Multiplicity and concentration of dual solutions for a Helmholtz system
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909998986035200 |
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| author | Qiu, Ruowen Yuan, Fei Zhao, Fukun |
| author_facet | Qiu, Ruowen Yuan, Fei Zhao, Fukun |
| contents | In this paper, we are concerned with the nonlinear Helmholtz system of Hamiltonian type \begin{equation*} \left\{\begin{array}{l} -Δu-k^2 u=P(x)|v|^{p-2}v,\quad \text{in}\ \mathbb{R}^N, \\ -Δv-k^2v=Q(x)|u|^{q-2}u,\quad \text{in}\ \mathbb{R}^N, \end{array} \right. \end{equation*} where $N\geq3$, $P,Q: \mathbb{R}^N\rightarrow \mathbb{R}$ are two positive continuous functions, the exponents $p,q>2$ satisfy $\frac{1}{p}+\frac{1}{q}>\frac{N-2}{N}$. First, we obtained the existence of a ground state solution via a dual variational method. Moreover, the concentration behavior of such dual ground state solutions is established
as $k\rightarrow\infty$, where a rescaling technique and the generalized Birman-Schwinger operator are involved. In addition, we also investigated the relation between the number of solutions and the topology of the set of the global maxima of the functions $P$ and $Q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_16754 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multiplicity and concentration of dual solutions for a Helmholtz system Qiu, Ruowen Yuan, Fei Zhao, Fukun Analysis of PDEs 35J50, 35J47 In this paper, we are concerned with the nonlinear Helmholtz system of Hamiltonian type \begin{equation*} \left\{\begin{array}{l} -Δu-k^2 u=P(x)|v|^{p-2}v,\quad \text{in}\ \mathbb{R}^N, \\ -Δv-k^2v=Q(x)|u|^{q-2}u,\quad \text{in}\ \mathbb{R}^N, \end{array} \right. \end{equation*} where $N\geq3$, $P,Q: \mathbb{R}^N\rightarrow \mathbb{R}$ are two positive continuous functions, the exponents $p,q>2$ satisfy $\frac{1}{p}+\frac{1}{q}>\frac{N-2}{N}$. First, we obtained the existence of a ground state solution via a dual variational method. Moreover, the concentration behavior of such dual ground state solutions is established as $k\rightarrow\infty$, where a rescaling technique and the generalized Birman-Schwinger operator are involved. In addition, we also investigated the relation between the number of solutions and the topology of the set of the global maxima of the functions $P$ and $Q$. |
| title | Multiplicity and concentration of dual solutions for a Helmholtz system |
| topic | Analysis of PDEs 35J50, 35J47 |
| url | https://arxiv.org/abs/2601.16754 |