Schrodinger-Maxwell systems on compact Riemannian manifolds

Abstract

In this paper we are focusing to the following Schr\"odinger-Maxwell system $(\mathcal{SM}_{\Psi(\lambda,\cdot)}^{e})$: \[ \begin{cases} -\Delta_{g}u+\beta(x)u+eu\phi=\Psi(\lambda,x)f(u) & \mathrm{in}\ M -\Delta_{g}\phi+\phi=qu^{2} & \mathrm{\mathrm{in}\ M} \end{cases} \] where $(M,g)$ is a 3-dimensional compact Riemannian manifold without boundary, $e,q>0$ are positive numbers, $f:\mathbb{R}\to\mathbb{R}$ is a continuous function, $\beta\in C^{\infty}(M)$ and $\Psi\in C^{\infty}(\mathbb{R}_{+}\times M)$ are positive functions. By various variational approaches, existence of multiple solutions of the problem $(\mathcal{SM}_{\Psi(\lambda,\cdot)}^{e})$ is established.