Bound state solutions for some non-autonomous asymptotically cubic Schrödinger–Poisson systems

Abstract

We are concerned with the following Schrodinger–Poisson systems $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u+u+K(x)\phi (x)u=q(x)f(u), &{}\quad x\in {\mathbb {R}}^3, \\ -\Delta \phi =K(x)u^2, &{} \quad x\in {\mathbb {R}}^3, \end{array} \right. \end{aligned}$$ where f is asymptotically cubic, $$\lim _{|x|\rightarrow \infty }K(x)=0$$ and $$\lim _{|x|\rightarrow \infty }q(x)=q_{\infty }>0$$ . We establish the existence of bound state solutions to this problem by using the method developed in Szulkin and Weth [20, 21].