Normalized multi-bump solutions for Choquard equation involving sublinear case

TL;DR

This paper establishes the existence and asymptotic behavior of normalized multi-bump solutions for a Choquard equation with a sublinear nonlinearity, using a novel variational approach, especially as the parameter epsilon approaches zero.

Contribution

It introduces a new variational method to find normalized multi-bump solutions for the Choquard equation in the sublinear case, extending previous results.

Findings

Existence of multi-bump solutions focused on local maxima of potential Q(x).

Solutions concentrate as epsilon approaches zero.

Results include the sublinear case p<2, expanding prior work.

Abstract

In this paper, we study the existence of normalized multi-bump solutions for the following Choquard equation \begin{equation*} -\epsilon^2\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.