On a fourth order nonlinear Helmholtz equation

TL;DR

This paper investigates solutions to a fourth order nonlinear Helmholtz equation with mixed dispersion in Euclidean space, employing the dual method to establish existence and qualitative properties of solutions.

Contribution

It introduces a novel application of the dual method to a fourth order nonlinear Helmholtz equation with periodic coefficients, advancing solution existence theory.

Findings

Existence of solutions established using the dual method.

Solutions exhibit specific qualitative properties.

The approach extends previous methods to higher-order equations.

Abstract

In this paper, we study the mixed dispersion fourth order nonlinear Helmholtz equation $\Delta^2 u -\beta \Delta u + \alpha u= \Gamma|u|^{p-2} u$ in $\mathbb R^N$ for positive, bounded and $\mathbb Z^N$-periodic functions $\Gamma$. Using the dual method of Evequoz and Weth, we find solutions to this equation and establish some of their qualitative properties.