Schrödinger-Maxwell systems with interplay between coefficients and data

Abstract

In this paper, we prove existence of a solution $(u,\varphi)$ in $W_0^{1,2}(\Omega) \times W_0^{1,2}(\Omega)$ for the Schrödinger-Maxwell type system $$ (*)\quad \left\{ \begin{array}{cl} -{\rm div} (M(x) \nabla u) + a(x)\,\varphi \,|u|^{r-2} u = f(x)\,, & \mbox{in $\Omega$,} \\ -{\rm div}(N(x) \nabla \varphi) = a(x) |u|^{r}\,, & \mbox{in $\Omega$,} \\ u = \varphi = 0\,, & \mbox{on $\partial\Omega$,} \end{array} \right. $$ with $M$ and $N$ bounded, uniformly elliptic matrices, $r > 1$, $a(x)$ in $L^{1}(\Omega)$ and $f$ such that $|f(x)| \leq Q\,a(x)$ for some positive constant $Q$.