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|^{ \sigma}|\nabla u|^{ p-2} \nabla u)= |x|^{ -\tau} u^q |\nabla u|^m \quad \mbox{ in } \ \Omega^*:= \Omega \setminus \{ 0 \}. \end{ equation} Here, $\Omega\subseteq \mathbb R^N$ denotes a domain containing the origin with $N\geq 2$, whereas $m,q\in [0,\infty)$, $1<p\leq N+\sigma$ 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.
Categories: math.AP