A Singular System of Schrödinger-Maxwell Equations

Abstract

We study existence of solutions for the singular system of Schrödinger-Maxwell equations

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{l} u \in W^{1,2}_{0}(\Omega ):\, -{{\,\text {div}\,}}(A(x)\nabla u) + \psi ^{\theta }\,u^{r-1} = f(x), \\ \psi \in W^{1,2}_{0}(\Omega ):\, -{{\,\text {div}\,}}(B(x)\nabla \psi ) = \dfrac{u^{r}}{\psi ^{1-\theta }}. \end{array} \right. \end{aligned} \end{aligned}$$

Here \(r > 1\), \(0< \theta < 1\), and \(f(x) \ge 0\) belongs to suitable Lebesgue spaces. We will also prove that the solution \((u,\psi )\) is a saddle point of a suitable functional.