Multiple cylindrically symmetric solutions of nonlinear Maxwell equations

Abstract

In this paper, we study the following nonlinear time-harmonic Maxwell equations \begin{equation}\label{equation 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, $x\in\mathbb{R}^{3}$, $1< q< {p}/({p-1})< 2< p< 6$, $P(x),Q(x)\in C\left(\mathbb{R}^{3},\mathbb{R}\right)$. Under some special cylindrical symmetric conditions for $\varepsilon(x)$, $P(x)$ and $Q(x)$, we obtain infinite many cylindrically symmetric solutions of \eqref{equation 0.1} by using variational method and fountain theorems without $\tau$-upper semi-continuity.