Ground-state solutions for the electrostatic nonlinear Klein-Gordon-Maxwell system

In this paper, we study the nonlinear Klein-Gordon equation coupled with the Maxwell equation in the electrostatic case: {-Delta u + [m(2) - (e phi + omega)(2)]u = f (u), in R(3), Delta phi = e(e phi + omega)u(2), in R(3), (P) where m, e, omega > 0. Benci and Fortunato (2002) [3] and D'Aprile and Mugnai (2004) [6], showed that, for any u is an element of H(1) (R(3)), the second equation of problem (P) has a unique solution phi(u) is an element of D(1,2)(R(3)), the map Lambda : u. H(1)(R(3)) -> phi(u) is an element of D(1,2)(R(3)) is continuously differentiable, and phi(u) is an element of [-omega/e, 0]. Furthermore, we prove that max {- omega/e - phi(u), phi(u)} <= psi(u) <= 0, where psi(u) = Lambda'(u)[u]/2. Then, we consider the ground-state solution of problem (P) with f (u) = vertical bar u vertical bar(p-2)u, 2 < p < 6. (C) 2011 Elsevier Ltd. All rights reserved.