Gradient estimates for nonlinear elliptic equations with a gradient-dependent nonlinearity

Abstract

In this paper, we obtain gradient estimates of the positive solutions to weighted p-Laplacian type equations with a gradient-dependent nonlinearity of the form 0.1

$${\rm div }( \vert x \vert ^\sigma \vert \nabla u \vert ^{p-2}\nabla u) = \vert x \vert ^{-\tau }u^q \vert \nabla u \vert ^m\quad {\rm in}\;\Omega^*: = \Omega {\rm \setminus }\{ 0\} .$$

Here, $\Omega \subseteq {\open R}^N$ denotes a domain containing the origin with $N\ges 2$, whereas $m,q\in [0,\infty )$, $1<p\les N+\sigma $ and $q>\max \{p-m-1,\sigma +\tau -1\}$. The main difficulty arises from the dependence of the right-hand side of (0.1) on x, u and $ \vert \nabla u \vert $, without any upper bound restriction on the power m of $ \vert \nabla u \vert $. Our proof of the gradient estimates is based on a two-step process relying on a modified version of the Bernstein's method. As a by-product, we extend the range of applicability of the Liouville-type results known for (0.1).