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 \[ {cases} ∇×(∇× u)+λ u=(Iα|u|^{2^{{}}α})|u|^{2^{{}}α-2}u & in\ Ω,\\ ν× u=0 & on\ ∂Ω, {cases} \] where Ω ⊂ R³ is a bounded domain, either convex or with C1,1 boundary, ν is the exterior normal, λ <0 is a real parameter, 2α=3+α with 0<α <3 is the upper critical exponent due to the Hardy–Littlewood–Sobolev inequality. By introducing some suitable Coulomb spaces involving curl operator W^α,2α₀(curl;Ω ) , we are able to 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.