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 $N\geq 2$, 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 $2\leq p<2^*$ we give the optimal parameter range in which nonnegative solutions of (0.1) in $\mathbf{ L}^\infty_{ Loc}(\mathbb{ R}^N)$ must be radially symmetric, and give a complete classification for these solutions in this range.
Categories: math.AP