论文标题
Liouville型定理用于涉及Grushin操作员的椭圆系统稳定解决方案
Liouville type theorems for stable solutions of elliptic system involving the Grushin operator
论文作者
论文摘要
我们检查退化椭圆系统 $$ - δ_{s} u = v^p，\ quad-Δ__{s} v = u^θ，\ quad u，v> 0 \ quad \ quad \ mbox {in} \; \ Mathbb {r}^n = \ Mathbb {r}^{n_1} \ times \ times \ mathbb {r}^{n_2}，\ quad \ quad \ mbox {wery} \; \; \; \; \; \; s \ geq 0 \; \; \; \ mbox {and} \; \; p，θ> 0。$$我们证明，系统没有平滑稳定解决方案提供$ p，θ> 0 $和$ n_s <2 +α +β，$ $α= \ frac {2（p+1）} {pθ-1} \ quad \ mbox {and} \ quadβ= \ frac {2（θ+1）} 此结果是\ cite {my}中某些结果的扩展。特别是，我们建立了新的$ u $和$ v $ \;（请参阅命题1.1）的新的积分估计，这对于处理案件$ 0 <p <1至关重要。$ $
We examine the degenerate elliptic system $$-Δ_{s} u = v^p, \quad -Δ_{s} v= u^θ, \quad u,v>0 \quad\mbox{in }\; \mathbb{R}^N=\mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, \quad\mbox{where }\;\;\;\; s \geq 0\;\; \mbox{and} \;\;p,θ>0.$$ We prove that the system has no smooth stable solution provided $p,θ>0$ and $N_s< 2 + α+ β,$ where $$α= \frac{2(p+1)}{pθ- 1} \quad\mbox{and} \quad β= \frac{2(θ+1)}{pθ- 1}.$$ This result is an extension of some result in \cite{ MY}. In particular, we establish a new the integral estimate for $u$ and $v$ \;(see Proposition 1.1), which is crucial to deal with the case $0 < p < 1.$
