Symmetry and classification of positive solutions of some weighted elliptic equations

We study the weighted elliptic equation \begin{equation} -div(|x|^{-2a}\nabla u)=|x|^{-bp}|u|^{p-2}u~~~\mbox{in}~$\mathbb{R}$^N ~~~~~~~~~~~~~~~~~~~~(0.1)\end{equation} with $N\ge 2$, which arises from the Caffarelli-Kohn-Nirenberg inequalities. Under the assumptions of finite energy and $a_1+a_2=N-2$, for nonnegative solutions we prove the equivalence between equation (0.1) with $a=a_1$ and equation (0.1) with $a=a_2$. Without finite energy assumptions, for $2\le p<2^{\ast }$ we give the optimal parameter range in which nonnegative solutions of (0.1) in ${\mathbf{L}}_{Loc}^{\mathrm{\infty }}({\mathbb{R}}^N)$ must be radially symmetric, and give a complete classification for these solutions in this range.