Complex solutions and stationary scattering for the nonlinear Helmholtz equation

TL;DR

This paper investigates stationary scattering solutions to the nonlinear Helmholtz equation in higher dimensions, establishing existence of complex solutions with outgoing radiation conditions using topological and bifurcation methods.

Contribution

It introduces a novel approach employing topological fixed point and bifurcation theory to prove existence of solutions without variational or maximum principle methods.

Findings

Existence of complex-valued solutions with outgoing radiation condition.

Development of a priori bounds for the integral equation.

Application of topological methods to nonlinear Helmholtz problems.

Abstract

We study a stationary scattering problem related to the nonlinear Helmholtz equation $-\Delta u - k^2 u = f(x,u) \ \ \text{in $\mathbb{R}^N$,}$ where $N \ge 3$ and $k>0$. For a given incident free wave $\varphi \in L^\infty(\mathbb{R}^N)$, we prove the existence of complex-valued solutions of the form $u=\varphi+u_{\text{sc}}$, where $u_{\text{sc}}$ satisfies the Sommerfeld outgoing radiation condition. Since neither a variational framework nor maximum principles are available for this problem, we use topological fixed point theory and global bifurcation theory to solve an associated integral equation involving the Helmholtz resolvent operator. The key step of this approach is the proof of suitable a priori bounds.