Normalized vector solutions of nonlinear Schr\"odinger systems

Given $\mu>0$ we look for solutions $\lambda\in\mathbb{R}$ and $v_1,\dots,v_k\in H^1(\mathbb{R}^N)$ of the system $$\begin{cases} \displaystyle -\Delta v_i+ \lambda v_i+V_i(x)v_i = \sum_{\substack{j=1}}^k\beta_{ij} v_iv_j^2 &\text{ in } \mathbb{R}^N, \text{ } i=1,\dots,k,\newline \displaystyle \int_{\mathbb{R}^N} \left(v_1^2+\dots+v_k^2 \right)\mathrm{d} x = \mu, \end{cases}$$ where $N=1,2,3$, $V_i:\mathbb R^N\to \mathbb R$ and $\beta_{ij}\in\mathbb{R}$ satisfy $\beta_{ij}=\beta_{ji}$ and $\beta_{ii}>0$. Under suitable assumptions on the $\beta_{ij}$'s, given a non-degenerate critical point $\xi_0$ of a suitable linear combination of the potentials $V_i$, we build solutions whose components concentrate at $\xi_0$ as the prescribed global mass $\mu$ is either large (when $N=1$) or small (when $N=3$) or it approaches some critical threshold (when $N=2$).