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