Existence of cylindrically symmetric ground states to a nonlinear curl-curl equation with non-constant coefficients

2016/06/14 by Hirsch, Andreas, Reichel, Wolfgang · 1 citation
#35Q60 #58E15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary: 35J20 #Secondary: 47J30

paper · doi:10.48550/arxiv.1606.04415

Abstract

We consider the nonlinear curl-curl problem ∇×∇× U + V(x) U=f(x,|U|²)U in ℝ³ 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, new rearrangement type inequalities from Brock and the recent extension of the Nehari-manifold technique by Szulkin and Weth.

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• Breather solutions to nonlinear Maxwell equations with retarded material laws

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