Solutions to a nonlinear Maxwell equation with two competing nonlinearities in $\mathbb{R}^3$

TL;DR

This paper investigates solutions to a nonlinear Maxwell equation in three dimensions with two competing nonlinearities, establishing a link to Schrödinger equations and proving the existence of least energy solutions under certain conditions.

Contribution

It introduces a novel equivalence between cylindrically symmetric Maxwell solutions and Schrödinger equations with singular potentials, and proves the existence of least energy solutions.

Findings

Existence of cylindrically equivariant solutions to the Maxwell equation.

Equivalence between Maxwell solutions and Schrödinger equation solutions.

Existence of least energy solutions in the considered setting.

Abstract

We are interested in the nonlinear, time-harmonic Maxwell equation $$ \nabla \times (\nabla \times \mathbf{E} ) + V(x) \mathbf{E} = h(x, \mathbf{E})\mbox{ in } \mathbb{R}^3 $$ with sign-changing nonlinear term $h$, i.e. we assume that $h$ is of the form $$ h(x, \alpha w) = f(x, \alpha) w - g(x, \alpha) w $$ for $w \in \mathbb{R}^3$, $|w|=1$ and $\alpha \in \mathbb{R}$. In particular, we can consider the nonlinearity consisting of two competing powers $h(x, \mathbf{E}) = |\mathbf{E}|^{p-2}\mathbf{E} - |\mathbf{E}|^{q-2}\mathbf{E}$ with $2 < q < p < 6$. Under appriopriate assumptions, we show that weak, cylindrically equivariant solutions of the special form are in one-to-one correspondence with weak solutions to a Schr\"odinger equation with a singular potential. Using this equivalence result we show the existence of the least energy solution among cylindrically equivariant solutions of the particular form to the Maxwell equation, as well as to the Schr\"odinger equation.