Existence of Solutions for Modified Schrödinger-Poisson System with Critical Nonlinearity in $\mathbb{R}^{3}$

Abstract

In this paper, we study the existence and multiplicity of semiclassical solutions of a modified version of the Schrödinger-Poisson system with critical nonlinearity in $\mathbb{R}^{3}$. Under some given conditions which are given in Section 1, we prove that the problem has at least one nontrivial solution provided that $\epsilon \leq \varepsilon$ and that for any $n^{*} \in \mathbb{N}$, it has at least $n^{*}$ pairs of solutions if $\epsilon \leq \varepsilon_{n^{*}}$, where $\varepsilon$ and $\varepsilon_{n^{*}}$ are sufficiently small positive numbers. Moreover, these solutions $u_{\epsilon} \to 0$ in $H^{1}(\mathbb{R}^{3})$ as $\epsilon \to 0$.