Dual variational methods for static Nonlinear Maxwell's Equations

Abstract

We prove the existence of a ground state and infinitely many geometrically distinct solutions for static nonlinear Maxwell's equations on ${\mathbb{R}}^3$. Our existence result relies on a variant of the Symmetric Mountain Pass Theorem that applies to periodic as well as vanishing nonlinearities. It is applied in a dual variational setting and thus provides an alternative approach with respect to the direct variational method introduced by Mederski.