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

Abstract

We consider the Schrödinger-Poisson system \begin{eqnarray}\left\{\begin{array} [c]{ll} -Δu+V(x) u+|u|^{p-2}u=λϕu, & \mbox{in}\mathbb{R}^{3},\\ -Δϕ= 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 $\lambda \varphi u$. 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.

19 pages

References in corpus (1)

• Standing waves with large frequency for 4-superlinear Schrödinger-Poisson systems