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

Abstract

In this paper, we consider the following nonhomogeneous Schrödinger-Poisson system $$\begin{cases}-∆u+u+\eta \phi u=u^5+\lambda f(x), \ & x\in\mathbb{R}^3,\\ -∆\phi=u^2, \ & x\in\mathbb{R}^3, \end{cases}$$where $\eta\ne 0,$ $λ>0$ is a real parameter and $f∈L^{\frac{6}{5}}(\mathbb{R}^3)$ is a nonzero nonnegative function. By using the Mountain Pass theorem and variational method, for $λ$ small, we show that the system with $\eta >0$ has at least two positive solutions, the system with $\eta<0$ has at least one positive solution. Our result generalizes and improves some recent results in the literature.