论文标题
关于涉及Grushin操作员的加权椭圆系统的解决方案的分类
On the classification of solutions to a weighted elliptic system involving the Grushin operator
论文作者
论文摘要
我们在这里调查以下加权退化椭圆系统\ begin {align*}-Δ_{s} u = \ big（1+ \ | \ | \ Mathbf {x}} \ |^{2（s+1）} \ big） \ big（1+ \ | \ MathBf {x} \ |^{2（s+1）} \ big） \ Mathbb {r}^n：= \ Mathbb {r}^{n_1} \ times \ times \ mathbb {r}^{n_2}。其中$δ_{s} =δ__{x}+| x |^{2S}δ_{y}，$是Grushin运算符，$ s \ geq 0，$ $ $ $ $ $ $ $ a $ \ | \ Mathbf {x} \ | = \ big（| x |^{2（s+1）}+| y |^2 \ big）^{\ frac {1} {2（s+1）}}，\; \ MathBf {X}：=（x，x，y）\ in \ Mathbb {r}^n：= \ m athbb {r}^{n_1} {n_1} \ times \ times \ times \ times \ mathbb {r}^{n_2 {n_2}。 \ cite {cow，hfh，hu，fa，dp}。结果，我们获得了加权的grushin方程\ begin {align*}-Δ_{s} u = \ big（1+\ | \ | \ Mathbf {x} \ |^{2（s+1）} \ big）^{ \ Quad U> 0 \ Quad \ mbox {in} \; \; \ Mathbb {r}^n。 \ end {align*}
We investigate here the following weighted degenerate elliptic system \begin{align*} -Δ_{s} u = \Big(1+\|\mathbf{x}\|^{2(s+1)}\Big)^{\fracα{2(s+1)}} v^p, \quad -Δ_{s} v = \Big(1+\|\mathbf{x}\|^{2(s+1)}\Big)^{\fracα{2(s+1)}}u^θ, \quad u,v>0\quad\mbox{in }\; \mathbb{R}^N:=\mathbb{R}^{N_1}\times \mathbb{R}^{N_2}. \end{align*} where $Δ_{s}=Δ_{x}+|x|^{2s}Δ_{y},$ is the Grushin operator, $s \geq 0,$ $α\geq 0$ and $1<p\leqθ.$ Here $$\|\mathbf{x}\|=\Big(|x|^{2(s+1)}+|y|^2\Big)^{\frac{1}{2(s+1)}}, \;\mbox{and}\;\; \mathbf{x}:=(x, y)\in \mathbb{R}^N:=\mathbb{R}^{N_1}\times \mathbb{R}^{N_2}.$$ In particular, we establish some new Liouville-type theorems for stable solutions of the system, which recover and considerably improve upon the known results \cite{cow, Hfh, HU, Fa, DP}. As a consequence, we obtain a nonexistence result for the weighted Grushin equation \begin{align*} -Δ_{s} u =\Big(1+\|\mathbf{x}\|^{2(s+1)}\Big)^{\fracα{2(s+1)}} u^p,\;\; \quad u>0 \quad \mbox{in }\;\; \mathbb{R}^N. \end{align*}
