Existence of solutions for a class of quasilinear Schrödinger equations on \({\mathbb{R}}\)
 DaBin Wang^{1}Email author and
 Kuo Yang^{1}
Received: 14 October 2015
Accepted: 12 November 2015
Published: 25 November 2015
Abstract
In this paper, we study the existence of nontrivial solution for a class of quasilinear Schrödinger equations in \({\mathbb{R}}\) with the nonlinearity asymptotically linear and, furthermore, the potential indefinite in sign. The tool used in this paper is the direct variation method.
Keywords
MSC
1 Introduction and main result
This problem has been studied by many people in the recent years [3–6], in which the existence of solutions has been established. Besides, Alves et al. [7] consider the existence and concentration of positive solutions as \(\varepsilon\rightarrow0\) for a related equation with \(\varepsilon ^{2}\). At the same time, many authors considered the corresponding high dimensional equations; see [1, 2, 8–11] and the references therein. However, in all the above papers, we notice that the potential was assumed positive definite. In this paper, we will investigate the nontrivial solution for equation (1.1) with the potential indefinite in sign; the tool used in our paper, which is different from the literature mentioned above, is the direct variation method. It is noticed that the ideas in this article come from the paper of Chen and Wang [12], where a SchrödingerPoisson system was considered.
To stated our main result, one needs to describe the eigenvalue of Schrödinger operator \(\Delta+V\):
 (V):

\(V\in C({\mathbb{R}})\) bounded from below and there exists an integer \(k\geq1\) such that \(\lambda_{k}<0<\lambda_{k+1}\).
 (f_{1}):

\(f\in C^{1}({\mathbb{R}}\times {\mathbb{R}})\) and there exist constants \(p>2\) and \(c>0\) such that$$\biglf(x,t) \bigr\leq c \bigl(1+t^{p1} \bigr), \quad\forall x\in {\mathbb{R}}, t \in {\mathbb{R}}. $$
 (f_{2}):

\(f(x,t)=o(t)\) as \(t\rightarrow0\) uniformly in \(x\in {\mathbb{R}}\).
 (f_{3}):

There exists \(0< h<\lambda_{\infty}\) such that \(F(x,t)=\int ^{t}_{0}f(x,s)\,ds\leq\frac{1}{2}ht^{2}\) for all \(x\in {\mathbb{R}}\) and \(t \in {\mathbb{R}}\).
Our main result of this paper is as follows.
Theorem 1.1
Assume (V), (f_{1})(f_{3}) are satisfied, then problem (1.1) has at least one nontrivial solution.
Second, the potential V in this paper is indefinite in sign, so under our conditions (f_{1})(f_{3}), the mountain pass lemma cannot be valid in the same way as in [17, 18]. Furthermore, the competing effect of \(\int_{{\mathbb{R}}}u^{\prime}^{2}u^{2}\,dx\) with the nonlinear term gives rise to much greater difficulty. For results as regards the potential change sign, we also mention [19] and the references in therein.
2 Proof of main result
For any \(r\in[2, \infty]\), the embedding \(E\hookrightarrow L^{r}({\mathbb{R}})\) is continuous.
Proof of Theorem 1.1
Step 1: To proof I is coercive. Suppose it is not true, then there exist \(M>0\) and \(\u_{n}\ \rightarrow\infty\) such that \(I(u_{n})\leq M\).
By using direct variation method, Steps 1 and 2 show that I has a global minimizer. From Step 3, the global minimizer of I is not zero. Therefore, we get a nontrivial solution of problem (1.1). □
Declarations
Acknowledgements
The research was supported by the Natural Science Foundation of China (11561043).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Liu, JQ, Wang, ZQ: Soliton solutions for quasilinear Schrödinger equations. I. Proc. Am. Math. Soc. 131, 441448 (2002) View ArticleGoogle Scholar
 Liu, JQ, Wang, ZQ, Guo, YX: Multibump solutions for quasilinear elliptic equations. J. Funct. Anal. 262, 40404102 (2012) MATHMathSciNetView ArticleGoogle Scholar
 Poppenberg, M, Schmitt, K, Wang, ZQ: On the existence of soliton solutions to quasilinear Schrödinger equations. Calc. Var. Partial Differ. Equ. 14, 329344 (2002) MATHMathSciNetView ArticleGoogle Scholar
 Alves, MJ, Carrião, PC, Miyagaki, OH: Nonautonomous perturbations for a class of quasilinear elliptic equations on \({\mathbb{R}}\). J. Math. Anal. Appl. 344, 186203 (2008) MATHMathSciNetView ArticleGoogle Scholar
 Ambrosetti, A, Wang, ZQ: Positive solutions to a class of quasilinear elliptic equations on \({\mathbb{R}}\). Discrete Contin. Dyn. Syst. 9, 5568 (2003) MATHMathSciNetView ArticleGoogle Scholar
 Chen, JQ, Guo, BL: Multiple nodal bound states for a quasilinear Schrödinger equations. J. Math. Phys. 46, 123502 (2005) MathSciNetView ArticleGoogle Scholar
 Alves, CO, Miyagaki, OH, Soares, SHM: On the existence and concentration of positive solutions to a class of quasilinear elliptic problems on \({\mathbb{R}}\). Math. Nachr. 284, 17841795 (2011) MATHMathSciNetView ArticleGoogle Scholar
 Colin, M, Jeanjean, L: Solutions for a quasilinear Schrödinger equations: a dual approach. Nonlinear Anal. 56, 213226 (2004) MATHMathSciNetView ArticleGoogle Scholar
 do Ó, JM, Miyagaki, OH, Soares, SHM: Soliton solutions for quasilinear Schrödinger equations: the critical exponential case. Nonlinear Anal. 67, 33573372 (2007) MATHMathSciNetView ArticleGoogle Scholar
 Liu, JQ, Wang, YQ, Wang, ZQ: Soliton solutions for quasilinear Schrödinger equations. II. J. Differ. Equ. 187, 473493 (2003) MATHView ArticleGoogle Scholar
 Liu, JQ, Wang, YQ, Wang, ZQ: Solutions for quasilinear Schrödinger equations via Nehari method. Commun. Partial Differ. Equ. 29, 879901 (2004) MATHView ArticleGoogle Scholar
 Chen, SW, Wang, CL: Existence of multiple nontrivial solutions for a SchrödingerPoisson system. J. Math. Anal. Appl. 411, 787793 (2014) MathSciNetView ArticleGoogle Scholar
 Reed, M, Simon, B: Methods of Modern Mathematical Physics, II: Fourier Analysis, SelfAdjointness. Academic Press, New York (1975) MATHGoogle Scholar
 Reed, M, Simon, B: Methods of Modern Mathematical Physics, IV: Analysis of Operators. Academic Press, New York (1978) MATHGoogle Scholar
 Shen, ZP, Han, ZQ: Existence of solutions to quasilinear Schrödinger equations with indefinite potential. Electron. J. Differ. Equ. 2015, 91 (2015) MathSciNetView ArticleGoogle Scholar
 Bartsch, T, Wang, ZQ: Existence and multiplicity results for some superlinear elliptic problems on \({\mathbb{R}}^{n}\). Commun. Partial Differ. Equ. 20, 17251741 (1995) MATHMathSciNetView ArticleGoogle Scholar
 Zhang, J, Tang, X, Zhang, W: Infinitely many solutions of quasilinear Schrödinger equation with signchanging potential. J. Math. Anal. Appl. 420, 17621775 (2014) MATHMathSciNetView ArticleGoogle Scholar
 Zhang, J, Tang, X, Zhang, W: Existence of infinitely many solutions for a quasilinear elliptic equation. Appl. Math. Lett. 37, 131135 (2014) MathSciNetView ArticleGoogle Scholar
 Chen, JQ: Multiple positive solutions to a class of modified nonlinear Schrödinger equations. J. Math. Anal. Appl. 415, 525542 (2014) MATHMathSciNetView ArticleGoogle Scholar