Real solutions to the nonlinear Helmholtz equation with local nonlinearity

TL;DR

This paper establishes the existence of real solutions to the nonlinear Helmholtz equation with local nonlinearities, including infinitely many solutions in the radial case, and relates these to standing wave solutions of nonlinear Klein-Gordon equations.

Contribution

It develops a variational framework to prove solutions without symmetry assumptions and extends results to the radial case with infinitely many solutions.

Findings

Existence of nontrivial solutions for compactly supported nonlinearities.

Infinitely many solutions in the radial case.

Solutions correspond to standing waves of nonlinear Klein-Gordon equations.

Abstract

In this paper, we study real solutions of the nonlinear Helmholtz equation $$ - \Delta u - k^2 u = f(x,u),\qquad x\in \R^N $$ satisfying the asymptotic conditions $$ u(x)=O(|x|^{\frac{1-N}{2}}) \quad \text{and} \quad \frac{\partial^2 u}{\partial r^2}(x)+k^2 u(x)) =o(|x|^{\frac{1-N}{2}}) \qquad \text{as $r=|x| \to \infty$.} $$ We develop the variational framework to prove the existence of nontrivial solutions for compactly supported nonlinearities without any symmetry assumptions. In addition, we consider the radial case in which, for a larger class of nonlinearities, infinitely many solutions are shown to exist. Our results give rise to the existence of standing wave solutions of corresponding nonlinear Klein-Gordon equations with arbitrarily large frequency.