Existence and non-existence of solitary waves for the critical Klein–Gordon equation coupled with Maxwellʹs equations Original Research Article

In this paper we investigate 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 non-existence result by a Pohožaev-type argument and then, adding a suitable perturbation, we recover existence of at least a radially symmetric solution. We overcome the lack of compactness relying on the Brezis–Nirenberg method.