On a critical time-harmonic Maxwell equation in nonlocal media

Abstract

In this paper, we study the existence of solutions for a critical time–harmonic Maxwell equation in nonlocal media \[ \begin{cases} \nabla\times(\nabla\times u)+\lambda u=\left(I_{\alpha}\ast|u|^{2^{{\ast}}_{\alpha}}\right)|u|^{2^{{\ast}}_{\alpha}-2}u & \mathrm{in}\ \Omega,\\ \nu\times u=0 & \mathrm{on}\ \partial\Omega, \end{cases} \] where $\Omega \subset \mathbb {R}^{3}$ is a bounded domain, either convex or with $\mathcal {C}^{1,1}$ boundary, $\nu$ is the exterior normal, $\lambda <0$ is a real parameter, $2^{\ast }_{\alpha }=3+\alpha$ 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 $W^{\alpha,2^{\ast }_{\alpha }}_{0}(\mathrm {curl};\Omega )$ , 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.