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

TL;DR

This paper establishes the existence of cylindrically symmetric ground state solutions for a nonlinear curl-curl equation with critical exponent in \\mathbb{R}^3, under certain spectral conditions on the potential, expanding understanding of solutions in electromagnetic models.

Contribution

It proves the existence of ground state solutions for a critical curl-curl equation with cylindrical symmetry, considering spectral conditions and extending results to a broader range of exponents.

Findings

Existence of nontrivial solutions when 0 not in the spectrum of the associated operator.

Ground state solutions exist for any p in (2,6) if the spectrum is positive.

Solutions are cylindrically symmetric and critical in the nonlinear setting.

Abstract

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 $p\in(2,6)$.