论文标题
liouville的结果和超临界源梯度项的准椭圆方程的溶液的渐近学
Liouville results and asymptotics of solutions of a quasilinear elliptic equation with supercritical source gradient term
论文作者
论文摘要
我们考虑椭圆形的quasilinear方程 - $δ$ m u = u p | $ \ nabla $ u | q在r n中，带有q $ \ ge $ m和p> 0，1 <m <n。我们的主要结果是liouville型属性，即，r n中的所有正c 1解决方案都是恒定的。我们还给出了它们的渐近行为：外部结构域R n \ b r0中的所有解决方案都是有界的。 B r0 \ {0}中的解决方案可以作为B r0中的连续功能扩展。 r n \ {0}中的解决方案具有有限的限制l $ \ ge $ 0 as | x | $ \ rightarrow $ $ \ infty $。我们的主要论点是伯恩斯坦对溶液梯度的估计，结合了梯度满足的方程式的精确的Osserman类型估计。
We consider the elliptic quasilinear equation --$Δ$ m u = u p |$\nabla$u| q in R N with q $\ge$ m and p > 0, 1 < m < N. Our main result is a Liouville-type property, namely, all the positive C 1 solutions in R N are constant. We also give their asymptotic behaviour : all the solutions in an exterior domain R N \B r0 are bounded. The solutions in B r0 \ {0} can be extended as a continuous functions in B r0. The solutions in R N \ {0} has a finite limit l $\ge$ 0 as |x| $\rightarrow$ $\infty$. Our main argument is a Bernstein estimate of the gradient of a power of the solution, combined with a precise Osserman's type estimate for the equation satisfied by the gradient.
