Dual variational methods for time-harmonic nonlinear Maxwell's equations

TL;DR

This paper introduces dual variational methods to establish the existence of infinitely many solutions for time-harmonic nonlinear Maxwell's equations, utilizing a novel symmetric mountain pass theorem that bypasses the Palais-Smale condition.

Contribution

It develops a new dual variational framework and a modified mountain pass theorem to prove multiple solutions without the Palais-Smale condition.

Findings

Proves existence of infinitely many solutions on bounded domains.

Establishes solutions on ^3 (3D space).

Introduces a new variational approach for nonlinear Maxwell's equations.

Abstract

We prove the existence of infinitely many nontrivial solutions for time-harmonic nonlinear Maxwell's equations on bounded domains and on $\mathbb{R}^3$ using dual variational methods. In the dual setting we apply a new version of the Symmetric Mountain Pass Theorem that does not require the Palais-Smale condition.