Asymptotic symmetry and local behaviors of solutions to a class of anisotropic elliptic equations Chun-Hsiung HsiaChang-Shou LinZhi-Qiang Wang 35J6035B33Hardy-Sobolev inequalityCaffarelli-Kohn-Nirenberg inequalitymoving planes method Let $0 \le a < (N-2)/2$, $a \le b < a+1$, $p = 2N/(N-2+2(b-a))$, and $B_1 = B_1(0)$ be the unit ball in $\mathbb{R}^N$, where $N \ge 3$. We first prove that any positive solution $u(x) \in C^2(B_1\setminus\{0\})$ of the equation \begin{equation}\label{eq:0.2} \begin{cases} -\Div(|x|^{-2a}\nabla u) = |x|^{-bp}u^{p-1}\quad\mbox{for }x \in B_1\setminus\{0\},\\ 0 \mbox{ is not a removable singularity of }u(x), \end{cases} \end{equation} is asymptotically symmetric with respect to the origin, i.e., \[ u(x) = u_0(|x|)(1+O(|x|^{\epsilon}))\quad\mbox{as }|x| \to 0, \] where $u_0(x) = u_0(|x|) \in C^2(\mathbb{R}^N\setminus\{0\})$ is an entire solution of \eqref{eq:0.2} and $\epsilon > 0$. Equation \eqref{eq:0.2} is arising from the celebrated Caffarelli-Kohn-Nirenberg inequality. For $a = b < 0$ and $p = 2N/(N-2)$, we show there is no positive solution of the equation \begin{equation}\label{eq:0.3} \begin{cases} -\Div(|x|^{-2a}\nabla u) = |x|^{-bp}u^{p-1} &\mbox{in }\Omega,\\ u = 0 &\mbox{on }\partial\Omega, \end{cases} \end{equation} in $D^{1,2}_a(\Omega)$, where $\Omega\subseteq\mathbb{R}_{+}^N$ is a cone domain satisfying $\Omega = \mathbb{R}_{+}\times\omega$ with $\omega\subset S^{N-1}$ being star-shaped with respect to the north pole on $S^{N-1}$. Indiana University Mathematics Journal 2011 text pdf 10.1512/iumj.2011.60.4376 10.1512/iumj.2011.60.4376 en Indiana Univ. Math. J. 60 (2011) 1623 - 1654 state-of-the-art mathematics http://iumj.org/access/