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

In this paper, we obtain gradient estimates of the positive solutions to weightedp-Laplacian type equations with a gradient-dependent nonlinearity of the form0.1$${\rm div }( \vert x \vert ^\sigma \vert \nabla u \vert ^{p-2}\nabla u) = \vert x \vert ^{-\tau }u^q \vert \nabla u \vert ^m\quad {\rm in}\;\Omega^*: = \Omega {\rm \setminus }\{ 0\} .$$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\}$. The main difficulty arises from the dependence of the right-hand side of (0.1) onx,uand$ \vert \nabla u \vert $, without any upper bound restriction on the powermof$ \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).