Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms

  title={Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms},
  author={Caihong Chang and Bei Hu and Zhengce Zhang},
  journal={Nonlinear Analysis},
2 Citations

On the Liouville property for fully nonlinear equations with superlinear first-order terms

We consider in this note one-side Liouville properties for viscosity solutions of various fully nonlinear uniformly elliptic inequalities, whose prototype is F (x,Du) ≥ Hi(x, u,Du) in R , where Hi

A priori estimates and Liouville type results for quasilinear elliptic equations involving gradient terms

. In this article we study local and global properties of positive solutions of − ∆ m u = | u | p − 1 u + M |∇ u | q in a domain Ω of R N , with m > 1, p, q > 0 and M ∈ R . Following some ideas used



Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems

In this paper we consider positive supersolutions of the elliptic equation $-\triangle u = f(u) |\nabla u|^q$, posed in exterior domains of $\mathbb{R}^N$ ($N\ge 2$), where $f$ is continuous in

A priori estimates and existence for quasi-linear elliptic equations

AbstractWe study the boundary value problem of quasi-linear elliptic equation $$\begin{array}{rl} {\rm div}(|\nabla u|^{m-2} \nabla u) + B(z,u,\nabla u) = 0 &\quad {\rm in}\, \Omega,\\ u = 0 &\quad

Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient

We study local and global properties of positive solutions of $-{\Delta}u=u^p]{\left |{\nabla u}\right |}^q$ in a domain ${\Omega}$ of ${\mathbb R}^N$, in the range $1<p+q$, $p\geq 0$, $0\leq q< 2$.

Asymptotic behavior of global solutions to a class of heat equations with gradient nonlinearity

The paper is devoted to investigating a semilinear parabolic equation with a nonlinear gradient source term: \begin{document}$ u_t = u_{xx}+x^m|u_x|^p, \ \ t>0, \ \ 0 where \begin{document}$ p>m+2

A Liouville-Type Theorem for an Elliptic Equation with Superquadratic Growth in the Gradient

Abstract We consider the elliptic equation -Δ⁢u=uq⁢|∇⁡u|p{-\Delta u=u^{q}|\nabla u|^{p}} in ℝn{\mathbb{R}^{n}} for any p>2{p>2} and q>0{q>0}. We prove a Liouville-type theorem, which asserts that any

Classification of certain qualitative properties of solutions for the quasilinear parabolic equations

In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation $${u_t} - div\left( {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \right) = - {\left| u

Discontinuous critical Fujita exponents for the heat equation with combined nonlinearities

We consider the nonlinear heat equation $u_t-\Delta u =|u|^p+b |\nabla u|^q$ in $(0,\infty)\times \R^n$, where $n\geq 1$, $p>1$, $q\geq 1$ and $b>0$. First, we focus our attention on positive

Asymptotic behavior of solutions for a free boundary problem with a nonlinear gradient absorption

This paper deals with the free boundary problem for a parabolic equation, $$u_t-u_{xx}=u^{p}-\lambda |u_x|^{q}$$ut-uxx=up-λ|ux|q, $$t>0$$t>0, $$01$$p,q>1. It is well known that global existence or

Nonexistence results and estimates for some nonlinear elliptic problems

AbstractHere we study the local or global behaviour of the solutions of elliptic inequalities involving quasilinear operators of the type $$L_{\mathcal{A}^u } = - div\left[ {\mathcal{A}\left(

Liouville Results and Asymptotics of Solutions of a Quasilinear Elliptic Equation with Supercritical Source Gradient Term

Abstract We consider the elliptic quasilinear equation -Δm⁢u=up⁢|∇⁡u|q{-\Delta_{m}u=u^{p}\lvert\nabla u\rvert^{q}} in ℝN{\mathbb{R}^{N}}, q≥m{q\geq m} and p>0{p>0}, 1<m<N{1<m<N}. Our main result is a