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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.06097 |
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Table of 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.