Multiple localized solutions of high topological type for semiclassical nonlinear Schrödinger-Poisson system

Abstract

We investigate localized solutions and their concentration behaviors of singularly perturbed Schrödinger-Poisson system \begin{equation*} \left\{ \begin{aligned} & -\varepsilon^{2}\Delta v+V(x)v+\phi v=P(x)f(v), \ x\in\mathbb{R}^{3}\\ & -\varepsilon^{2}\Delta \phi=v^{2}, \ x\in\mathbb{R}^{3} . \end{aligned} \right. \end{equation*} Here, $V, P\in C^{1}(\mathbb{R}^{3},\mathbb{R})$. Without the Ambrosetti-Rabinowitz type condition and $f$ only belongs to $C(\mathbb{R},\mathbb{R})$, by using the penalization techniques in [31] and the Nehari manifold method in [40,41], we study multiple localized solutions of high topological type for above system.