标题： 具有归一化向量解的非线性薛定谔系统 显示英文标题

标题： Normalized vector solutions of nonlinear Schrödinger systems

摘要： 给定$\mu>0$，我们寻找系统\[ \begin{cases} \displaystyle -\Delta v_i+ \lambda v_i+V_i(x)v_i = \sum_{\substack{j=1}}^k\beta_{ij} v_iv_j^2 &\text{ in } \mathbb{R}^N, \text{ } i=1,\dots,k,\newline \displaystyle \int_{\mathbb{R}^N} \left(v_1^2+\dots+v_k^2 \right)\mathrm{d} x = \mu, \end{cases}\]的解$ \lambda\in\mathbb{R}$和$v_1,\dots,v_k\in H^1(\mathbb{R}^N)$，其中$N=1,2,3$、$V_i:\mathbb R^N\to \mathbb R$和$\beta_{ij}\in\mathbb{R}$满足$\beta_{ij}=\beta_{ji}$和$\beta_{ii}>0$。 在关于$\beta_{ij}$的合适假设下，给定一个非退化的临界点$\xi_0$，它是某些势能的合适线性组合$V_i$的临界点，我们构造解的分量在$\xi_0$聚焦的解，当规定的整体质量$\mu$要么很大（当$N=1$时），要么很小（当$N=3$时），或者它接近某个临界阈值（当$N=2$时）。 显示英文摘要

摘要： 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 $N=2$).