# 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}
}
• 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 $m,q\in [0,\infty )$, $1\max \{p-m-1… 1 Citations Liouville Results and Asymptotics of Solutions of a Quasilinear Elliptic Equation with Supercritical Source Gradient Term We consider the elliptic quasilinear equation −∆ m u = u p |∇u| q in R N with q ≥ m and p > 0, 1 < m < N. Our main result is a Liouville-type property, namely, all the positive C 1 solutions in R N ## References SHOWING 1-10 OF 38 REFERENCES Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification • Mathematics • 2016 We generalise and sharpen several recent results in the literature regarding the existence and complete classification of the isolated singularities for a broad class of nonlinear elliptic equations Elliptic equations with nonlinear absorption depending on the solution and its gradient • Mathematics • 2014 We study positive solutions of equation (E1)$-\Delta u + u^p|\nabla u|^q= 0$($0\leq p$,$0\leq q\leq 2$,$p+q>1$) and (E2)$-\Delta u + u^p + |\nabla u|^q =0$($p>1$,$1<q\leq 2$) in a smooth Isolated singularities of positive solutions to the weighted p-Laplacian • Mathematics • 2016 In this paper we consider the weighted p-Laplacian $$-\mathrm{div}(|x|^\alpha |\nabla u|^{p-2}\nabla u)+ |x|^\gamma |u|^{q-1}u=0$$-div(|x|α|∇u|p-2∇u)+|x|γ|u|q-1u=0 in $$B{\setminus }\{0\}$$B\{0}, Singular solutions of some quasilinear elliptic equations • Mathematics • 1986 AbstractWe study isolated singularities of the quasilinear equation $$(*) - div (|\nabla u|^{p - 2} \nabla u) + |u|^{q - 1} u = 0$$ in an open set of ℝN, where 1 < p ≦ N, p -1 ≦ q < N(p — 1)/ (N Elliptic Partial Differential Equations of Second Order • Physics • 1997 We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations Local and global properties of solutions of quasilinear Hamilton-Jacobi equations • Mathematics • 2014 We study some properties of the solutions of (E)$\;-\Gd_p u+|\nabla u|^q=0$in a domain$\Gw \sbs \BBR^N$, mostly when$p\geq q>p-1\$. We give a universal priori estimate of the gradient of the
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