Multiple Positive Solutions for a Nonhomogeneous Schrö dinger-Poisson System with Critical Exponent

Abstract

In this paper, we consider the following nonhomogeneous Schrödinger- Poisson system

$\left\{\begin{array}{ll}-\Delta u+u+\eta \phi u=u^{5}+\lambda f(x), & x \in \mathbb{R}^{3} \\-\Delta \phi=u^{2}, & x \in \mathbb{R}^{3}\end{array}\right.$

where $\eta \neq 0$, λ>0 is a real parameter and $f \in L^{\frac{6}{5}}\left(\mathbb{R}^{3}\right)$ is a nonzero nonnegative function. By using the Mountain Pass theorem and variational method, for λ small, we show that the system with η >0 has at least two positive solutions, the system with η <0 has at least one positive solution. Our result generalizes and improves some recent results in the literature.

Keywords

Schrödinger-Poisson system / critical exponent / variational method / positive solu- tions