论文标题
liouville型定理和具有非线性梯度术语的准椭圆方程的解决方案的存在
Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms
论文作者
论文摘要
本文涉及具有非线性梯度项的准椭圆方程的阳性弱解的两种特性。首先，我们为涉及$ M $ -M $ -LAPLACIAN操作员\ begin \ begin \ begin {equation*}-Δ__{m} u = u^q | \ nabla u | $ n \ geq1 $，$ m> 1 $和$ p，q \ geq0 $。伯恩斯坦梯度估计的技术被用来研究该案例$ p <m $。此外，还建立了在亚批判性范围下的liouville型定理\ begin \ begin {equation*} q（n-m）+p（n-1）+p（n-1）<n（m-1）\ end \ end {equation {等式*}。然后，我们使用学位参数来获得$-Δ_Mu = f（x，u，u，\ nabla u）$的非线性dirichlet问题的积极弱解决方案，其中$ f $满足某些结构条件。我们的证明是基于先验估计的，该估计将通过在半空间中使用liouville型定理一起实现。作为另一个应用程序，证明了一些新的Harnack不平等现象。
This paper is concerned with two properties of positive weak solutions of quasilinear elliptic equations with nonlinear gradient terms. First, we show a Liouville-type theorem for positive weak solutions of the equation involving the $m$-Laplacian operator \begin{equation*} -Δ_{m}u=u^q|\nabla u|^p\ \ \ \mathrm{in}\ \mathbb{R}^N, \end{equation*} where $N\geq1$, $m>1$ and $p,q\geq0$. The technique of Bernstein gradient estimates is ultilized to study the case $p<m$. Moreover, a Liouville-type theorem for supersolutions under subcritial range of exponents \begin{equation*} q(N-m)+p(N-1)<N(m-1) \end{equation*} is also established. Then, we use a degree argument to obtain the existence of positive weak solutions for a nonlinear Dirichlet problem of the type $-Δ_m u = f(x,u,\nabla u)$, with $f$ satisfying certain structure conditions. Our proof is based on a priori estimates, which will be accomplished by using a blow-up argument together with the Liouville-type theorem in the half-space. As another application, some new Harnack inequalities are proved.
