Existence, nonexistence, symmetry and uniqueness of ground state for critical Schrödinger system involving Hardy term

Abstract

We study the following elliptic system with critical exponent: \begin{displaymath} \begin{cases}-Δu_j-\frac{λ_j}{|x|^2}u_j=u_j^{2^*-1}+\sum\limits_{k\neq j}β_{jk}α_{jk}u_j^{α_{jk}-1}u_k^{α_{kj}},\;\;x\in\R^N, u_j\in D^{1,2}(\R^N),\quad u_j>0 \;\; \hbox{in} \quad \R^N\setminus \{0\},\quad j=1,...,r.\end{cases}\end{displaymath} Here $N\ge 3,r\ge 2,2^{\ast }=\frac{2N}{N-2},{\lambda }_j\in (0,\frac{(N-2{)}^2}{4})$ for all $j=1,...,r$; ${\beta }_{jk}={\beta }_{kj}$; \; ${\alpha }_{jk}>1,{\alpha }_{kj}>1,$ satisfying ${\alpha }_{jk}+{\alpha }_{kj}=2^{\ast }$ for all $k\ne j$. Note that the nonlinearities $u_j^{2^{\ast }-1}$ and the coupling terms all are critical in arbitrary dimension $N\ge 3$. The signs of the coupling constants $\bb_{ij}$'s are decisive for the existence of the ground state solutions. We show that the critical system with $r\ge 3$ has a positive least energy solution for all ${\beta }_{jk}>0$. However, there is no ground state solutions if all ${\beta }_{jk}$ are negative. We also prove that the positive solutions of the system are radially symmetric. Furthermore, we obtain the uniqueness theorem for the case $r\ge 3$ with $N=4$ and the existence theorem when $r=2$ with general coupling exponents.

39 pages