A Note on the Existence Results for Schrödinger–Maxwell System with Super-Critical Nonlinearitie

Abstract

The paper considers the following Schrodinger–Maxwell system with supercritical nonlinearitie, 0.1$$ \textstyle\begin{cases} -\Delta u+K(x) \phi u =|u|^{p-1}u+h(x), \quad \mbox{in} \varOmega , -\Delta \phi = K(x)u^{2}, \quad \mbox{in} \varOmega , \phi = u=0, \quad \mbox{in} \partial \varOmega , \end{cases} $$ where $\varOmega \subset \mathbb{R}^{3}$ is a bounded domain with smooth boundary, $1< p \mbox{and} K$, $h\in {L^{\infty }} (\varOmega )$. We prove the existence of at least one non-trivial weak solution. This result is already known for the subcritical case. In this paper, we extend it to the supercritical values of $p$ as well. We use a new variational principle to prove our result.