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

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 Pohozaev-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. (C) 2004 Elsevier Ltd. All rights reserved.