论文标题
$ΔU + au^{p + 1} = 0 $和liouville定理的梯度估计值
Gradient Estimates For $Δu + au^{p+1}=0$ And Liouville Theorems
论文作者
论文摘要
简而言之，我们使用一种统一的方法来考虑以下非线性椭圆方程$ΔU + au + au + au + au + au + au^{p + 1} = 0 $定义在完整的非政策的riemannian歧管折叠$（m，g）$ a> 0 $ a> 0 $ a> $ a> $ a> $ a> $ a> $ a> $ p <\ frac $ p <\ frac {4} $和$ p <0} $ p> p> p <0} $ and或p> p> p> n} $ and或p.对于情况，$ a> 0 $，除了$ \ dim（m）= 4 $的情况外，这大大改善了先前已知结果，并为Case $ \ dim（m）\ leq 2 $补充了结果。对于$ a <0 $和$ p> 0 $的情况，我们还大大提高了以前的相关结果。当$（m，g）$的RICCI曲率是非负值时，我们还获得了上述方程式的liouville型定理。
In this short note, we use a unified method to consider the gradient estimates of the positive solution to the following nonlinear elliptic equation $Δu + au^{p+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ where $a > 0$ and $ p <\frac{4}{n}$ or $a < 0$ and $p >0$ are two constants. For the case $a>0$, this improves considerably the previous known results except for the cases $\dim(M)=4$ and supplements the results for the case $\dim(M)\leq 2$. For the case $a<0$ and $p>0$, we also improve considerably the previous related results. When the Ricci curvature of $(M,g)$ is nonnegative, we also obtain a Liouville-type theorem for the above equation.
