On critical Klein-Gordon-Maxwell systems with super-linear nonlinearities

This paper is concerned with two classes of critical Klein-Gordon-Maxwell systems as follows {-Delta u + V(x)u - (2 omega + phi)phi u = mu f(u) + u(5), x is an element of R-3, Delta phi = (omega + phi)u(2), x is an element of R-3 and {-epsilon(2)Delta u + V(x)u - (2 omega + phi)phi u = g(x, u) + K(x)u(5), x is an element of R-3, Delta phi = (omega + phi)u(2), x is an element of R-3, where mu > 0, epsilon > 0, V, K is an element of C(R-3, R+), f is an element of C(R, R) and g is an element of C(R-3 x R, R) are super-linear at infinity. Instead of the expression "mu sufficiently large" in the existing works, we give a certain range mu >= mu(0) under which the first system admits a ground state solution when V is positive and periodic. Under some mild conditions on V, K and g, we show that the second system has at least one nontrivial solution provided epsilon is an element of (0, epsilon(0)], where the bound epsilon(0) > 0 is expressed explicitly in terms of V, K and g. (C) 2020 Elsevier Ltd. All rights reserved.