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 \begin{equation} \label{one} {\rm div} (|x|^σ|\nabla u|^{p-2} \nabla u)= |x|^{-τ} u^q |\nabla u|^m \quad \mbox{in } \ Ω^*:= Ω\setminus \{ 0 \}. \end{equation} Here, $\Omega \subseteq {\mathbb{R}}^N$ denotes a domain containing the origin with $N\ge 2$, whereas $m,q\in [0,\infty )$, $1<p\le N+\sigma$ and $q>\max \{p-m-1,\sigma +\tau -1\}$. The main difficulty arises from the dependence of the right-hand side of the equation on $x$, $u$ and $|\nabla u|$, without any upper bound restriction on the power $m$ of $|\nabla u|$. 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 our problem.

12 pages