Corpus ID: 236469125

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

  title={On the Liouville property for fully nonlinear equations with superlinear first-order terms},
  author={Marco Cirant and Alessandro Goffi},
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 has superlinear growth in the gradient variable. After a brief survey on the existing literature, we discuss the validity or the failure of the Liouville property in the model cases H1(u,Du) = u + |Du| , H2(u,Du) = u|Du| and H3(x,Du) = ±u|Du| − b(x) ·Du, where q ≥ 0, γ > 1 and b is a suitable velocity… Expand
Liouville results for fully nonlinear equations modeled on H\"ormander vector fields: II. Carnot groups and Grushin geometries
The paper treats second order fully nonlinear degenerate elliptic equations having a family of subunit vector fields satisfying a full-rank bracket condition. It studies Liouville properties forExpand


Optimal Liouville theorems for supersolutions of elliptic equations with the Laplacian
In this paper we consider the question of nonexistence of positive supersolutions of the equation −1u = f (u) in exterior domains of RN , where f is continuous and positive in (0,+1). When N # 3,Expand
Liouville results for fully nonlinear equations modeled on Hörmander vector fields: I. The Heisenberg group
This paper studies Liouville properties for viscosity sub- and supersolutions of fully nonlinear degenerate elliptic PDEs, under the main assumption that the operator has a family of generalizedExpand
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 NExpand
Liouville properties and critical value of fully nonlinear elliptic operators
We prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms toExpand
A Liouville-Type Theorem for an Elliptic Equation with Superquadratic Growth in the Gradient
Abstract We consider the elliptic equation - Δ ⁢ u = u q ⁢ | ∇ ⁡ u | p {-\Delta u=u^{q}|\nabla u|^{p}} in ℝ n {\mathbb{R}^{n}} for any p > 2 {p>2} and q > 0 {q>0} . We prove a Liouville-type theorem,Expand
Nonexistence of positive supersolutions to some nonlinear elliptic problems
Abstract In this paper we obtain Liouville type theorems for positive supersolutions of the elliptic problem − Δ u + | ∇ u | q = λ f ( u ) in exterior domains of R N . Here q > 1 and the function fExpand
Abstract In this paper we prove global bounds on the spatial gradient of viscosity solutions to second order linear and nonlinear parabolic equations in ( 0 , T ) × R N . Our assumptions include theExpand
Fundamental solutions and liouville type theorems for nonlinear integral operators
Abstract In this article we study basic properties for a class of nonlinear integral operators related to their fundamental solutions. Our goal is to establish Liouville type theorems: non-existenceExpand
A Liouville comparison principle for solutions ofsingular quasilinear elliptic second-order partial differentialinequalities
We compare entire weak solutions $u$ and $v$ of quasilinear partial differential inequalities on $R^n$ without any assumptions on their behaviour at infinity and show among other things, that theyExpand
On the equivalence of stochastic completeness, Liouville and Khas'minskii condition in linear and nonlinear setting
Set in Riemannian enviroment, the aim of this paper is to present and discuss some equivalent characterizations of the Liouville property relative to special operators, in some sense modeled afterExpand