→C01(Ω)2 ❝♦♠
H(τ, U) = (I−(−∆)−1) (AU +G(U) + (1−τ)F(x) +τ F1(x)).
◆♦t❡ q✉❡✱
H(0, U) =(I−(−∆)−1)(AU +G(U) +F(x))=I−TF
❡
H(1, U) =(I−(−∆)−1)(AU +G(U) +F1(x))
=I −TF1. ❆❧é♠ ❞✐ss♦✱ s❡❣✉❡ ❞❛ ❡st✐♠❛t✐✈❛ ❛ ♣r✐♦r✐✱ ▲❡♠❛ ✭✶✳✶✮✱ q✉❡ t♦❞❛ s♦❧✉çã♦ ❞❡
⎧ ⎪ ⎨ ⎪ ⎩
−∆U =AU +G(U) + (1−τ)F(x) +τ F1(x) x∈Ω
U = 0 x∈∂Ω
é ✉♥✐❢♦r♠❡♠❡♥t❡ ❧✐♠✐t❛❞❛ ❡♠ C1
0(Ω)2✳ ❙❡♥❞♦ ❛ss✐♠✱ ♣❛r❛ R > 0 s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱
t❡♠♦s q✉❡ H(τ, U) = 0 ♣❛r❛ t♦❞♦ (τ, U) ∈ [0,1]×∂BC1
❙✐st❡♠❛ ●r❛❞✐❡♥t❡ ✶✻
❛ ❤♦♠♦t♦♣✐❛ H é ❛❞♠✐ssí✈❡❧ ✭✈❡❥❛ ❆♣ê♥❞✐❝❡ ❉✮✱ ❡ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ✐♥✈❛r✐â♥❝✐❛ ♣♦r
❤♦♠♦t♦♣✐❛ ❞♦ ❣r❛✉ t♦♣♦❧ó❣✐❝♦✱ t❡♠♦s q✉❡
deg(I−TF, BC1
0(Ω)2(0, R),0) = deg(I−TF1, BC01(Ω)2(0, R),0). ❋❛ç❛♠♦s γ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ t❛❧ q✉❡ ♦ ▲❡♠❛ ✭✶✳✸✮ s❡❥❛ ❛♣❧✐❝á✈❡❧ ♣❛r❛
m(x) =pup+−1 ❡ k(x) =qv+q−1,
♣❛r❛ t♦❞❛ (u, v) s♦❧✉çã♦ ❛r❜✐trár✐❛ ❞❡ ✭✶✳✶✮ ❝♦♠ F = F1✳ ❆♣❧✐❝❛♥❞♦ ❡♥tã♦ ♦ ▲❡♠❛ ✭✶✳✸✮
♣❛r❛t= 1✱ ♦❜t❡♠♦s q✉❡(u, v)é ♥ã♦ ❞❡❣❡♥❡r❛❞❛✳ ❆❧é♠ ❞✐ss♦✱ ♦ í♥❞✐❝❡ ❞❡ I−TF1 ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦ ❛tr❛✈és ❞❛ ❤♦♠♦t♦♣✐❛ ❞❛❞❛ ❡♠ ✭✶✳✶✾✮ ❡ ❡st❡ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ í♥❞✐❝❡ ❞❛ s♦❧✉çã♦ tr✐✈✐❛❧ ❞❡ ✭✶✳✶✾✮ ♣❛r❛ t = 0✳ ❯s❛♥❞♦ ♦ ▲❡♠❛ ✭✶✳✹✮ ♥ós ❞❡❞✉③✐♠♦s q✉❡ ❡ss❡ í♥❞✐❝❡ é −1✱
✭✈❡❥❛ t❛♠❜é♠ ❚❡♦r❡♠❛ ✭❉✳✽✮✮✳ P♦rt❛♥t♦
deg(I−TF1, BC01(Ω)2(0, R),0) =
(−1)= 0,
♥♦t❡ q✉❡ ❛ s♦♠❛ ❛❝✐♠❛ é ✜♥✐t❛✱ ✈❡❥❛ ❞❡t❛❧❤❡s ❡♠ ✭❉✳✷✮✳ ❙❡♥❞♦ ❛ss✐♠✱
deg(I−TF, BC1
0(Ω)2(0, R),0)= 0
❡ ♣♦r ✜♠✱ ✉s❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ s♦❧✉çã♦ ❞♦ ❣r❛✉ t♦♣♦❧ó❣✐❝♦ ❝♦♥❝❧✉í♠♦s q✉❡ ❡①✐st❡
U ∈BC1
0(Ω)2(0, R)t❛❧ q✉❡ (I−TF)(U) = 0✱ ✐st♦ é✱U é s♦❧✉çã♦ ❞❡ ✭✶✳✶✮✳
✶✳✸✳ ❘❡❣✉❧❛r✐❞❛❞❡
❙❡❥❛Ω∈RN ✉♠ ❞♦♠í♥✐♦ s✉❛✈❡ ❡ ❧✐♠✐t❛❞♦✱ ❝♦♠ N ≥3✳ ❱❛♠♦s ❞✐s❝✉t✐r ❛ r❡❣✉❧❛r✐❞❛❞❡
❞♦ s❡❣✉✐♥t❡ s✐st❡♠❛ ❡❧í♣t✐❝♦
⎧ ⎪ ⎨ ⎪ ⎩
−∆U =H(x, U(x)) x∈Ω
U(x) = 0 x∈∂Ω,
✭✶✳✷✶✮
❡♠ q✉❡U(x) =
⎛ ⎝ u(x)
v(x)
⎞
⎠❡H(x, U(x)) =
⎛
⎝ h(x, u, v)
k(x, u, v)
⎞
⎠✳ ❆s ❢✉♥çõ❡sh, k: Ω×R×R−→ Rsã♦ ❝♦♥tí♥✉❛s ❡ ❡①✐st❡♠ F(x) = (f(x), g(x))∈Lr(Ω)×Lr(Ω) ❡C > 0 t❛✐s q✉❡
❙✐st❡♠❛ ●r❛❞✐❡♥t❡ ✶✼
❝♦♠ 1< s <2∗−1❡ ❛❧❣✉♠ r > N✳ ❙❡❥❛ U = (u, v)∈H1
0(Ω)×H01(Ω) s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✭✶✳✷✶✮✳ ❱❛♠♦s ♠♦str❛r q✉❡ u, v ∈
W2,r(Ω) ❡ q✉❡ ❡①✐st❡ ❝♦♥st❛♥t❡ C >0✱ t❛❧ q✉❡
U2,r ≤C(Fγr +Uη),
❝♦♠ γ, η > 1✳ P❛r❛ ✐ss♦✱ ✈❛♠♦s ✉s❛r ✉♠ ♠ét♦❞♦ ❞♦ t✐♣♦ ❜♦♦tstr❛♣✱ q✉❡ é ✉♠ ♠ét♦❞♦
❞❡ ✐t❡r❛çõ❡s ✉s❛♥❞♦ s❡q✉ê♥❝✐❛s ❞❡ ✐♠❡rsõ❡s ❡♥tr❡ ♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ Wk,p(Ω) ❡Lq(Ω)✳
❖❜s❡r✈❡ q✉❡✱
Ω|
H(x, U)|2∗/s ≤
Ω|
F(x)|2∗/s+C
Ω|
U|2∗. ✭✶✳✷✷✮
❉❡s❡❥❛♠♦s q✉❡ ❛s ✐♥t❡❣r❛✐s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛♥t❡r✐♦r s❡❥❛♠ ✜♥✐t❛s✳ P❡❧❛ ✐♠❡rsã♦ ❞❡ ❙♦✲ ❜♦❧❡✈✱ H1
0(Ω) ֒→ L2
∗
(Ω)✱ t❡♠♦s q✉❡ |U| ∈ L2∗
(Ω)✳ ❉❡♥♦t❛♥❞♦ p1 := 2∗/s✱ r❡st❛ s❛❜❡r s❡ |F(x)| ∈Lp1(Ω)✳
❙❡ N ≥ 4 ❡♥tã♦ 2∗ ≤ N✱ ❡st❡ ❢❛t♦ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❤✐♣ót❡s❡ s > 1 ✐♠♣❧✐❝❛ q✉❡ p1 := 2
∗
s <2∗ ≤N < r✳ ❊ ❛ss✐♠ t❡♠♦s q✉❡ |F| ∈Lp1(Ω)✳ ❆❣♦r❛ s❡ N = 3 ❡♥tã♦ p1 ♣♦❞❡
s❡r ♠❛✐♦r ♦✉ ♠❡♥♦r q✉❡ r✳ ❙❡♥❞♦ ❛ss✐♠✱ ✈❛♠♦s ❞✐✈✐❞✐r ❛ ❞❡♠♦♥str❛çã♦ ❡♠ ❞♦✐s ❝❛s♦s✿
❈❛s♦ ✶✿ N = 3 ❡ p1 ≥ r✳ ◆❡ss❡ ❝❛s♦ ❝♦♥s✐❞❡r❛♠♦s r ♥♦ ❧✉❣❛r ❞❡ p1 ♥❛ ✐♥t❡❣r❛❧ ❡♠
✭✶✳✷✷✮✱ ❞❛í
Ω|
H(x, U)|r ≤
Ω|
F(x)|r+C
Ω|
U|rs ✭✶✳✷✸✮
❝♦♠♦p1 = 2
∗
s ≥r t❡♠♦s q✉❡ rs≤2
∗ ❧♦❣♦ |U| ∈Lrs ❡ ♣♦rt❛♥t♦ ❝♦♥❝❧✉í♠♦s q✉❡ |H| ∈Lr✳
❆♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✭❇✳✶✸✮✱ ❝♦♥❝❧✉í♠♦s q✉❡ u, v ∈W2,r(Ω) ❡ q✉❡
U2,r ≤CHr.
❯t✐❧✐③❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✷✸✮✱ r❡s✉❧t❛
U2,r ≤C(Fr+Usrs).
❊♥✜♠✱ ❝♦♠♦ 1< rs≤2∗✱ ♦❜t❡♠♦s
❙✐st❡♠❛ ●r❛❞✐❡♥t❡ ✶✽
❈❛s♦ ✷✿ N ≥ 3 ❡ p1 < r✳ ◆❡ss❡ ❝❛s♦✱ ♦❜s❡r✈❛♠♦s q✉❡ |F(x)| ∈ Lp1(Ω) ❡ ❞❛ ❞❡s✐✲
❣✉❛❧❞❛❞❡ ✭✶✳✷✷✮✱ ❝♦♥❝❧✉í♠♦s q✉❡ |H| ∈Lp1✳ ❆♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✭❇✳✶✸✮✱ ❝♦♥❝❧✉í♠♦s q✉❡
u, v ∈W2,p1(Ω) ❡ q✉❡
U2,p1 ≤CHp1.
❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✷✷✮✱ r❡s✉❧t❛ q✉❡
U2,p1 ≤CFp1 +Us2∗
,
❡ ♣❡❧❛ ✐♠❡rsã♦ ❞❡ H1
0(Ω) ❡♠ L2
∗
(Ω)✱ ❝♦♥❝❧✉í♠♦s q✉❡
U2,p1 ≤C
Fp1 +U
s
. ✭✶✳✷✹✮
❆❣♦r❛ ♣r❡❝✐s❛♠♦s ❛♥❛❧✐s❛r três s✐t✉❛çõ❡s✿
❈❛s♦ ✷✐✿ ❙❡2p1 > N ❡♥tã♦ ♣❡❧♦ ✐t❡♠ ✭✐✐✐✮ ❞♦ ❚❡♦r❡♠❛ ✭❇✳✶✷✮✱ t❡♠♦s q✉❡ W2,p1(Ω)֒→
C0,α(Ω)✱ ♣❛r❛α <1✳ ❆ss✐♠✱
Ω|
H(x, U)|r≤
Ω|
F(x)|r+C
Ω|
U|rs<∞, ✭✶✳✷✺✮
♣♦rt❛♥t♦✱ |H| ∈Lr(Ω)✳ P❡❧♦ ❚❡♦r❡♠❛ ✭❇✳✶✸✮✱ r❡s✉❧t❛ q✉❡ |U| ∈W2,r(Ω) ❡
U2,r ≤CHr,
✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✷✺✮✱ r❡s✉❧t❛ q✉❡
U2,r ≤C(Fr+Usrs)
≤CFr+UsC0,α(Ω)
.
P♦rt❛♥t♦✱
U2,r ≤C
Fr+U s
2,p1
.
❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ ✭✶✳✷✹✮✱ ♦❜t❡♠♦s
❙✐st❡♠❛ ●r❛❞✐❡♥t❡ ✶✾
❈❛s♦ ✷✐✐✿ ❙❡ 2p1 =N ❡♥tã♦ ♣❡❧♦ ❚❡♦r❡♠❛ ✭❇✳✶✷✮✭✐✐✮ ✈❛❧❡ ❛ ✐♠❡rsã♦ W2,p1(Ω) ֒→Lσ
♣❛r❛ t♦❞♦ p1 < σ <∞✳ ❯s❛♥❞♦ ❡st❛ ✐♠❡rsã♦ ♣❛r❛ σ =rs✱ t❡♠♦s q✉❡ |U| ∈Lrs✱ ❧♦❣♦
Ω|
H(x, U)|r≤
Ω|
F(x)|r+C
Ω|
U|rs<∞, ✭✶✳✷✻✮
♣♦rt❛♥t♦✱ |H| ∈Lr(Ω)✳ P❡❧♦ ❚❡♦r❡♠❛ ✭❇✳✶✸✮✱ r❡s✉❧t❛ q✉❡ |U| ∈W2,r(Ω) ❡
U2,r ≤CHr.
❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✷✻✮✱ r❡s✉❧t❛ q✉❡
U2,r ≤C(Fr+Usrs)
≤Fr+Us2,p1.
❊♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✷✹✮✱ ♦❜t❡♠♦s
U2,r ≤CFr+Fsr+Us2.
❈❛s♦ ✷✐✐✐✿ ❙❡ 2p1 < N ❡♥tã♦ ❛♣❧✐❝❛♠♦s ♦ ❚❡♦r❡♠❛ ✭❇✳✶✷✮✭✐✮ ❡ ❝♦♥❝❧✉✐♠♦s q✉❡ u, v ∈
Lq1(Ω)✱ ❡♠ q✉❡ q
1 = NN p−21p1✳ ◆♦t❡ q✉❡✱
Ω|
H(x, U)|q1/s
≤
Ω|
F(x)|q1/s+C
Ω|
U|q1.
❊♥tã♦ r❡♣❡t✐♠♦s ♦ ♣r♦❝❡ss♦✱ ❛❣♦r❛ ❝♦♠ p2 =q1/s✳
❱❛♠♦s ✐t❡r❛r ♦ ♣r♦❝❡ss♦ k ✈❡③❡s ♣❛r❛ ♦❜t❡r ♥ú♠❡r♦s pm ❡ qm ❝♦♠ m= 1, ...k✱ t❛✐s q✉❡ pm =
qm−1
s ❡ qm =
N pm
(N −2pm) .
◆♦t❡ q✉❡✱ ♦ ♥ú♠❡r♦ ❞❡ ✐t❡r❛çõ❡s é ✜♥✐t♦✳ ■st♦ é✱ ❡①✐st❡ k > 0 t❛❧q✉❡ 2pk > N✳ ❉❡
❢❛t♦✱ ❝♦♠♦ s <2∗−1✱ t❡♠♦s q✉❡ p2
p1
= q1 2∗ =
N
N s−2.2∗ >
N
N(2∗−1)−2.2∗ = 1. P♦rt❛♥t♦✱
p2
p1
❙✐st❡♠❛ ●r❛❞✐❡♥t❡ ✷✵
♣❛r❛ ❛❧❣✉♠δ >0✳ ❆❣♦r❛❀
p3
p2
= q2
q1
= p2
p1
N−2p1
N−2p2
> p2 p1
= 1 +δ.
❞❡ ♦♥❞❡ s❡ ❝♦♥❝❧✉✐ q✉❡ p3 > p2(1 +δ)✳ ▼❛s p2 = (1 + δ)p1 ♣♦rt❛♥t♦ p3 > (1 + δ)2p1✳
■t❡r❛♥❞♦ ♦ ♣r♦❝❡ss♦ t❡♠♦s
pk>(1 +δ)k−1p1.
▲♦❣♦✱ ♣❛r❛ ❛❧❣✉♠ k s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ t❡♠♦s 2pk > N✳ P♦rt❛♥t♦✱ ♦ ♥ú♠❡r♦ ❞❡
✐t❡r❛çõ❡s é ✜♥✐t♦✳
❙❡♥❞♦ ❛ss✐♠✱ ❝♦♥❝❧✉í♠♦s q✉❡ U ∈W2,r(Ω)×W2,r(Ω) ❡ q✉❡
U2,r ≤C(Fγr +Uη),
❈❛♣ít✉❧♦ ✷
❙✐st❡♠❛ ❍❛♠✐❧t♦♥✐❛♥♦
◆❡st❡ ❝❛♣ít✉❧♦ ✈❛♠♦s ❡st✉❞❛r ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❡❧í♣t✐❝❛s ❞♦ t✐♣♦ ❤❛♠✐❧t♦♥✐❛♥♦
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−∆u=au+bv+vp++f(x) x∈Ω
−∆v =cu+av+uq++g(x) x∈Ω
u=v = 0 x∈∂Ω,
✭✷✳✶✮
❡♠ q✉❡Ω∈RN é ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ s✉❛✈❡ ❝♦♠ N ≥3❀a, b, c∈Rsã♦ t❛✐s q✉❡ b, c >0❀
❛s ❢✉♥çõ❡s f, g ∈ Lr ❝♦♠ r > N ❡ ❞❡♥♦t❛♠♦s w
+ = max{w,0}✳ ❖s ❡①♣♦❡♥t❡s p ❡ q sã♦
t❛✐s q✉❡1< p, q < NN+1−1✳
◆♦ss♦ ♦❜❥❡t✐✈♦✱ é ❡♥❝♦♥tr❛r ❡st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐ ♣❛r❛ ♣♦ssí✈❡✐s s♦❧✉çõ❡s ❞❡st❡ s✐st❡♠❛ ❡ ♠♦str❛r ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s✳
◆♦t❡ q✉❡✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ ♣♦❞❡♠♦s t♦♠❛r b=c♥♦ s✐st❡♠❛ ✭✷✳✶✮✳ ❉❡ ❢❛t♦✱
s❡(u, v) é s♦❧✉çã♦ ❞❡
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−∆u=au+bv/δ+v+p/δ+f(x)/δ x∈Ω −∆v =cδu+av+δquq
++g(x) x∈Ω
u=v = 0 x∈∂Ω,
❝♦♠ δ >0✱ ❡♥tã♦ (δu, v) é s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ✭✷✳✶✮✳ ❆ss✐♠ ❝♦♠♦ ❢❡✐t♦ ❡♠ ❬✷✸❪✱ ♣♦❞❡♠♦s
❡s❝♦❧❤❡r δ =b/c ❡ ♦❜t❡♠♦s ❛ ❞✐❛❣♦♥❛❧ s❡❝✉♥❞ár✐❛ ❝♦♠ ✈❛❧♦r❡s ✐❣✉❛✐s ❛ √bc✳ ❆ ♠❡♥♦s
❞❡ r❡❡s❝❛❧❛r ♦s ✈❛❧♦r❡s ♣♦❞❡♠♦s tr❛❜❛❧❤❛r ❛❣♦r❛ ❝♦♠ ♦ s✐st❡♠❛
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−∆u=au+bv+vp++f(x) x∈Ω −∆v =bu+av+uq++g(x) x∈Ω
u=v = 0 x∈∂Ω.
✭✷✳✷✮
❙✐st❡♠❛ ❍❛♠✐❧t♦♥✐❛♥♦ ✷✷
➱ ❝♦♥✈❡♥✐❡♥t❡ ❡s❝r❡✈❡r♠♦s ♦ s✐st❡♠❛ ✭✷✳✷✮ ♥❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧✿
⎧ ⎪ ⎨ ⎪ ⎩
−∆U =M U+G(U) +F(x) x∈Ω
U = 0 x∈∂Ω,
✭✷✳✸✮
♦♥❞❡
U =
⎛ ⎝ u
v
⎞
⎠, M =
⎛ ⎝ a b
b a
⎞
⎠∈M2x2(R), G(U) =
⎛ ⎝ v
p
+
uq+
⎞
⎠ e F(x) =
⎛ ⎝ f(x)
g(x)
⎞ ⎠.
❖s ❛✉t♦✈❛❧♦r❡s ❞❛ ♠❛tr✐③ M sã♦ ξ=a+b ❡ η=a−b✳
❈♦♥s✐❞❡r❛♠♦s λ1 < λ2 ≤ λ3 ≤ ... ≤ λn ≤ ... ♦s ❛✉t♦✈❛❧♦r❡s ❞❡ (−∆, H01(Ω)) ❡
φ1, φ2, ..., φn, ... ❛s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛✉t♦❢✉♥çõ❡s ❡ t❛❧ q✉❡ ❛ ♥♦r♠❛ L2 ❞❡ φ1 s❡❥❛ ♥♦r♠❛❧✐✲
③❛❞❛ ✐❣✉❛❧ ❛ 1✳
✷✳✶✳ ❊st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐
P❛r❛ ❡♥❝♦♥tr❛r♠♦s ❡st✐♠❛t✐✈❛s ❛ ♣r✐♦r✐ ♣❛r❛ ♣♦ssí✈❡✐s s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛ ✭✷✳✷✮✱ ♥♦s ❜❛s❡❛♠♦s ❡♠ ❛r❣✉♠❡♥t♦s ❡♥❝♦♥tr❛❞♦s ❡♠ ❬✼❪ ❡ ❬✶✵❪✳ ■♥✐❝✐❛❧♠❡♥t❡ ❡♥❝♦♥tr❛r❡♠♦s ✉♠❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ ♣♦ssí✈❡✐s s♦❧✉çõ❡s U ❞♦ s✐st❡♠❛ ✭✷✳✸✮✱ ♥♦ ❡s♣❛ç♦ W2,p+1p (Ω)×W2,
q+1
q (Ω)✳
❱❛♠♦s s✉♣♦r q✉❡λ1é ✉♠❛ ❛✉t♦✈❛❧♦r ❞❛ ♠❛tr✐③M✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡ q✉❡λ1 =a+b✱
❙❡❣✉❡ ❞♦s r❡s✉❧t❛❞♦s ❞❛❞♦s ♥♦ ❆♣ê♥❞✐❝❡❈q✉❡ Φ = (φ1, φ1) é ✉♠❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛
⎧ ⎪ ⎨ ⎪ ⎩
−∆U =M U x∈Ω
U = 0 x∈∂Ω.
✭✷✳✹✮
✷✳✶ ▲❡♠❛✳ ❙✉♣♦♥❤❛ q✉❡ λ1 =a+b✳ ❈♦♥s✐❞❡r❡ f, g∈Lr✱ r > N✱ s❛t✐s❢❛③❡♥❞♦
Ω
φ1f +
Ω
φ1g <0; ✭✷✳✺✮
❡ 1< p, q < NN+1−1. ❚♦❞❛ ♣♦ssí✈❡❧ s♦❧✉çã♦ U ∈H ❞❡ ✭✷✳✸✮ s❛t✐s❢❛③
UC1
❙✐st❡♠❛ ❍❛♠✐❧t♦♥✐❛♥♦ ✷✸
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ U ∈ H s♦❧✉çã♦ ❞❡ ✭✷✳✸✮✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ✭✷✳✸✮ ♣♦r Φ ❡ ✐♥t❡❣r❛♥❞♦✱
♦❜t❡♠♦s
Ω−
∆U.Φ =
Ω
M U.Φ +
Ω
G(U).Φ +
Ω
F(x).Φ
❝♦♠♦M é ❛✉t♦❛❞❥✉♥t❛ ❡ Φé s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✭✷✳✹✮✱ r❡s✉❧t❛
Ω
G(U).Φ =−
Ω
F(x).Φ≤CF(x)r ✭✷✳✼✮
P♦❞❡♠♦s ❞❡❝♦♠♣♦r U =tΦ +U1 t❛❧ q✉❡ Φ, U1H = 0✳ ◆♦t❡ q✉❡
(Φ, U1)H = 0 ⇒
Ω
(∇φ1∇u1+∇φ1∇v1) = 0 ⇒λ1
Ω
(φ1u1+φ1v1) = 0 ⇒
Ω
Φ·U1 = 0.
✐st♦ é✱ U1 é ♦rt♦❣♦♥❛❧ ❛ Φ t❛♠❜é♠ ❡♠ L2(Ω)× L2(Ω)✳ P♦rt❛♥t♦✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❡st❛
❞❡❝♦♠♣♦s✐çã♦ ♣♦r Φ ❡ ✐♥t❡❣r❛♥❞♦✱ r❡s✉❧t❛
Ω
U ·Φ =t
Ω
Φ·Φ +
Ω
U1·Φ = t
Ω
Φ·Φ,
❧♦❣♦
t =C
Ω
U ·Φ =C
Ω
(uφ1+vφ1)
≤C
Ω
(u+φ1+v+φ1).
❆♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✱ t❡♠♦s
t≤
Ω
uq+φ1
1/q
Ω
φ1
1/q′
+
Ω
vp+φ1
1/p
Ω
φ1
1/p′
❡♠ q✉❡✱p ❡ p′ sã♦ ❡①♣♦❡♥t❡s ❝♦♥❥✉❣❛❞♦s✱ ♦ ♠❡s♠♦ ♣❛r❛ q ❡q′✳ P♦rt❛♥t♦
❙✐st❡♠❛ ❍❛♠✐❧t♦♥✐❛♥♦ ✷✹
❆♥á❧♦❣♦ ❛♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ s❡ t >0 ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✽✮ ♥♦s ❞á ✉♠ ❧✐♠✐t❛çã♦ ♣❛r❛ |t|✳
❉❡ss❛ ❢♦r♠❛✱ ✈❛♠♦s ❞✐✈✐❞✐r ❛ ❞❡♠♦♥str❛çã♦ ❡♠ ❞♦✐s ❝❛s♦s✳
❈❛s♦ ✶✿ t ≥ 0✳ ◆❡ss❡ ❝❛s♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✽✮ ♥♦s ❞á ✉♠❛ ❧✐♠✐t❛çã♦ ♣❛r❛ |t|✱
❝♦♥s❡q✉❡♥t❡♠❡♥t❡ r❡st❛ ❡♥❝♦♥tr❛r♠♦s ✉♠❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ U1✳
❱❛♠♦s ❞❡✜♥✐r ♦s s❡❣✉✐♥t❡s ❡s♣❛ç♦s✿
L=Lp+1p ×L q+1
q ❡ W =W2, p+1
p ×W2, q+1
q ,
r❡s♣❡❝t✐✈❛♠❡♥t❡ ❝♦♠ s✉❛s ♥♦r♠❛s
UL=up+1
p +v q+1
q ❡ UW =u2, p+1
p +v2, q+1
q .
◆♦t❡ q✉❡ U1 s❛t✐s❢❛③ ❛ ❡q✉❛çã♦
−∆U1 =M U1+G(U) +F(x),
♣❛ss❛♥❞♦ ❛ ♥♦r♠❛ ❞♦ ❡s♣❛ç♦ L ❡♠ ❛♠❜♦s ♦s ❧❛❞♦s✱ t❡♠♦s
∆U1+M U1L =G(U) +F(x)L≤ G(U)L+F(x)L.
P♦rt❛♥t♦✱
∆U1 +M U1L≤ v p
+p+1
p +u
q
+q+1
q +F(x)L
≤
Ω
v+p+1
p p+1
+
Ω
uq++1
q q+1
+F(x)r. ✭✷✳✾✮
P♦❞❡♠♦s ❡s❝r❡✈❡r✱
Ω
v+p+1 =
Ω
v+pαφα1φ−1αv
p(1−α)+1 +
♣❛r❛ 0 < α < 1 ❛ s❡r ❞❡t❡r♠✐♥❛❞♦ ♣♦st❡r✐♦r♠❡♥t❡✳ P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✱ t❡♠♦s
q✉❡
Ω
vp++1 ≤
Ω
v+pφ1
α⎛
⎝
Ω
vp+
1 1−α
+
φ
α
1−α
1
⎞ ⎠
1−α
