Multiple cylindrically symmetric solutions of nonlinear Maxwell equations - Wen, Yanyun; Zhao, Peihao (Topological Methods in Nonlinear Analysis, 2023)
In this paper, we study the following nonlinear time-harmonic Maxwell equations\begin{equation}\label{equation 0.1} abla×( abla × E)-ω^2ε(x)E=P(x)|E|^{p-2}E+Q(x)|E|^{q-2}E,\end{equation}where $ε(x)$ 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 $ε(x)$, $P(x)$ and $Q(x)$,we obtain infinite many cylindrically symmetric solutions of \eqref{equation 0.1}by using variational method and fountain theorems without $τ$-upper semi-continuity.