一类拟线性SchrO¨dinger方程正解的存在性 Existence of Positive Solutions for Quasilinear Schr¨dinger Equations

doi:10.12677/PM.2023.133072

设为首页加入收藏期刊导航网站地图

期刊菜单

文章导航

PureMathematicsnØêÆ,2023,13(3),669-682

PublishedOnlineMarch2023inHans.https://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2023.133072

˜a[‚5Schr¨odinger•§)•35

±±±¯¯¯

þ°nóŒÆnÆ§þ°

ÂvFÏµ2023c222F¶¹^FÏµ2023c323F¶uÙFÏµ2023c330F

Á‡

©?ØXe˜a[‚5Schr¨odinger•§

−∆u+V(x)u+

∆(u

)u= f(u),x∈R

Ù¥V(x):R

→R• ³¼ê,γ>0,…N≥3.γ∈(0,γ

)ž,·‚þã¯K

).d ³¼êV(x) ≡V

∞

>0,·‚3H

)∩C

)þy²²;»•)u

•35,

…γ→0

ž,÷vu

→u

,Ù¥u

´±eŒ‚5¯KÄ):

−∆u+V

∞

u= f(u),x∈R

'…c

[‚5Schr¨odinger•§§C©•{§L

∞

-O§MorseS“

ExistenceofPositiveSolutionsfor

QuasilinearSchr¨odingerEquations

MinZhou

CollegeofScience,UniversityofShanghaiforScienceandTechnology,Shanghai

Received:Feb.22

,2023;accepted:Mar.23

,2023;published:Mar.30

,2023

©ÙÚ^:±¯.˜a[‚5Schr¨odinger•§)•35[J].nØêÆ,2023,13(3):669-682.

DOI:10.12677/pm.2023.133072

±¯

Abstract

ThispaperfocusesonaclassofquasilinearSchr¨odingerequations:

−∆u+V(x)u+

∆(u

)u= f(u),x∈R

whereV(x) : R

→Risagivenpotential,γ>0andN≥3.Firstly,weobtainapositive

solutionfortheaboveprobleminγ∈(0,γ

).ThepotentialfunctionV(x)isconsidered

inV(x) ≡V

∞

>0.Weprovetheexistenceofapositiveclassicalradialsolutionu

and

uptoasubsequence,u

→u

inH

) ∩C

)asγ→0

,whereu

istheground

stateofthefollowingsemilinearproblem:

∆u+V

∞

u= f(u),x∈R

Keywords

QuasilinearSchr¨odingerEquations,VariationalMethods,L

∞

-Estimate,Mores

Iteration

2023byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).

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

1.Úó

©ïÄXe[‚5Schr¨odinger•§7Å)•35:

i∂

z= −∆z+W(x)z−l(|z|

)z+

[∆ρ(|z|

)]ρ

(|z|

)z,x∈R

(1)

Ù¥z:R

×R→C,W:R

→R´˜‡‰½ ³,γ´˜‡¢~ê,l,ρ•¢¼ê.¯¤±•,

•§(1)7Å)÷ve[‚5Schr¨odinger•§

−∆u+V(x)u+

[∆ρ(|u|

)]ρ

(|u|

)u= f(u),x∈R

,(2)

DOI:10.12677/pm.2023.133072670nØêÆ

±¯

Ù¥V(x) = W(x)−E,f(t) := l(|t|

)t•#š‚5‘./X(2)[‚5Schr¨odinger•§®2•A

^uÔnõ‡+•(„[1]).~Xρ(t) = 1ž,•§òz•XeŒ‚5œ¹:

−∆u+V(x)u= l(u),x∈R

3L30cp,þã•§2•ïÄ,ë„©z[2,3].ρ(t) = tž,=

−∆u+V(x)u+

∆(u

)u= f(u),x∈R

,(3)

§[lfNÔnÆ¥‡6N•§;ρ(t) =(1+t)

1/2

ž,þã•§ïápõÇ‡á

-1ì3ÔŸ¥gÏ•§,ë„©z[4,5].

3LA›cp,kNõÆö|^C©•{,Œþ'u•§(2))•35Úõ-

5(J.γ<0ž,˜•¡ÏLå•zy²(3))•35.ù•¡¤JŒ±ë„©

z[6–8].,˜•¡,~êγ“LØÓÔnA,ÏdïÄ|γ|→0ž,Ä)•359ÙìC1

•´›©-‡.3©z[1]¥,AdachiïÄ±e•§:

−∆u+λu+

∆(u

)u= |u|

p−2

u,x∈R

,(4)

(JL²,3˜mH

)∩C

)þ,•§(4)Ä)u

÷vγ→0

−

ž,ku

→u

,Ù¥u

Xe•§•˜Ä):

−∆u+λu= |u|

p−2

u,x∈R

3©z[9]¥§ly²3p∈(2,4)œ¹e•§(4))ìC1•,õAdachi<

3[1]¥‰Ñ(J.3©z[10–13]¥Adachi<A^»©ÛÚC©•{,š‚5‘ä

.O•ÚH

‡.O•ž,•§(4)Ä)ìC1•.

'u¯K(3)ïÄÌ‡8¥3γ<0.éuγ>0,)•35(J„'.5¿,γ>0,

·‚ØU†^[6]¥CþC†{ïÄ¯K.•ŽÑù˜(J§ÏLC©•{(Ü?E|,

Alves3©z[7]¥y²e•§)•35

−∆u+V(x)u+

∆(u

)u= |u|

p−2

u,x∈R

,(5)

Ù¥2 <p<2

∗

,γ>0•v¢ê.3©z[14]¥,lïá´k.O•š‚5‘[‚

5Schr¨odinger•§)•35.éuγ>0Úγ→0

œ¹e,•§(5))ìC1•ÿ™•

Ä.

©Ì‡óŠ´éγ>0ïÄ(3)Ä)•35.

©òïÄ±eäk˜„š‚5üëêý•§)

−∆u+V(x)u+

∆(u

)u= f(u),x∈R

(6)

•35,Ù¥N≥3,γ>0,V(x) ∈C(R

,R),…÷v:

)x∈R

ž,V(x) ≥V

>0;

DOI:10.12677/pm.2023.133072671nØêÆ

±¯

)lim

|x|→∞

V(x) = V

∞

…V(x) ≤V

∞

éuš‚5‘f,f∈C

),¿÷v

)éut≤0,kf(t) = 0,…•3p∈(2,2

∗

),¦

|f(t)|≤C(1+|t|

p−1

Ù¥2

∗

N−2

•.•ê;

)

lim

t→0

f(t)

= 0;

)•3θ>2,¦éut>0,¤á

0 <θF(t) ≤tf(t),

Ù¥F(t) =

f(s)ds.

•§(6)éAg,•¼I•

I(u) =

(1−γu

)|∇u|

dx+

V(x)u

dx−

F(u)dx,

Ù3˜mH

)¥ŒUØ´û½Â,¤±©Ì‡(J´:Äk,duI(u)3H

)¥

ŒUØ´û½Â,ÏdUþ•¼I(u)Ø´1w,•5Ã¯K´XÛyÌÜ©5§

=1−γu

>0;Ùg,3‡˜mR

¥,;—5˜„Ø¤á.

É©z[7]Ú[14]éu,·‚é•¼I?1±e?

−div(g

(u)∇u)+g(u)g

(u)|∇u|

+V(x)u= f(u),x∈R

,(7)

Ù¥g(t)=

1−γt

,γ>0,k|t|<

√

3γ

.d(7)ªk˜‡)u

,,ÏLMorseS“f)

˜—L

∞

-O(†γÃ').·‚òy²,γvž,u´¯K(6)).Äk|^MorseS“,‰

Ñu

-‰ê˜—k.5,ƒA^u

˜—O5y²Ù3˜mH

)þrÂñ5.˜

·‚UÙ3˜mH

þÂñ5, ·‚ÒŒ±A^ýO5u

3˜mH

)∩C

)þ

Âñ5.

©Ì‡(JXe

½½½nnn1.1b^‡(V

), (V

)9(f

)−(f

)¤á, K•3γ

>0 ,¦¯K(6) éu¤kγ∈[0,γ

)Ñ

•3)u

,…•3†γÃ'~êC>0,¦

∞

≤C.

2.ý•£

É©z[14]éu,·‚ïÄ±e[‚5Schr¨odinger•§

−div(g

(u)∇u)+g

(u)g

(u)|∇u|

+V(x)u= f(u),x∈R

(8)

DOI:10.12677/pm.2023.133072672nØêÆ

±¯

Ù¥g

(t) : [0,+∞) →R,½ÂXe

(t) =

(

1−γt

,if0 ≤t<

√

3γ

√

2γt

√

,ift≥

√

3γ

éut<0,kg

(t) = g

(−t),g

(t) ∈C

(R,(

1/6,1]),…3[0,+∞)þü~.

G

(t)=

(s)ds.d½Â,G

(t)´‰½Û¼ê,…•3‡¼êG

−1

(t).N´y²,

−1

(t)äkXe5Ÿ.

ÚÚÚnnn2.1

(1)lim

t→0

−1

(t)

= 1;

(2)lim

t→+∞

−1

(t)

√

(3)éut≥0,kt≤G

−1

(t) ≤

√

6t;

(4)éut≥0,k−

≤

(t)

(t) ≤0.

´••§(8)Uþ•¼•

I(u) =

(u)|∇u|

dx+

V(x)|u|

dx−

F(u)dx.(9)

·‚•{´y²•§(9)š²….:u…|u(x)|

∞

√

3γ

•35,§´•§(9)š²…).e

÷v|u(x)|

∞

√

3γ

,uB••§(6)˜‡š²…).

ƒ,XeCþ

v= G

(u) =

(s)ds,

•§(9)Uþ•¼Œ±U¤Xe

(v) =

|∇v|

dx+

V(x)|G

−1

(v)|

dx−

F(G

−1

(v))dx.

dÚn2.1Ú^‡(V

),(V

)9(f

) −(f

),·‚Œ•§J

(v)3˜mH

)þ´û½Â,J

∈

),R)…éuv∈H

(v)ϕ=

∇v∇ϕdx+

V(x)

−1

(v)

−1

(v))

ϕdx−

f(G

−1

(v))

−1

(v))

ϕdx.

ÚÚÚnnn2.2ev∈H

)´J

.:,Ku= G

−1

(v) ∈H

)…u´•§(8)f).

yyy²²²bv∈H

)´J

.:,dÚn2.1Œ±†OŽu=G

−1

(v)∈H

).éu¤

kϕ∈H

),k

∇v∇ϕdx+

V(x)

−1

(v)

−1

(v))

ϕdx−

f(G

−1

(v))

−1

(v))

ϕdx= 0,

DOI:10.12677/pm.2023.133072673nØêÆ

±¯

ϕ= g

(u)ψÙ¥ψ∈C

∞

),Kk

∇v∇ug

(u)ψdx+

∇v∇ψg

(u)dx+

V(x)uψdx−

f(u)ψdx= 0,



−div(g

(u)∇u)+g

(u)g

(u)|∇u|

+V(x)u−f(u)



ψdx= 0.

nþ,¦•§(8)š²…)•35,•I‡ïÄXe•§š²…)•35

−∆v+V(x)

−1

(v)

−1

(v))

f(G

−1

(v))

−1

(v))

,x∈R

.(10)

3.½n1.1y²

½ÂH

)˜m‰ê•

kuk=



(|∇u|

)dx



1/2

9Xe•§

−∆v+V

∞

−1

(v)

−1

(v))

f(G

−1

(v))

−1

(v))

(11)

ÙéAUþ•¼Xe

∞

(v) =



|∇v|

∞

−1

(v)|



dx−

F(G

−1

(v))dx.

½Â

∞

= inf{J

∞

(v)|v∈H

)\{0},J

∞

(v) = 0}.

•y²þã¯K,·‚Äk£Berestycki-Lions[2]ÚColin-Jeanjean[6]éXe•§A‡

²;(Ø

−∆v= k(v),x∈R

.(12)

•§(12)éAUþ•¼•

J(v) =

|∇v|

dx−

K(v)dx,

Ù¥K(s) =

k(t)dt.

k(s)÷v±e^‡ž

DOI:10.12677/pm.2023.133072674nØêÆ

±¯

)k(s) ∈C(R,R);

)−∞<liminf

s→0

k(s)

≤limsup

s→0

k(s)

= −C<0;

)lim

s→+∞

|k(s)|

∗

−1

= 0;

)•3s

>0,¦K(s

) >0,

Œ±±e·K(„[2,15,16]).

···KKK3.1b=inf{J(w)|w••§(12)š²…f)}.e(k

)−(k

)¤á,Kb>0…¼

êw∈H

).éu?¿x∈R

,•3´»η∈C([0,1],H

)),¦η(t)(x) >0…

max

t∈[0,1]

J(η(t)) = J(w).

Ïdd•§(11)9−∆v= k(v)Œ

k(s) =

f(G

−1

(s))

−1

(s))

−V

∞

−1

(s)

g(G

−1

(s))

´k(s)÷v(k

) −(k

).Ïd,•3w

∞

∈H

)¦d

∞

>0.d,·‚Œ±é´»η∈

C([0,1],H

)),éu¤kx∈R

,t∈(0,1]…η(0) = 0,kJ

∞

(η(1)) <0,w

∞

∈η([0,1])9

max

t∈[0,1]

∞

(η(t)) = J

∞

•ïÄJ

,½Â

=inf

η∈Γ

max

t∈[0,1]

(η(t)),

Ù¥Γ = {η∈C([0,1],H

))|η(0) = 0,J

(η(1)) <0}.

e5òyJ

ì´½nAÛ^‡,¿y²PSSk.5.

ÚÚÚnnn3.1b(f

)−(f

),(V

)9(V

)¤á.K•3ρ>09e∈H

),¦éukvk= ρ,k

(v) >0,

…J

(e) <0

yyy²²²dÚn2.1,(V

),(f

)9Sobolevi\,k

(v) =

|∇v|

dx+

V(x)|G

−1

(v)|

dx−

F(G

−1

(v))dx

≥

|∇v|

dx+

V(x)|v|

dx−



ε|G

−1

(v)|

−1

(v)|



≥min{1,V

}



|∇v|

+|v|



dx−C

|v|

≥min{1,V

}

kvk

−Ckvk

Ïd,p>2ž,J

(v)3v= 0?kÛÜ•Š.

DOI:10.12677/pm.2023.133072675nØêÆ

±¯

,˜•¡,d^‡(f

)Œ,éut>0k

F(t) ≥Ct

Ù¥w∈C

∞

),supp(w) = B

…w(x) ≥0,

(tw) =

|∇w|

dx+

V(x)|G

−1

(tw)|

dx−

F(G

−1

(tw))dx

|∇w|

dx+

V(x)|G

−1

(tw)|

dx−

F(G

−1

(tw))dx

≤

|∇w|

dx+

V(x)|G

−1

(tw)|

dx−

−1

(tw))

−C

)dx

≤

|∇w|

dx+3t

∞

|w|

dx−Ct

|w|

dx+C.

duθ>2,t→∞ž,kJ

(tw) →−∞

ÚÚÚnnn3.2(f

)−(f

),(V

)9(V

)¤á.KJ

PSS´k..

yyy²²²{v

}⊂H

)•J

PSS,=

) =

|∇v

dx+

V(x)|G

−1

−

F(G

−1

))dx= d+o

(1).

(13)

Kéu?¿ϕ∈H

),hJ

),ϕi= o

(1)kϕk,=



∇v

∇ϕ+V(x)

−1

)

−1

))



dx−

f(G

−1

))

−1

))

ϕdx= o

(1)kϕk.(14)

ϕ= G

−1

)),dÚn2.1-(4)Œ

|∇(G

−1

)))|≤



−1

)

−1

))

−1

))



|∇v

|≤|∇v

|.(15)

,˜•¡,dÚn2.1-(3),·‚k

−1

))|≤

√

6|v

|.(16)

(Ü(15)Ú(16)Œ

−1

))k≤

√

6kv

·‚Œ±ÀJϕ= G

−1

)).ÒkhJ

),G

−1

))i= o

(1)kv

k·‚k

DOI:10.12677/pm.2023.133072676nØêÆ

±¯

(1)kv



−1

)

−1

))

−1

))



|∇v

V(x)|G

−1

dx−

f(G

−1

))G

−1

)dx

≤

|∇v

dx+

V(x)|G

−1

dx−

f(G

−1

))G

−1

)dx.

(17)

Ïd,d(f

),(13),(14),(17)9Ún2.1,·‚Œ

θd+o

(1)+o

(1)kv

k= θJ

)−hJ

),G

−1

))i

≥

θ−2

|∇v

dx+

θ−2

V(x)|G

−1

≥

θ−2

min{1,V

}kv

=kv

k˜—k..

/Ï©z[17]¥(Ø,·‚J

PSSk.±eL«.

···KKK3.2(f

) −(f

),(V

)9(V

)¤á,{v

}⊂H

)•J

PSS.K•3{v

}fS

(•^{v

}L«).éuêm∈N∪{0},S{y

}⊂R

∈H

(1)éui6= j,n→∞ž,k|y

|→∞…|y

−y

|→∞,

(2)3˜mH

)þ,kv

) = 0,

(3)éu0 ≤i≤m,kw

6= 0…(J

∞

)

) = 0,

(4)kv

−v

−

i=0

(·−y

)k→0,

(5)J

) →J

i=0

∞

•|^·K3.2J

PSS;5,·‚I‡y²±eÚn.

ÚÚÚnnn3.3(f

)−(f

),(V

)9(V

).Kd

∞

yyy²²²Šâ·K3.1,•3w

∞

∈H

)¦J

∞

) = d

∞

>0.·‚Œ±é´»η∈C([0,1],H

))§

¦éu¤kx∈R

9t∈(0,1],η(0) = 0,J

∞

(η(1)) <0,w

∞

∈η([0,1]),kη(t)(x) >0…

max

t∈[0,1]

∞

(η(t)) = J

∞

Ù¥½:w

∞

…η,éu?¿t∈(0,1],k

(η(t)) <J

∞

(η(t)),

ddŒ

d≤max

t∈[0,1]

(η(t)) <max

t∈[0,1]

∞

(η(t)) = d

∞

ùÒ¤y².

e5(ÜÚn3.3Ú·K3.2,y²J

kì´.:.

ÚÚÚnnn3.4b^‡(f

)−(f

),(V

)9(V

)¤á.KJ

k.:.

yyy²²²dÚn3.1Ú3.2§•3˜‡k.PSS{v

}⊂H

).d·K3.2•3m∈N∪{0}Úv

∈

DOI:10.12677/pm.2023.133072677nØêÆ

±¯

),¦3˜mH

)þk

) = 0

…

) →J

i=1

∞

Ù¥{w

}

i=1

´J

∞

š²….:.

) <0,@BØ^y².eJ

) ≥0,K

=lim

n→∞

) = J

i=1

∞

) ≥md

∞

dÚn3.3,Œm= 0.Ïdd·K3.2-(4),·‚Œ

→v

inH(R

ÚÚÚnnn3.5(f

)−(f

),(V

)9(V

)¤á,v

•J

÷vJ

)=d

˜‡.:.K•3C>0

(†γÕá),¦

≤Cd

.(18)

yyy²²²Šâ^‡(f

),·‚Œ±±eO

θd

= θJ

)−hJ

),G

−1

))i

|∇v

dx+

V(x)|G

−1

dx−θ

F(G

−1

))dx

−

∇v

∇



−1

))



dx−

V(x)|G

−1

f(G

−1

))G

−1

)dx

≥

θ−2

|∇v

dx+

θ−2

V(x)|G

−1

≥

θ−2

min{1,V

}kv

ù`²kv

≤Cd

d,•y²kv

-‰ê˜—O.•ÄXeUþ•¼

∞

(v) = 3



|∇v|

∞



dx−C

|v|

dx+C

Ù¥θ∈(2,2

∗

),·‚^c

∞

L«Uþ•¼P

∞

ì´Y²†γÕá.duJ

(v)≤P

∞

(v),•ÄXe8

DOI:10.12677/pm.2023.133072678nØêÆ

±¯

= {η∈C([0,1],H

)|η(0) = 0,P

∞

(η(1)) <0}.

Ù¥Γ

⊂Γ,Ïdk

=inf

η∈Γ

max

t∈[0,1]

(η(t)) ≤inf

η∈Γ

max

t∈[0,1]

(η(t))

≤inf

η∈Γ

max

t∈[0,1]

∞

(η(t)) := c

∞

Ïd,)v

7L÷vXeO

≤

2θc

∞

(θ−2)min{1,V

}

·‚5¿,L

∞

-‰êØ‡L

√

3γ

¯K(8)f)•´¯K(6)f).¤±e˜Ú·‚òïÄJ



.:L

∞

O,ù‡y²•´MorseS“˜‡A^.

ÚÚÚnnn3.6XJv

∈H

)´¯K(8)f),Kv

∈L

∞

)…•3˜‡Õáuγ~êC

∗

>0,

¦|v

∞

≤C

∗

−1

L«G

−1

…^vL

«v

.v∈H

)´−∆v+V(x)

−1

(v)

g(G

−1

(v))

f(G

−1

(v))

g(G

−1

(v))

f),=éu?¿ϕ∈H

),k

∇v∇ϕdx+

V(x)

−1

(v)

g(G

−1

(v))

ϕdx=

f(G

−1

(v))

g(G

−1

(v))

ϕdx,(19)

dÚn3.4,Œv>0.T>0,¿½Â

(

v,0 <v≤T,

T,v≥T.

3•§(19)¥,-ϕ= v

2(η−1)

v(η>1),Œ

|∇v|

·v

2(η−1)

dx+2(η−1)

{x|v(x)<T}

2(η−1)−1

v|∇v|

dx+

V(x)

−1

(v)

g(G

−1

(v))

2(η−1)

vdx

f(G

−1

(v))

g(G

−1

(v))

2(η−1)

vdx.

dÚn2.1-(3)(Üþã•§†ý1‘´šK,éu?¿η>0Œ

|∇v|

2(η−1)

dx≤

f(G

−1

(v))

g(G

−1

(v))

2(η−1)

vdx

≤C

−1

(v)|

p−1

g(G

−1

(v))

2(η−1)

vdx

≤C

2(η−1)

dx.

(20)

DOI:10.12677/pm.2023.133072679nØêÆ

±¯

,˜•¡,|^SobolevØª,·‚k



(vv

η−1

)

∗



∗

≤C

|∇(vv

η−1

≤C

|∇v|

2(η−1)

dx+C(η−1)

|∇v|

2(η−1)

≤Cη

|∇v|

2(η−1)

dx,

ùp·‚^(a+b)

≤2(a

)9η

≥(η−1)

+1.

dªf(20),H¨olderØª9Sobolevi\½n



(vv

η−1

)

∗



∗

≤Cη

p−2

2(η−1)

≤Cη



∗



p−2

∗



(vv

η−1

)

∗

−p+2



∗

−p+2

∗

≤Cη

kvk

p−2



η22

∗

−p+2



∗

−p+2

∗

ùp·‚¦^0 ≤v

≤v.e5,Pζ=

∗

−p+2

,



(vv

η−1

)

∗



∗

≤Cη

kvk

p−2

|v|

2η

ηζ

dFatouÚnŒ

|v|

η2

∗

≤(Cη

kvk

p−2

)

2η

|v|

ηζ

.(21)

·‚½Âη

n+1

ζ= 2

∗

,Ù¥n= 0,1,2,...Úη

∗

+2−p

.d(21)k

|v|

∗

≤(Cη

kvk

p−2

)

2η

|v|

∗

≤(Ckvk

p−2

)

2η

|v|

∗

dMorseS“

|v|

∗

≤(Ckvk

p−2

)

2η

i=0

(

∗

)

(η

)

i=0

(

∗

)

(

∗

)

i=0

∗

)

|v|

∗

Ïdk

|v|

∞

≤Ckvk

∗

−2

∗

−p

≤C

∗

½½½nnn1.1yyy²²²dÚn3.69Ún2.1-(3)k

∞

= |G

−1

∞

≤

√

6|v

∞

DOI:10.12677/pm.2023.133072680nØêÆ

±¯

•3γ

>0,é?¿0 <γ<γ

∞

= |G

−1

∞

≤

√

6|v

∞

≤

√

∗

≤

√

3γ

Ïd,é?¿γ∈(0,γ

),u

= G

−1

)´¯K(6)).

ë•©z

[1]Adachi,S.andWatanabe,T.(2012)AsymptoticPropertiesofGroundStatesofQuasilinear

Schr¨odingerEquationswithH

-SubcriticalExponent.AdvancedNonlinearStudies,12,255-

279.https://doi.org/10.1515/ans-2012-0205

[2]Berestycki, H.andLions,P.L. (1983)NonlinearScalar Field EquationsI.ArchiveforRational

MechanicsandAnalysis,82,313-346.https://doi.org/10.1007/BF00250555

[3]Lions,P.L.(1984)TheConcentrationCompactnessPrincipleintheCalculusofVariations.

TheLocallyCompactCase.PartIand II.Annalesdel’InstitutHenriPoincar´e:AnalyseNon

Lin´eaire,109-145,223-283.https://doi.org/10.1016/s0294-1449(16)30422-x

[4]Borovskii,A.andGalkin,A.(1983)DynamicalModulationofanUltrashortHigh-Intensity

LaserPulseinMatter.JournalofExperimentalandTheoreticalPhysics,77,562-573.

[5]Brandi,H.,Manus,C.,Mainfray,G.,Lehner,T.andBonnaud,G.(1993)Relativisticand

PonderomotiveSelf-FocusingofaLaserBeaminaRadiallyInhomogeneousPlasma.Physics

ofFluids,5,3539-3550.https://doi.org/10.1063/1.860828

[6]Colin,M.andJeanjean,L.(2004)SolutionsforaQuasilinearSchr¨odingerEquations:ADual

Approach.NonlinearAnalysis,56,213-226.https://doi.org/10.1016/j.na.2003.09.008

[7]Alves,C.,Wang,Y.andShen,Y.(2015)SolitonSolutionsforaClassofQuasilinear

Schr¨odingerEquationswithaParameter.JournalofDiﬀerentialEquations,259,318-343.

https://doi.org/10.1016/j.jde.2015.02.030

[8]Liu,J., Wang,Y. and Wang,Z.-Q.(2003)SolitonSolutionsforQuasilinearSchr¨odinger Equa-

tionsII.JournalofDiﬀerentialEquations,187,473-493.

https://doi.org/10.1016/S0022-0396(02)00064-5

[9]Wang,Y.andShen,Y.(2018)ExistenceandAsymptoticBehaviorofPositiveSolutionsfora

ClassofQuasilinearSchr¨odingerEquations.AdvancedNonlinearStudies,18,131-150.

https://doi.org/10.1515/ans-2017-6026

[10]Adachi,S.,Shibata,M.andWatanabe,T.(2014)AsymptoticBehaviorofPositiveSolutions

fora ClassofQuasilinear EllipticEquationswithGeneralNonlinearities. Communicationson

PureandAppliedAnalysis,13,97-118.https://doi.org/10.3934/cpaa.2014.13.97

DOI:10.12677/pm.2023.133072681nØêÆ

±¯

[11]Adachi,S.,Shibata,M.andWatanabe,T.(2020)UniquenessofAsymptoticLimitofGround

StatesforaClassofQuasilinearSchr¨odingerEquationwithH

-CriticalGrowthinR

.Appli-

cableAnalysis,101,671-691.https://doi.org/10.1080/00036811.2020.1757079

[12]Adachi,S.,Shibata,M.andWatanabe,T.(2014)Blow-UpPhenomenaandAsymptoticPro-

ﬁlesofGroundStatesofQuasilinearEllipticEquationswithH

-SupercriticalNonlinearities.

JournalofDiﬀerentialEquations,256,1492-1514.https://doi.org/10.1016/j.jde.2013.11.004

[13]Adachi, S.,Shibata,M.andWatanabe, T.(2019) AsymptoticProperty of Ground Statesfora

ClassofQuasilinearSchr¨odingerEquationwithH

-CriticalGrowth.Calculus of Variations and

PartialDiﬀerentialEquations,58, ArticleNo.88.https://doi.org/10.1007/s00526-019-1527-y

[14]Wang,Y.andLi,Z.(2018)ExistenceofSolutionstoQuasilinearSchr¨odingerEquationsIn-

volvingCriticalSobolevExponent.TaiwaneseJournalofMathematics,22,401-420.

https://doi.org/10.11650/tjm/8150

[15]Jeanjean,L.(1999)OntheExistenceofBoundedPalais-SmaleSequencesandApplicationto

aLandesman-LazerTypeProblem.ProceedingsoftheRoyalSocietyofEdinburgh.SectionA:

Mathematics,129,787-809.https://doi.org/10.1017/S0308210500013147

[16]Jeanjean,L.andTanaka,K.(2003)ARemarkonLeastEnergySolutionsinR

.Proceedings

oftheAmericanMathematicalSociety,131,2399-2408.

https://doi.org/10.1090/S0002-9939-02-06821-1

[17]Adachi,S.andWatanabe,T.(2011)G-InvariantPositiveSolutionsforaQuasilinear

Schr¨odingerEquation.AdvancesinDiﬀerentialEquations,16,289-324.

https://doi.org/10.57262/ade/1355854310

DOI:10.12677/pm.2023.133072682nØêÆ