Gradient estimates for nonlinear elliptic equations with a gradientdependent nonlinearity
Abstract
In this paper, we obtain gradient estimates of the positive solutions to weighted $p$Laplacian type equations with a gradientdependent nonlinearity of the form \begin{equation} \label{one} {\rm div} (x^{\sigma}\nabla u^{p2} \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 $q>\max\{pm1,\sigma+\tau1\}$. The main difficulty arises from the dependence of the righthand 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 twostep process relying on a modified version of the Bernstein's method. As a byproduct, we extend the range of applicability of the Liouvilletype results known for our problem.
 Publication:

arXiv eprints
 Pub Date:
 January 2018
 arXiv:
 arXiv:1802.00109
 Bibcode:
 2018arXiv180200109C
 Keywords:

 Mathematics  Analysis of PDEs;
 35J60;
 35B53
 EPrint:
 12 pages