On a critical Schrödinger system involving Hardy terms

Abstract

This paper concerns an elliptic system with critical exponents:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_j-\frac{\lambda _j}{|x|^2}u_j=u_j^{2^*-1}+\sum \limits _{k\ne j}\beta _{jk}\alpha _{jk}u_j^{\alpha _{jk}-1}u_k^{\alpha _{kj}},\;\;x\in {{\mathbb {R}}}^N,\\ u_j\in D^{1,2}({{\mathbb {R}}}^N),\quad u_j>0 \;\; \hbox {in} \quad {{\mathbb {R}}}^N\setminus \{0\},\quad j=1, \ldots ,r,\end{array}\right. } \end{aligned}$$

where $N\ge 3, r\ge 2, 2^*=\frac{2N}{N-2}, \lambda _j\in (0, \frac{(N-2)^2}{4})$ for all $j=1, \ldots ,r$; $\beta _{jk}=\beta _{kj}$; $\alpha _{jk}>1, \alpha _{kj}>1,$ and $\alpha _{jk}+\alpha _{kj}=2^*$ for all $k\ne j$. Note that the nonlinearities $u_j^{2^*-1}$ and the coupling terms are all critical in arbitrary dimension $N\ge 3$. The signs of the coupling constants $\beta _{ij}$ are decisive for the existence of the ground-state solutions. We show that the critical system with $r\ge 3$ has a positive ground-state solution for all $\beta _{jk}>0$ with some constraint on $\lambda _j$. However, there is no ground-state solution when all $\beta _{jk}$ are negative. It is also proved that the positive solution of the system is radially symmetric. Furthermore, we obtain an uniqueness theorem for the case $r\ge 3$ with $N=4$ and an existence theorem for the case $r=2$ with general coupling exponents.