Multiple positive bound states for critical Schrödinger-Poisson systems

Abstract

Using variational methods we prove some results about existence and multiplicity of positive bound states of to the following Schrödinger-Poisson system: $$ \left\{ \begin{array}{l} \vspace{2mm} -Δu+V(x)u+K(x)ϕ(x)u=u^5 -Δϕ=K(x)u^2\qquad x\in\R^3 \end{array}\right.\quad\quad (SP) $$ We remark that $(SP)$ exhibits a "double" lack of compactness because of the unboundedness of ${\mathbb{R}}^3$ and the critical growth of the nonlinear term and that in our assumptions ground state solutions of $(SP)$ do not exist.