Gradient estimates for the p-laplace equation with a singular source

In this paper we will employ the method of “DeGiorgi–Nash–Moser” to establish a \(L^\infty \) $L^{\infty }$ local estimate for the gradient of local weak solutions to p-Laplacian equation 1.1 \(\begin{aligned} -\text{ div }\left\{ |\nabla u|^{p-2}\nabla u\right\} =f(x,u,\nabla u)\,. \end{aligned}\) $$\begin{array}{c}\hfill -\text{div}\left\{{|\mathrm{\nabla }u|}^{p-2},\mathrm{\nabla },u\right\}=f(x,u,\mathrm{\nabla }u).\end{array}$$ In this work we extend to \(1<p<2\) $1<p<2$ the local estimates of the \(\Vert \nabla u\Vert _\infty \) ${\Vert \mathrm{\nabla }u\Vert }_{\infty }$ of the local weak solutions of (1.1), obtained by Bhattacharya, DiBenedetto and Manfredi in Limits as \(p\rightarrow \infty \) $p\to \infty$ of \(\Delta _p u_p=f\) ${\mathrm{\Delta }}_p{u}_p=f$ and related extremal problems (1989), for \(p>2\) $p>2$ .