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

TL;DR

This work analyzes the nonlinear fourth-order Helmholtz equation with mixed dispersion $\Delta^2 u - \beta k^2\,\Delta u + \alpha k^4 u = W(x)\,|u|^{p-2}u$ in $\mathbb{R}^N$ in the high-frequency limit $k\to\infty$, for a nonnegative weight $W$ with $W_\infty < W_0$. It develops a dual variational framework using the Helmholtz resolvent and a factorization $L=\Delta^2-\beta\Delta+\alpha=(-\Delta-a_1)(-\Delta-a_2)$ to define a Mountain Pass functional whose critical points yield solutions; it proves the existence and concentration of dual ground states near the global maximizers $M$ of $W$, and establishes multiplicity via Lusternik–Schnirelmann category. A key off-diagonal energy estimate controls nonlocal interactions and enables Palais–Smale compactness below the limiting energy $c_\infty$, ensuring precise concentration profiles as $k\to\infty$ and convergence to limit-ground states solving $\Delta^2 u - \beta\Delta u + \alpha u = W_0\,|u|^{p-2}u$. The results provide a rigorous description of high-frequency concentration and multiplicity for a nonlocal, higher-order Helmholtz-type problem and extend dual variational techniques beyond periodic weights.

Abstract

In this paper, we study the nonlinear Helmholtz equation with mixed dispersion \begin{equation*} Δ^2 u-βk^2\, Δu+αk^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) $α<0$, $β\in\mathbb{R}$; \qquad (b) $α>0$, $β<-2\sqrtα$; \qquad (c) $α=0$, $β<0$, \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.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Remark 1
• Lemma 2.5
• Remark 2
• Lemma 3.1