Regularity of the scattering matrix for nonlinear Helmholtz eigenfunctions

Abstract

We study the nonlinear Helmholtz equation (∆ - λ2)u = ±|u|p‒1u on Rn, λ > 0, p ∈ N odd, and more generally (∆g C V - λ2)u = N [u], where ∆g is the (positive) Laplace–Beltrami operator on an asymptotically Euclidean or conic manifold, V is a short range potential, and N [u] is a more general polynomial nonlinearity. Under the conditions (p - 1)(n - 1)=2 > 2 and k > (n - 1)=2, for every f 2 Hk(Sωn‒1) of sufficiently small norm, we show there is a nonlinear Helmholtz eigenfunction taking the form 'Equation Presented' for some b ∈ Hk(Sωn‒1) and ϵ > 0. That is, the nonlinear scattering matrix f → b preserves Sobolev regularity, which is an improvement over the authors’ previous work (2020) with Z..