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Solutions on Asymptotically Periodic Elliptic System with New Conditions

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Abstract

This paper is concerned with the following elliptic system:

$$\left\{\begin{array}{l@{\quad}l@{\quad}l}-\Delta u+U_{1}(x)u =F_{u}(x,u,v)&\mbox{in}&\mathbb{R}^{N},\\ -\Delta v+U_{2}(x)v=F_{v}(x,u,v)& \mbox{in}&\mathbb{R}^{N},\\ u, \ v \in H^{1}(\mathbb{R}^{N}).\end{array}\right.$$

Assuming that the potential U i (x) are periodic in x and 0 lies in a spectral gap of \({\sigma (-\Delta + U_{i}), i=1,2}\), two types of ground state solutions are obtained with some new super-quadratic conditions on nonlinearity F which are weaker that some well known ones. For the case that U i (x) and F(x,u,v) are asymptotically periodic in x, a nontrivial solution is established by using a generalized linking theorem and some new techniques.

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Authors and Affiliations

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

    Dongdong Qin & Xianhua Tang

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  1. Dongdong Qin
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  2. Xianhua Tang
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Correspondence to Dongdong Qin.

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Qin, D., Tang, X. Solutions on Asymptotically Periodic Elliptic System with New Conditions. Results. Math. 70, 539–565 (2016). https://doi.org/10.1007/s00025-015-0491-x

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  • DOI: https://doi.org/10.1007/s00025-015-0491-x

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