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Normalized multi-bump solutions for Choquard equation involving sublinear case (2505.06097v1)

Published 9 May 2025 in math.AP

Abstract: In this paper, we study the existence of normalized multi-bump solutions for the following Choquard equation \begin{equation*} -\epsilon2\Delta u +\lambda u=\epsilon{-(N-\mu)}\left(\int_{\mathbb{R}N}\frac{Q(y)|u(y)|p}{|x-y|{\mu}}dy\right)Q(x)|u|{p-2}u, \text{in}\ \mathbb{R}N, \end{equation*} where $N\geq3$, $\mu\in (0,N)$, $\epsilon>0$ is a small parameter and $\lambda\in\mathbb{R}$ 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 $\epsilon>0$. The asymptotic behavior of the solutions as $\epsilon\rightarrow0$ are also explored. It is worth noting that our results encompass the sublinear case $p<2$, which complements some of the previous works.

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