Regularizing effect for a system of Schrödinger–Maxwell equations

Abstract

AbstractWe prove some existence results for the following Schrödinger–Maxwell system of elliptic equations:\left\{\begin{aligned} &\displaystyle{-}\div(M(x)\nabla u)+A\varphi|u|^{r-2}u=% f,&&\displaystyle u\in W_{0}^{1,2}(\Omega),\\ &\displaystyle{-}\div(M(x)\nabla\varphi)=|u|^{r},&&\displaystyle\varphi\in W_{% 0}^{1,2}(\Omega).\end{aligned}\right.In particular, we prove the existence of a finite energy solution {(u,\varphi)} if {r>2^{*}} and f does not belong to the “dual space” {L^{\frac{2N}{N+2}}(\Omega)}.