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

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 $λ>0$ 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 $-Δ+V$. This system is strongly indefinite, i.e., the operator $-Δ+V$ has infinite-dimensional negative and positive spaces and it has a competitive interplay of the nonlinearities $|u|^{p-2}u$ and $λϕu$. 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 $λ>0,$ this system has a nontrivial solution.

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