Uncountably Many Solutions for Nonlinear Helmholtz and Curl-Curl Equations

Abstract

We obtain uncountably many solutions of nonlinear Helmholtz and curl-curl equations on the entire space using a fixed point approach. The constructed solutions are mildly localized as they lie in the essential spectrum of the corresponding linear operator. As a new auxiliary tool a limiting absorption principle for the curl-curl operator is proved.

Keywords

• nonlinear helmholtz equations
• curl-curl equations
• limiting absorption principles
• herglotz waves
• 35q60
• 35q61
• 35j91