Multiple solutions for a Schrödinger-Poisson-Slater equation with external Coulomb potential

Abstract

Consider the following Schrödinger-Poisson-Slater system,

$\left\{ \begin{gathered} - \Delta u + (\omega - \tfrac{\beta } {{|x|}})u + \lambda \varphi (x)u = |u|^{p - 1} u,x \in \mathbb{R}^3 , \hfill \\ - \Delta \varphi = u^2 ,u \in H^1 (\mathbb{R}^3 ), \hfill \\ \end{gathered} \right. $

where ω > 0, λ > 0 and β > 0 are real numbers, p ∈ (1, 2). For β = 0, it is known that problem (P) has no nontrivial solution if λ > 0 suitably large. When β > 0, −β/|x| is an important potential in physics, which is called external Coulomb potential. In this paper, we find that (P) with β > 0 has totally different properties from that of β = 0. For β > 0, we prove that (P) has a ground state and multiple solutions if λ > c _p,ω , where c _p,ω > 0 is a constant which can be expressed explicitly via ω and p.