Normalized vector solutions of nonlinear Schrödinger 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 $$\{\begin{array}{ll}{\displaystyle -\mathrm{\Delta }{v}_i+\lambda v_i+V_i(x)v_i=\sum _{\begin{array}{c}j=1\end{array}}^k{\beta }_{ij}{v}_i{v}_j^2}& \text{ in }{\mathbb{R}}^N,i=1,\dots ,k,\\ {\displaystyle {\int }_{{\mathbb{R}}^N}(v_1^2+\cdots +v_k^2)\mathrm{d}x=\mu ,}\end{array}$$ 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 $N=2$).