Normalized multi-bump solutions for Choquard equation involving sublinear case

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 $N\ge 3$, $\mu \in (0,N)$, $\epsilon >0$ 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 $\epsilon \to 0$ are also explored. It is worth noting that our results encompass the sublinear case $p<2$, which complements some of the previous works.