Title: Asymptotic symmetry and local behaviors of solutions to a class of anisotropic elliptic equations
Authors: Chun-Hsiung Hsia, Chang-Shou Lin and Zhi-Qiang Wang
Issue: Volume 60 (2011), Issue 5, 1623-1654
Abstract:
![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 $S^{N-1}$.](https://iumj.s3-us-west-2.amazonaws.com/abstracts/4376.png)
Title: Asymptotic symmetry and local behaviors of solutions to a class of anisotropic elliptic equations
Authors: Chun-Hsiung Hsia, Chang-Shou Lin and Zhi-Qiang Wang
Issue: Volume 60 (2011), Issue 5, 1623-1654
Abstract: