Multiple standing waves of Helmholtz equation with mixed dispersion concentrating in the high frequency limit
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911389098967040 |
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| author | Chen, Shaoxiong Yuan, Fei Zhao, Fukun Zhou, Jiazheng |
| author_facet | Chen, Shaoxiong Yuan, Fei Zhao, Fukun Zhou, Jiazheng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_14657 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multiple standing waves of Helmholtz equation with mixed dispersion concentrating in the high frequency limit Chen, Shaoxiong Yuan, Fei Zhao, Fukun Zhou, Jiazheng Analysis of PDEs 35J20, 35J05 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. |
| title | Multiple standing waves of Helmholtz equation with mixed dispersion concentrating in the high frequency limit |
| topic | Analysis of PDEs 35J20, 35J05 |
| url | https://arxiv.org/abs/2601.14657 |