Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space

2014 ◽  
Vol 52 (3-4) ◽  
pp. 641-659 ◽  
Author(s):  
Alexander Quaas ◽  
Aliang Xia
Keyword(s):  
Half Space ◽  
Elliptic Equations ◽  
Fractional Laplacian ◽  
Nonlinear Elliptic Equations ◽  
Liouville Type ◽  
Nonlinear Elliptic ◽  
Liouville Type Theorems

Liouville-type theorems for elliptic equations in half-space with mixed boundary value conditions

2016 ◽  
Vol 8 (1) ◽  
pp. 193-202 ◽  
Author(s):  
Abdellaziz Harrabi ◽  
Belgacem Rahal
Keyword(s):  
Elliptic Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Nonlinear Elliptic Equations ◽  
Monotonicity Formula ◽  
Compact Set ◽  
Liouville Type ◽  
Nonlinear Elliptic ◽  
Liouville Type Theorems ◽  
Nonexistence Of Solutions

Abstract In this paper we study the nonexistence of solutions, which are stable or stable outside a compact set, possibly unbounded and sign-changing, of some nonlinear elliptic equations with mixed boundary value conditions. The main methods used are the integral estimates and the monotonicity formula.

Liouville-type Theorems for Fully Nonlinear Elliptic Equations and Systems in Half Spaces

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Guozhen Lu ◽  
Jiuyi Zhu
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
Nonlinear Elliptic Equations ◽  
Decay Estimates ◽  
Fully Nonlinear Elliptic Equations ◽  
Liouville Type ◽  
Fully Nonlinear Equations ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Liouville Type Theorems

AbstractThe main purpose of this paper is to establish Liouville-type theorems and decay estimates for viscosity solutions to a class of fully nonlinear elliptic equations or systems in half spaces without the boundedness assumptions on the solutions. Using the blow-up method and doubling lemma of [18], we remove the boundedness assumption on solutions which was often required in the proof of Liouville-type theorems in the literature. We also prove the Liouville-type theorems for supersolutions of a system of fully nonlinear equations with Pucci extremal operators in half spaces. Liouville theorems and decay estimates for high order elliptic equations and systems have also been established by the authors in an earlier work [15] when no boundedness assumption was given on the solutions.

Asymptotic behaviour and symmetry of positive solutions to nonlinear elliptic equations in a half-space

2016 ◽  
Vol 146 (6) ◽  
pp. 1243-1263 ◽  
Author(s):  
Lei Wei
Keyword(s):  
Half Space ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Unique Positive Solution ◽  
Bounded Smooth Domain ◽  
Nonlinear Elliptic ◽  
Bounded Smooth ◽  
Image Position ◽  
General Nonlinearity

We consider the following equation:where d(x) = d(x, ∂Ω), θ > –2 and Ω is a half-space. The existence and non-existence of several kinds of positive solutions to this equation when , f(u) = up(p > 1) and Ω is a bounded smooth domain were studied by Bandle, Moroz and Reichel in 2008. Here, we study exact the behaviour of positive solutions to this equation as d(x) → 0+ and d(x) → ∞, respectively, and the symmetry of positive solutions when , Ω is a half-space and f(u) is a more general nonlinearity term than up. Under suitable conditions for f, we show that the equation has a unique positive solution W, which is a function of x1 only, and W satisfies

scholarly journals Nonlinear elliptic equations on the upper half space

2020 ◽  
pp. 2050085
Author(s):  
Sufang Tang ◽  
Lei Wang ◽  
Meijun Zhu
Keyword(s):  
Boundary Condition ◽  
Half Space ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Symmetric Functions ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Nonlinear Elliptic Equations ◽  
Nonlinear Elliptic

In this paper, we shall classify all positive solutions of [Formula: see text] on the upper half space [Formula: see text] with nonlinear boundary condition [Formula: see text] on [Formula: see text] for parameters [Formula: see text] and [Formula: see text]. We will prove that for [Formula: see text] or [Formula: see text], [Formula: see text] (and [Formula: see text]) all positive solutions are functions of last variable; for [Formula: see text] (and [Formula: see text]) positive solutions must be either some functions depending only on last variable, or radially symmetric functions.

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

2019 ◽  
Vol 150 (3) ◽  
pp. 1361-1376
Author(s):  
Joshua Ching ◽  
Florica C. Cîrstea
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Gradient Estimates ◽  
Main Difficulty ◽  
Liouville Type ◽  
Step Process ◽  
Nonlinear Elliptic ◽  
Liouville Type Results ◽  
Image Position ◽  
The Right

AbstractIn this paper, we obtain gradient estimates of the positive solutions to weightedp-Laplacian type equations with a gradient-dependent nonlinearity of the form0.1$${\rm div }( \vert x \vert ^\sigma \vert \nabla u \vert ^{p-2}\nabla u) = \vert x \vert ^{-\tau }u^q \vert \nabla u \vert ^m\quad {\rm in}\;\Omega^*: = \Omega {\rm \setminus }\{ 0\} .$$Here,$\Omega \subseteq {\open R}^N$denotes a domain containing the origin with$N\ges 2$, whereas$m,q\in [0,\infty )$,$1<p\les N+\sigma $and$q>\max \{p-m-1,\sigma +\tau -1\}$. The main difficulty arises from the dependence of the right-hand side of (0.1) onx,uand$ \vert \nabla u \vert $, without any upper bound restriction on the powermof$ \vert \nabla u \vert $. Our proof of the gradient estimates is based on a two-step process relying on a modified version of the Bernstein's method. As a by-product, we extend the range of applicability of the Liouville-type results known for (0.1).

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