In this paper, we study the sign-changing radial solutions of the following coupled Schr\"odinger system \begin{ equation} \left\{ \begin{ array}{ lr} -{ \Delta}u_j+\lambda_j u_j=\mu_j u_j^3+\sum_{ i\neq j}\beta_{ ij} u_i^2 u_j \,\,\,\,\,\,\,\, \mbox{ in }B_1 ,\nonumber u_j\in H_{ 0,r}^1(B_1)\mbox{ for }j=1,\cdots,N.\nonumber \end{ array} \right. \end{ equation} Here, $\lambda_j,\,\mu_j>0$ and $\beta_{ ij}=\beta_{ ji}$ are constants for $i,j=1,\cdots,N$ and $i\neq j$. $B_1$ denotes the unit ball in the Euclidean space $\mathbb{ R}^3$ centred at the origin. For any $P_1,\cdots,P_N\in\mathbb{ N}$, we prove the uniqueness of the radial solution $(u_1,\cdots,u_j)$ with $u_j$ changes its sign exactly $P_j$ times for any $j=1,\cdots,N$ in the following case: $\lambda_j\geq1$ and $|\beta_{ ij}|$ are small for $i,j=1,\cdots,N$ and $i\neq j$. New Liouville-type theorems and boundedness results are established for this purpose.
Categories: math.AP