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

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 $Q$.