Existence and symmetry breaking of ground state solutions for Schrödinger–Poisson systems

Abstract

We study the Schrödinger–Poisson system:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+u+\lambda \phi u=a\left( x\right) \left| u\right| ^{p-2}u &{} \text { in }{{\mathbb {R}}}^{3}, \\ -\Delta \phi =u^{2} &{} \ \text {in }{{\mathbb {R}}}^{3}, \end{array} \right. \end{aligned}$$

where parameter $\lambda >0$, $2<p<3$ and $a\left( x\right)$ is a positive continuous function in ${{\mathbb {R}}}^{3}$. Assuming that $a\left( x\right) \ge \lim _{\left| x\right| \rightarrow \infty }a\left( x\right) =a_{\infty }>0$ and other suitable conditions, we explore the energy functional corresponding to the system which is bounded below on $H^{1}\left( {{\mathbb {R}}}^{3}\right)$ and the existence and multiplicity of positive (ground state) solutions for $\left[ \frac{A\left( p\right) }{p} a_{\infty }\right] ^{2/\left( p-2\right) }<\lambda \le \left[ \frac{A\left( p\right) }{p}a_{1}\right] ^{2/\left( p-2\right) },$ where $A\left( p\right) :=2^{\left( 6-p\right) /2}\left( 3-p\right) ^{3-p}\left( p-2\right) ^{\left( p-2\right) }$ and $a_{\infty }<a_{1}<a_{\max }:=\sup _{x\in {{\mathbb {R}}} ^{3}}a\left( x\right) .$ More importantly, when $a\left( x\right) =a\left( \left| x\right| \right)$ and $a\left( 0\right) =a_{\max },$ we establish the existence of non-radial ground state solutions.