Heisenberg群上含梯度项微分不等式的Keller-Osserman型条件
Keller-Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group
摘要：
本文的目的是研究Heisenberg群上一类含梯度项的微分不等式的非负整体解的定性性质。本文主要研究了两类不等式$Δ^ϕu \ge f(u)l(|\nabla u|)$和$Δ^ϕu \ge f(u) - h(u) g(|\nabla u|)$，其中$f,l,h,g$是非负连续函数，满足一定的单调性.算子$Δ^ϕ$，称为$ϕ$ -拉普拉斯算子，可以被看作是$p$ -拉普拉斯算子的自然推广，最近被许多作者考虑在这种设置中。引入两个新的Keller-Osserman型条件，证明了Liouville定理，这两个条件都是在$\erre^m$中研究原型微分不等式$Δu \ge f(u)$时出现的经典条件的推广.此外，我们显示我们的条件时，我们专门的情况下，$p$ -拉普拉斯的锐度。不用说，我们的结果在欧几里得空间中也是一样的，只是作了一些明显的小修改。
英文摘要：
The aim of this paper is to study the qualitative behaviour of non-negative entire solutions of certain differential inequalities involving gradient terms on the Heisenberg group. We focus our investigation on the two classes of inequalities of the form $Δ^ϕu \ge f(u)l(|\nabla u|)$ and $Δ^ϕu \ge f(u) - h(u) g(|\nabla u|)$, where $f,l,h,g$ are non-negative continuous functions satisfying certain monotonicity properties. The operator $Δ^ϕ$, called the $ϕ$-Laplacian, can be viewed as a natural generalization of the $p$-Laplace operator recently considered by various authors in this setting. We prove some Liouville theorems introducing two new Keller-Osserman type conditions, both extending the classical one which appeared long ago in the study of the prototype differential inequality $Δu \ge f(u)$ in $\erre^m$. Furthermore, we show sharpness of our conditions when we specialize to the case of the $p$-Laplacian. Needless to say, our results continue to hold, with the obvious minor modifications, also in the Euclidean space.
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