Existence and asymptotics of nonlinear Helmholtz eigenfunctions

Abstract

We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form \begin{equation*} (Δ- λ^2) u = N[u], \end{equation*} where $\Delta =-{\sum }_j{\partial }_j^2$ is the Laplacian on ${\mathbb{R}}^n$ with sign convention that it is positive as an operator, $\lambda$ is a positive real number, and $N[u]$ is a nonlinear operator that is a sum of monomials of degree $\ge p$ in $u$, $\overline{u}$ and their derivatives of order up to two, for some $p\ge 2$. Nonlinear Helmholtz eigenfunctions with $N[u]=\pm |u{|}^{p-1}u$ were first considered by Gutiérrez. Such equations are of interest in part because, for certain nonlinearities $N[u]$, they furnish standing waves for nonlinear evolution equations, that is, solutions that are time-harmonic. We show that, under the condition $(p-1)(n-1)/2>2$ and $k>(n-1)/2$, for every $f\in H^{k+2}({\mathbb{S}}^{n-1})$ of sufficiently small norm, there is a nonlinear Helmholtz function taking the form \begin{equation*} u(r, ω) = r^{-(n-1)/2} \Big( e^{-iλr} f(ω) + e^{+iλr} g(ω) + O(r^{-ε}) \Big), \text{ as } r \to \infty, \quad ε> 0, \end{equation*} for some $g\in H^k({\mathbb{S}}^{n-1})$. Moreover, we prove the result in the general setting of asymptotically conic manifolds.

36 pages, 2 figures