Vector Solutions with Prescribed Component-Wise Nodes for a Schrödinger System (Analysis in Theory and Applications, 2019)
For the Schrödinger system$$\left\{\begin{array}{ll}-\Delta u_j +\lambda_j u_j =\sum\limits_{i=1}^k \beta_{ij} u_i^2 u_j\quad \mbox{in}\ \ \ \ \mathbb R^N,\\ u_j(x)\to0\quad\text{ as }\ \ |x|\to\infty,  j=1,\cdots,k,\end{array}\right.$$where $k\geq 2$ and $N=2, 3$, we prove that for any $\lambda_j>0$ and $\beta_{jj}>0$ and any positive integers $p_j$, $j=1,2,\cdots,k$, there exists $b>0$ such that if $\beta_{ij}=\beta_{ji}\leq b$ for all $i eq j$ then there exists a radial solution $(u_1,u_2,\cdots,u_k)$ with $u_j$ having exactly $p_j-1$ zeros. Moreover, there exists a positive constant $C_0$ such that if $\beta_{ij}=\beta_{ji}\leq b\ (i eq j)$ then any solution obtained satisfies$$\sum_{i,j=1}^k|\beta_{ij}|\int_{\mathbb R^N}u_i^2u_j^2\leq C_0.$$Therefore, the solutions exhibit a trend of phase separations as $\beta_{ij}\to-\infty$ for $i eq j.$