On a critical time-harmonic Maxwell equation in nonlocal media

In this paper, we study the existence of solutions for a critical time-harmonic Maxwell equation in nonlocal media ( del x (del x u) + lambda u = I-alpha* |u|(2)(alpha)& lowast;|u|(2)(alpha)& lowast;(-2)u in Omega, nu x u = 0 on partial derivative Omega, where Omega subset of R-3 is a bounded domain, either convex or with C-1,C-1 boundary, nu is the exterior normal, lambda < 0 is a real parameter, 2(alpha)(& lowast;) = 3+ alpha with 0 < alpha < 3 is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality. By introducing some suitable Coulomb spaces involving curl operator W-0(alpha)alpha,2 & lowast; (curl; Omega), we are able to alpha obtain the ground state solutions of the curl-curl equation via the method of constraining Nehari-Pankov manifold. Correspondingly, some sharp constants of the Sobolev-like inequalities with curl operator are obtained by a nonlocal version of the concentration-compactness principle.