Multiplicity of semiclassical states for Schrödinger–Poisson systems with critical frequency

Abstract

We are concerned with the Schrodinger–Poisson system $$\begin{aligned} \left\{ \begin{array}{lll} -\,\epsilon ^2\Delta u+V(x)u+K(x)\phi u=u^5,&{}\quad x\in {\mathbb {R}}^3,\\ -\,\Delta \phi =K(x)u^2, &{}\quad x\in {\mathbb {R}}^3, \end{array}\right. \end{aligned}$$where $$\epsilon >0$$ is a parameter, $$V\in L^{\frac{3}{2}}({\mathbb {R}}^3)$$ and $$K\in L^{2}({\mathbb {R}}^3)$$ are nonnegative functions and V is assumed to be zero in some region of $${\mathbb {R}}^3$$, which means it is of the critical frequency case. By virtue of a global compactness lemma and Lusternik–Schnirelman theory, we show the multiplicity of high energy semiclassical states.