一类分数阶薛定谔－泊松系统非平凡解的存在性 Existence of Nontrivial Solution for a Class of Fractional Schro¨dinger-Poisson System

doi:10.12677/AAM.2023.124177

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AdvancesinAppliedMathematicsA^êÆ?Ð,2023,12(4),1704-1712

PublishedOnlineApril2023inHans.https://www.hanspub.org/journal/aam

https://doi.org/10.12677/aam.2023.124177

˜a©êÅ½™õÑtXÚš²…)

•35

ŠŠŠïïï___

=²nóŒÆnÆ§[‹=²

ÂvFÏµ2023c324F¶¹^FÏµ2023c418F¶uÙFÏµ2023c427F

Á‡

©ïÄ˜aäkCÒ©êÅ½™õÑtXÚ











−(∆)

u+u+k(x)φu= a(x)|u|

p−1

u,x∈R

−(∆)

φ= k(x)u

,x∈R

š²…)•35,Ù¥

3s+4t

s+t

<p<

3+2s

3−2s

,s,t∈(0,1)…4s+ 2t>3,a(x)∈C(R

)CÒ…

lim

|x|→∞

a(x)=a

∞

<0,k(x)∈C(R

)∩L

4s+2t−3

).A^ì´Ún,©TXÚ–•3

˜‡š²…).

'…c

©êÅ½™õÑtXÚ§CÒ§š²…)

ExistenceofNontrivialSolutionforaClass

ofFractionalSchr¨odinger-PoissonSystem

JuanxiaMeng

©ÙÚ^:Šï_.˜a©êÅ½™-ÑtXÚš²…)•35[J].A^êÆ?Ð,2023,12(4):1704-1712.

DOI:10.12677/aam.2023.124177

Šï_

CollegeofScience,LanzhouUniversityofTechnology,LanzhouGansu

Received:Mar.24

,2023;accepted:Apr.18

,2023;published:Apr.27

,2023

Abstract

Inthispaper,weareconcernedwiththeexistenceofnontrivialsolutionforaclassof

fractionalSchr¨odinger-Poissonsystem:











−(∆)

u+u+k(x)φu= a(x)|u|

p−1

u,x∈R

−(∆)

φ= k(x)u

,x∈R

where

3s+4t

s+t

<p<

3+2s

3−2s

,s,t∈(0,1)and4s+2t>3,a(x) ∈C(R

)isasign-changingfunction

withlim

|x|→∞

a(x) = a

∞

<0,k(x) ∈C(R

)∩L

4s+2t−3

).Byusingmountainpasstheorem,

weobtainthatthissystemhasatleastonenontrivialsolution.

Keywords

FractionalSchr¨odinger-PoissonSystem,Sign-ChangingWeight,NontrivialSolution

2023byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).

http://creativecommons.org/licenses/by/4.0/

1.Úó

Cc5,Nõ©z•ÄXe©êÅ½™õÑtXÚ











(−∆)

u+V(x)u+K(x)φu= f(x,u),x∈R

(−∆)

φ= K(x)u

,x∈R

(1)

DOI:10.12677/aam.2023.1241771705A^êÆ?Ð

Šï_

d?s,t∈(0,1),©êLaplacianŽf(−∆)

½Â•

(−∆)

v(x) = C

N,s

P.V.

v(x)−v(y)

|x−y|

N+2s

dy,v∈S(R

d?P.V.L«…ÜÌŠ,C

N,s

IOz~ê,S(R

) ´d¯„P~¼ê|¤–]¼ê˜m.5

¿©êLaplacian Žf(−∆)

´3©z[1,2]ÄgÚ\.•õk'(−∆)

&E,žë•[3]9

Ù¥ë•©z.c,k'XÚ(1)?ØýŒõê´k')½CÒ)•35,õ)5(

J,X[4–15].Ù¥,©z[5,6,12,14]•ÄXÚ(1) ½aq¯KCÒ)•35;©z[4,10]ïÄ

XÚ(1) 3.^‡epUþ)•35;©z[7,13]?ØXÚ(1) )•3598¥5;©

z[8,9,11,15]•XÚ(1) Ä)•35½õ-5.,,ÏLÎnƒ'©zuy:éäkC

Ò©êÅ ½™-ÑtXÚ)•35¯KïÄ%š~.,˜•¡,·‚5¿,{¡Ÿ

3[16]¥•Ä±eÅ½™-ÑtXÚ











−∆u+u+φu= a(x)|u|

p−1

u,x∈R

−∆φ= k(x)u

,x∈R

(2)

Ù¥3≤p<5,a(x)∈C(R

) CÒ…lim

|x|→∞

a(x)=a

∞

<0,k(x)∈C(R

)∩L

).|^ì´Ú

n[17],Šö¼Å½™-ÑtXÚ(2)–k˜‡š²…)•35(J.É±þ©zéu,

©•ÄXe©êÅ½™-ÑtXÚ











−(∆)

u+u+k(x)φu= a(x)|u|

p−1

u, x∈R

−(∆)

φ= k(x)u

, x∈R

(3)

š²…)•35,Ù¥

3s+4t

s+t

<p<

3+2s

3−2s

,s,t∈(0,1) …4s+2t>3,a(x) Úk(x) ÷vµ

(A1)a(x) ∈C(R

,R)•˜aCÒ¼ê…÷va

∞

= lim

|x|→∞

a(x) <0.

(A2)k(x) ∈C(R

,R),k(x) ≥0 …k(x) ∈L

4s+2t−3

©Ì‡(J•Xe½n:

½n1.1.b(A1),(A2)¤á,K©êÅ½™-ÑtXÚ(3) –•3˜‡š²…).

2½½½nnn1.1yyy²²²

éu½u∈H

),½ÂD

t,2

)þ‚5Žf

(v) =

k(x)u

vdx,

DOI:10.12677/aam.2023.1241771706A^êÆ?Ð

Šï_

@oÒk

(v)|≤

k(x)u

|v|dx

≤(

k(x)

4s+2t−3

dx)

4s+2t−3

(

3−2s

)

3−2s

(

∗

)

∗

≤Ckk(x)k

4s+2t−3

)

kuk

)

kvk

t,2

)

ÏddRieszL«½nŒ•,•3•˜φ

∈D

t,2

),¦

hφ

,vi

t,2

)

= L

(v), ∀v∈D

t,2

(−∆)

vdx=

k(x)u

vdx, ∀v∈D

t,2

•Ò´`φ

´(3)¥1‡•§f).òφ

‘\1˜‡•§Ò

−(∆)

u+u+k(x)φ

u= a(x)|u|

p−1

u.(4)

Ïd¦)•§(3)du¦)•§(4).•§(4))éAUþ•¼

F(u) =

|(−∆)

dx+

k(x)φ

−

p+1

a(x)|u|

p+1

dx, u∈H

)

.:.

½ÂΦ:H

) →D

t,2

)•Φ(u) = φ

,Kke¡Ún.

Ún2.1

(i)Φ ëY.

(ii)Φ òk.8Nk.8.

(iii)e{u

}3H

)¥k.…u

u,@ok

k(x)φ

dx→

k(x)φ

dx.

y:(i )é¤ku∈H

),k

|= kφ

t,2

)

= kΦ(u)k

t,2

)

DOI:10.12677/aam.2023.1241771707A^êÆ?Ð

Šï_

¤á.¤±,•y²Φ ëY,•Iy:u7→L

´ëY.

(v)−L

(v)|≤

k(x)|v||u

−u

|dx

≤Ckkk

4s+2t−3

)

kvk

t,2

)

−u

3−2s

)

Ï•v´?¿,n→∞,k|L

(v)−L

(v)|→0 .

(ii)Ï•kφ

t,2

)

= |L

|≤Ckkk

4t+2s−3

)

kuk

)

.¤±Ún¥(ii)¤á.

(iii)3©z[16]¥®²y²φ

φ

,e¡y²

k(x)φ

dx→

k(x)φ

dx.

Äk·‚5¿duφ

φ

,¤±·‚k

k(x)(φ

−φ

dx→0.(5)

e¡y²

k(x)φ

−u

|dx→0.(6)

Ï•k(x)∈L

4s+2t−3

) …ëY,Ïdé∀ε>0,∃ρ= ρ(ε)>0,¦|x|≥ρžk(x)≤ε,u´

·‚

|x|≥ρ

k(x)φ

−u

|dx≤ε[

|x|≥ρ

dx+

|x|≥ρ

dx]

≤εCkφ

t,2

)

(ku

+kuk

)

≤Cε.

,˜•¡,bs,t∈(0,1),*XJ4s+ 2t>3,Kk2≤

3+2t

3−2s

,ÏdH

)→

3+2t

).k:

|x|≤ρ

k(x)φ

−u

|dx

≤kkk

∞

)

[

|x|≤ρ

−u

|dx]

≤kkk

∞(R

)

Ckφ

t,2

)

[

|x|≤ρ

−u

3+2t

dx]

3+2t

→0.

ùÒy²(6),nÜ(5)Ú(6)·‚

k(x)φ

dx→

k(x)φ

dx.Úny..

DOI:10.12677/aam.2023.1241771708A^êÆ?Ð

Šï_

Ún2.2.•¼F÷v(PS)^‡.

y:u

⊂H

)÷v|F(u

)|≤cÚF

) →0 ,@ok

|(−∆)

dx+

k(x)φ

dx−

a(x)|u

p+1

dx= o(1)ku

k(7)

|(−∆)

dx+

k(x)φ

dx−

p+1

a(x)|u

p+1

dx≤c.

|^þ¡üª?˜Ú

[

−

p+1

]

|(−∆)

dx+[

−

p+1

]

k(x)φ

≤c+o(1)ku

Ï•p>

3s+4t

s+t

,¤±k

[

−

p+1

]

|(−∆)

dx≤c+o(1)ku

dþª•u

k..Ïd·‚Œ±b½u

u,e¡y²u

→u.•Iy²ku

k→kuk.d•§(7)

·‚k

|(−∆)

dx+

k(x)φ

−

(x)|u

p+1

(x)|u

p+1

dx+o(1).

dulim

|x|→∞

a(x) = a

∞

<0 ,Ïda

(x)k;|8,|^¢ËÅi\½n•

(x)|u

p+1

dx=

(x)|u|

p+1

dx+o(1).

¤±

|(−∆)

dx+

k(x)φ

dx+

−

(x)|u

p+1

|(−∆)

dx+

k(x)φ

dx+

−

(x)|u|

p+1

dx+o(1).(8)

·‚äó

−

(x)|u|

p+1

dx≤liminf

n→∞

−

(x)|u

p+1

dx.

DOI:10.12677/aam.2023.1241771709A^êÆ?Ð

Šï_

¯¢þd

−

(x)||u

p+1

−|u|

p+1

−|u

−u|

p+1

|dx

≤ka(x)k

∞

)

||u

p+1

−|u|

p+1

−|u

−u|

p+1

|dx

= o(1)

•

−

(x)|u

p+1

dx=

−

(x)|u|

p+1

dx+

−

(x)|u

−u|

p+1

dx+o(1),

•Ò´äó¤á.

XJu

9u,@oku

k<kuk,2d

k(x)φ

dx→

k(x)φ

dxÚ

−

(x)|u|

p+1

dx≤

liminf

n→∞

−

(x)|u

p+1

dx.•

|(−∆)

dx+

k(x)φ

dx+

−

(x)|u

p+1

|(−∆)

dx+

k(x)φ

dx+

−

(x)|u|

p+1

dx+o(1),

ù†(8)ªgñ.Ïdu

→u,y..

½n1y²:Äky²•3α,ρ>0 ¦F

∂B

>α>0.¯¢þd¢ËÅi\½nk

F(u) ≥

kuk

−Ckak

∞

kuk

p+1

)

dþªŒ••3ρ>0 ,¦F

∂B

>α>0 .

·‚2y²•3η∈H

) ,kηk>ρ,¦F(η)<0 ,¯¢þ,À¼êϕ

=θ

s+t

ϕ(θx)∈

),θ∈R

,ϕ

6= 0,¦suppϕ

⊂suppa

,@oÒk

F(θ

s+t

ϕ(θx)) ≤

4s+2t−3

kϕk

+kkk

∞

4s+2t−3

dx−

(p+1)(s+t)−3

p+1

|ϕ|

p+1

dx.

Ï•p>

3s+4t

s+t

,¤±(p+1)(s+t)−3 >4s+2t−3.θ→+∞ž,F(θ

s+t

ϕ(θx)) →−∞.Ïd,

é,‡θ

¿©Œž,-η= ϕ

= θ

s+t

ϕ(θ

x),¤±kF(η) <0.½Â

Γ = {γ∈C([0,1],H

)) : γ(0) = 0,γ(1) = η},

c=inf

γ∈Γ

sup

θ∈[0,1]

F(γ(θ)),

…).

DOI:10.12677/aam.2023.1241771710A^êÆ?Ð

Šï_

3(((ØØØ

´é˜‡k.(PS)S,¦TS÷v(PS)^‡,ddŒ±y²š²…)•35.

ë•©z

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