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Existence of solutions for periodic elliptic system with general superlinear nonlinearity

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Abstract

This paper is concerned with the following elliptic system:

$$ \left\{ \begin{array}{lll}-\triangle u+V(x)u=W_{v}(x, u, v), \quad\, x \in \mathbb{R}^{N}, \\ -\triangle v+V(x)v=W_{u}(x, u, v), \quad\, x \in \mathbb{R}^{N}, \\ u, v \in H^{1}(\mathbb{R}^{N}), \end{array}\right.$$

where V and W are 1-periodic in x, and W(x,s,t) is super-quadratic in \({|as+bt|}\) as \({|as+bt|\rightarrow \infty}\) a.e. \({x\in \mathbb{R}^N}\) for some a, b > 0. By using a generalized linking theorem established by Li and Szulkin, we are able to obtain the existence of nontrivial solutions under some more generic assumptions on the nonlinearity.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan People’s Republic of China

    Fangfang Liao, X. H. Tang & Jian Zhang

  2. Department of Mathematics, Xiangnan University, Chenzhou, 423000, Hunan People’s Republic of China

    Fangfang Liao

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  1. Fangfang Liao
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  2. X. H. Tang
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  3. Jian Zhang
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Correspondence to X. H. Tang.

Additional information

This work is partially supported by the NNSF (No: 11171351) and SRFDP (No: 20120162110021) of China, Scientific Research Fund of Hunan Provincial Education Department (12C0895) and the Construct Program of the Key Discipline in Hunan Province.

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Liao, F., Tang, X.H. & Zhang, J. Existence of solutions for periodic elliptic system with general superlinear nonlinearity. Z. Angew. Math. Phys. 66, 689–701 (2015). https://doi.org/10.1007/s00033-014-0425-6

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  • DOI: https://doi.org/10.1007/s00033-014-0425-6

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