Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms

TL;DR

This paper establishes Liouville-type theorems for positive solutions of quasilinear elliptic equations with nonlinear gradient terms, and uses these results to prove existence and Harnack inequalities for related boundary value problems.

Contribution

It introduces new Liouville-type theorems for equations involving the m-Laplacian with gradient nonlinearities and applies them to prove existence of solutions and Harnack inequalities.

Findings

Liouville-type theorem for the m-Laplacian with gradient terms.

Existence of positive weak solutions for nonlinear Dirichlet problems.

New Harnack inequalities for solutions of these equations.

Abstract

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*} -\Delta_{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 $-\Delta_m u = f(x,u,\nabla u)$, 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.