论文标题
liouville定理用于无限laplacian，带有梯度和kpp型方程
Liouville theorems for infinity Laplacian with gradient and KPP type equation
论文作者
论文摘要
在本文中，我们证明了涉及无限拉普拉斯的非线性方程式的新的liouville类型结果，其形式为$Δ^Δ^undty u + q（x）\ cdot \ cdot \ nabla {u} | \ nabla {u} | \ nabla {u} | \ nabla {u} \ mathbb {r}^d，$$，其中$γ\在[0，2] $和$δ^γ_\ infty $中是$（3-γ）$ - 与Infinity Laplacian相关的同质运算符。在假设下，$ \ liminf_ {| x | \ to \ infty} \ lim_ {s \ to0} f（x，x，s）/s^{3-γ}> 0 $ and $ q $是无限度的连续函数，在无限限度下消失了，我们构建了一个正面的解决方案，构建了一个正界解决方案，以及$ f（x，x，x）/s $ f（x，s^$ s $ s^3-} $ s^3-} $ s^-s^-s^-s^3-}通过改善具有非线性梯度的无穷laplacian操作员的滑动方法，具有独特性。否则，如果$ \ limsup_ {| x | \ to \ infty} \ sup _ {[δ_1，Δ_2]} f（x，s）<0 $，则不存在的结果还提供了一些合适的条件。为此，我们开发了新型技术，以克服因梯度术语的无限拉普拉斯和非线性的堕落而引起的困难。我们的方法是基于新的规律性结果，强大的最大原则以及霍普夫的无限拉普拉斯（Infinity Laplacian）的引理，涉及梯度和潜力。我们还构建了一些示例来说明我们的结果。我们进一步研究了相应的非线性操作员的主要特征值的一些更深的定性特性，$Δ 兴趣。此处获得的结果可被视为在[1、2、11、24、48、52]中获得的liouville型结果的急剧扩展。
In this paper, we prove new Liouville type results for a nonlinear equation involving infinity Laplacian with gradient of the form $$Δ^γ_\infty u + q(x)\cdot \nabla{u} |\nabla{u}|^{2-γ} + f(x, u)\,=\,0\quad \text{in}\; \mathbb{R}^d,$$ where $γ\in [0, 2]$ and $Δ^γ_\infty$ is a $(3-γ)$-homogeneous operator associated with the infinity Laplacian. Under the assumptions $\liminf_{|x|\to\infty}\lim_{s\to0}f(x,s)/s^{3-γ}>0$ and $q$ is a continuous function vanishing at infinity, we construct a positive bounded solution to the equation and if $f(x,s)/s^{3-γ}$ decreasing in $s$, we further obtain the uniqueness by improving sliding method for infinity Laplacian operator with nonlinear gradient. Otherwise, if $\limsup_{|x|\to\infty}\sup_{[δ_1,δ_2]}f(x,s)<0$, then nonexistence result holds provided additionally some suitable conditions. To this aim, we develop novel techniques to overcome the difficulties stemming from the degeneracy of infinity Laplacian and nonlinearity of the gradient term. Our approach is based on a new regularity result, the strong maximum principle, and Hopf's lemma for infinity Laplacian involving gradient and potential. We also construct some examples to illustrate our results. We further investigate some deeper qualitative properties of the principal eigenvalue of the corresponding nonlinear operator $$Δ^γ_\infty u + q(x)\cdot \nabla{u} |\nabla{u}|^{2-γ} + c(x)u^{3-γ},$$ with Dirichlet boundary condition in smooth bounded domains, which may be of independent interest. The results obtained here could be considered as sharp extension of the Liouville type results obtained in [1, 2, 11, 24, 48, 52].
