Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell's equations

In this paper we investigate the existence and non-existence of solutions for a nonlinear elliptic system arising from the coupling of the nonlinear Klein-Gordon equation with the Maxwell equations, when the nonlinearity exhibits critical growth. We prove a nonexistence result by a Pohozaev type argument and then, adding a suitable perturbation, we recover the existence of at least a radially symmetric solution. We overcome the lack of compactness relying on the Brezis-Nirenberg method.