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

AbstractIn this paper, we study a nonlinear Schrödinger–Poisson system $$\left\{ \begin{array}{ll} -\Delta u+V_{\lambda }( x) u+\mu K( x) \phi u=f (x, u) & \text{in}\;\mathbb{R}^{3}, \\ -\Delta \phi =K ( x ) u^{2} & \text{in}\;\mathbb{R}^{3},\end{array}\right.$$-Δu+Vλ(x)u+μK(x)ϕu=f(x,u)inR3,-Δϕ=K(x)u2inR3,where $${\mu > 0}$$μ>0 is a parameter, $${V_{\lambda }}$$Vλ is allowed to be sign-changing and f is an indefinite function. We require that $${V_{\lambda }:=\lambda V^{+}-V^{-}}$$Vλ:=λV+-V- with V+ having a bounded potential well Ω whose depth is controlled by λ and $${V^{-} \geq 0}$$V-≥0 for all $${x\in \mathbb{R} ^{3}}$$x∈R3. 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.