Multiple standing waves for the nonlinear Helmholtz equation concentrating in the high frequency limit

TL;DR

This paper investigates the existence and multiplicity of solutions to a nonlinear Helmholtz equation at high frequencies, showing solutions concentrate at maximum points of the coefficient function as frequency increases.

Contribution

It establishes the existence of multiple solutions for large frequencies and describes their concentration behavior in relation to the coefficient function Q.

Findings

Solutions concentrate at maximum points of Q as frequency increases

Multiple solutions exist for large frequency values

Solutions are associated with ground states of a dual equation

Abstract

This paper studies for large frequency number $k>0$ the existence and multiplicity of solutions of the semilinear problem $$ -\Delta u -k^2 u=Q(x)|u|^{p-2}u\quad\text{ in }\mathbb{R}^N, \quad N\geq 2. $$ The exponent $p$ is subcritical and the coefficient $Q$ is continuous, nonnegative and satisfies the condition $$ \limsup_{|x|\to\infty}Q(x)<\sup_{x\in\mathbb{R}^N}Q(x). $$ In the limit $k\to\infty$, sequences of solutions associated to ground states of a dual equation are shown to concentrate, after rescaling, at global maximum points of the function $Q$.