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

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 div|x|s|.u|p-2. u = |x|-tuq|.u|m in O* := O \ {0}. (0.1) Here, O. RN denotes a domain containing the origin with N2, whereas m, q. [0,8), 1 < pN + s and q > max{p - m - 1, s + t - 1}. The main difficulty arises from the dependence of the right-hand side of (0.1) on x, u and |.u|, without any upper bound restriction on the power m of |.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 (0.1).