Normalized multi-bump solutions for Choquard equation involving sublinear case
He Zhang, Shuai Yao, Haibo Chen
TL;DR
This work addresses the existence of normalized multi-bump solutions for a nonlocal Choquard equation with a small parameter $\epsilon$ under an $L^2$ mass constraint, including the challenging sublinear regime $p<2$. A novel variational framework is developed, employing a tail-minimizing flow and deformation arguments to construct constrained critical points without relying on penalization or AR conditions, and a concentration-compactness type decomposition is established to identify the bump profiles. The authors prove the existence of a family of normalized multi-bump solutions concentrating at isolated local maxima of the potential $Q$ as $\epsilon\to 0$, with the Lagrange multipliers $\lambda_{\epsilon}$ converging to a limit $\lambda_0$ and the energy converging to $\sigma_0$, thereby extending prior results to the sublinear case and non-uniqueness scenarios. These results advance the understanding of semiclassical, mass-constrained states in nonlocal nonlinear optics and quantum models, and provide a versatile variational toolkit for nonlocal constrained problems.
Abstract
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.