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

  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},
  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


Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification
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
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
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
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
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
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
Existence and classification of singular solutions to nonlinear elliptic equations with a gradient term
Let Ω be a domain in R with N ≥ 2 and 0 ∈ Ω. For 0 0 and m+q > 1, we obtain a complete classification of the behaviour near 0 (as well at∞ if Ω = R ) for all positive C(Ω\{0}) solutions of the
Entire solutions of completely coercive quasilinear elliptic equations, II
Abstract A famous theorem of Sergei Bernstein says that every entire solution u = u ( x ) , x ∈ R 2 , of the minimal surface equation div { D u 1 + | D u | 2 } = 0 is an affine function; no
The problem of dirichlet for quasilinear elliptic differential equations with many independent variables
  • J. Serrin
  • Mathematics
    Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1969
This paper is concerned with the existence of solutions of the Dirichlet problem for quasilinear elliptic partial differential equations of second order, the conclusions being in the form of
Isolated singularities for weighted quasilinear elliptic equations
Abstract We classify all the possible asymptotic behavior at the origin for positive solutions of quasilinear elliptic equations of the form div ( | ∇ u | p − 2 ∇ u ) = b ( x ) h ( u ) in Ω ∖ { 0 } ,