Multiple positive bound states for critical Schr\"odinger-Poisson systems

TL;DR

This paper uses variational methods to establish the existence and multiplicity of positive bound states in a critical Schr"odinger-Poisson system, addressing challenges from unbounded domain and critical nonlinear growth.

Contribution

It proves new results on positive bound states for a critical Schr"odinger-Poisson system, highlighting multiplicity and existence without ground states.

Findings

Existence of multiple positive bound states.

Overcoming lack of compactness due to unbounded domain.

Handling critical nonlinear growth in the system.

Abstract

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