Existence and Multiplicity of Solutions for Semilinear Elliptic Systems with Periodic Potential

  • Guofeng Che
  • Haibo ChenEmail author
  • Liu Yang


In this paper, we consider the following semilinear elliptic systems:
$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u+V(x)u=F_{u}(x, u, v),\quad \text{ in } \mathbb {R}^{N},\\ -\Delta v+V(x)v=F_{v}(x, u, v),\quad \text{ in } \mathbb {R}^{N},\\ \end{array} \right. \end{aligned}$$
where \(V:\mathbb {R}^{N}\rightarrow \mathbb {R},~F_{u}(x,u,v)\) and \(F_{v}(x,u,v)\) are periodic in x. We assume that 0 is a right boundary point of the essential spectrum of \(-\triangle +V\). Under appropriate assumptions on \(F_{u}(x, u, v)\) and \(F_{v}(x, u, v)\), we prove the above system has a ground-state solution by using the Nehari-type technique in a strongly indefinite setting. Furthermore, the existence of infinitely many geometrically distinct solutions is obtained via variational methods. Recent results from the literature are improved and extended.


Semilinear elliptic systems Strongly indefinite functional Ground state Nehari–Pankov manifold Variational methods 

Mathematics Subject Classification

35B38 35J20 


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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsHengyang Normal UniversityHengyangPeople’s Republic of China

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