Pedro Manuel Sánchez Aguilar

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛

■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

Pr♦❜❧❡♠❛s ❊❧í♣t✐❝♦s ❙❡♠✐❧✐♥❡❛r❡s ❝♦♠

❉❡♣❡♥❞ê♥❝✐❛ ❞♦ ●r❛❞✐❡♥t❡

♣♦r

P❡❞r♦ ▼❛♥✉❡❧ ❙á♥❝❤❡③ ❆❣✉✐❧❛r

❇r❛sí❧✐❛

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

Pr♦❜❧❡♠❛s ❊❧í♣t✐❝♦s ❙❡♠✐❧✐♥❡❛r❡s ❝♦♠

❉❡♣❡♥❞ê♥❝✐❛ ❞♦ ●r❛❞✐❡♥t❡

♣♦r

P❡❞r♦ ▼❛♥✉❡❧ ❙á♥❝❤❡③ ❆❣✉✐❧❛r

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡

▼❊❙❚❘❊ ❊▼ ▼❆❚❊▼➪❚■❈❆

❇r❛sí❧✐❛✱ ✷✸ ❞❡ ❛❣♦st♦ ❞❡ ✷✵✶✸

❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛✿

Pr♦❢a_{✳ ❉r❛✳ ▼❛♥✉❡❧❛ ❈❛❡t❛♥♦ ▼❛rt✐♥s ❞❡ ❘❡③❡♥❞❡ ✭❯♥❇✮✕❖r✐❡♥t❛❞♦r❛}

Pr♦❢✳ ❉r✳ ❯❜❡r❧â♥❞✐♦ ❇❛t✐st❛ ❙❡✈❡r♦ ✭❯❋P❇✮

Pr♦❢✳ ❉r✳ ❘✐❝❛r❞♦ ❘✉✈✐❛r♦ ✭❯♥❇✮

❆❣r❛❞❡❝✐♠❡♥t♦s

❖❜r✐❣❛❞♦✱ ♠❡✉ ❉❡✉s✱ ♣♦r ❞❛r✲♠❡ ❢♦rç❛s ♣❛r❛ ❧✐❞❛r ❝♦♠ ❛s ❛❞✈❡rs✐❞❛❞❡s ❡ ❛ss✐♠ ❝✉♠♣r✐r ❝♦♠ ♠❛✐s ✉♠ ♦❜❥❡t✐✈♦ ❡♠ ♠✐♥❤❛ ✈✐❞❛✳

➚ ♣r♦❢❡ss♦r❛ ▼❛♥✉❡❧❛ ❈❛❡t❛♥♦ ▼❛rt✐♥s ❞❡ ❘❡③❡♥❞❡✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡ ❛♣♦✐♦ ♥❡st❡ tr❛❜❛❧❤♦✱ ♣❡❧♦s ❝♦♥s❡❧❤♦s ❡✱ s♦❜r❡t✉❞♦✱ ♣♦r s✉❛ ❛♠✐③❛❞❡✳

❆♦s ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✱ ♣r♦❢❡ss♦r❡s ❯❜❡r❧â♥❞✐♦ ❇❛t✐st❛ ❙❡✈❡r♦✱ ❘✐❝❛r❞♦ ❘✉✈✐❛r♦✱ ❈❛r❧♦s ❆❧❜❡rt♦ P❡r❡✐r❛ ❞♦s ❙❛♥t♦s✱ ♣❡❧❛s ♦♣♦rt✉♥❛s ❝♦rr❡çõ❡s ❡ s✉❣❡st♦✄❡s✳

❆♦s ♠❡✉s ♣❛✐s✱ ●❧❛❞②s ❚❡♦❞♦♥✐❧❛ ❆❣✉✐❧❛r ❱✐t❡r✐ ❡ ❍❡r♠❡♥❡❣✐❧❞♦ ❙á♥❝❤❡③ ❱❡❣❛✱ ♣❡❧❛s ♣❛❧❛✈r❛s ❞❡ ❛❧❡♥t♦ ❡ ♣♦r t❡r❡♠ ❢♦♠❡♥t❛❞♦ ❡♠ ♠✐♠ ♦ ❞❡s❡❥♦ ❞❡ s✉♣❡r❛çã♦ ♥❛ ✈✐❞❛✳

❆♦ ❈◆Pq✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❞✉r❛♥t❡ t♦❞♦ ♦ ♣❡rí♦❞♦✳

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦✱ ✐❧✉str❛♠♦s ❛ ✉t✐❧✐③❛çã♦ ❞❡ ❞♦✐s ♠ét♦❞♦s ❞✐❢❡r❡♥t❡s ✕ s✉❜ ❡ s✉♣❡rs♦❧✉çã♦ ❡ ✈❛r✐❛❝✐♦♥❛❧ ✕ ♥❛ ♦❜t❡♥çã♦ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ ✉♠❛ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s ❡❧í♣t✐❝♦s s❡♠✐❧✐♥❡❛r❡s ❝✉❥❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ ❛♣r❡s❡♥t❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ t❡r♠♦ ❣r❛❞✐❡♥t❡✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ s✉❜ ❡ s✉♣❡rs♦❧✉çã♦✱ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✱ t❡r♠♦ ❣r❛❞✐❡♥t❡✱ ♣r♦❜❧❡♠❛s s✐♥❣✉❧❛r❡s✱ s♦❧✉çã♦ ❣r♦✉♥❞ st❛t❡✳

❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ ✐❧❧✉str❛t❡ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t✇♦ ❞✐✛❡r❡♥t ♠❡t❤♦❞s ✕ ❧♦✇❡r ❛♥❞ ✉♣♣❡r s♦❧✉t✐♦♥ ❛♥❞ ✈❛r✐❛t✐♦♥❛❧ ✕ t♦ ♦❜t❛✐♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ s♦❧✉t✐♦♥ ❢♦r ❛ ❝❧❛ss ♦❢ s❡♠✐❧✐♥❡❛r ❡❧❧✐♣t✐❝ ♣r♦❜❧❡♠s ✇❤♦s❡ ♥♦♥❧✐♥❡❛r✐t② ♣r❡s❡♥ts ❞❡♣❡♥❞❡♥❝❡ ♦♥ t❤❡ ❣r❛❞✐❡♥t t❡r♠✳

❑❡② ✇♦r❞s✿ ❧♦✇❡r ❛♥❞ ✉♣♣❡r s♦❧✉t✐♦♥✱ ▼♦✉♥t❛✐♥ P❛ss✱ ❣r❛❞✐❡♥t t❡r♠✱ s✐♥❣✉❧❛r ♣r♦❜❧❡♠s✱ ❣r♦✉♥❞ st❛t❡ s♦❧✉t✐♦♥✳

❙✉♠ár✐♦

◆♦t❛çõ❡s ✶

■♥tr♦❞✉çã♦ ✹

✶ ◆♦çõ❡s ♣r❡❧✐♠✐♥❛r❡s ❡ r❡s✉❧t❛❞♦s ❛✉①✐❧✐❛r❡s ✶✸

✷ ❙♦❧✉çõ❡s ❣r♦✉♥❞ st❛t❡ ♣❛r❛ ❛ ❡q✉❛çã♦ ❞❡ ▲❛♥❡✲❊♠❞❡♥✲❋♦✇❧❡r s✐♥❣✉❧❛r

❝♦♠ t❡r♠♦ ❞❡ ❝♦♥✈❡❝çã♦ s✉❜❧✐♥❡❛r ✷✹

✷✳✶ ❊①✐stê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❡♠ ❞♦♠í♥✐♦s ❧✐♠✐t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷ ❊①✐stê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ❣r♦✉♥❞ st❛t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✸ ❊q✉❛çõ❡s ❡❧í♣t✐❝❛s s❡♠✐❧✐♥❡❛r❡s ❝♦♠ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ ❣r❛❞✐❡♥t❡ ✈✐❛

té❝♥✐❝❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ✹✷

✸✳✶ ❙♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✈❛r✐❛❝✐♦♥❛❧ ✈✐❛ té❝♥✐❝❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ✳ ✳ ✹✺ ✸✳✷ ❙♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ♥ã♦ ✈❛r✐❛❝✐♦♥❛❧ ❛tr❛✈és ❞❡ ✉♠ ♠ét♦❞♦ ✐t❡r❛t✐✈♦ ✳ ✺✻

✹ ❆♣ê♥❞✐❝❡ ✺✾

◆♦t❛çõ❡s

◆❡st❡ tr❛❜❛❧❤♦✱ ❢❛③❡♠♦s ✉s♦ ❞❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s✿

• Ω⊂RN é ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❝♦♠ ❢r♦♥t❡✐r❛ s✉❛✈❡❀

• | · | r❡♣r❡s❡♥t❛ ❛ ♥♦r♠❛ ❡✉❝❧✐❞✐❛♥❛ ❡♠ RN_{, N ≥1}❀

• Br ={x∈RN :|x|< r}, r >0❀

• ❉❛❞♦s ❛❜❡rt♦s Ω0, Ω ⊂RN✱ ❡s❝r❡✈❡♠♦s Ω0 ⊂⊂Ω q✉❛♥❞♦ Ω0 ❡stá ❝♦♠♣❛❝t❛♠❡♥t❡

❝♦♥t✐❞♦ ❡♠ Ω✱ ✐st♦ é✱ Ω0 é ✉♠ ❝♦♠♣❛❝t♦ ❝♦♥t✐❞♦ Ω❀

• λ1 é ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ❞♦ ♣r♦❜❧❡♠❛

(

−∆ϕ =λϕ ❡♠ Ω,

ϕ = 0 s♦❜r❡ ∂Ω, (AV)

❡ ϕ1 é ✉♠❛ λ1✲❛✉t♦❢✉♥çã♦ ♣♦s✐t✐✈❛ ❞❡ (AV)❀

• ❯♠ ✈❡t♦r ❞❛ ❢♦r♠❛ α = (α1, . . . , αN)✱ ♦♥❞❡ ❝❛❞❛ ❝♦♠♣♦♥❡♥t❡ αi ∈ N∪ {0}✱ é ❝❤❛♠❛❞♦ ✉♠ ♠✉❧t✐✲í♥❞✐❝❡ ❞❡ ♦r❞❡♠

|α|=α1+· · ·+αN;

• ❉❛❞♦ ✉♠ ♠✉❧t✐✲í♥❞✐❝❡ α✱ ❞❡✜♥✐♠♦s

Dαu:= ∂

|α|_u

∂α1x

1. . . ∂αNxN

,

q✉❛♥❞♦ ❛ ❞❡r✐✈❛❞❛ ♠✐st❛ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❡①✐st❡❀

◆♦t❛çõ❡s ✷

• C(Ω) =

(

u∈C(Ω) :u é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥tí♥✉❛ ❡♠ s✉❜❝♦♥❥✉♥t♦s

❧✐♠✐t❛❞♦s ❞❡Ω

)

,

❝✉❥❛ ♥♦r♠❛ é kukC(Ω) = sup

x∈Ω

|u(x)|;

• C0(Ω) sã♦ ❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❞❡ s✉♣♦rt❡ ❝♦♠♣❛❝t♦ ❡♠ Ω❀

• Ck_{(Ω) ={u: Ω−→R:u é k ✈❡③❡s ❝♦♥t✐♥✉❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡❧ }✱k ∈N∪ {0}❀}

• Ck_{(Ω) =}

(

u∈Ck_{(Ω) :D}α_{u ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥tí♥✉❛ s♦❜r❡ s✉❜❝♦♥❥✉♥t♦s} ❧✐♠✐t❛❞♦s ❞❡ Ω, ♣❛r❛ t♦❞♦ |α| ≤k

)

.

❆ss✐♠ s❡ u ∈ Ck_{(Ω)✱ ❡♥tã♦ D}α_{u ❡st❡♥❞❡✲s❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ❛té Ω ♣❛r❛ ❝❛❞❛ ♠✉❧t✐✲} í♥❞✐❝❡ α✱ |α| ≤k✳

kuk_Ck_(Ω) =

X

|α|≤k

kDα_uk C(Ω);

• C∞_{(Ω) =} ∞

\

k=0

Ck_{(Ω) ❡ C}∞_{(Ω) =}

∞

\

k=0

Ck_(Ω)❀

• Ck

0(Ω) =Ck(Ω)∩C0(Ω) ❡ C0∞(Ω) =C∞(Ω)∩C0(Ω)❀

• ❙❡❥❛ k∈N_{∪ {0}} ❡_{0< γ ≤1}✳ _Ck,γ_{(Ω) ❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ❞❡ ❍ö❧❞❡r ❞❛❞♦ ♣♦r}

Ck,γ_{(Ω) ={u∈C}k_{(Ω) : [D}α_u]

C0,γ_(Ω) <∞, ♣❛r❛ t♦❞♦ ♠✉❧t✐✲í♥❞✐❝❡|α| ≤k},

♦♥❞❡ [Dα_u]

C0,γ_(Ω)= sup

x,y∈Ω

x6=y

|Dα_u(x)−Dα_u(y)|

|x−y|γ ✱ ❝✉❥❛ ♥♦r♠❛ é

kuk_Ck,γ_(Ω) =kuk_Ck_(Ω)+

X

|α|=k

[Dα_u]

C0,γ_(Ω), ♣❛r❛ t♦❞♦u∈Ck,γ(Ω);

• ❯♠❛ ❢✉♥çã♦ u : Ω−→R é ❝❤❛♠❛❞❛ ▲✐♣s❝❤✐t③ s❡ s❛t✐s❢❛③ _{|u(x)−u(y)| ≤C|x−y|}✱

♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡C ❡ ♣❛r❛ t♦❞♦sx, y ∈Ω✳ ◆❡st❡ ❝❛s♦✱ ❡s❝r❡✈❡♠♦su∈Lip(Ω)❀

• u+_{(x) = max{u(x),0}✱ u}−_{(x) = max{−u(x),0}❀}

• ❖ ❡s♣❛ç♦ Lp_{(Ω) é ❞❡♥♦t❛❞♦ ♣♦r}

Lp(Ω) ={f : Ω−→R_:f é ♠❡♥s✉rá✈❡❧ ❡

Z

Ω

◆♦t❛çõ❡s ✸

❡♠ q✉❡ 1≤p <∞ ❡ Ω⊂RN é ✉♠ ❛❜❡rt♦ ❝♦♥❡①♦✱ ♠✉♥✐❞♦ ❝♦♠ ❛ ♥♦r♠❛ ❞❛❞❛ ♣♦r

kfkp =

Z

Ω

|f|p dx

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❛1≤p≤ ∞❡k∈N_{∪ {0}}✱ ❞❡♥♦t❛♠♦s ♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈_Wk,p_(Ω)❝♦♠♦ s❡♥❞♦

Wk,p_{(Ω) :={u∈L}p_{(Ω) :D}α_u∈Lp_{(Ω) ♣❛r❛ t♦❞♦ ♠✉❧t✐✲í♥❞✐❝❡ α t❛❧ q✉❡ |α| ≤k},} ❝♦♠ ♥♦r♠❛ ❞❛❞❛ ♣♦r

kukWk,p =

                

 X

|α|≤k

Z

Ω

|Dα_u|p

 

1

p

, s❡ 1≤p <∞,

X

|α|≤k

kDαuk∞, s❡p=∞;

• ❉❡♥♦t❛♠♦s ♣♦r X∗ _{♦ ❡s♣❛ç♦ ❞✉❛❧ ❞❡ X✱ ❝♦♠ ♥♦r♠❛ ❞❛❞❛ ♣♦r}

kIkX∗ = sup x∈X

kxkX=1

■♥tr♦❞✉çã♦

◆❛s ú❧t✐♠❛s ❞é❝❛❞❛s✱ ♦s ♣r♦❜❧❡♠❛s ❡❧í♣t✐❝♦s tê♠ ❝❤❛♠❛❞♦ ❛t❡♥çã♦ ❞♦s ♣❡sq✉✐s❛❞♦r❡s ❡♠ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s P❛r❝✐❛✐s ♣❡❧❛s ❛♣❧✐❝❛çõ❡s ❡♠ ❞✐✈❡rs❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✳ ❊♠ ✈ár✐❛s ❞❡st❛s ❛♣❧✐❝❛çõ❡s✱ é ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ s❛❜❡r s❡ ❛ ❡q✉❛çã♦ q✉❡ ♠♦❞❡❧❛ ❝❡rt♦ ❢❡♥ô♠❡♥♦ ♣♦ss✉✐ s♦❧✉çã♦✳ ❈♦♠ ❡st❛ ✜♥❛❧✐❞❛❞❡✱ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞❛s té❝♥✐❝❛s q✉❡ ❞❡t❡r♠✐♥❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❞❡ ♣r♦❜❧❡♠❛s ❡❧í♣t✐❝♦s✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♥♦s ❝♦♥❝❡♥tr❛♠♦s ❡♠ ❞✉❛s té❝♥✐❝❛s q✉❡ ♠♦str❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ❞❡ ✉♠ t✐♣♦ ❡s♣❡❝í✜❝♦ ❞❡ ♣r♦❜❧❡♠❛s✳

❖ ▼ét♦❞♦ ❞❡ ❙✉❜ ❡ ❙✉♣❡rs♦❧✉çã♦ é ✉♠ ❞♦s ♠ét♦❞♦s ♠❛✐s ✉s❛❞♦s ♣❛r❛ ❡st✉❞❛r ♣r♦❜❧❡♠❛s ❡❧í♣t✐❝♦s ❡ r❡q✉❡r ❝❡rt❛ ❤❛❜✐❧✐❞❛❞❡ ❡♠ ❝❛❞❛ ❝❛s♦✳ ❙❛❜❡♠♦s q✉❡ s❡ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ♠✉❞❛ ❞❡ s✐♥❛❧ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦✱ ❡♥tã♦ ❡❧❛ ❤á ❞❡ t❡r ✉♠ ③❡r♦✳ ❆té ❝❡rt♦ ♣♦♥t♦✱ ♦ ♠ét♦❞♦ ❞❡ s✉❜ ❡ s✉♣❡rs♦❧✉çã♦ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠❛ ❡①t❡♥sã♦ ❞❡st❡ ❢❛t♦✳ ❆ ✐❞❡✐❛ ❞❡st❡ ♠ét♦❞♦ ❝♦♥s✐st❡ ❡♠ ❞❡t❡r♠✐♥❛r ✉♠❛ s✉❜s♦❧✉çã♦ u ❡ ✉♠❛ s✉♣❡rs♦❧✉çã♦ u ❞♦

♣r♦❜❧❡♠❛ ❡♠ q✉❡stã♦ ❡ ❛♣❧✐❝❛r ✉♠ t❡♦r❡♠❛ ❞❡ ❝♦♠♣❛r❛çã♦ ♣❛r❛ ✈❡r✐✜❝❛r q✉❡ u ≤ u✱ ♦

q✉❡ ♥♦s ♣❡r♠✐t❡ ❞❡t❡r♠✐♥❛r ✉♠❛ s♦❧✉çã♦ u t❛❧ q✉❡ u≤u≤u✳

❖s ▼ét♦❞♦s ❱❛r✐❛❝✐♦♥❛✐s s✉r❣✐r❛♠ ♥♦ sé❝✉❧♦ ❳❱■■■ ❡ ❢♦r❛♠ ❊✉❧❡r ❡ ▲❛❣r❛♥❣❡ q✉❡ ♦ tr❛♥s❢♦r♠❛r❛♠ ❡♠ ✉♠❛ t❡♦r✐❛ ♠❛t❡♠át✐❝❛ r✐❣♦r♦s❛✳ ❖ ♠ét♦❞♦ ✈❛r✐❛❝✐♦♥❛❧ ❢♦✐ ❛♣❧✐❝❛❞♦✱ ❛♣ós ❛ s✉❛ ❞❡s❝♦❜❡rt❛✱ s♦❜r❡t✉❞♦ ♥❛ ❢ís✐❝❛✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❡♠ ♠❡❝â♥✐❝❛✱ ❡ t♦r♥♦✉✲s❡ ✉♠❛ ❞✐s❝✐♣❧✐♥❛ ♠❛t❡♠át✐❝❛ ✐♥❞❡♣❡♥❞❡♥t❡✱ ❝♦♠ s❡✉s ♣ró♣r✐♦s ♠ét♦❞♦s ❞❡ ♣❡sq✉✐s❛✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡s✱ ✈❡❥❛ ❬✹✵❪✳ ❯♠ ♠ét♦❞♦ ✈❛r✐❛❝✐♦♥❛❧ ✐♠♣♦rt❛♥t❡ ♥❛ s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❡❧í♣t✐❝♦s s❡♠✐❧✐♥❡❛r❡s é ❛ ❚❡♦r✐❛ ❞♦ P♦♥t♦ ❈rít✐❝♦✱ q✉❡ ❛ ❣r♦ss♦ ♠♦❞♦ ❝♦♥s✐st❡ ❡♠ ✈❡r ❝❛❞❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❝♦♠♦ ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❢✉♥❝✐♦♥❛❧✳ P♦r s✉❛ ✈❡③✱ ✉♠ ❞♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛ t❡♦r✐❛ ❞♦ P♦♥t♦ ❈rít✐❝♦✱ ❡ q✉❡ s❡rá ✉s❛❞♦ ♥❡st❡ tr❛❜❛❧❤♦✱ é ♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✱ ❞❡✈✐❞♦ ❛ ❆♠❜r♦s❡tt✐ ❡ ❘❛❜✐♥♦✇✐t③ ❬✼❪✳ ❊st❡ t❡♦r❡♠❛ ❣❛r❛♥t❡✱ s♦❜ ❤✐♣ót❡s❡s ❛❞❡q✉❛❞❛s✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ ♣♦♥t♦s ❝rít✐❝♦s ❞❡ ♠✐♥✐♠❛① ♣❛r❛ ♦ ❢✉♥❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ❡st✉❞❛❞♦✳

■♥tr♦❞✉çã♦ ✺

♣❛r❛ ❡st✉❞❛r ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ ❡❧í♣t✐❝♦ s❡♠✐❧✐♥❡❛r ❝✉❥❛ ♥ã♦✲ ❧✐♥❡❛r✐❞❛❞❡ ❛♣r❡s❡♥t❛ ✉♠ t❡r♠♦ ❣r❛❞✐❡♥t❡✱ t❛♠❜é♠ ❝❤❛♠❛❞♦ ❞❡ t❡r♠♦ ❞❡ ❝♦♥✈❡❝çã♦✳

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

◆♦ ❈❛♣ít✉❧♦ ✶✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ ♥♦çõ❡s q✉❡ ❝♦♠♣õ❡♠ ♦ ♠❡✐♦ s♦❜r❡ ♦ q✉❛❧ ❞❡s❡♥✈♦❧✈❡r❡♠♦s ♦ ♥♦ss♦ tr❛❜❛❧❤♦✳ ◆♦s ❧✐♠✐t❛♠♦s ❛ ❛♣r❡s❡♥t❛r✱ ❞❡ ♠❛♥❡✐r❛ r❡s✉♠✐❞❛✱ ❛s ♥♦çõ❡s ❢✉♥❞❛♠❡♥t❛✐s q✉❡ ❢♦r❛♠ ✉t✐❧✐③❛❞❛s✱ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛✱ ♥♦ ❞❡❝♦rr❡r ❞❡st❡ t❡①t♦✳ ❊♥✉♥❝✐❛♠♦s✱ t❛♠❜é♠✱ ❛❧❣✉♥s t❡♦r❡♠❛s ❝❧áss✐❝♦s q✉❡ ♥♦s ❛✉①✐❧✐❛rã♦ ♥❛ ♦❜t❡♥çã♦ ❞♦s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s✳

❖s ♣ró①✐♠♦s ❞♦✐s ❝❛♣ít✉❧♦s ❞❡st❡ tr❛❜❛❧❤♦ ❛❜♦r❞❛♠✱ ❞❡ ❢♦r♠❛ ❞❡t❛❧❤❛❞❛✱ ❝♦♠♦ ♦s ♠ét♦❞♦s ❞❡ s✉❜ ❡ s✉♣❡rs♦❧✉çã♦ ❡ ✈❛r✐❛❝✐♦♥❛❧✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞♦s ♣❛r❛ ♦❜t❡r ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣r♦❜❧❡♠❛ ❡❧í♣t✐❝♦ s❡♠✐❧✐♥❡❛r✳

◆♦ ❈❛♣ít✉❧♦ ✷✱ ❡st✉❞❛♠♦s ♦ r❡s✉❧t❛❞♦ ❞❡ ▼❛r✐✉s ●❤❡r❣✉ ❡ ❱✐❝❡♥t✐✉ ❘➔❞✉❧❡s❝✉ ❬✸✷❪✱ q✉❡ ❡♠ ✷✵✵✼ ❞❡t❡r♠✐♥❛r❛♠ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛

    

−∆u=p(x)(g(u) +f(u) +|∇u|a_{) ❡♠ R}N_,

u >0 ❡♠ RN_,

u(x)−→0 q✉❛♥❞♦ |x| → ∞,

✭✶✮

♦♥❞❡ N ≥3, 0< a < 1✱ ❡ p:RN _{−→(0,∞)} é ✉♠❛ ❢✉♥çã♦ ❍ö❧❞❡r ❝♦♥tí♥✉❛ ❞❡ ❡①♣♦❡♥t❡ 0< γ <1 t❛❧ q✉❡

(p1)

Z ∞

1

t φ(t) dt <∞✱ ♦♥❞❡ φ(r) = max

|x|=rp(x)✳

❆ss✉♠✐r❡♠♦s t❛♠❜é♠ q✉❡ g ∈C1_{((0,∞)) é ✉♠❛ ❢✉♥çã♦ ❞❡❝r❡s❝❡♥t❡ ❡ ♣♦s✐t✐✈❛ t❛❧ q✉❡}

(g1) lim

t→0g(t) = +∞✱

❡ ❛ ❢✉♥çã♦ f : [0,∞) −→ [0,∞) é C1_{((0,∞))✱ ♥ã♦✲❞❡❝r❡s❝❡♥t❡ ❡ ♣♦s✐t✐✈❛ ❡♠ (0,∞) ❡}

s❛t✐s❢❛③

(f1) ❛ ❢✉♥çã♦ t7−→

f(t)

t é ♥ã♦✲❝r❡s❝❡♥t❡✱ ♣❛r❛ t♦❞♦ t∈(0,∞)❀

(f2) lim

t→0+ f(t)

t = +∞ ❡ tlim→∞

f(t)

t = 0✳

❆ ✐❞❡✐❛ ❞❡s❡♥✈♦❧✈✐❞❛ ❡♠ ❬✸✷❪ ❝♦♥s✐st❡ ❡♠ ❛♣❧✐❝❛r ♦ ♠ét♦❞♦ ❞❡ s✉❜ ❡ s✉♣❡rs♦❧✉çã♦ ♣❛r❛ ♦

♣r♦❜❧❡♠❛ _

   

−∆u=p(x)(g(u) +f(u) +|∇u|a_{) ❡♠ Ω,}

u >0 ❡♠ Ω,

u(x) = 0 s♦❜r❡ ∂Ω,

■♥tr♦❞✉çã♦ ✻

♦ q✉❡ ♥♦s ♣❡r♠✐t❡ ❣❡r❛r ✉♠❛ s❡q✉ê♥❝✐❛ ♠♦♥ót♦♥❛ ❞❡ s♦❧✉çõ❡s (un) ♣❛r❛ ❛ ❢❛♠í❧✐❛ ❞❡ ♣r♦❜❧❡♠❛s ❞♦ t✐♣♦ (2)✱ s♦❜r❡ Bn✱ ❧✐♠✐t❛❞❛ t❛♥t♦ s✉♣❡r✐♦r q✉❛♥t♦ ✐♥❢❡r✐♦r♠❡♥t❡✳ ❆ss✐♠✱

♣♦r ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛❞rã♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ (un) ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ q✉❡ ❝♦♥✈❡r❣❡ ✉♥✐❢♦r♠❡♠❡♥t❡ ❛ ✉♠❛ ❢✉♥çã♦ ❡♠ C2,α_{(Ω)✳ P♦st❡r✐♦r♠❡♥t❡✱ ♣♦r ✉♠ ♣r♦❝❡ss♦} ❞✐❛❣♦♥❛❧✱ s❡❣✉❡ q✉❡ (un) ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ q✉❡ ❝♦♥✈❡r❣❡ ✉♥✐❢♦r♠❡♠❡♥t❡ s♦❜r❡ s✉❜❝♦♥❥✉♥t♦s ❛❜❡rt♦s ❧✐♠✐t❛❞♦s ❞❡ RN ❛ ✉♠❛ ❢✉♥çã♦ _{u ∈ C}2,α

loc(R

N_{)✱ q✉❡ é s♦❧✉çã♦ ❞♦}

♣r♦❜❧❡♠❛ (1)✳ ❚❛❧ ♣r♦❝❡❞✐♠❡♥t♦ s❡rá ✉t✐❧✐③❛❞♦ ♣❛r❛ ❞❡♠♦♥str❛r♠♦s ♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧

❞♦ ❈❛♣ít✉❧♦ ✷✱ ❛ s❛❜❡r✿

❚❡♦r❡♠❛ ✵✳✶✳ ❙✉♣♦♥❤❛ q✉❡ (f1)✱ (f2)✱ (g1) ❡ (p1) s❡❥❛♠ s❛t✐s❢❡✐t❛s✳ ❊♥tã♦ ♦ ♣r♦❜❧❡♠❛

(1) t❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦ ❡♠ C_loc2,α(RN₎✳

➱ ✐♠♣♦rt❛♥t❡ ♠❡♥❝✐♦♥❛r q✉❡ ❛ ❡q✉❛çã♦ (1) é ❞♦ t✐♣♦ ▲❛♥❡✲❊♠❞❡♥✲❋♦✇❧❡r✳ ❊q✉❛çõ❡s

❞❡st❡ t✐♣♦ s❡ ♦r✐❣✐♥❛r❛♠ ❛ ♣❛rt✐r ❞❡ t❡♦r✐❛s s♦❜r❡ ❛ ❞✐♥â♠✐❝❛ ❞❡ ❣❛s❡s ❡♠ ❛str♦❢ís✐❝❛ ✭✈❡❥❛ ❬✷✷❪✮✱ s✉r❣✐♥❞♦ t❛♠❜é♠ ♥♦ ❡st✉❞♦ ❞❛ ♠❡❝â♥✐❝❛ ❞♦s ✢✉✐❞♦s✱ ♠❡❝â♥✐❝❛ r❡❧❛t✐✈íst✐❝❛✱ ❢ís✐❝❛ ♥✉❝❧❡❛r ❡ ♥♦ ❡st✉❞♦ ❞❛ q✉í♠✐❝❛✳ P❛r❛ ✉♠❛ ❛❜♦r❞❛❣❡♠ ♠❛✐s ❞❡t❛❧❤❛❞❛ ❞❡st❡ t✐♣♦ ❞❡ ❡q✉❛çõ❡s✱ r❡❢❡r✐♠♦s ❛♦ ❧❡✐t♦r ♦ tr❛❜❛❧❤♦ ❞❡ ❲♦♥❣ ❬✹✾❪✳ ❆ ❡q✉❛çã♦ ❞❡ ▲❛♥❡✲❊♠❞❡♥✲ ❋♦✇❧❡r t❡♠ s✐❞♦ ❡st✉❞❛❞❛ ♣♦r ♠✉✐t♦s ❛✉t♦r❡s q✉❡ ✉t✐❧✐③❛♠ ✈ár✐♦s ♠ét♦❞♦s ❡ té❝♥✐❝❛s✳ ❊♥tr❡ ❡❧❛s✱ ❝✐t❛♠♦s ❛ t❡♦r✐❛ ❞♦ ♣♦♥t♦ ❝rít✐❝♦✱ ❛ t❡♦r✐❛ ❞♦ ♣♦♥t♦ ✜①♦ ❡ ❛ t❡♦r✐❛ ❞♦ ❣r❛✉ t♦♣♦❧ó❣✐❝♦✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s✱ ✈❡❥❛ ❬✷✱ ✺✱ ✸✸✱ ✸✼✱ ✸✾❪✳

❋✉♥çõ❡s q✉❡ s❛t✐s❢❛③❡♠ ✉♠❛ ❝♦♥❞✐çã♦ ❝♦♠♦ (g1) sã♦ ❞✐t❛s s✐♥❣✉❧❛r❡s✳ Pr♦❜❧❡♠❛s

❡♥✈♦❧✈❡♥❞♦ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡s s✐♥❣✉❧❛r❡s s✉r❣❡♠ ❡♠ ✈ár✐❛s s✐t✉❛çõ❡s ❢ís✐❝❛s✱ ♣r❡s❡♥t❡s ♥❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ ❡❧étr✐❝❛ ✭❋✉❧❦s ❡ ▼❛②❜❡❡✱ ✶✾✻✵ ❬✷✸❪✮✱ ♥❛ t❡♦r✐❛ ❞♦s ✢✉✐❞♦s ♣s❡✉❞♦♣❧ást✐❝♦s ✭❈❛❧❧❡❣❛r✐ ❡ ◆❛s❤♠❛♥✱ ✶✾✽✵ ❬✶✵❪✮✱ ❡♠ s✉♣❡r❢í❝✐❡s ♠í♥✐♠❛s s✐♥❣✉❧❛r❡s ✭❈❛✛❛r❡❧❧✐✱ ❍❛r❞t ❡ ❙✐♠♦♥✱ ✶✾✽✹ ❬✾❪✮✱ ❡♠ ♣r♦❝❡ss♦s ❞❡ r❡❛çã♦✲❞✐❢✉sã♦✱ ♥❛ ♦❜t❡♥çã♦ ❞❡ ❞✐✈❡rs♦s í♥❞✐❝❡s ❣❡♦❢ís✐❝♦s ❡ ❡♠ ♣r♦❝❡ss♦s ✐♥❞✉str✐❛✐s✱ ❡♥tr❡ ♦✉tr♦s✳

❊♠ r❡❧❛çã♦ ❛ s♦❧✉çõ❡s ❣r♦✉♥❞ st❛t❡ ♣❛r❛ ❡q✉❛çõ❡s ❡❧í♣t✐❝❛s s✐♥❣✉❧❛r❡s✱ ✐st♦ é✱ s♦❧✉çõ❡s ♣♦s✐t✐✈❛s ❞❡✜♥✐❞❛s ❡♠ t♦❞♦ ♦ ❡s♣❛ç♦ ❡ t❡♥❞❡♥❞♦ ❛ ③❡r♦ ♥♦ ✐♥✜♥✐t♦✱ ❡①✐st❡♠ ♠✉✐t♦s tr❛❜❛❧❤♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ q✉❛♥❞♦ ❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ ♥ã♦ ❛♣r❡s❡♥t❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ ❣r❛❞✐❡♥t❡ ❞❛ s♦❧✉çã♦✳ ❊♥tr❡ ❡❧❡s✱ ✐♥✐❝✐❛♠♦s ❝✐t❛♥❞♦ ♦s q✉❡ ❞❡t❡r♠✐♥❛r❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦ u∈C_loc2,α(RN₎♣❛r❛ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛

    

−∆u=p(x)g(u) ❡♠ RN_,

u >0 ❡♠ RN_,

u(x)−→0 q✉❛♥❞♦ |x| → ∞,

■♥tr♦❞✉çã♦ ✼

♦♥❞❡ p:RN _{−→(0,∞)} é ✉♠❛ ❢✉♥çã♦ ❧♦❝❛❧♠❡♥t❡ ❍ö❧❞❡r ❝♦♥tí♥✉❛ q✉❡ s❛t✐s❢❛③

Z ∞

0

tψ(t)dt <∞, s❡♥❞♦ ψ(t) = max{p(x) :|x|=t}, t >0,

❡ g ∈C1_{((0,∞),(0,∞)) s❛t✐s❢❛③ ♣❡❧♦ ♠❡♥♦s ❞✉❛s ❞❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿}

(g1) g é ♥ã♦✲❝r❡s❝❡♥t❡✱ (g2) lim

t→0+g(t) = ∞✱

(g3) g é ❧✐♠✐t❛❞❛ ♣❛r❛ t s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ (g4) lim

t→0+ g(t)

t =∞,

(g5)

g(t)

t+c é ❞❡❝r❡s❝❡♥t❡ ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ c≥0✱ (g6) limt→∞

g(t)

t = 0✳

P♦r ❡①❡♠♣❧♦✱ ▲❛✐r ❡ ❙❤❛❦❡r ❬✸✺❪ ❡♠ ✶✾✾✻ ❡ ❩❤❛♥❣ ❬✺✺❪ ❡♠ ✶✾✾✼ ♠♦str❛r❛♠ q✉❡ ♦ ♣r♦❜❧❡♠❛

(3) ❛❞♠✐t❡ s♦❧✉çã♦ s❡g s❛t✐s❢❛③(g1)❡(g2)✳ ❊♠ ✶✾✾✾✱ ❈îrst❡❛ ❡ ❘➔❞✉❧❡s❝✉ ❬✶✶❪ ♠♦str❛r❛♠

q✉❡ ♦ ♣r♦❜❧❡♠❛ (3) ❛❞♠✐t❡ s♦❧✉çã♦ q✉❛♥❞♦ g ♥ã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♠♦♥ót♦♥❛✱ ♣♦ré♠

s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s (g3), (g4)❡ (g5)✱ ❝♦♠ c >0✳ ❊♠ ❬✷✶❪✱ ❋❡♥❣ ❡ ▲✐✉ ❡st❛❜❡❧❡❝❡r❛♠✱ ❡♠

✷✵✵✹✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ❣r♦✉♥❞ st❛t❡ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛(3) q✉❛♥❞♦ ❣ s❛t✐s❢❛③ ❛s

❝♦♥❞✐çõ❡s(g2)❡(g3)✳ ❊♠ ✷✵✵✻✱ ●♦♥ç❛❧✈❡s ❡ ❙❛♥t♦s ❬✷✽❪ ❡st❛❜❡❧❡❝❡r❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛

s♦❧✉çã♦ ♣❛r❛ (3) s♦❜ ❛ ❤✐♣ót❡s❡ q✉❡ g s❛t✐s❢❛③ (g4), (g5) ❡ (g6)✱ ❝♦♠ c = 0✳ ❋✐♥❛❧♠❡♥t❡

❩❤❛♥❣ ❬✺✻❪ ♠♦str♦✉✱ ❡♠ ✷✵✵✼✱ q✉❡(3) t❡♠ s♦❧✉çã♦ s♦❜ ❛ ❝♦♥❞✐çã♦ q✉❡ ❣ s❛t✐s❢❛ç❛ s♦♠❡♥t❡ (g4)❡ (g6)✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s✱ r❡❢❡r✐♠♦s ❛♦ ❧❡✐t♦r ♦s tr❛❜❛❧❤♦s ❬✸✽✱ ✸✵✱ ✹✷❪✳

Pr♦❜❧❡♠❛s ❡♠ q✉❡ ❛ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡ t❡♠ ✉♠ t❡r♠♦ ❣r❛❞✐❡♥t❡ ✭♦✉ ❝♦♥✈❡❝t✐✈♦✮ s✉r❣❡♠ ❡♠ t❡♦r✐❛ ❞❡ ❝♦♥tr♦❧❡ ❡st♦❝ást✐❝♦ ✭▲❛sr② ❡ ▲✐♦♥s✱ ✶✾✽✾ ❬✸✻❪✮✱ ♥♦ ❡st✉❞♦ ❞❡ ✉♠ ❝❛♠♣♦ ❡❧❡tr♦♠❛❣♥ét✐❝♦ ✭❙t✉❛rt✱ ✶✾✾✶ ❬✹✹❪✱ ❙t✉❛rt ❡ ❩❤♦✉✱ ✶✾✾✻ ❬✹✺❪✮✱ ❡♠ ✉♠ ♠❡✐♦ ♥ã♦✲❧✐♥❡❛r✱ ❡♥tr❡ ♦✉tr♦s✳

❈♦♠ r❡❧❛çã♦ ❛ ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ t❡r♠♦ ❞❡ ❝♦♥✈❡❝çã♦✱ ♣♦❞❡♠♦s ❝✐t❛r ❉✐♥✉ ❬✶✺❪✱ q✉❡ ❡♠ ✷✵✵✸ ❣❛r❛♥t✐✉ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ❝❧áss✐❝❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛

    

−∆u+q(x)|∇u|a _=p(x)u−γ _{❡♠ R}N_,

u >0 ❡♠ RN_,

u(x)−→0 q✉❛♥❞♦ |x| → ∞,

✭✹✮

♦♥❞❡ N ≥3, a, γ >0 ❡p, q ∈C_loc0,α(RN_{) sã♦ t❛✐s q✉❡ p >0, q≥0 ❡}

Z ∞

0

rΦ(r)dr <∞, ♦♥❞❡ Φ(r) = max

|x|=rp(x).

■♥tr♦❞✉çã♦ ✽

❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ (1)✱ r❡❧❛t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✳ ❆✐♥❞❛ ❡♠ ✷✵✵✼✱ ❡st❡

tr❛❜❛❧❤♦ ❢♦✐ ♠❡❧❤♦r❛❞♦ ♣♦r ❳✉❡ ❡ ❩❤❛♥❣ ❬✺✶❪✱ q✉❡ ❝♦♥s✐❞❡r❛r❛♠ (1) s❡♠ ❡①✐❣✐r q✉❛❧q✉❡r

♠♦♥♦t♦♥✐❝✐❞❛❞❡ s♦❜r❡ g ❡ f ✱ ♠❛s s✉♣♦♥❞♦

lim

s→0+ g(s)

s =∞, slim→∞

g(s)

s = 0, slim→0+ f(s)

s =∞ e slim→∞

f(s)

s = 0.

P♦st❡r✐♦r♠❡♥t❡✱ ❳✉❡ ❡ ❙❤❛♦ ❬✺✷✱ ✺✸❪ ♠❡❧❤♦r❛r❛♠ ♦ tr❛❜❛❧❤♦ ❞❡ ❉✐♥✉ ❝♦♥s✐❞❡r❛♥❞♦p(x)g(u)

❡ f(x, u)✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡♠ ✈❡③ ❞❡ p(x)u−γ _{❡♠ (4)✱ s♦❜ ❝♦♥❞✐çõ❡s ❛❞❡q✉❛❞❛s ♣❛r❛ g ❡}

f✳

●♦♥ç❛❧✈❡s ❡ ❙✐❧✈❛✱ ❬✷✾❪ ❡♠ ✷✵✶✵✱ tr❛❜❛❧❤❛r❛♠ r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❡ ♥ã♦ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛

    

−∆u=λp(x)(g(u) +|∇u|a_{) ❡♠ R}N_,

u >0 ❡♠ RN_,

u(x)−→0 q✉❛♥❞♦ |x| → ∞,

(P+)

♦♥❞❡ N ≥3, 0< a <1 ❡ λ >0 é ✉♠ ♣❛râ♠❡tr♦✱ p :RN _{−→(0,∞) é ❧♦❝❛❧♠❡♥t❡ ❍ö❧❞❡r} ❝♦♥tí♥✉❛ ❡ s❛t✐s❢❛③ _{Z ∞}

0

tφ(t)dt < ∞ ♦✉

Z ∞

0

tφ(t)dt=∞,

♦♥❞❡ φ(r) = max

|x|=rp(x), r≥0✳ ❆❧é♠ ❞✐ss♦✱ g : (0,∞)−→(0,∞)é C

1 _{❡ s❛t✐s❢❛③}

lim

s→0+ g(s)

s =ρ0, slim→∞

g(s)

s =ρ∞, ♦♥❞❡ ρ0 ∈(0,∞] ❡ ρ∞ ∈[0,∞].

P❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ r6= 2✱ ❲✉ ❡ ❨❛♥❣ ❬✺✵❪✱ ❡♠ ✷✵✶✵✱ ♠♦str❛r❛♠ ❛ ♥ã♦ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛

s♦❧✉çã♦ r❛❞✐❛❧♠❡♥t❡ s✐♠étr✐❝❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛

    

−div(|∇u|r−2_{∇u) =p(x)(g(u) +f(u) +|∇u|}a_{) ❡♠ R}N_,

u >0 ❡♠ RN_,

u(x)−→0 q✉❛♥❞♦ |x| → ∞,

♦♥❞❡ p ❡ f sã♦ ❢✉♥çõ❡s r❛❞✐❛✐s✱ f ∈ C_loc0,α((0,∞),(0,∞))✱ N ≥ 3✱ a ≥ 0✱ g ∈

C1_{((0,∞),(0,∞))❡ s✉❜❧✐♥❡❛r ♥❛ ♦r✐❣❡♠ ❡ ♥♦ ✐♥✜♥✐t♦✳}

■♥tr♦❞✉çã♦ ✾

s♦❧✉çã♦ u∈C1(RN_)∩C2₍RN _{\ {0})}r❛❞✐❛❧♠❡♥t❡ s✐♠étr✐❝❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛

    

−div(|∇u|r−2_{∇u) +b(x)|∇u|}p−1 _{=a(u)f(u) ❡♠ R}N_,

u >0 ❡♠ RN_,

u(x)−→0 q✉❛♥❞♦ |x| → ∞,

♦♥❞❡ N ≥3✱ 1< r < N✱ a✱ b:RN _{−→(0,∞)} sã♦ ❢✉♥çõ❡s r❛❞✐❛✐s ❝♦♥tí♥✉❛s ❡_a s❛t✐s❢❛③

          

Z ∞

0

sr−11a(s)r−11ds <∞, s❡ 1< r≤2,

Z ∞

0

s(r

−₂₎N+1

r−1 a(s)ds <∞, s❡ 2≤r < N,

s❡♥❞♦ f : (0,∞)−→(0,∞) ❞❡ ❝❧❛ss❡C1_{✱ s✐♥❣✉❧❛r ♥♦ ③❡r♦✱ lim}

s→∞f(s)/s

r−1 _{= 0 ❡ ❛ ❢✉♥çã♦}

s 7−→f(s)/sr−1 _{é ♥ã♦ ❝r❡s❝❡♥t❡ ❡♠ (0,∞)✳}

◆♦ ❈❛♣ít✉❧♦ ✸ ❞❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ✉♠ r❡s✉❧t❛❞♦ ♣✐♦♥❡✐r♦ ❞❡✈✐❞♦ ❛ ❉❥❛✐r♦ ❞❡ ❋✐❣✉❡✐r❡❞♦✱ ▼❛r✐♦ ●✐r❛r❞✐ ❡ ▼✐❝❤❡❧❡ ▼❛t③❡✉ ❬✶✻❪✱ q✉❡ ❡♠ ✷✵✵✹ ❝♦♥s✐❞❡r❛r❛♠ ❛ s♦❧✉❜✐❧✐❞❛❞❡

❞♦ ♣r♦❜❧❡♠❛ ₍

−∆u=f(x, u,∇u) ❡♠ Ω,

u= 0 s♦❜r❡ ∂Ω, ✭✺✮

♦♥❞❡ Ω é ✉♠ ❞♦♠í♥✐♦ s✉❛✈❡ ❧✐♠✐t❛❞♦ ❡♠ RN_{, N ≥ 3}✳ ❊st❡ t✐♣♦ ❞❡ ❡q✉❛çõ❡s ♥ã♦ t❡♠

s✐❞♦ ❡①t❡♥s✐✈❛♠❡♥t❡ ❡st✉❞❛❞♦ ♣♦r ♠ét♦❞♦s ✈❛r✐❛❝✐♦♥❛✐s ❝♦♠♦ ♥♦ ❝❛s♦ ❡♠ q✉❡ ♥ã♦ ❡①✐st❡ ❛ ♣r❡s❡♥ç❛ ❞♦ ❣r❛❞✐❡♥t❡✳ ❆ r❛③ã♦ é q✉❡✱ ❝♦♥tr❛r✐❛♠❡♥t❡ ❛♦ ú❧t✐♠♦ ❝❛s♦✱ ❛ ❡q✉❛çã♦ ❡♠

(5) ♥ã♦ é ✈❛r✐❛❝✐♦♥❛❧✳ ❆ss✐♠✱ ❛ t❡♦r✐❛ ❞♦ ♣♦♥t♦ ❝rít✐❝♦ ♥ã♦ é ❛❞❡q✉❛❞❛ ♣❛r❛ ✉♠ ❛t❛q✉❡

❞✐r❡✐t♦ ❛♦ ♣r♦❜❧❡♠❛✳ ❆ té❝♥✐❝❛ ✉s❛❞❛ ♥❡st❡ tr❛❜❛❧❤♦ ❝♦♥s✐st❡ ❡♠ ❛ss♦❝✐❛r✱ ❛♦ ♣r♦❜❧❡♠❛

(5)✱ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛

(

−∆u=f(x, u,∇ω) ❡♠ Ω,

u= 0 s♦❜r❡ ∂Ω, ✭✻✮

♦♥❞❡ ω ∈H1

0(Ω) ♥ã♦ ❞❡♣❡♥❞❡ ❞❡u✳

❆❣♦r❛ ♦ ♣r♦❜❧❡♠❛ (6) é ✈❛r✐❛❝✐♦♥❛❧ ❡ ♣♦❞❡♠♦s tr❛tá✲❧♦ ♣♦r ♠ét♦❞♦s ✈❛r✐❛❝✐♦♥❛✐s✳

❊s♣❡❝✐✜❝❛♠❡♥t❡✱ ✉t✐❧✐③❛r❡♠♦s ❛ té❝♥✐❝❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ♣❛r❛ ❛❜♦r❞❛r ♦ ♣r♦❜❧❡♠❛

(6)✳ P❛r❛ ❡st❡ ✜♠✱ ❛tr✐❜✉✐r❡♠♦s ❤✐♣ót❡s❡s s♦❜r❡f ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ♦ ♣r♦❜❧❡♠❛ (6) ♣♦ss❛

s❡r tr❛t❛❞♦ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❞❡ ❆♠❜r♦s❡tt✐ ❡ ❘❛❜✐♥♦✇✐t③ ❬✼❪✳ ◆♦ss♦ ♣r✐♠❡✐r♦ ❝♦♥❥✉♥t♦ ❞❡ ❤✐♣ót❡s❡s s♦❜r❡ ❛ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡ f é ♦ s❡❣✉✐♥t❡✿

■♥tr♦❞✉çã♦ ✶✵

✭f1✮ lim

t→0

f(x, t, ξ)

t = 0 ✉♥✐❢♦r♠❡♠❡♥t❡✱ ♣❛r❛ x∈Ω, ξ ∈R

N_❀

✭f2✮ ❡①✐st❡♠ ❝♦♥st❛♥t❡s a1 >0❡ p∈(1,NN+2−2) t❛✐s q✉❡

|f(x, t, ξ)| ≤a1(1 +|t|p), ♣❛r❛ t♦❞♦sx∈Ω, t∈R, ξ ∈RN;

✭f3✮ ❡①✐st❡♠ ❝♦♥st❛♥t❡s θ >2 ❡t0 >0 t❛✐s q✉❡

0< θF(x, t, ξ)≤tf(x, t, ξ), ♣❛r❛ t♦❞♦sx∈Ω, |t| ≥t0, ξ ∈RN,

♦♥❞❡

F(x, t, ξ) =

Z t

0

f(x, s, ξ)ds;

✭f4✮ ❡①✐st❡♠ ❝♦♥st❛♥t❡s a2, a3 >0t❛✐s q✉❡

F(x, t, ξ)≥a2|t|θ−a3, ♣❛r❛ t♦❞♦sx∈Ω, t∈R, ξ ∈RN.

❙♦❜ ❡st❡ ❝♦♥❥✉♥t♦ ❞❡ ❤✐♣ót❡s❡s✱ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ♣❛r❛ ♠♦str❛r ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿

❚❡♦r❡♠❛ ✵✳✷✳ ❙✉♣♦♥❤❛ q✉❡ (f0)−(f4) s❡❥❛♠ s❛t✐s❢❡✐t❛s✳ ❊♥tã♦ ❡①✐st❡♠ ❝♦♥st❛♥t❡s

♣♦s✐t✐✈❛s c1 ❡ c2 t❛✐s q✉❡✱ ♣❛r❛ ❝❛❞❛ ω∈H01(Ω)✱ ♦ ♣r♦❜❧❡♠❛ (6) t❡♠ ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ uω t❛❧ q✉❡

c1 ≤ kuωk ≤c2.

❆❧é♠ ❞✐ss♦✱ (6) ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❡ ✉♠❛ s♦❧✉çã♦ ♥❡❣❛t✐✈❛✳

P❛r❛ ♦❜t❡r ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛(5)✱ é ♥❡❝❡ssár✐♦ ❛❞✐❝✐♦♥❛r ❛ s❡❣✉✐♥t❡ ❤✐♣ót❡s❡✿ (f5) ❛ ❢✉♥çã♦f s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s ▲✐♣s❝❤✐t③ ❧♦❝❛✐s✿

(i) |f(x, t′

, ξ)−f(x, t′′

, ξ)| ≤L1 |t

′

−t′′

|, ♣❛r❛ t♦❞♦s x∈Ω, t′

, t′′

∈[0, ρ1], |ξ| ≤ρ2✱

(ii) |f(x, t, ξ′

)−f(x, t, ξ′′

)| ≤L2|ξ

′

−ξ′′

|, ♣❛r❛ t♦❞♦sx∈Ω, t∈[0, ρ1], |ξ

′

|, |ξ′′

| ≤ρ2✱

♦♥❞❡ρ1 ❡ρ2❞❡♣❡♥❞❡♠ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞❡p✱N✱θ✱a1✱a2✱a3 ❞❛❞♦s ♥❛s ❤✐♣ót❡s❡s ❛♥t❡r✐♦r❡s✳

❖ ❚❡♦r❡♠❛ ✵✳✷ ♥♦s ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛(un)⊂H₀1(Ω) ❞❡ s♦❧✉çõ❡s ♣❛r❛ ❛ ❢❛♠í❧✐❛ ❞❡ ♣r♦❜❧❡♠❛s

(

−∆un =f(x, un,∇un−1) ❡♠ Ω,

■♥tr♦❞✉çã♦ ✶✶

♦ q✉❡ ♥♦s ♣❡r♠✐t❡✱ ♠❡❞✐❛♥t❡ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛❞rã♦✱ ♠♦str❛r q✉❡ ❡①✐st❡ u ∈ H01(Ω)

♣♦s✐t✐✈❛ s❛t✐s❢❛③❡♥❞♦ ♦ ♣r♦❜❧❡♠❛(5)✳ ▼❡❞✐❛♥t❡ ✉♠ ❛r❣✉♠❡♥t♦ ❜♦♦tstr❛♣✱ ♠♦str❛♠♦s q✉❡

u∈C2,α_{(Ω)✱ ✐st♦ é✱ ♠♦str❛♠♦s ♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ❞♦ ❈❛♣ít✉❧♦ ✸✱ ❛ s❛❜❡r✿}

❚❡♦r❡♠❛ ✵✳✸✳ ❙✉♣♦♥❤❛ q✉❡ ❛s ❝♦♥❞✐çõ❡s (f0)−(f5) s❡❥❛♠ ✈á❧✐❞❛s✳ ❊♥tã♦ ♦ ♣r♦❜❧❡♠❛(5)

t❡♠ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❡ ✉♠❛ s♦❧✉çã♦ ♥❡❣❛t✐✈❛✱ ❞❡s❞❡ q✉❡

λ−₁1L1+λ

−1₂

1 L2 <1,

♦♥❞❡ λ1 é ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ❞❡ (−∆, H01(Ω))✳ ❆❧é♠ ❞✐ss♦✱ ❛ s♦❧✉çã♦ ♦❜t✐❞❛ é ❞❡ ❝❧❛ss❡

C2_✳

❱ár✐♦s tr❛❜❛❧❤♦s tr❛t❛♠ ♦ ♣r♦❜❧❡♠❛(5)✉s❛♥❞♦ s✉❜ ❡ s✉♣❡rs♦❧✉çã♦✱ ♦ ❣r❛✉ t♦♣♦❧ó❣✐❝♦✱

t❡♦r❡♠❛s ❞❡ ♣♦♥t♦ ✜①♦ ❡ ♠ét♦❞♦ ❞❡ ●❛❧❡r❦✐♥✳ ❱❡❥❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❬✸✱ ✹✱ ✶✽✱ ✹✶✱ ✺✹❪✳ ❊♠ ❬✶✻❪ ♦s ❛✉t♦r❡s ❞❡s❡♥✈♦❧✈❡r❛♠ ✉♠ ♠ét♦❞♦ ❝♦♠♣❧❡t❛♠❡♥t❡ ❞✐❢❡r❡♥t❡ ❞❡ t✐♣♦ ✈❛r✐❛❝✐♦♥❛❧✳ ❈♦♠ ❜❛s❡ ♥❛ s✉♣♦s✐çã♦ q✉❡ f t❡♠ ✉♠ ❝r❡s❝✐♠❡♥t♦ s✉❜❝rít✐❝♦ ❝♦♠ ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦

s✉♣❡r❧✐♥❡❛r ♥❛ ♦r✐❣❡♠ ❡ ♥♦ ✐♥✜♥✐t♦ ❝♦♠ r❡s♣❡✐t♦ à s❡❣✉♥❞❛ ✈❛r✐á✈❡❧✱ ♦❜t✐✈❡r❛♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ♣♦s✐t✐✈❛ ❡ ✉♠❛ s♦❧✉çã♦ ♥❡❣❛t✐✈❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ (5)✱ ✉s❛♥❞♦ ♦ ❚❡♦r❡♠❛

❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛ ❡ ✉♠❛ té❝♥✐❝❛ ✐t❡r❛t✐✈❛✳

❆ ♠❡s♠❛ ✐❞❡✐❛ ❞❡ ❬✶✻❪ ❢♦✐ ✉t✐❧✐③❛❞❛✱ ❛✐♥❞❛ ♥♦ ❛♥♦ 2004✱ ♣♦r ●✐r❛r❞✐ ❡ ▼❛t③❡✉ ❬✷✻❪✱

q✉❡ ❡st✉❞❛r❛♠ ♦ ♠❡s♠♦ ♣r♦❜❧❡♠❛ (5) s✉♣♦♥❞♦ q✉❡f s❛t✐s❢❛③ (f0),(f1)✱(f3)−(f5)❝♦♠ ❛

❤✐♣ót❡s❡ ❛❞✐❝✐♦♥❛❧✿

( ˜f2) ❡①✐st❡♠ ❝♦♥st❛♥t❡s a1 >0❡ p∈(1,N_N+2₋₂), r∈(0,1)t❛✐s q✉❡

|f(x, t, ξ)| ≤a1(1 +|t|p)(1 +|ξ|r), ♣❛r❛ t♦❞♦sx∈Ω, t∈R, ξ ∈RN.

◆❛ ❛♣❧✐❝❛çã♦ ❞❛ té❝♥✐❝❛✱ ❛ ♥♦✈✐❞❛❞❡ ❞❡st❡ tr❛❜❛❧❤♦ ❝♦♥s✐st❡ ❡♠ ❝♦♥s✐❞❡r❛r ✉♠ ❝♦♥✈❡♥✐❡♥t❡ tr✉♥❝❛♠❡♥t♦ ❞❛ ❢✉♥çã♦ f✱ ♣❛r❛ ♦ q✉❛❧ ♥ã♦ ❤á ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ ❣r❛❞✐❡♥t❡ ♥♦ ✐♥✜♥✐t♦✳

❘❡❢❡r✐♠♦s ❛✐♥❞❛✱ ❛♦ ❧❡✐t♦r✱ ♦s tr❛❜❛❧❤♦s ❬✷✺✱ ✷✼❪ ❞♦s ♠❡s♠♦s ❛✉t♦r❡s✳

❊♠ ✷✵✵✽✱ ●✐♦✈❛♥② ❋✐❣✉❡✐r❡❞♦ ❬✶✾❪ ✉t✐❧✐③♦✉ ❡st❡ ♠❡s♠♦ ♠ét♦❞♦ ♣❛r❛ ❡q✉❛çõ❡s ❡❧í♣t✐❝❛s q✉❛s✐❧✐♥❡❛r❡s ❡ ♠♦str♦✉ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛

(

−∆pu+|u|p−2u=f(u,|∇u|p−2∇u),

u∈W1,p_(RN_{), u(x)>0, ♣❛r❛ t♦❞♦ x∈R}N_,

♦♥❞❡ 1 < p < N ❡ f : R_× RN _{−→ R é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ q✉❡ ✈❡r✐✜❝❛ ❝♦♥❞✐çõ❡s} ❛♥á❧♦❣❛s ❛ (f1), (f3)−(f5) ❡ s❛t✐s❢❛③✿

■♥tr♦❞✉çã♦ ✶✷

• ❡①✐st❡ q∈(p, p∗_{) t❛❧ q✉❡ lim} |s|→+∞

|f(s,|ξ|p−2_ξ)|

|s|q−1 = 0 ♣❛r❛ t♦❞♦ξ ∈R

N_❀

• s7−→ f(s,|ξ|

p−2_ξ)

sp−1 é ❝r❡s❝❡♥t❡ ♣❛r❛ ❝❛❞❛ s >0✱ ♣❛r❛ t♦❞♦ ξ∈R

N_✳

❊♠ ✷✵✶✵✱ ❡st❡ ♠❡s♠♦ ♠ét♦❞♦ ❢♦✐ ❛♣❧✐❝❛❞♦ ♣♦r ❚❡♥❣ ❡ ❩❤❛♥❣ ❬✹✻❪ ♣❛r❛ ✐♥✈❡st✐❣❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ ❡♥✈♦❧✈❡♥❞♦ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ✐♠♣✉❧s✐✈❛s✳ ❊♠ ✷✵✶✷✱ ▲✐✉✱ ❙❤✐ ❡ ❲❡✐ ❬✸✶❪ ♠♦str❛r❛♠✱ ♠❡❞✐❛♥t❡ ❛ t❡♦r✐❛ ❞❡ ▼♦rs❡ ❡ ✉♠❛ té❝♥✐❝❛ ✐t❡r❛t✐✈❛✱ q✉❡ ♦ ♣r♦❜❧❡♠❛ (5) ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦ s♦❜ ❛ s✉♣♦s✐çã♦ ❞❡ q✉❡ f

t❡♠ ✉♠ ❝r❡s❝✐♠❡♥t♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❧✐♥❡❛r ♥♦ ③❡r♦ ❡ ♥♦ ✐♥✜♥✐t♦ ❝♦♠ r❡❧❛çã♦ à s❡❣✉♥❞❛ ✈❛r✐á✈❡❧✳

◆♦ ❈❛♣ít✉❧♦ ✹ ❞❡st❡ tr❛❜❛❧❤♦ ❡♥❝♦♥tr❛♠✲s❡ ❛ ✈❡r✐✜❝❛çã♦ ❞❡ ❛❧❣✉♠❛s ❞❛s ❛✜r♠❛çõ❡s ❢❡✐t❛s ♥♦s ❈❛♣ít✉❧♦s ✷ ❡ ✸ ❡ ❛s ❞❡♠♦♥str❛çõ❡s ❞❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s té❝♥✐❝♦s✳

❈❛♣ít✉❧♦

1

◆♦çõ❡s ♣r❡❧✐♠✐♥❛r❡s ❡ r❡s✉❧t❛❞♦s

❛✉①✐❧✐❛r❡s

◆❡st❡ ❝❛♣ít✉❧♦ ❡♥✉♥❝✐❛♠♦s ❛s ♣r✐♥❝✐♣❛✐s ❞❡✜♥✐çõ❡s ❡ t❡♦r❡♠❛s ✉t✐❧✐③❛❞♦s ♥♦ ❞❡❝♦rr❡r ❞❡st❡ tr❛❜❛❧❤♦✳

❉❡✜♥✐çã♦ ✶✳✶✳ ❬✶❪ ❙❡❥❛♠ 1≤p≤ ∞ ❡ k ∈N✳ ❖ ❡s♣❛ç♦ _Wk,p

0 (Ω) é ❞❡✜♥✐❞♦ ❝♦♠♦ s❡♥❞♦

♦ ❢❡❝❤♦ ❞❡ C∞

0 (Ω) ♥❛ ♥♦r♠❛k · kWk,p✱ ✐st♦ é✱

W₀k,p(Ω) =C∞

0 (Ω)

k·k_{W k,p}

.

❖❜s❡r✈❛çã♦ ✶✳✷✳ ❖ ❡s♣❛ç♦ W₀1,2(Ω) s❡rá ❞❡♥♦t❛❞♦ ♣♦r H1

0(Ω) ❡ s✉❛ ♥♦r♠❛ ❞❛❞❛ ♣♦r

kuk:=kuk_W1,2 0 =

Z

Ω

|∇u|2dx

1 2

, ♣❛r❛ t♦❞♦u∈W₀1,2(Ω).

❉❡✜♥✐çã♦ ✶✳✸✳ ❬✹✽❪ ❙❡❥❛X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳ ❯♠❛ ❛♣❧✐❝❛çã♦I :X −→Ré ❝❤❛♠❛❞❛

✉♠ ❢✉♥❝✐♦♥❛❧✳ ❆❝❡r❝❛ ❞❡ ✉♠ ❢✉♥❝✐♦♥❛❧ I :U −→ R✱ ♦♥❞❡ _U é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❞❡

X✱ ❞✐③❡♠♦s q✉❡✿

(i) t❡♠ ❞❡r✐✈❛❞❛ ❞❡ ●❛t❡❛✉① T ∈X∗ _{❡♠ u∈U s❡✱ ♣❛r❛ t♦❞♦ h∈X✱}

lim

t→0

1

t[I(u+th)−I(u)− hT, thi] = 0.

Pr❡❧✐♠✐♥❛r❡s ✶✹

(ii) t❡♠ ❞❡r✐✈❛❞❛ ❞❡ ❋ré❝❤❡t T ∈X∗ ❡♠ u∈U s❡

lim

t→0

1

khk[I(u+h)−I(u)− hT, hi] = 0;

(iii) ♣❡rt❡♥❝❡ ❛ C1_{(U,R) s❡ ❛ ❞❡r✐✈❛❞❛ ❞❡ ❋ré❝❤❡t ❞❡ I ❡①✐st❡ ❡ é ❝♦♥tí♥✉❛ ❡♠U❀}

(iv) ✉♠ ♣♦♥t♦ ❝rít✐❝♦ u ❞❡ I é ✉♠ ♣♦♥t♦ ❡♠ q✉❡I′(u) = 0✱ ✐st♦ é✱

hI′(u), ϕi= 0, ♣❛r❛ t♦❞♦ ϕ∈X.

◆♦ ❈❛♣ít✉❧♦ ✸✱ sã♦ ✉t✐❧✐③❛❞♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ❉♦♠✐♥❛❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❡ ❛❧❣✉♠❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❛✉①✐❧✐❛r❡s✱ q✉❡ s❡❣✉❡♠ ❡♥✉♥❝✐❛❞❛s ❛❜❛✐①♦✳

❚❡♦r❡♠❛ ✶✳✹✳ (❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ❉♦♠✐♥❛❞❛ ❞❡ ▲❡❜❡s❣✉❡ )❬✹✼❪

❙❡❥❛ Ω⊂RN ✉♠ s✉❜❝♦♥❥✉♥t♦ ♠❡♥s✉rá✈❡❧✳ ❙✉♣♦♥❤❛ q✉❡ _(fn) é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s

♠❡♥s✉rá✈❡✐s s♦❜r❡ Ωt❛✐s q✉❡ fn −→f✱ q✳t✳♣✳ ❡♠ Ω✱ q✉❛♥❞♦ n→ ∞✳ ❙❡ ❡①✐st❡ φ∈L1(Ω) t❛❧ q✉❡ |fn| ≤φ q✳t✳♣✳ ❡♠ Ω✱ ♣❛r❛ t♦❞♦ n✱ ❡♥tã♦

Z

Ω

fn −→

Z

Ω

f, q✉❛♥❞♦ n→ ∞.

❚❡♦r❡♠❛ ✶✳✺✳ ❬✽❪ ❙❡❥❛♠ (fn) ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ Lp(Ω) ❡ f ∈ Lp(Ω)✱ t❛✐s q✉❡

kfn−fkp −→0, q✉❛♥❞♦ n → ∞✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ (fnk) t❛❧ q✉❡

✶✳ fnk −→f q✳t✳♣✳ ❡♠ Ω✱ q✉❛♥❞♦ nk → ∞❀

✷✳ |fnk| ≤φ(x) q✳t✳♣✳ ❡♠ Ω✱ ♣❛r❛ t♦❞♦ k✱ ❝♦♠ φ ∈L

p_(Ω)✳

❚❡♦r❡♠❛ ✶✳✻✳ ❬✶❪ ❙❡ 1≤p < ∞ ❡ a✱ b ≥0✱ ❡♥tã♦ (a+b)p ≤2p−1(ap +bp).

❚❡♦r❡♠❛ ✶✳✼✳ (❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣)❬✷✵❪ ❙❡❥❛♠ 1< p, q <∞, 1_p + 1_q = 1✳ ❊♥tã♦

ab≤ a

p

p + bq

q , ♣❛r❛ t♦❞♦s a, b >0.

❚❡♦r❡♠❛ ✶✳✽✳ (❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r)❬✷✵❪ ❙✉♣♦♥❤❛ q✉❡ 1≤p, q ≤ ∞, 1_p+1_q = 1✳

❊♥tã♦✱ s❡ u∈Lp_{(Ω) ❡ v ∈L}q_{(Ω)✱ t❡♠♦s q✉❡}

Z

Ω

Pr❡❧✐♠✐♥❛r❡s ✶✺

P❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ♣r♦❝❡ss♦ ❞✐❛❣♦♥❛❧ ❡ ♦ ❛r❣✉♠❡♥t♦ ❜♦♦tstr❛♣✱ ❢❡✐t♦s ♥♦s ❈❛♣ít✉❧♦s ✷ ❡ ✸✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♥❡❝❡ss✐t❛♠♦s ❞❛s ✐♠❡rsõ❡s ❞❡ ❙♦❜♦❧❡✈ ❡ ❍ö❧❞❡r✳

❉❡✜♥✐çã♦ ✶✳✾✳ ❬✷✵❪ ❙❡❥❛♠ X ❡ Y ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ t❛✐s q✉❡ X ⊂ Y✳ ❉✐③❡♠♦s q✉❡ X

❡stá ✐♠❡rs♦ ❝♦♥t✐♥✉❛♠❡♥t❡ ❡♠ Y✱ ❡ ❡s❝r❡✈❡♠♦s X ֒→Y✱ ❞❡s❞❡ q✉❡

kxkY ≤CkxkX, ♣❛r❛ t♦❞♦ x∈X ❡ ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡C > 0.

❉✐③❡♠♦s q✉❡ X ❡stá ✐♠❡rs♦ ❝♦♠♣❛❝t❛♠❡♥t❡ ❡♠ Y✱ ❡ ❡s❝r❡✈❡♠♦s Xcpct֒→Y✱ ❞❡s❞❡ q✉❡ X ֒→Y ❡ ❝❛❞❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❡♠ X é ♣r❡❝♦♠♣❛❝t❛ ❡♠ Y✳

❚❡♦r❡♠❛ ✶✳✶✵✳ (■♠❡rsã♦ ❞❡ ❍ö❧❞❡r )❬✶❪ ❙❡❥❛♠ k ∈N_{∪ {0}} ❡ _{0< ν < γ ≤1}✱ ❡♥tã♦

✶✳ Ck,γ_(Ω)cpct_֒→Ck_(Ω)❀ ✷✳ Ck,γ_(Ω)cpct_֒→Ck,ν_(Ω)✳

❚❡♦r❡♠❛ ✶✳✶✶✳ (■♠❡rsã♦ ❞❡ ❙♦❜♦❧❡✈)❬✶✼❪ ❙❡❥❛♠ Ω ⊂ RN ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦

s❛t✐s❢❛③❡♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ❝♦♥❡✱ m >0✱ j ≥0 ❡ 1≤p <∞✳ ❊♥tã♦✱

✶✳ ❙❡ m < N p✱

Wj+m,p(Ω) ֒→Wj,q(Ω), ♣❛r❛ t♦❞♦ p≤q ≤ N p

N −mp;

✷✳ ❙❡ m = N p✱

Wj+m,p(Ω)֒→Wj,q(Ω), ♣❛r❛ t♦❞♦ p≤q <∞;

✸✳ ❙❡ m > N

p > m−1 ❡ Ω t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ▲✐♣s❝❤✐t③ ❧♦❝❛❧✱

Wj+m,p(Ω)֒→Cj,λ(Ω), ♣❛r❛ 0< λ≤m− N

p.

❖❜s❡r✈❛çã♦ ✶✳✶✷✳ ❯♠ ❞♦♠í♥✐♦ Ω s❛t✐s❢❛③ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ❝♦♥❡ s❡ ❡①✐st❡ ✉♠ ❝♦♥❡

❧✐♠✐t❛❞♦ K t❛❧ q✉❡ q✉❛❧q✉❡r x ∈ Ω é ♦ ✈ért✐❝❡ ❞❡ ✉♠ ❝♦♥❡ Kx ❝♦♥❣r✉❡♥t❡ ❛ K ❡ ✐♥t❡✐r❛♠❡♥t❡ ❝♦♥t✐❞♦ ❡♠ Ω✳

❚❡♦r❡♠❛ ✶✳✶✸✳ (■♠❡rsã♦ ❝♦♠♣❛❝t❛)❬✶✼❪ ❙❡❥❛♠ Ω ⊂ RN ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦

s❛t✐s❢❛③❡♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ❝♦♥❡✱ m≥1✱ j ≥0 ❡ 1≤p < ∞✳ ❊♥tã♦

✶✳ ❙❡ m < N p✱

Pr❡❧✐♠✐♥❛r❡s ✶✻

✷✳ ❙❡ m = N_p✱

Wj+m,p_(Ω)cpct_֒→Wj,q_{(Ω), ♣❛r❛ t♦❞♦ 1≤q <∞;} ✸✳ ❙❡ m > N_p > m−1 ❡ Ω t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ▲✐♣s❝❤✐t③ ❧♦❝❛❧✱

Wj+m,p_(Ω)cpct_֒→Cj,λ_{(Ω), ♣❛r❛ 0< λ < m−} N

p.

◆❡st❡ tr❛❜❛❧❤♦✱ ❝♦♥s✐❞❡r❛♠♦s ♦ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❧✐♥❡❛r ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠

L✱ t❡♥❞♦ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❢♦r♠❛s✿

Lu=−

N

X

i,j=1

aij_(x)u xixj+

N

X

i=1

bi_(x)u

xi +c(x)u, ✭✶✳✶✮

Lu=

N

X

i,j=1

(ai,j_(x)u xi +b

i_{(x)u) +} N

X

i=1

ci_(x)u

xi +d(x)u, ✭✶✳✷✮

♦♥❞❡ ♦s ❝♦❡✜❝✐❡♥t❡s aij_{, b}i_{, c: Ω−→Rsã♦ ❢✉♥çõ❡s ❞❛❞❛s✳}

❉❡✜♥✐çã♦ ✶✳✶✹✳ ❬✷✹❪ ❖ ♦♣❡r❛❞♦rL s❡rá ❞✐t♦ ♥❛ ❢♦r♠❛ ♥ã♦ ❞✐✈❡r❣❡♥t❡ s❡ é ❞❛❞♦ ♣♦r(1.1)

❡ ♥❛ ❢♦r♠❛ ❞✐✈❡r❣❡♥t❡ s❡ é ❞❛❞♦ ♣♦r (1.2)✳

❖❜s❡r✈❛çã♦ ✶✳✶✺✳ ❙❡ ♥ã♦ ❢♦r ✐♥❞✐❝❛❞♦ ♦ ❝♦♥trár✐♦✱ L ❡st❛rá ♥❛ ❢♦r♠❛ ♥ã♦ ❞✐✈❡r❣❡♥t❡✳

❉❡✜♥✐çã♦ ✶✳✶✻✳ ❬✷✹❪ ❉✐③❡♠♦s q✉❡ ♦ ♦♣❡r❛❞♦r L é ❡❧í♣t✐❝♦ ♥♦ ♣♦♥t♦ x ∈ Ω s❡ ❛ ❢♦r♠❛

q✉❛❞rát✐❝❛ ❛ss♦❝✐❛❞❛ à ♠❛tr✐③ A(x) = [(aij_{(x))]é ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛✱ ✐st♦ é✱ s❡ λ(x) ❞❡♥♦t❛} ♦ ♠❡♥♦r ❛✉t♦✈❛❧♦r ❞❡ A✱ ❡♥tã♦

n

X

ij=1

aij_(x)ξ

iξj ≥λ(x)|ξ|2 >0,

♣❛r❛ t♦❞♦ ξ = (ξ1, . . . , ξn) ∈ RN\{0}✳ ❖ ♦♣❡r❛❞♦r L é ❡❧í♣t✐❝♦ ❡♠ Ω s❡ ❢♦r ❡❧í♣t✐❝♦ ❡♠ ❝❛❞❛ ♣♦♥t♦ ❞❡ Ω✳ ❋✐♥❛❧♠❡♥t❡✱ ❞✐③❡♠♦s q✉❡ L é ✉♥✐❢♦r♠❡♠❡♥t❡ ❡❧í♣t✐❝♦ ❡♠ Ω s❡ ❡①✐st❡

θ0 >0 t❛❧ q✉❡ λ(x)≥θ0 ♣❛r❛ t♦❞♦ x∈Ω✳

❆❣♦r❛✱ ♣♦❞❡♠♦s ❛♣r❡s❡♥t❛r ❛❧❣✉♥s t❡♦r❡♠❛s ❝❧áss✐❝♦s q✉❡ ♥♦s ❛✉①✐❧✐❛rã♦ ♥❛ ♦❜t❡♥çã♦ ❞♦s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s ❞❡st❡ tr❛❜❛❧❤♦✳

▲❡♠❛ ✶✳✶✼✳ (▲❡♠❛ ❞❡ ❍♦♣❢)❬✷✹❪ ❙✉♣♦♥❤❛ q✉❡ B ⊂ RN é ✉♠❛ ❜♦❧❛ ❛❜❡rt❛✱ _L é ✉♠

♦♣❡r❛❞♦r ✉♥✐❢♦r♠❡♠❡♥t❡ ❡❧í♣t✐❝♦ ❡♠ B✱ u ∈C2_{(B) ❡ Lu ≥0 ❡♠ B ✳ ❙✉♣♦♥❤❛ ❛✐♥❞❛ q✉❡}

Pr❡❧✐♠✐♥❛r❡s ✶✼

(i) s❡ c= 0 ❡♠ B ❡ ❡①✐st❡ ❛ ❞❡r✐✈❛❞❛ ♥♦r♠❛❧ ∂u_∂η(x0)✱ ❡♥tã♦ ∂u_∂η(x0)>0✳

(ii) s❡ c≤0 ❡♠ B ❡ u(x0)≥0 ❡♥tã♦ ✈❛❧❡ ♦ ♠❡s♠♦ r❡s✉❧t❛❞♦ ❞♦ ✐t❡♠ ❛❝✐♠❛✳

❚❡♦r❡♠❛ ✶✳✶✽✳ ❬✷✹❪ ❙✉♣♦♥❤❛ q✉❡ u ∈ C2_{(Ω), f ∈ C}0,α_{(Ω) s❛t✐s❢❛③❡♠ ∆u =f ❡♠ Ω✳} ❊♥tã♦ u∈C2,α_(Ω)✳

❚❡♦r❡♠❛ ✶✳✶✾✳ (❆❣♠♦♥✱ ❉♦✉❣❧✐s✱ ◆✐r❡♥❜❡r❣)❬✻❪ ❙✉♣♦♥❤❛ q✉❡ Ω⊂RN é ✉♠ ❞♦♠í♥✐♦

❞❡ ❝❧❛ss❡ C2 _{❝♦♠ ∂Ω ❧✐♠✐t❛❞❛✱ f ∈L}p_{(Ω), 1< p <∞ ❡ u∈H}1

0(Ω) é s♦❧✉çã♦ ❢r❛❝❛ ❞❡

(

−∆u=f ❡♠ Ω, u= 0 ❡♠ ∂Ω.

❊♥tã♦ u∈W2,p_{(Ω) ❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C =C(Ω, p)>0 t❛❧ q✉❡}

kukW2,p_(Ω) ≤Ckfk_Lp_(Ω).

❚❡♦r❡♠❛ ✶✳✷✵✳ (❙❝❤❛✉❞❡r)❬✶✼❪ ❙✉♣♦♥❤❛ q✉❡ Ω⊂RN é ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❞❡ ❝❧❛ss❡

C2,γ_{✱ f ∈C}0,γ_{(Ω)✱ 0< γ <1 ❡ u∈H}1

0(Ω)∩C0,γ(Ω) é s♦❧✉çã♦ ❢r❛❝❛ ❞❡

(

−∆u=f ❡♠ Ω, u= 0 ❡♠ ∂Ω.

❊♥tã♦ u∈C2,γ_{(Ω) ❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C =C(Ω, γ)>0 t❛❧ q✉❡}

kukC2,γ_(Ω) ≤Ckfk_C0,γ_(Ω).

❚❡♦r❡♠❛ ✶✳✷✶✳ (❊st✐♠❛t✐✈❛ ✐♥t❡r✐♦r ❞❡ ❙❝❤❛✉❞❡r)❬✶✼❪ ❙❡❥❛ L ✉♠ ♦♣❡r❛❞♦r

✉♥✐❢♦r♠❡♠❡♥t❡ ❡❧í♣t✐❝♦ ❝♦♠

max{kaijk_C0,γ_(Ω), kbik_C0,γ_(Ω), kck_C0,γ_(Ω) :i, j = 1, . . . , n} ≤α.

❊♥tã♦ ♣❛r❛ Ω0, Ω1✱ ❝♦♠ Ω0 ⊂⊂Ω1✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C =C(N, γ, θ0, α)>0 t❛❧ q✉❡

kukC2,γ_(Ω

0) ≤C{kLukC0,γ(Ω1)+kukC(Ω1)}, ♣❛r❛ t♦❞♦ u∈C

2,γ_(Ω).

▲❡♠❛ ✶✳✷✷✳ (❊st✐♠❛t✐✈❛ ✐♥t❡r✐♦r Lp_{)❬✶❪ ❙❡❥❛♠ Ω}

0, Ω ❞♦♠í♥✐♦s ❧✐♠✐t❛❞♦s ❞❡RN ❝♦♠

Ω0 ⊂ Ω✳ ❙✉♣♦♥❤❛ q✉❡ L é ✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❧✐♥❡❛r ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠

Pr❡❧✐♠✐♥❛r❡s ✶✽

❝♦♥st❛♥t❡ K t❛❧ q✉❡

kwkW2,q_(Ω

0) ≤K(kLwkLq(Ω)+kwkLq(Ω)),

♣❛r❛ t♦❞♦ w∈W2,q_(Ω)✳

❖ t❡♦r❡♠❛ ❡♥✉♥❝✐❛❞♦ ❛ s❡❣✉✐r ❡♥❝♦♥tr❛✲s❡ ❡♠ ●✐❧❜❛r❣ ❡ ❚r✉❞✐♥❣❡r ❬✷✹❪ ✭❚❡♦r❡♠❛ ✽✳✶✾✮ ❡ s❡rá ✉t✐❧✐③❛❞♦ ♣❛r❛ ❞❡♠♦♥str❛r q✉❡ ❛ s♦❧✉çã♦ ♦❜t✐❞❛ ♥♦ ❚❡♦r❡♠❛ ✵✳✷ é ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈❛ ✭♦✉ ❡str✐t❛♠❡♥t❡ ♥❡❣❛t✐✈❛✮✳

❚❡♦r❡♠❛ ✶✳✷✸✳ ❬✷✹❪ ❙❡❥❛ L ✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ♥❛ ❢♦r♠❛ ❞✐✈❡r❣❡♥t❡ s❛t✐s❢❛③❡♥❞♦✿

✶✳ ❊①✐st❡♠ ❝♦♥st❛♥t❡s Λ ❡ ν t❛✐s q✉❡

N

X

i,j=1

|aij(x)|2 ≤Λ2, λ−2

N

X

i=1

(|bi(x)|2+|ci(x)|2) +λ−1|d(x)|2 ≤ν2,

✷✳ _Z

(d(x)v(x)−bi_v

xi) dx≤0, ♣❛r❛ t♦❞♦ v ≥0, v ∈C

1 0(Ω),

❡ u ∈ W1,2_{(Ω) s❛t✐s❢❛③❡♥❞♦ Lu ≥ 0 ❡♠ Ω✳ ❊♥tã♦✱ s❡ ♣❛r❛ ❛❧❣✉♠❛ ❜♦❧❛ B ⊂⊂ Ω t❡♠♦s}

q✉❡ sup

B

u(x) = sup

Ω

u(x)≥0✱ ❛ ❢✉♥çã♦ u é ❝♦♥st❛♥t❡ ❡♠ Ω✳

◆♦ss♦ ♣ró①✐♠♦ ♦❜❥❡t✐✈♦ é ❡♥✉♥❝✐❛r ♦ ❚❡♦r❡♠❛ ❞❛ ❡st✐♠❛t✐✈❛ ✐♥t❡r✐♦r ❣r❛❞✐❡♥t❡ ❞❡ ▲❛❞②③❡♥s❦❛②❛ ❡ ❯r❛❧✬ts❡✈❛ ❬✸✹❪ q✉❡ ♥♦s ♣❡r♠✐t❡ ♦❜t❡r✱ ♥♦ ❈❛♣ít✉❧♦ ✷ ❞❡st❡ t❡①t♦✱ ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡n♣❛r❛ ❛ s❡q✉ê♥❝✐❛ (|∇un|)✱ ♦♥❞❡ ❝❛❞❛un é s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛

(2) s♦❜r❡Bn✱ n≥1✳ P❛r❛ ✐st♦✱ ❝♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦ ♥❛ ❢♦r♠❛ ❞✐✈❡r❣❡♥t❡✿

d dxi

ai(x, u,∇u) +a(x, u,∇u) = 0, ✭✶✳✸✮ ♦♥❞❡ a(x, s, p) = a(x1, . . . , xN, s, p1, . . . , pN) ❡ ai(x, s, p) = ai(x1, . . . , xN, s, p1, . . . , pN)✱

i= 1, . . . , N✱ sã♦ ❢✉♥çõ❡s ❞❛❞❛s✳

❉❡✜♥✐çã♦ ✶✳✷✹✳ ❬✸✹❪ ❯♠❛ ❢✉♥çã♦u∈Wm,p_{(Ω) é ❝❤❛♠❛❞❛ s♦❧✉çã♦ ❧✐♠✐t❛❞❛ ❣❡♥❡r❛❧✐③❛❞❛} ❞❛ ❡q✉❛çã♦ (1.3)✱ ❞❡s❞❡ q✉❡ max

Ω

|u|<∞ ❡

Z

Ω

[ai(x, u,∇u)ηxi −a(x, u,∇u)η] dx= 0,

Pr❡❧✐♠✐♥❛r❡s ✶✾

❚❡♦r❡♠❛ ✶✳✷✺✳ (❊st✐♠❛t✐✈❛ ✐♥t❡r✐♦r ❣r❛❞✐❡♥t❡ ❞❡ ▲❛❞②③❡♥s❦❛②❛ ❡ ❯r❛❧✬ts❡✈❛)

❬✸✹❪ ❈♦♥s✐❞❡r❡ (1.3)✱ ♦♥❞❡ ❛s ❢✉♥çõ❡s a(x, u,∇u) ❡ ai(x, u,∇u), i = 1, . . . , N sã♦ ♠❡♥s✉rá✈❡✐s ♣❛r❛ x ∈Ω, u ❡ p ❛r❜✐trár✐♦s✱ ❡ ai(x, u,∇u) sã♦ ❞✐❢❡r❡♥❝✐á✈❡✐s ❝♦♠ r❡s♣❡✐t♦ ❛ x, u, p✳ ❆❧é♠ ❞✐ss♦✱ s❛t✐s❢❛③❡♠

✶✳ υ(|u|)(1 +|p|)m−2_|ξ|2 _≤ ∂ai(x, u, p)

∂pj

ξiξj ≤µ(|u|)(1 +|p|)m−2|ξ|2✱ ♣❛r❛ ξ∈R N

✳

✷✳ N

X

i=1

|∂ai

∂u|+|ai|

(1 +|p|) +

N

X

i,j+1

_∂x∂ai_j

≤µ(|u|)(1 +|p|)m✱

♦♥❞❡ m ≥ 1 ❡ υ, µ sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❞❡✜♥✐❞❛s ♣❛r❛ t ≥ 0 t❛✐s q✉❡ υ é ♣♦s✐t✐✈❛ ♥ã♦

❝r❡s❝❡♥t❡ ❡ µ ♥ã♦ ❞❡❝r❡s❝❡♥t❡✳ ❙❡❥❛♠ u ✉♠❛ s♦❧✉çã♦ ❣❡♥❡r❛❧✐③❛❞❛ ❧✐♠✐t❛❞❛ ❞❛ ❡q✉❛çã♦

(1.3) ❡♠ W1,2_{(Ω) ❡ Ω}

0 ⊂⊂Ω t❛✐s q✉❡

✸✳ Z

Ω0

(1 +|∇u|)m−2

N

X

i,j=1

u2xixj dx <∞✱

✹✳ Z

Ω0

|∇u|m+2_{dx <∞✳}

❊♥tã♦✱ ♦ max

Ω0

|∇u| é ❧✐♠✐t❛❞♦ ♣♦r ✉♠❛ ❡①♣r❡ssã♦ ❡♠ t❡r♠♦s ❞❡ max

Ω |u|✱ m✱ υ(maxΩ |u|)✱

µ(max

Ω |u|) ❡ ❛ ❞✐stâ♥❝✐❛ ❞❡ Ω0 ❛ ∂Ω✳

❊♥✉♥❝✐❛r❡♠♦s✱ ❛❣♦r❛✱ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❛✉t♦✈❛❧♦r ❞♦ ❧❛♣❧❛❝✐❛♥♦ ❝♦♠ ❝♦♥❞✐çã♦ ❞❡ ❢r♦♥t❡✐r❛ ❞❡ ❉✐r✐❝❤❧❡t✳

❚❡♦r❡♠❛ ✶✳✷✻✳ ❬✷✵❪ ❙❡❥❛♠ Ω⊂RN _{❝♦♠ ❢r♦♥t❡✐r❛ ❞❡ ❝❧❛ss❡ C}∞ _{❡ λ∈R✳ ❙❡ ϕ∈H}1 0(Ω)

é ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ₍

−∆ϕ =λϕ ❡♠ Ω,

ϕ= 0 ❡♠ ∂Ω, ✭✶✳✹✮

❡♥tã♦ ϕ ∈C∞_(Ω)✳

❚❡♦r❡♠❛ ✶✳✷✼✳ ❬✷✵❪✳

(i) ❖ ♣r♦❜❧❡♠❛ (1.4) ♣♦ss✉✐ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❛✉t♦✈❛❧♦r❡s

0< λ1 < λ2 ≤λ3· · ·, t❛❧ q✉❡ λk −→ ∞ q✉❛♥❞♦ k→ ∞. ❆❧é♠ ❞✐ss♦✱ λ1 = inf

v6=0

v∈H1 0(Ω)

kvk2

kvk2 2

Pr❡❧✐♠✐♥❛r❡s ✷✵

(ii) ❙❡ϕ1 é ✉♠❛ ❛✉t♦❢✉♥çã♦ ❛ss♦❝✐❛❞❛ ❛♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦rλ1❞♦ ♣r♦❜❧❡♠❛(1.4)✱ ❡♥tã♦

ϕ1 >0 ❡♠ Ω✳

❆♣r❡s❡♥t❛♠♦s✱ ❛❣♦r❛✱ ❛❧❣✉♥s r❡s✉❧t❛❞♦s s♦❜r❡ ❢✉♥❝✐♦♥❛✐s ❧✐♥❡❛r❡s ❡ ♦ ❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛✱ ❞❡✈✐❞♦ ❛ ❆♠❜r♦s❡tt✐✲❘❛❜✐♥♦✇✐t③ ❬✼❪✱ q✉❡ é ❛ ❢❡rr❛♠❡♥t❛ ❡ss❡♥❝✐❛❧ ❞♦ ❈❛♣ít✉❧♦ ✸ ❞❡st❡ tr❛❜❛❧❤♦✳

Pr♦♣♦s✐çã♦ ✶✳✷✽✳ ❬✹✽❪ ❙❡I t❡♠ ❞❡r✐✈❛❞❛ ❞❡ ●❛t❡❛✉① ❝♦♥tí♥✉❛ ❡♠U✱ ❡♥tã♦I ∈C1_(U,R)✳

❉❡✜♥✐çã♦ ✶✳✷✾✳ ❬✼❪ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳ ❉✐③❡♠♦s q✉❡ I ∈C1_{(X,R) s❛t✐s❢❛③ ❛}

❝♦♥❞✐çã♦ ❞❡ P❛❧❛✐s✲❙♠❛❧❡ ♥♦ ♥í✈❡❧ c((P S)c)✱ s❡ t♦❞❛ s❡q✉ê♥❝✐❛ (un)⊂X s❛t✐s❢❛③❡♥❞♦

lim

n→∞I(un) =c ❡ nlim→∞kI

′

(un)kX∗ = 0 ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡✳

❚❡♦r❡♠❛ ✶✳✸✵✳ (❚❡♦r❡♠❛ ❞♦ P❛ss♦ ❞❛ ▼♦♥t❛♥❤❛)❬✼❪ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤

❡ I ∈C1_{(X,R) t❛❧ q✉❡ I(0) = 0 ❡}

(I1) ❡①✐st❡♠ ρ✱ α >0 t❛✐s q✉❡ I(v)≥α✱ ♣❛r❛ t♦❞♦ v ∈∂Bρ(0);

(I2) ❡①✐st❡ ❡∈X t❛❧ q✉❡ k❡kX > ρ ❡ I(❡)≤0✳ ❙✉♣♦♥❤❛ q✉❡ I s❛t✐s❢❛ç❛ (P S)c✱ ❝♦♠

c:= inf

γ∈Γtmax∈[0,1]I(γ(t)),

♦♥❞❡ Γ :={γ ∈C([0,1], X) :γ(0) = 0 ❡ γ(1) =❡}✳ ❊♥tã♦ ❡①✐st❡ u6= 0 t❛❧ q✉❡ I(u) = c

❡ I′(u) = 0✳

Pr♦♣♦s✐çã♦ ✶✳✸✶✳ ❬✽❪ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳ ❙❡ xn ⇀ x ❢r❛❝❛♠❡♥t❡ ❡♠ X ❡ s❡

In−→I ❢♦rt❡♠❡♥t❡ ❡♠X∗✱ q✉❛♥❞♦ n→ ∞✱ ❡♥tã♦ hIn, xni −→ hI, xi✱ q✉❛♥❞♦ n → ∞✳ ❋✐♥❛❧♠❡♥t❡✱ t❡r♠✐♥❛♠♦s ❡st❡ ❝❛♣ít✉❧♦ ❡♥✉♥❝✐❛♥❞♦ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s té❝♥✐❝♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❈❛♣ít✉❧♦ ✷ ❞❡st❡ tr❛❜❛❧❤♦✳

▲❡♠❛ ✶✳✸✷✳ (❊①✐st❡♥❝❡✱ ●♦♥ç❛❧✈❡s ❡ ❙❛♥t♦s)❬✷✽❪ ❙❡❥❛ Ω⊂RN ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦

❝♦♠ ❢r♦♥t❡✐r❛ s✉❛✈❡✳ ❙✉♣♦♥❤❛ q✉❡ b ∈ C0,α_{(Ω) ❝♦♠ b(x) > 0✱ ♣❛r❛ t♦❞♦ x ∈ Ω✱ ❡}

g ∈C1_{((0,∞),(0,∞))s❛t✐s❢❛③}

✶✳ lim

s→0+ g(s)

s =∞ ❡ slim→∞

g(s)