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

TL;DR

The paper develops a dual variational framework for time-harmonic nonlinear Maxwell equations, leveraging a Helmholtz decomposition to reduce curl-curl problems to variational equilibria of a dual functional. It proves a generalized symmetric mountain-pass theorem without requiring the Palais–Smale condition, and applies it to show that the nonlinear Neumann problem on bounded domains has a ground state and infinitely many bound states, while the homogeneous problem on $\mathbb{R}^3$ with isotropic constants admits dual ground states and infinitely many $L^p$-solutions. The approach uses a Limiting Absorption Principle, a Div-Curl-Lemma, and a PS-attracting compactness framework to obtain multiplicity and geometric distinctness of solutions. Together, these results extend dual variational methods to curl-curl Maxwell operators and provide a robust path from abstract critical-point theory to concrete Maxwell-type boundary-value problems. The work highlights the utility of dual formulations in handling strongly indefinite curl-curl problems without relying on Nehari manifolds.

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.

Theorems & Definitions (53)

• Theorem 1
• Theorem 2
• Definition 3
• Definition 4
• Theorem 5
• Remark 6
• Proposition 7
• proof
• Proposition 8
• proof