Ground states for Schrödinger–Poisson type systems

Abstract

In this paper we consider the following elliptic system in ${\mathbb{R}^3}$

$$\qquad\left\{\begin{array}{ll}-\Delta u+u+\lambda K(x)\phi u=a(x)|u|^{p-1}u \quad &x \in {\mathbb{R}}^{3}\\ -\Delta \phi=K(x)u^{2} \quad &x \in {\mathbb{R}}^{3}\end{array}\right.$$

where λ is a real parameter, ${p\in (1, 5)}$ if λ < 0 while ${p\in (3, 5)}$ if λ > 0 and K(x), a(x) are non-negative real functions defined on ${\mathbb{R}^3}$ . Assuming that ${\lim_{|x|\rightarrow+\infty}K(x)=K_{\infty} >0 }$ and ${\lim_{|x|\rightarrow+\infty}a(x)=a_{\infty} >0 }$ and satisfying suitable assumptions, but not requiring any symmetry property on them, we prove the existence of positive ground states, namely the existence of positive solutions with minimal energy.