Abstract:

In this paper, the existence of ground state and bound state solutions for the following Schrödinger-Poisson system

$\begin{align*} \left\{\begin{array}{ll} -\Delta u+V(x)u+\phi (x)u=|u|^{p-2}u \quad \text{in} \quad\mathbb{R}^{3},\\ -\Delta \phi (x)=u^{2} \quad \text{in} \quad\mathbb{R}^{3} \end{array} \right. \end{align*}$

is studied, where $V$ is a nonstabilizing continuous potential and $p\in(4,6)$. It is proved that the shell equation has a ground state solution by using the concentration-compactness principle when the potential function $V$ is almost periodic on $\mathbb{R}^{3}$. Moreover, a bound state solution is obtained when $V(x)=V(x^{1}, x^{2}, x^{3})$ is $T_{i}$-periodic in $x^{i}$ for $i=2,3$ and almost periodic in $x^{1}$ uniformly with respect to $(x^{2},x^{3})\in [0,T_{2}]\times[0,T_{3}]$.

Key words: Schrödinger-Poisson system, nonstabilizing potential, almost periodic function, concentration-compactness principle