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

@article{Ching2020GradientEF,
  title={Gradient estimates for nonlinear elliptic equations with a gradient-dependent nonlinearity},
  author={Joshua Ching and Florica C. C{\^i}rstea},
  journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics},
  year={2020},
  volume={150},
  pages={1361 - 1376}
}
  • Joshua Ching, F. Cîrstea
  • Published 1 February 2018
  • Mathematics, Physics
  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Abstract 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 0.1$${\rm div }( \vert x \vert ^\sigma \vert \nabla u \vert ^{p-2}\nabla u) = \vert x \vert ^{-\tau }u^q \vert \nabla u \vert ^m\quad {\rm in}\;\Omega^*: = \Omega {\rm \setminus }\{ 0\} .$$Here, $\Omega \subseteq {\open R}^N$ denotes a domain containing the origin with $N\ges 2$, whereas $m,q\in [0,\infty )$, $1\max \{p-m-1… 
Liouville Results and Asymptotics of Solutions of a Quasilinear Elliptic Equation with Supercritical Source Gradient Term
We consider the elliptic quasilinear equation −∆ m u = u p |∇u| q in R N with q ≥ m and p > 0, 1 < m < N. Our main result is a Liouville-type property, namely, all the positive C 1 solutions in R N

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