Dual variational methods for static Nonlinear Maxwell's Equations

TL;DR

The paper addresses static nonlinear Maxwell equations with spatially varying permeability $\mu$ and nonlinear polarization $f$ in $\mathbb{R}^3$. It develops a dual variational framework by introducing the polarization $P$ and the inverse nonlinearity $\psi$, yielding a dual functional $J$ whose critical points correspond to solutions of the original system. Under periodic or vanishing nonlinearities (conditions $(A')_{per}$ or $(A')_{van}$), it proves the existence of a ground state and infinitely many geometrically distinct bound states via a symmetric mountain-pass-type critical-point theorem and PS-compactness arguments. This approach extends existence results to non-constant $\mu$ and decaying coefficients, offering an alternative to direct variational methods and highlighting potential technical advantages of the dual formulation.

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.

Theorems & Definitions (33)

• Theorem 1
• Remark 2
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof
• Lemma 6
• proof