Multiple standing waves of Helmholtz equation with mixed dispersion concentrating in the high frequency limit

TL;DR

This paper investigates the existence and concentration behavior of multiple standing wave solutions to a nonlinear Helmholtz equation with mixed dispersion, focusing on high-frequency limits and the influence of the potential's maximum points.

Contribution

It establishes the existence of dual ground state solutions and multiple solutions that concentrate at the global maxima of the potential function as frequency increases.

Findings

Solutions concentrate at the global maxima of W as k→∞

Existence of multiple solutions related to maximum points of W

Precise characterization of solution concentration behavior

Abstract

In this paper, we study the nonlinear Helmholtz equation with mixed dispersion \begin{equation*} \Delta^2 u-\beta k^2\, \Delta u+\alpha k^4 u=W(x)\, |u|^{p-2}u~\text{in}~\mathbb{R}^N, \end{equation*} where the weight function $W(x)$ is continuous, nonnegative, and satisfies \[ \limsup_{|x|\to\infty} W(x) \;<\; \sup_{x\in\mathbb{R}^N} W(x). \] Within each of the following parameter ranges, \begin{center} (a) $\alpha<0$, $\beta\in\mathbb{R}$; \qquad (b) $\alpha>0$, $\beta<-2\sqrt{\alpha}$; \qquad (c) $\alpha=0$, $\beta<0$, \end{center} After a suitable rescaling, we obtain the existence of dual ground state solutions, which concentrate along the global maximizers of $W$ as $k\to\infty$. In addition, we establish the existence of multiple solutions associated with the set of global maximum points of $W$, and we further characterize the precise concentration behavior of these solutions.