Cylindrically symmetric ground state solutions for curl–curl equations with critical exponent

Abstract

We study the following nonlinear critical curl–curl equation

$$\begin{aligned} \nabla \times \nabla \times U +V(x)U=|U|^{p-2}U+ |U|^4U,\quad x\in \mathbb {R}^3, \end{aligned}$$

(0.1)

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 $L^\infty ({\mathbb {R}}^3)$. When $0\not \in \sigma (-\Delta +\frac{1}{r^2}+V)$ and $p\in (4,6)$, we prove the existence of nontrivial solution for (0.1), which is indeed a ground state solution in a suitable cylindrically symmetric space. In particular, if $\sigma (-\Delta +\frac{1}{r^2}+V)>0$, a ground state solution is obtained for any $p\in (2,6)$.