Existence of a nontrivial solution for a strongly indefinite periodic Schrodinger-Poisson system

Abstract

We consider the Schr\"odinger-Poisson system \begin{eqnarray}\left\{\begin{array} [c]{ll} -\Delta u+V(x) u+|u|^{p-2}u=\lambda \phi u, & \mbox{in}\mathbb{R}^{3},\\ -\Delta\phi= u^{2}, & \mbox{in}\mathbb{R}^{3}. \end{array} \right.\nonumber \end{eqnarray} where $\lambda>0$ is a parameter, $3< p<6$, $V\in C(\mathbb{R}^{3}) $ is $1$-periodic in $x_j$ for $j = 1,2,3$ and 0 is in a spectral gap of the operator $-\Delta+V$. This system is strongly indefinite, i.e., the operator $-\Delta+V$ has infinite-dimensional negative and positive spaces and it has a competitive interplay of the nonlinearities $|u|^{p-2}u$ and $\lambda \phi u$. Moreover, the functional corresponding to this system does not satisfy the Palai-Smale condition. Using a new infinite-dimensional linking theorem, we prove that, for sufficiently small $\lambda>0,$ this system has a nontrivial solution.