Existence and asymptotic behavior of standing waves of the nonlinear Helmholtz equation in the plane

TL;DR

This paper investigates the existence and asymptotic properties of standing wave solutions to a nonlinear Helmholtz equation in the plane, considering both decaying and periodic coefficients, and establishes a nonlinear far-field relation.

Contribution

It proves the existence of real-valued solutions for the nonlinear Helmholtz equation with various coefficient types and derives a nonlinear far-field relation, advancing understanding of such wave phenomena.

Findings

Existence of real-valued $W^{2,p}$ solutions for decaying and periodic $Q$

Derivation of a nonlinear far-field relation for solutions

Solutions exist for $p \\geq 6$ in the plane

Abstract

In this paper we study the semilinear elliptic problem $$ -\Delta u -k^2u=Q|u|^{p-2}u\quad\text{ in }\mathbb{R}^2, $$ where $k>0$, $p\geq 6$ and $Q$ is a bounded function. We prove the existence of real-valued $W^{2,p}$-solutions, both for decaying and for periodic coefficient $Q$. In addition, a nonlinear far-field relation is derived for these solutions.