# Existence of Cylindrically Symmetric Ground States to a Nonlinear Curl-Curl Equation with Non-Constant Coefficients

### Andreas Hirsch

Karlsruhe Institute of Technology (KIT), Germany### Wolfgang Reichel

Karlsruhe Institute of Technology (KIT), Germany

## Abstract

We consider the nonlinear curl-curl problem $\nabla\times\nabla\times U + V(x) U=f(x, |U|^2)U$ in $\mathbb R^3$ related to the nonlinear Maxwell equations with Kerr-type nonlinear material laws. We prove the existence of a symmetric ground-state type solution for a bounded, cylindrically symmetric coefficient $V$ and subcritical cylindrically symmetric nonlinearity $f$. The new existence result extends the class of problems for which ground-state type solutions are known. It is based on compactness properties of symmetric functions due to Lions [J. Funct. Anal. 41 (1981)(2), 236–275], new rearrangement type inequalities by Brock [Proc. Indian Acad. Sci. Math. Sci. 110 (2000), 157–204] and the recent extension of the Nehari-manifold technique from Szulkin and Weth [Handbook of Nonconvex Analysis and Applications (2010), pp. 597–632].

## Cite this article

Andreas Hirsch, Wolfgang Reichel, Existence of Cylindrically Symmetric Ground States to a Nonlinear Curl-Curl Equation with Non-Constant Coefficients. Z. Anal. Anwend. 36 (2017), no. 4, pp. 419–435

DOI 10.4171/ZAA/1595