Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.06097 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908356413751296 |
|---|---|
| author | Zhang, He Yao, Shuai Chen, Haibo |
| author_facet | Zhang, He Yao, Shuai Chen, Haibo |
| contents | In this paper, we study the existence of normalized multi-bump solutions for the following Choquard equation \begin{equation*} -ε^2Δu +λu=ε^{-(N-μ)}\left(\int_{\mathbb{R}^N}\frac{Q(y)|u(y)|^p}{|x-y|^μ}dy\right)Q(x)|u|^{p-2}u, \text{in}\ \mathbb{R}^N, \end{equation*} where $N\geq3$, $μ\in (0,N)$, $ε>0$ is a small parameter and $λ\in\mathbb{R}$ appears as a Lagrange multiplier. By developing a new variational approach, we show that the problem has a family of normalized multi-bump solutions focused on the isolated part of the local maximum of the potential $Q(x)$ for sufficiently small $ε>0$. The asymptotic behavior of the solutions as $ε\rightarrow0$ are also explored. It is worth noting that our results encompass the sublinear case $p<2$, which complements some of the previous works. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06097 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalized multi-bump solutions for Choquard equation involving sublinear case Zhang, He Yao, Shuai Chen, Haibo Analysis of PDEs In this paper, we study the existence of normalized multi-bump solutions for the following Choquard equation \begin{equation*} -ε^2Δu +λu=ε^{-(N-μ)}\left(\int_{\mathbb{R}^N}\frac{Q(y)|u(y)|^p}{|x-y|^μ}dy\right)Q(x)|u|^{p-2}u, \text{in}\ \mathbb{R}^N, \end{equation*} where $N\geq3$, $μ\in (0,N)$, $ε>0$ is a small parameter and $λ\in\mathbb{R}$ appears as a Lagrange multiplier. By developing a new variational approach, we show that the problem has a family of normalized multi-bump solutions focused on the isolated part of the local maximum of the potential $Q(x)$ for sufficiently small $ε>0$. The asymptotic behavior of the solutions as $ε\rightarrow0$ are also explored. It is worth noting that our results encompass the sublinear case $p<2$, which complements some of the previous works. |
| title | Normalized multi-bump solutions for Choquard equation involving sublinear case |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.06097 |