Abstract
In this paper, we consider the following quasilinear Schrödinger system.
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: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: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 [6–8].
Let the case 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: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:
Whenk < 0,α > 1,β > 1, andα + 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 denotes the Lebesgue space with the normwhere 1 ≤ p < ∞. with the normwhere 1 ≤ p < ∞.with normsand
The embedding H1↪Lq is continuous and compact for q ∈ (2, 2*).
In (1.1), the Euler–Lagrange functional associated with Equation 1.1 is given by
For (u, v), constructing the variable like [16, 17], we have
Since h is strictly monotone, it has a well-defined inverse function f and u = f(z), v = f(w). Note thatand
Similarly, the same operation holds true for v = f(w).
Using the variable, (1.1) will becomewhere andon [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 is a weak solution for (1.1). The energy functional I(u, v) reduces to the following functional:
There are some properties of as follows, which are proved in [16, 17].
Lemma 2.1
(
t)
and its derivative satisfy the following properties:(i) ast → 0;
(ii) f(t) ≤ |t| for any;
(iii) for all;
(iv) for all;
(v) there exists a positive constantCsuch that
(vi) for all.
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.
being
non-negative and eitherA(
z,
w) → +
∞orB(
z,
w) → +
∞as. Assume that there are two points(
z1,
w1), (
z2,
w2) ⊂
Xsuch thatwhereΓ
λ= {
γ∈
C([0, 1] × [0, 1],
X):
γ(0, 0) = (
z1,
w1),
γ(1, 1) = (
z2,
w2)}
. Then, for almost everyλ∈
L,there is a sequence{(
zn,
wn)} ⊂
Xsuch that(i) (zn, wn) is bounded;
(ii) Φλ(z, w) → cλ;
(iii) in the dualX−1ofX.
Moreover, the mapλ → cλis non-increasing and continuous from the left.
Let λ ∈ L be an arbitrary but fixed value where exists, where is the derivative of cλ with respect to λ. Let {λn} ⊂ L be a strictly increasing sequence such that λn → λ. To prove Theorem 3.1, we will show the following lemmas:
Lemma 3.1. There exists a sequence of path {γn} ⊂ Γ andsuch that
(i) ifγn(t1, t2) satisfies
(ii) .
Proof. The proof is standard; see [1].
Lemma 3.1means that there exists a sequence of paths {γn} ⊂ Γ such thatfor all 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 . Now, for α > 0, we definewhere K is given in lemma 3.1.
Lemma 3.2For allα > 0,
Proof. We assume that (3.2) does not hold. Then, there exists α > 0 such that for any (z, w) ∈ Fα, we obtain
Without loss of generality, we can assume that
A classical deformation argument then says that there exists ϵ ∈ [0, α] and a homeomorphism η: X → X such that
Let {γn} ⊂ Γ be the sequence obtained in lemma 3.1. We choose and fix sufficiently large in order that
By lemma 3.1 and (3.4), η(γm) ∈ Γ. Now if (z, w) = γm(t1, t2) satisfiesthen (3.5) implies that
If (z, w) = γm(t1, t2) satisfiesby lemma 3.1 and (3.7), we obtain (z, w) such that with Φλ(z, w) ≤ cλ + ϵ. From (3.6), we obtain
Combining (3.8) with (3.9), we obtainwhich 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 , which is contained in the ball of radius K + 1 centered at the origin. Hence, this is proved.
Let , we define the following energy functional:where λ ∈ L. Moreover, letand
Letting , then A(z, w) → +∞ and B(z, w) ≥ 0.
By a standard argument in [18, 19], we have the following Pohožaev-type identity:
Lemma 3.3If (z, w) ∈ H1is a critical point of (3.10), then (z,w) satisfiesPλ(z, w) = 0, where
Similar to [9], we obtain the following lemma:
Lemma 3.4
Φ
λ(
z,
w)
meet the conditions as follows:(i) there exists (z, w) ∈ H1 \{(0, 0)} such that Φλ(z, w) < 0 for allλ ∈ L;
(ii) forcλ, we obtain
∈
L, whereBy Theorem 3.1, it is easy to know that for every , there exists a bounded sequence (zn, wn) ⊂ H1 such that Φλ(zn, wn) → cλ and .
Lemma 3.5If (zn, wn) ⊂ H1is the sequence obtained above, then for almost every, there exists (zλ, wλ) ∈ H1 \{(0, 0)} such that Φλ(zλ, wλ) → cλand.Proof. Since (zn, wn) is bounded in H1, up to a subsequence, there exists (zλ, wλ) ∈ H1 such thatSince , by the Lebesgue dominated convergence theorem, it is easy to get , that is, , as shown in [23]. Similar to [22, 24, 25], there exists C > 0 such thatBy Hölder inequality and Lemma 2.1(ii) and (iv), we deduce thatwhere and . Similarly, we obtainFollowing (3.12), 3.13, 3.14, and .3.15, we obtainwhich 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 , there is a (zλ, wλ) ∈ H1 such that
By Lemma 3.5, we obtain
Thus, there exists such that
Next, we prove that is bounded in H1. From Lemma 3.4it follows that
By Lemma 2.1 (v) and Sobolev inequality, it follows thatand
Therefore,
Combining (3.17) and (3.18), we infer that there exists C > 0 such that
Thus, there exists C > 0 independent of n such that
Next, we can assume that the limit of exists. By Theorem 3.1, we know that λ → cλ is continuous from the left. Thus, we obtain
Then, by using the fact thatandfor any and , it follows that
Up to a subsequence, there exists a subsequence 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
By Lemma 3.3, it follows that
According to (3.17), we have m ≥ 0. Let (zn, wn) be a sequence such that
Similar to Lemma 3.5, we can prove that there exists (z′, w′) ∈ H1 such thatwhich implies that is a ground-state solution of (1.1). The proof is complete.
Statements
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.
JeanjeanLOn 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.
2.
WangYHuangXGround states of Nehari-Pohožaev type for a quasilinear Schrödinger system with superlinear reaction. Electron Res Archive (2023) 31(4):2071–94. 10.3934/era.2023106
3.
ChenCYangH. Multiple Solutions for a Class of Quasilinear Schrödinger Systems in $\mathbb{R}^{N}$. Bull Malays Math Sci Soc (2019) 42:611–36. 10.1007/s40840-017-0502-z
4.
ChenJZhangQ. Positive solutions for quasilinear Schrödinger system with positive parameter. Z Angew Math Phys (2022) 73–144. 10.1007/S00033-022-01781-1
5.
ChenJZhangQGround state solution of Nehari-Pohožaev type for periodic quasilinear Schrödinger system. J Math Phys (2020) 61:101510. 10.1063/5.0014321
6.
LangeHToomireBZweifelPFTime-dependent dissipation in nonlinear Schrödinger systems. J Math Phys (1995) 36:1274–83. 10.1063/1.531120
7.
LaedkeEWSpatschekKHStenfloLEvolution theorem for a class of perturbed envelope soliton solutions. J Math Phys (1983) 24:2764–9. 10.1063/1.525675
8.
RitchieBRelativistic self-focusing and channel formation in laser-plasma interactions. Phys Rev E (1994) 50:687–9. 10.1103/physreve.50.r687
9.
ChenJHuangXChengBZhuCSome results on standing wave solutions for a class of quasilinear Schrödinger equations. J Math Phys (2019) 60:091506. 10.1063/1.5093720
10.
LiuJQWangZQSoliton solutions for quasilinear Schrödinger equations I. Proc Amer Math Soc (2002) 131(2):441–8. 10.1090/s0002-9939-02-06783-7
11.
SeveroUBSymmetric and nonsymmetric solutions for a class of quasilinear Schrödinger equations. Adv Nonlinear Stud (2008) 8:375–89. 10.1515/ans-2008-0208
12.
MoameniA. Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbb{R}^{N}$. J Differential Equations (2006) 229:570–87. 10.1016/j.jde.2006.07.001
13.
AlvesCOde Morais FilhoDC. Existence and concentration of positive solutions for a Schrödinger logarithmic equation. Z Angew Math Phys (2018) 69–144. 10.1007/s00033-018-1038-2
14.
ChenJChenBHuangXGround state solutions for a class of quasilinear Schrödinger equations with Choquard type nonlinearity. Appl Math Lett (2020) 102:106141. 10.1016/j.aml.2019.106141
15.
YangXZhangWZhaoFExistence and multiplicity of solutions for a quasilinear Choquard equation via perturbation method. J Math Phys (2018) 59:081503. 10.1063/1.5038762
16.
ColinMJeanjeanLSolutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal (2004) 56:213–26. 10.1016/j.na.2003.09.008
17.
LiuJQWangYWangZQSoliton solutions for quasilinear Schrödinger equations, II. J Differential Equations (2003) 187:473–93. 10.1016/s0022-0396(02)00064-5
18.
ChenJZhangQExistence of positive ground state solutions for quasilinear Schrödinger system with positive parameter. Appl Anal (2022) 102:2676–91. 10.1080/00036811.2022.2033232
19.
WillemMMinimax theorems. Berlin: Birkhauser (1996).
20.
ChenSWuXExistence of positive solutions for a class of quasilinear Schrödinger equations of Choquard type. J Math Anal Appl (2019) 475:1754–77. 10.1016/j.jmaa.2019.03.051
21.
do ȮJMMiyagakiOHSoaresSHMSoliton solutions for quasilinear Schrödinger equations with critical growth. J Differential Equations (2010) 248:722–44. 10.1016/j.jde.2009.11.030
22.
FangXSzulkinAMultiple solutions for a quasilinear Schrödinger equation. J Differential Equations (2013) 254:2015–32. 10.1016/j.jde.2012.11.017
23.
LiGOn the existence of nontrivial solutions for quasilinear Schrödinger systems. Boundary Value Probl (2022) 2022:40. 10.1186/s13661-022-01623-z
24.
WuXMultiple solutions for quasilinear Schrödinger equations with a parameter. J Differential Equations (2014) 256:2619–32. 10.1016/j.jde.2014.01.026
25.
ZhangJTangXZhangWInfinitely many solutions of quasilinear Schrödinger equation with sign-changing potential. J Math Anal Appl (2014) 420:1762–75. 10.1016/j.jmaa.2014.06.055
Summary
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
Volume
12 - 2024
Edited by
Pietro Prestininzi, Roma Tre University, Italy
Reviewed by
Jianhua Chen, Nanchang University, China
Li Guofa, Qujing Normal University, China
Updates
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, jinshizhangjing@163.com
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.