# Oscillating solutions for nonlinear Helmholtz Equations

## TL;DR

This paper establishes the existence and asymptotic behavior of radially symmetric oscillating solutions for nonlinear Helmholtz equations, including generalizations to nonautonomous and nonradial cases, with applications to standing waves in Klein-Gordon and Schrödinger equations.

## Contribution

It provides new existence results for oscillating solutions of nonlinear Helmholtz equations, extending to nonautonomous and nonradial cases, and links these solutions to standing waves in quantum equations.

## Key findings

- Existence of radially symmetric oscillating solutions.
- Asymptotic behavior characterized at infinity.
- Results applicable to Klein-Gordon and Schrödinger equations.

## Abstract

Existence results for radially symmetric oscillating solutions for a class of nonlinear autonomous Helmholtz equations are given and their exact asymptotic behavior at infinity is established. Some generalizations to nonautonomous radial equations as well as existence results for nonradial solutions are found. Our theorems prove the existence of standing waves solutions of nonlinear Klein-Gordon or Schr\"odinger equations with large frequencies.