  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
$L^\infty(\R^3)$. When $0\not\in \sigma(-\Delta+\frac{1}{r^2}+V)$ 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(-\Delta+\frac{1}{r^2}+V)>0$, a ground state
solution is obtained for any $p\in(2,6)$.
