Schr\"odinger-Maxwell systems on Hadamard manifolds

In this paper we study nonlinear Schr"odinger-Maxwell systems on $n-$dimensional Hadamard manifolds, $3\le n\le 5.$ The main difficulty resides in the lack of compactness of such manifolds which is recovered by exploring suitable isometric actions. By combining variational arguments, some existence, uniqueness and multiplicity of isometry-invariant weak solutions are established for the Schr"odinger-Maxwell system depending on the behavior of the nonlinear term.