Normalized vector solutions of nonlinear Schrödinger systems

Abstract

Given $\hat{I}¼>0$ we look for solutions $\hat{I}»\in \mathbb{R}$ and $v_1,\dots ,v_k\in H^1({\mathbb{R}}^N)$ of the system \[ \begin{cases} \displaystyle -Δv_i+ λv_i+V_i(x)v_i = \sum_{\substack{j=1}}^kβ_{ij} v_iv_j^2 &\text{ in } \mathbb{R}^N, \text{ } i=1,\dots,k,\newline \displaystyle \int_{\mathbb{R}^N} \left(v_1^2+\dots+v_k^2 \right)\mathrm{d} x = μ, \end{cases}\] where $N=1,2,3$, $V_i:{\mathbb{R}}^N\to \mathbb{R}$ and ${\hat{I}}_{ij}^2\in \mathbb{R}$ satisfy ${\hat{I}}_{ij}^2={\hat{I}}_{ji}^2$ and ${\hat{I}}_{ii}^2>0$. Under suitable assumptions on the ${\hat{I}}_{ij}^2$'s, given a non-degenerate critical point $\hat{I}{¾}_0$ of a suitable linear combination of the potentials $V_i$, we build solutions whose components concentrate at $\hat{I}{¾}_0$ as the prescribed global mass $\hat{I}¼$ is either large (when $N=1$) or small (when $N=3$) or it approaches some critical threshold (when $N=2$).

23 pages, 1 figure