Cylindrically Symmetric Ground State Solutions for Curl-Curl Equations with Critical Exponent

We study the following nonlinear critical curl-curl equation \begin{equation}\label{eq0.1}\nabla\times \nabla\times U +V(x)U=|U|^{p-2}U+ |U|^4U,\quad x\in $\mathbb{R}$^3,\end{equation} where $V(x)=V(r,x_3)$ with $r=\sqrt{x_1^2+x_2^2}$ is 1-periodic in $x_3$ direction and belongs to $L^{\mathrm{\infty }}({\mathbb{R}}^3)$. When $0\in ̸\sigma (-\mathrm{\Delta }+\frac{1}{r^2}+V)$ and $p\in (4,6)$, we prove the existence of nontrivial solution for (\ref{eq0.1}), which is indeed a ground state solution in a suitable cylindrically symmetric space. Especially, if $\sigma (-\mathrm{\Delta }+\frac{1}{r^2}+V)>0$, a ground state solution is obtained for any $p\in (2,6)$.