On the periodic and asymptotically periodic nonlinear Helmholtz equation

TL;DR

This paper proves the existence of multiple periodic and asymptotically periodic standing wave solutions for a nonlinear Helmholtz equation with periodic or asymptotically periodic coefficients, expanding understanding of solution structures in nonlinear wave equations.

Contribution

It establishes the existence of infinitely many solutions for the nonlinear Helmholtz equation with periodic coefficients and demonstrates solutions also exist when coefficients are asymptotically periodic.

Findings

Existence of infinitely many $L^p$-standing wave solutions for periodic $Q$.

Existence of at least one nontrivial solution for asymptotically periodic $Q$.

Abstract

In the first part of this paper, the existence of infinitely many $L^p$-standing wave solutions for the nonlinear Helmholtz equation $$ -\Delta u -\lambda u=Q(x)|u|^{p-2}u\quad\text{ in }\mathbb{R}^N $$ is proven for $N\geq 2$ and $\lambda>0$, under the assumption that $Q$ be a nonnegative, periodic and bounded function and the exponent $p$ lies in the Helmholtz subcritical range. In a second part, the existence of a nontrivial solution is shown in the case where the coefficient $Q$ is only asymptotically periodic.