Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain

We find solutions $E:Ω\to\mathbb{R}^3$ of the problem $\left\{\begin{aligned} &\nabla\times(\nabla\times E) + λE = \partial_E F(x,E) &&\quad \text{in}Ω\\ &ν\times E = 0 &&\quad \text{on}\partialΩ\end{aligned} \right.$ on a simply connected, smooth, bounded domain $Ω\subset\mathbb{R}^3$ with connected boundary and exterior normal $ν:\partialΩ\to\mathbb{R}^3$. Here $\nabla\times$ denotes the curl operator in $\mathbb{R}^3$, the nonlinearity $F:Ω\times\mathbb{R}^3\to\mathbb{R}$ is superquadratic and subcritical in $E$. The model nonlinearity is of the form $F(x,E)=Γ(x)|E|^p$ for $Γ\in L^\infty(Ω)$ 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.

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