Ground and Bound State Solutions of Semilinear Time-Harmonic Maxwell Equations in a Bounded Domain

Abstract

We find solutions ${E : \Omega \to \mathbb{R}^3}$ of the problem

$$\left\{\begin{aligned}&\nabla \times(\nabla \times E) + \lambda E = \partial_E F(x, E) &&\quad \text{in }\quad \Omega \\& \nu \times E = 0 &&\quad \text{on }\quad \partial \Omega \end{aligned} \right.$$

on a simply connected, smooth, bounded domain ${\Omega \subset \mathbb{R}^3}$ with connected boundary and exterior normal ${\nu : \partial \Omega \to \mathbb{R}^3}$. Here ${\nabla \times}$ denotes the curl operator in ${\mathbb{R}^3}$, the nonlinearity ${F : \Omega \times \mathbb{R}^3 \to \mathbb{R}}$ is superquadratic and subcritical in E. The model nonlinearity is of the form ${F(x, E)=\Gamma(x) |E|^p}$ for ${\Gamma \in L^\infty(\Omega)}$ positive, some 2 < p < 6. It need not be radial nor even in the E-variable. The problem comes from the time-harmonic Maxwell equations, the boundary conditions are those for Ω surrounded by a perfect conductor.