Semiclassical states for singularly perturbed Schrödinger-Poisson systems with a general Berestycki-Lions or critical nonlinearity

Abstract

This paper is concerned with two classes of singularly perturbed Schrödinger-Poisson systems of the form \begin{equation*} \begin{cases} -\varepsilon^2\triangle u+u+ \phi u=f(u), & x\in {\mathbb{R}}^{3},\\ -\triangle \phi=u^2, & x\in \mathbb{R}^3, \end{cases} \end{equation*} and \begin{equation*} \begin{cases} -\varepsilon^2\triangle u+V(x)u+ \phi u=g(x,u)+K(x)u^5, & x\in {\mathbb{R}}^{3},\\ -\triangle \phi=u^2,& x\in \mathbb{R}^3, \end{cases} \end{equation*} where $\varepsilon > 0$ is a small parameter. We prove that: (1) the first system admits a concentrating bounded state for small $\varepsilon$, where $f\in \mathcal{C}(\mathbb{R},\mathbb{R})$ satisfies Berestycki-Lions assumptions which are almost necessary; (2) there exists a constant $\varepsilon_0> 0$ determined by $V,K$ and $g$ such that for any $\varepsilon\in (0,\varepsilon_0]$ the second system has a nontrivial solution, where $V,K\in \mathcal{C}(\mathbb{R}^3,\mathbb{R})$, $V(x)\ge 0$, $K(x)> 0$, $g\in \mathcal{C}(\mathbb{R}^3\times \mathbb{R},\mathbb{R})$ is an indefinite function. Our results improve and complement the previous ones in the literature.