The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Zeitschrift für Analysis und ihre Anwendungen


Full-Text PDF (215 KB) | Metadata | Table of Contents | ZAA summary
Volume 36, Issue 4, 2017, pp. 419–435
DOI: 10.4171/ZAA/1595

Published online: 2017-10-09

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

Andreas Hirsch[1] and Wolfgang Reichel[2]

(1) Karlsruhe Institute of Technology (KIT), Germany
(2) Karlsruhe Institute of Technology (KIT), Germany

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].

Keywords: Curl-curl problem, nonlinear elliptic equations, cylindrical symmetry, variational methods

Hirsch Andreas, Reichel Wolfgang: Existence of Cylindrically Symmetric Ground States to a Nonlinear Curl-Curl Equation with Non-Constant Coefficients. Z. Anal. Anwend. 36 (2017), 419-435. doi: 10.4171/ZAA/1595