On a critical Maxwell equation in nonlocal media

TL;DR

This paper investigates the existence of solutions for a critical Maxwell equation in nonlocal media, employing advanced mathematical techniques to establish ground state solutions and sharp inequalities involving curl operators.

Contribution

It introduces Coulomb spaces and applies the Nehari-Pankov manifold method to find solutions, along with deriving sharp constants for Sobolev-like inequalities in a nonlocal context.

Findings

Existence of ground state solutions for the critical Maxwell equation.

Derivation of sharp constants for Sobolev-like inequalities with curl operator.

Development of a nonlocal concentration-compactness principle.

Abstract

In this paper, we study the existence of solutions for a critical time-harmonic Maxwell equation in nonlocal media. By introducing some suitable Coulomb spaces involving curl operator, 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.