On the nonlinear Schrödinger–Poisson systems with sign-changing potential

Abstract

In this paper, we study a nonlinear Schrodinger–Poisson system $$\left\{ \begin{array}{ll} -\Delta u+V_{\lambda }( x) u+\mu K( x) \phi u=f (x, u) & \text{in}\;\mathbb{R}^{3}, \\ -\Delta \phi =K ( x ) u^{2} & \text{in}\;\mathbb{R}^{3},\end{array}\right.$$ where \({\mu > 0}\) is a parameter, \({V_{\lambda }}\) is allowed to be sign-changing and f is an indefinite function. We require that \({V_{\lambda }:=\lambda V^{+}-V^{-}}\) with V+ having a bounded potential well Ω whose depth is controlled by λ and \({V^{-} \geq 0}\) for all \({x\in \mathbb{R} ^{3}}\). Under some suitable assumptions on K and f, the existence and the nonexistence of nontrivial solutions are obtained by using variational methods. Furthermore, the phenomenon of concentration of solutions is explored as well.