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

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.