Multiplicity of solutions for Schrödinger-Poisson system with critical exponent in $\mathbb{R}^{3}$

Abstract

<abstract> <p>In this paper, we study the following Schrödinger-Poisson system with critical exponent</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document} $ \begin{equation*} \begin{cases} -\Delta u-k(x)\phi u=\lambda h(x)|u|^{p-2}u+s(x)|u|^{4}u, \ ~~x\in\mathbb{R}^{3},\\ -\triangle\phi=k(x)u^{2}, \ \ \ \ \ \ \ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ x\in\mathbb{R}^{3}, \\ \end{cases} \end{equation*} $ \end{document} </tex-math> </disp-formula></p> <p>where $1 < p < 2$ and $\lambda > 0.$ Under suitable conditions on $k$, $h$ and $s$, we show that there exists $\lambda^{\ast}>0$ such that the above problem possesses infinitely many solutions with negative energy for each $\lambda\in(0, \lambda^{\ast})$. Moreover, we prove the existence of infinitely many solutions with positive energy. The main tools are the concentration compactness principle, $Z_{2}$ index theory and Fountain Theorem. These results extend some existing results in the literature.</p> </abstract>