Multiple cylindrically symmetric solutions of nonlinear Maxwell equations

Abstract

In this paper, we study the following nonlinear time-harmonic Maxwell equations\begin{equation}\tag*{(0.1)}\nabla\times(\nabla \times E)-\omega^2\varepsilon(x)E=P(x)|E|^{p-2}E+Q(x)|E|^{q-2}E,\end{equation}where $\varepsilon(x)$ is the permittivity of the material, $x\in\mathbb{R}^{3}$, $1< q< {p}/({p-1})< 2< p< 6$, $P(x),Q(x)\in C(\mathbb{R}^{3},\mathbb{R})$. Under some special cylindrical symmetric conditions for $\varepsilon(x)$, $P(x)$ and $Q(x)$,we obtain infinite many cylindrically symmetric solutions of (0.1) by using variational method and fountain theorems without $\tau$-upper semi-continuity.