Skip to main content

ORIGINAL RESEARCH article

Front. Phys., 01 May 2024
Sec. Complex Physical Systems

Existence of a ground-state solution for a quasilinear Schrödinger system

Xue ZhangXue Zhang1Jing Zhang,,
Jing Zhang1,2,3*
  • 1College of Mathematics Science, Inner Mongolia Normal University, Hohhot, Inner Mongolia, China
  • 2Key Laboratory of Infinite-Dimensional Hamiltonian System and Its Algorithm Application, Ministry of Education, Inner Mongolia Normal University, Hohhot, Inner Mongolia, China
  • 3Center for Applied Mathematics Inner Mongolia, Inner Mongolia Normal University, Hohhot, Inner Mongolia, China

In this paper, we consider the following quasilinear Schrödinger system.

Δu+u+k2Δ|u|2u=2αα+β|u|α2u|v|β,xRN,Δv+v+k2Δ|v|2v=2βα+β|u|α|v|β2v,xRN,

where k < 0 is a real constant, α > 1, β > 1, and α + β < 2*. We take advantage of the critical point theorem developed by Jeanjean (Proc. R. Soc. Edinburgh Sect A., 1999, 129: 787–809) and combine it with Pohožaev identity to obtain the existence of a ground-state solution, which is the non-trivial solution with the least possible energy.

1 Introduction

This article is concerned with the following quasilinear Schrödinger system:

Δu+u+k2Δ|u|2u=2αα+β|u|α2u|v|β,xRN,Δv+v+k2Δ|v|2v=2βα+β|u|α|v|β2v,xRN,(1.1)

where k < 0 is a real constant.

Many scholars have made significant contributions to the study of the quasilinear Schrödinger system. Wang and Huang proved the existence of ground-state solutions for a class of systems by establishing a suitable Nehari–Pohožaev-type constraint set and considering related minimization problems in [2]. The existence of infinitely many solutions was established for the quasilinear Schrödinger system by the symmetric Mountain Pass Theorem; see [3]. The existence of positive solutions was obtained by using the monotonicity trick and Morse iteration in [4]. Chen and Zhang proved the existence of ground-state solutions by minimization under a convenient constraint and concentration compactness lemma in [5].

The quasilinear Schrödinger system (1.1) is in part motivated by the following quasilinear Schrödinger equation:

iϵz=ϵΔz+Wxzl|z|2zkϵΔh|z|2h|z|2z,forxRN,N>2,(1.2)

where W(x) is a given potential, k is a real constant, and l and h are real functions that are essentially pure power forms. The quasilinear Schrödinger Equation 1.2 describes several physical phenomena with different h; see [68].

Let the case h(s)=s,l(s)=μsp12 and k > 0. Setting z(t, x) = exp(−iFt)u(x), one can obtain a corresponding equation of elliptic type which has the formal variational structure:

ϵΔu+VxuϵkΔ|u|2u=μ|u|p1u,u>0xRN,N>2,(1.3)

where V(x) = W(x) – F is the new potential function. The problem (1.3) has been studied by many academics. In [9], the existence results of multiple solutions were studied via dual approach techniques and variational methods when k > 0 was small enough. The existence of soliton solutions was established by a minimization argument; see [10]. The Mountain Pass Theorem is combined with the principle of symmetric criticality to establish the multiplicity of solutions in [11]. In [12], the author proved the existence of soliton solutions via making a change in variables and creating a suitable Orlicz space. The minimax principles for lower semicontinuous functionals were used to find solutions in [13].

In [14], the authors used the method developed by [1, 15] to divide the energy functional into two parts and established the existence of ground-state solutions for a type of quasilinear Schrödinger equation like 1.3. Inspired by [14], we try to find the existence of ground-state solutions for system 1.1. This achievement can enrich the relatively few existing results about this system.

The main result of this paper is the following:

Theorem 1.1. When k < 0, α > 1, β > 1, and α + β < 2*, then (1.1) has a ground-state solution.

This paper is organized as follows. In Section 2, preparation work is completed. In Section 3, we reformulate this problem and prove Theorem 1.1. In this paper, C is defined as different constants.

2 Reformulation of the problem and preliminaries

First, we explain that Lq(RN) denotes the Lebesgue space with the norm

up=RN|u|pdx1p,

where 1 ≤ p < . Lq=Lq(RN)×Lq(RN) with the norm

u,vp=RN|u|pdx1p+RN|v|pdx1p,

where 1 ≤ p < .

H1=u,v:u,vL2RN,u,vL2RN

with norms

u,v=u+v=RN|u|2+u2dx12+RN|v|2+v2dx12

and

u,v2=u2+v2.

The embedding H1Lq is continuous and compact for q ∈ (2, 2*).

In (1.1), the Euler–Lagrange functional associated with Equation 1.1 is given by

Iu,v=12RN1ku2|u|2dx+12RN|u|2dx+12RN1kv2|v|2dx+12RN|v|2dx2α+βRN|u|α|v|βdx.

For (u, v), constructing the variable like [16, 17], we have

dz=k1ku2du,z=hu=12ku1ku2+12lnku+1ku2,
dw=k1kv2dv,w=hv=12kv1kv2+12lnkv+1kv2.

Since h is strictly monotone, it has a well-defined inverse function f and u = f(z), v = f(w). Note that

huku,|u|1kk2u|u|,|u|1k,hu=k1ku2

and

fz1kz,|z|1k2k|z|z,|z|1k,
fz=1hu=1k1kv2=1k1kfz2.

Similarly, the same operation holds true for v = f(w).

Using the variable, (1.1) will become

1kΔz+fzfz=2αα+β|fz|α2fz|fw|β,xRN,1kΔw+fwfw=2βα+β|fz|α|fw|β2fw,xRN,(2.1)

where f:[0,)R and

f=1k1kf2

on [0, ), f(0) = 0, and f(−t) = f(t) on [0, ). From the above facts, if (z, w) is a weak solution for (2.1), then (u,v)=f(z),f(w) is a weak solution for (1.1). The energy functional I(u, v) reduces to the following functional:

ϕz,w=12RN1k|z|2dx+12RNf2zdx+12RN1k|w|2dx+12RNf2wdx2α+βRN|fz|α|fw|βdx.(2.2)

There are some properties of f:RR as follows, which are proved in [16, 17].

Lemma 2.1. The function f(t) and its derivative satisfy the following properties:

(i) f(t)t1 as t → 0;

(ii) f(t) ≤ |t| for any tR;

(iii) f(t)214|t| for all tR;

(iv) f2(t)2tf(t)f(t)f2(t) for all tR;

(v) there exists a positive constant C such that

|ft|C|t|,ift1,C|t|12,ift>1;

(vi) |f(t)f(t)|12 for all tR.

3 Proof of theorem 1.1

In this section, we will complete the proof of Theorem 1.1. First, we will recall the critical point theorem in [1], which is crucial for proving Theorem 1.1.

Theorem 3.1. Let X,(,) be a Banach space and LR+ an interval. Consider the following family of C1-functionals on X:

Φλz,w=Az,w+λBz,w,λL,

with B being non-negative and either A(z, w) → +∞ or B(z, w) → +∞ as (z,w). Assume that there are two points (z1, w1), (z2, w2) ⊂ X such that

cλ=infγΓλmaxt1,t20,1×0,1Φλγt1,t2>maxΦλz1,w1,Φλz2,w2for allλL,

where Γλ = {γC([0, 1] × [0, 1], X): γ(0, 0) = (z1, w1), γ(1, 1) = (z2, w2)}. Then, for almost every λL, there is a sequence {(zn, wn)} ⊂ X such that

(i) (zn, wn) is bounded;

(ii) Φλ(z, w) → cλ;

(iii) Φλ(zn,wn)0 in the dual X−1 of X.

Moreover, the map λcλ is non-increasing and continuous from the left.

Let λL be an arbitrary but fixed value where cλ exists, where cλ is the derivative of cλ with respect to λ. Let {λn} ⊂ L be a strictly increasing sequence such that λnλ. To prove Theorem 3.1, we will show the following lemmas:

Lemma 3.1. There exists a sequence of path {γn} ⊂ Γ and K=K(cλ)>0 such that

(i) γn(t1,t2)K if γn(t1, t2) satisfies

Φλγnt1,t2cλλλn;(3.1)

(ii) max(t1,t2)[0,1]Φλ(γn(t1,t2))cλ+(cλ+2)(λλn).

Proof. The proof is standard; see [1].

Lemma 3.1. means that there exists a sequence of paths {γn} ⊂ Γ such that

maxt1,t20,1×0,1Φλγnt1,t2cλ,

for all nN sufficiently large; starting from a level strictly below cλ, all the “top” of the path is contained in the ball centered at the origin of fixed radius K=K(cλ)>0. Now, for α > 0, we define

Fα=z,wX:z,wK+1 and |Φλz,wcλ|α,

where K is given in lemma 3.1.

Lemma 3.2. For all α > 0,

infΦλz,w:z,wFα=0.(3.2)

Proof. We assume that (3.2) does not hold. Then, there exists α > 0 such that for any (z, w) ∈ Fα, we obtain

Φλz,wα.(3.3)

Without loss of generality, we can assume that

0<α<12cλmaxΦλz1,w1,Φλz2,w2.

A classical deformation argument then says that there exists ϵ ∈ [0, α] and a homeomorphism η: XX such that

ηu=u,if|Φλz,wcλ|α,(3.4)
Φληz,wΦλz,w,z,wX,(3.5)
Φληz,wcλϵ,z,wX, satisfying z,wK and Φλz,wcλ+ϵ.(3.6)

Let {γn} ⊂ Γ be the sequence obtained in lemma 3.1. We choose and fix mN sufficiently large in order that

cλ+2λλmϵ.(3.7)

By lemma 3.1 and (3.4), η(γm) ∈ Γ. Now if (z, w) = γm(t1, t2) satisfies

Φλz,wcλλλm,

then (3.5) implies that

Φληz,wcλλλm.(3.8)

If (z, w) = γm(t1, t2) satisfies

Φλz,w>cλλλm,

by lemma 3.1 and (3.7), we obtain (z, w) such that (z,w)K with Φλ(z, w) ≤ cλ + ϵ. From (3.6), we obtain

Φληz,wcλϵcλλλm.(3.9)

Combining (3.8) with (3.9), we obtain

maxt1,t20,1×0,1Φληγmt1,t2cλλλm,

which contradicts the variational characterization of cλ.

Next, we prove theorem 3.1.

Proof. Since lemma 3.2 is true, there exists a PS sequence for Φλ at the level cλR, which is contained in the ball of radius K + 1 centered at the origin. Hence, this is proved.

Let L=12,1, we define the following energy functional:

Φλz,w=12RN1k|z|2+z2+1k|w|2+w2dxλRN12z2f2z+w2f2w+2α+β|fz|α|fw|βdx,(3.10)

where λL. Moreover, let

Az,w=12RN1k|z|2+z2+1k|w|2+w2dx

and

Bz,w=λRN12z2f2z+w2f2w+2α+β|fz|α|fw|βdx.

Letting (z,w)+, then A(z, w) → + and B(z, w) ≥ 0.

By a standard argument in [18, 19], we have the following Pohožaev-type identity:

Lemma 3.3. If (z, w) ∈ H1 is a critical point of (3.10), then (z,w) satisfies Pλ(z, w) = 0, where

Pλz,wN22RN1k|z|2+|w|2dx+N2RNf2z+f2wdx2Nλα+βRN|fz|α|fw|βdx.(3.11)

Similar to [9], we obtain the following lemma:

Lemma 3.4. Φλ(z, w) meet the conditions as follows:

(i) there exists (z, w) ∈ H1 \{(0, 0)} such that Φλ(z, w) < 0 for all λL;

(ii) for cλ, we obtain

cλ=infγΓmaxt1,t20,1×0,1Φλγt1,t2>maxΦλ0,0,Φλz,w,

for all λL, where

Γ=γC0,1×0,1,H1:γ0,0=0,0,γ1,1=z,w.

Proof. (i) Let (z, w) ∈ H1 \{(0, 0)} be fixed. For any λL=12,1, we obtain

Φλz,wΦ12z,w=12RN1k|z|2+|w|2dx+14RNz2+f2z+w2+f2wdx1α+βRN|fz|α|fw|βdx.

As [20, 21], we consider ϕ,φC0(R) such that 0 ≤ ϕ(x) ≤ 1, 0 ≤ φ(x) ≤ 1 and

ϕx=1,if|x|1,0,if|x|1,φx=1,if|x|1,0,if|x|1.

By Lemma 2.1 (ii) and (v), we obtain

|ftϕ|C|tϕ|Cftϕ.

By Lemma 2.1 (ii),

Φλt1ϕ,t2φ12RN1k|t1ϕ|2+t12ϕ2dx+12RN1k|t2φ|2+t22φ2dx1α+βRN|ft1ϕ|α|ft2φ|βdxt122RN1k|ϕ|2+ϕ2dx+t222RN1k|φ|2+φ2dxC|ft1|α+|ft2|βα+βRN|ϕ|α|φ|βdx.

It follows that Φλ(t1ϕ, t2φ) → − as (t1, t2) → (+, + ). Thus, there exists (t3, t4) > 0 such that Φλ(t3ϕ, t4φ) < 0. Thus, taking (z, w) = (t3ϕ, t4φ), we obtain Φλ(z, w) < 0 for all λL.

(ii) As [20, 22], there exists C > 0 and ρ1 > 0 small enough such that

RN1k|z|2+f2z+1k|w|2+f2wdxCz,w,

for (z,w)ρ1. From Lemma 2.1 (iii) and Hölder inequality, we obtain

Φλz,w12RN1k|z|2+f2zdx+12RN1k|w|2+f2wdx1α+βRN|fz|α|fw|βdxCz,w1α+βRN|fz|α|fw|βdxCz,wCzα1pwβ1pfor allz,wρ1,

where α1 = α or α2, β1 = β or β2, and (1p+1p)=1. It can conclude that Φλ has a strict local minimum at 0, and hence, cλ > 0.

By Theorem 3.1, it is easy to know that for every λ12,1, there exists a bounded sequence (zn, wn) ⊂ H1 such that Φλ(zn, wn) → cλ and Φλ(zn,wn)0.

Lemma 3.5. If (zn, wn) ⊂ H1 is the sequence obtained above, then for almost every λL=[12,1], there exists (zλ, wλ) ∈ H1 \{(0, 0)} such that Φλ(zλ, wλ) → cλ and Φλ(zλ,wλ)0.

Proof. Since (zn, wn) is bounded in H1, up to a subsequence, there exists (zλ, wλ) ∈ H1 such that

zn,wnzλ,wλinH1,
zn,wnzλ,wλ in Ls for all 2<s<2*,
znx,wnxzλx,wλx a. e. in RN.

Since Φλ(zn,wn)0, by the Lebesgue dominated convergence theorem, it is easy to get Φλ(zn,wn)Φλ(zλ,wλ), that is, Φλ(zλ,wλ)=0, as shown in [23]. Similar to [22, 24, 25], there exists C > 0 such that

RN1k|znzλ|2+fznfznfzλfzλznzλdxCznzλ2,(3.12)
RN1k|wnwλ|2+fwnfwnfwλfwλwnwλdxCwnwλ2.(3.13)

By Hölder inequality and Lemma 2.1(ii) and (iv), we deduce that

2αα+βRN|fzn|α2fznfzn|fwn|βznzλdx+2βα+βRN|fzn|α|fwn|β2fwnfwnwnwλdx2αα+βRN|zn|α1|wn|βznzλdx+2βα+βRN|zn|α|wn|β1wnwλdx2αα+βRN|zn|1β|wn|1α1dxα1βznzλp1+2αα+βRN|zn|1β1|wn|1αdxβ1αwnwλp20,(3.14)

where p1=1(α1)β and p2=1(β1)α. Similarly, we obtain

2αα+βRN|fzλ|α2fzλfzλ|fwλ|βznzλdx+2βα+βRN|fzλ|α|fwλ|β2fwλfwλwnwλdx0.(3.15)

Following (3.12), 3.13, 3.14, and .3.15, we obtain

0Φλzn,wnΦλzλ,wλ,znzλ,wnwλ=RN1k|znzλ|2+fznzλfznzλznzλdx+RN1k|wnwλ|2+fwnwλfwnwλwnwλdx2αα+βRN|fzn|α2fznfzn|fwn|β|fzλ|α2fzλfzλ|fwλ|βznzλdx2βα+βRN|fzn|α|fwn|β2fwnfwn|fzλ|α|fwλ|β2fwλfwλwnwλdxCznzλ2+Cwnwλ2+on1,(3.16)

which implies that (zn, wn) → (zλ, wλ) in H1. Thus, (zλ, wλ) is a non-trivial critical point of Φλ(z, w) with Φλ(zλ, wλ) = cλ.

Next, we prove Theorem 1.1.

Proof. At first, using Theorem 3.1, for arbitrary λL=12,1, there is a (zλ, wλ) ∈ H1 such that

zn,wnzλ,wλ0,0 in H1,
Φλzn,wncλ and Φλzn,wn0.

By Lemma 3.5, we obtain

Φλzλ,wλcλ and Φλzλ,wλ=0.

Thus, there exists λn12,1 such that

λn1,zλn,wλnH1,
Φλnzλn,wλn=0 and Φλnzλn,wλn=cλn.

Next, we prove that {(zλn,wλn)} is bounded in H1. From Lemma 3.4

Φλnzλn,wλn=c12,Φλnzλn,wλn=0,

it follows that

c12Φλnzλn,wλn=Φλnzλn,wλn1NPλnzλn,wλn=N22NRN2N21k|zλn|2+|wλn|2+f2zλn+f2wλndx.(3.17)

By Lemma 2.1 (v) and Sobolev inequality, it follows that

|zλn|1zλn2dxCRNf2zλndx,|wλn|1wλn2dxCRNf2wλwdx

and

|zλn|>1zλn2dx|zλn|>1zλn2*dxCRN|zλn|2dx2*2,
|wλn|>1wλn2dx|wλn|>1wλn2*dxCRN|wλn|2dx2*2.

Therefore,

RNzλn2+wλn2dx=|zλn|1zλn2dx+|zλn|>1zλn2dx+|wλn|1wλn2dx+|wλn|>1wλn2dxCRNf2zλndx+CRNf2wλwdx+CRN|zλn|2dx2*2+CRN|wλn|2dx2*2.(3.18)

Combining (3.17) and (3.18), we infer that there exists C > 0 such that

RNzλn2+wλn2dxC.

Thus, there exists C > 0 independent of n such that

zλn,wλn2=RN|zλn|2+zλn2dx+RN|wλn|2+wλn2dxC.

Next, we can assume that the limit of Φλn(zλn,wλn) exists. By Theorem 3.1, we know that λcλ is continuous from the left. Thus, we obtain

0limnΦλnzλn,wλnc12.

Then, by using the fact that

Φzλn,wλn=Φλnzλn,wλn+λn1αβRN2α+β|fzλn|α|fwλn|βdx

and

Φzλn,wλn,ϕ,ψ=Φλnzλn,wλn,ϕ,ψ+λn1βRN2α+β|fzλn|α1fzλnϕ|fwλn|βdx+λn1αRN2α+β|fzλn|α|fwλn|β1fwλnψdx,

for any ϕ,ψC0(RN) and (zλn,wλn)C, it follows that

limnΦzλn,wλn=c1,limnΦzλn,wλn=0.

Up to a subsequence, there exists a subsequence (zλn,wλn) denoted by (zn, wn) and (z0, w0) ∈ H1 such that (zn, wn) ⇀ (z0, w0) in H1. Using the same method as Lemma 3.5, we will obtain the existence of a non-trivial solution (z0, w0) for Φ and Φ′(z0, w0) = 0 and Φ(z0, w0) = c1.

To find ground-state solutions, we need to define that

minfΦz,w:z,w0,0,Φz,w=0.

By Lemma 3.3, it follows that

Pz,w=P1z,w=0.

According to (3.17), we have m ≥ 0. Let (zn, wn) be a sequence such that

Φzn,wn=0 and Φzn,wnm.

Similar to Lemma 3.5, we can prove that there exists (z′, w′) ∈ H1 such that

Φz,w=0 and Φz,w=m,

which implies that (u,v)=f(z),f(w) is a ground-state solution of (1.1). The proof is complete.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material; further inquiries can be directed to the corresponding authors.

Author contributions

XZ: writing–original draft and writing–review and editing. JZ: writing–original draft and writing–review and editing.

Funding

The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. JZ was supported by the Natural Science Foundation of Inner Mongolia Autonomous Region (nos 2022MS01001), the Key Laboratory of Infinite-dimensional Hamiltonian System and Its Algorithm Application (Inner Mongolia Normal University), the Ministry of Education (No. 2023KFZD01), the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region (No. NJYT23100), the Fundamental Research Funds for the Inner Mongolia Normal University (No. 2022JBQN072), and the Mathematics First-class Disciplines Cultivation Fund of Inner Mongolia Normal University (No. 2024YLKY14). XZ was supported by the Fundamental Research Funds for the Inner Mongolia Normal University (2022JBXC03) and the Graduate Students Research Innovation Fund of Inner Mongolia Normal University (CXJJS22100).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

References

1. Jeanjean L On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $\mathbb{R}^{N}$. Proc R Soc Edinb Sect A. (1999) 129:787–809.

CrossRef Full Text | Google Scholar

2. Wang Y, Huang X Ground states of Nehari-Pohožaev type for a quasilinear Schrödinger system with superlinear reaction. Electron Res Archive (2023) 31(4):2071–94. doi:10.3934/era.2023106

CrossRef Full Text | Google Scholar

3. Chen C, Yang H. Multiple Solutions for a Class of Quasilinear Schrödinger Systems in $\mathbb{R}^{N}$. Bull Malays Math Sci Soc (2019) 42:611–36. doi:10.1007/s40840-017-0502-z

CrossRef Full Text | Google Scholar

4. Chen J, Zhang Q. Positive solutions for quasilinear Schrödinger system with positive parameter. Z Angew Math Phys (2022) 73–144. doi:10.1007/S00033-022-01781-1

CrossRef Full Text | Google Scholar

5. Chen J, Zhang Q Ground state solution of Nehari-Pohožaev type for periodic quasilinear Schrödinger system. J Math Phys (2020) 61:101510. doi:10.1063/5.0014321

CrossRef Full Text | Google Scholar

6. Lange H, Toomire B, Zweifel PF Time-dependent dissipation in nonlinear Schrödinger systems. J Math Phys (1995) 36:1274–83. doi:10.1063/1.531120

CrossRef Full Text | Google Scholar

7. Laedke EW, Spatschek KH, Stenflo L Evolution theorem for a class of perturbed envelope soliton solutions. J Math Phys (1983) 24:2764–9. doi:10.1063/1.525675

CrossRef Full Text | Google Scholar

8. Ritchie B Relativistic self-focusing and channel formation in laser-plasma interactions. Phys Rev E (1994) 50:687–9. doi:10.1103/physreve.50.r687

PubMed Abstract | CrossRef Full Text | Google Scholar

9. Chen J, Huang X, Cheng B, Zhu C Some results on standing wave solutions for a class of quasilinear Schrödinger equations. J Math Phys (2019) 60:091506. doi:10.1063/1.5093720

CrossRef Full Text | Google Scholar

10. Liu JQ, Wang ZQ Soliton solutions for quasilinear Schrödinger equations I. Proc Amer Math Soc (2002) 131(2):441–8. doi:10.1090/s0002-9939-02-06783-7

CrossRef Full Text | Google Scholar

11. Severo UB Symmetric and nonsymmetric solutions for a class of quasilinear Schrödinger equations. Adv Nonlinear Stud (2008) 8:375–89. doi:10.1515/ans-2008-0208

CrossRef Full Text | Google Scholar

12. Moameni A. Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbb{R}^{N}$. J Differential Equations (2006) 229:570–87. doi:10.1016/j.jde.2006.07.001

CrossRef Full Text | Google Scholar

13. Alves CO, de Morais Filho DC. Existence and concentration of positive solutions for a Schrödinger logarithmic equation. Z Angew Math Phys (2018) 69–144. doi:10.1007/s00033-018-1038-2

CrossRef Full Text | Google Scholar

14. Chen J, Chen B, Huang X Ground state solutions for a class of quasilinear Schrödinger equations with Choquard type nonlinearity. Appl Math Lett (2020) 102:106141. doi:10.1016/j.aml.2019.106141

CrossRef Full Text | Google Scholar

15. Yang X, Zhang W, Zhao F Existence and multiplicity of solutions for a quasilinear Choquard equation via perturbation method. J Math Phys (2018) 59:081503. doi:10.1063/1.5038762

CrossRef Full Text | Google Scholar

16. Colin M, Jeanjean L Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal (2004) 56:213–26. doi:10.1016/j.na.2003.09.008

CrossRef Full Text | Google Scholar

17. Liu JQ, Wang Y, Wang ZQ Soliton solutions for quasilinear Schrödinger equations, II. J Differential Equations (2003) 187:473–93. doi:10.1016/s0022-0396(02)00064-5

CrossRef Full Text | Google Scholar

18. Chen J, Zhang Q Existence of positive ground state solutions for quasilinear Schrödinger system with positive parameter. Appl Anal (2022) 102:2676–91. doi:10.1080/00036811.2022.2033232

CrossRef Full Text | Google Scholar

19. Willem M Minimax theorems. Berlin: Birkhauser (1996).

Google Scholar

20. Chen S, Wu X Existence of positive solutions for a class of quasilinear Schrödinger equations of Choquard type. J Math Anal Appl (2019) 475:1754–77. doi:10.1016/j.jmaa.2019.03.051

CrossRef Full Text | Google Scholar

21. do Ȯ JM, Miyagaki OH, Soares SHM Soliton solutions for quasilinear Schrödinger equations with critical growth. J Differential Equations (2010) 248:722–44. doi:10.1016/j.jde.2009.11.030

CrossRef Full Text | Google Scholar

22. Fang X, Szulkin A Multiple solutions for a quasilinear Schrödinger equation. J Differential Equations (2013) 254:2015–32. doi:10.1016/j.jde.2012.11.017

CrossRef Full Text | Google Scholar

23. Li G On the existence of nontrivial solutions for quasilinear Schrödinger systems. Boundary Value Probl (2022) 2022:40. doi:10.1186/s13661-022-01623-z

CrossRef Full Text | Google Scholar

24. Wu X Multiple solutions for quasilinear Schrödinger equations with a parameter. J Differential Equations (2014) 256:2619–32. doi:10.1016/j.jde.2014.01.026

CrossRef Full Text | Google Scholar

25. Zhang J, Tang X, Zhang W Infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential. J Math Anal Appl (2014) 420:1762–75. doi:10.1016/j.jmaa.2014.06.055

CrossRef Full Text | Google Scholar

Keywords: quasilinear Schrödinger system, Pohožaev identity, ground-state solution, critical point theorem, Lebesgue dominated convergence theorem

Citation: Zhang X and Zhang J (2024) Existence of a ground-state solution for a quasilinear Schrödinger system. Front. Phys. 12:1386144. doi: 10.3389/fphy.2024.1386144

Received: 14 February 2024; Accepted: 26 March 2024;
Published: 01 May 2024.

Edited by:

Pietro Prestininzi, Roma Tre University, Italy

Reviewed by:

Jianhua Chen, Nanchang University, China
Li Guofa, Qujing Normal University, China

Copyright © 2024 Zhang and Zhang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Jing Zhang, amluc2hpemhhbmdqaW5nQDE2My5jb20=

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.