scholarly journals Radial symmetry for a generalized nonlinear fractional p-Laplacian problem

2021 ◽  
Vol 26 (2) ◽  
pp. 349-362
Author(s):  
Wenwen Hou ◽  
Lihong Zhang ◽  
Ravi P. Agarwal ◽  
Guotao Wang
Keyword(s):  
Positive Solutions ◽  
Direct Method ◽  
Radial Symmetry ◽  
Laplacian Operator ◽  
Negative Power ◽  
Laplacian Equation ◽  
Narrow Region ◽  
Method Of Moving Planes ◽  
Moving Planes

This paper first introduces a generalized fractional p-Laplacian operator (–Δ)sF;p. By using the direct method of moving planes, with the help of two lemmas, namely decay at infinity and narrow region principle involving the generalized fractional p-Laplacian, we study the monotonicity and radial symmetry of positive solutions of a generalized fractional p-Laplacian equation with negative power. In addition, a similar conclusion is also given for a generalized Hénon-type nonlinear fractional p-Laplacian equation.

Positive solutions to nonlinear systems involving fully nonlinear fractional operators

2018 ◽  
Vol 21 (2) ◽  
pp. 552-574 ◽  
Author(s):  
Pengcheng Niu ◽  
Leyun Wu ◽  
Xiaoxue Ji
Keyword(s):  
Positive Solutions ◽  
Direct Method ◽  
Radial Symmetry ◽  
Fractional Operators ◽  
Fully Nonlinear ◽  
Narrow Region ◽  
Method Of Moving Planes ◽  
Whole Space ◽  
Moving Planes ◽  
Fractional System

Abstract In this paper we consider the following fractional system $$\begin{array}{} \displaystyle \left\{ \begin{gathered} F(x,u(x),v(x),{\mathcal{F}_\alpha }(u(x))) = 0,\\ G(x,v(x),u(x),{\mathcal{G}_\beta }(v(x))) = 0, \\ \end{gathered} \right. \end{array}$$ where 0 < α, β < 2, 𝓕α and 𝓖β are the fully nonlinear fractional operators: $$\begin{array}{} \displaystyle {\mathcal{F}_\alpha }(u(x)) = {C_{n,\alpha }}PV\int_{{\mathbb{R}^n}} {\frac{{f(u(x) - u(y))}} {{{{\left| {x - y} \right|}^{n + \alpha }}}}dy} ,\\ \displaystyle{\mathcal{G}_\beta }(v(x)) = {C_{n,\beta }}PV\int_{{\mathbb{R}^n}} {\frac{{g(v(x) - v(y))}} {{{{\left| {x - y} \right|}^{n + \beta }}}}dy} . \end{array}$$ A decay at infinity principle and a narrow region principle for solutions to the system are established. Based on these principles, we prove the radial symmetry and monotonicity of positive solutions to the system in the whole space and a unit ball respectively, and the nonexistence in a half space by generalizing the direct method of moving planes to the nonlinear system.

scholarly journals Radial Symmetry and Monotonicity of Solutions to a System Involving Fractional p-Laplacian in a Ball

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Linfen Cao ◽  
Xiaoshan Wang ◽  
Zhaohui Dai
Keyword(s):  
Nonlinear System ◽  
Unit Ball ◽  
Positive Solutions ◽  
Direct Method ◽  
Radial Symmetry ◽  
Continuous Functions ◽  
Method Of Moving Planes ◽  
Moving Planes

In this paper, we study a nonlinear system involving the fractional p-Laplacian in a unit ball and establish the radial symmetry and monotonicity of its positive solutions. By using the direct method of moving planes, we prove the following result. For 0<s,t<1,p>0, if u and v satisfy the following nonlinear system -Δpsux=fvx;  -Δptvx=gux,  x∈B10;  ux,vx=0,  x∉B10. and f,g are nonnegative continuous functions satisfying the following: (i) f(r) and g(r) are increasing for r>0; (ii) f′(r)/rp-2, g′(r)/rp-2 are bounded near r=0. Then the positive solutions (u,v) must be radially symmetric and monotone decreasing about the origin.

Symmetry of solutions for a fractional p-Laplacian equation of Choquard type

2020 ◽  
Vol 31 (04) ◽  
pp. 2050026
Author(s):  
Phuong Le
Keyword(s):  
Direct Method ◽  
Full Range ◽  
Fractional Systems ◽  
New Approach ◽  
Laplacian Equation ◽  
Method Of Moving Planes ◽  
Moving Planes ◽  
Nonlocal Nonlinearities ◽  
Laplacian Equations ◽  
Decay Assumption

Let [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] be a positive solution of the equation [Formula: see text] We prove that if [Formula: see text] satisfies some decay assumption at infinity, then [Formula: see text] must be radially symmetric and monotone decreasing about some point in [Formula: see text]. Instead of using equivalent fractional systems, we exploit a generalized direct method of moving planes for fractional [Formula: see text]-Laplacian equations with nonlocal nonlinearities. This new approach enables us to cover the full range [Formula: see text] in our results.

scholarly journals Symmetry and Nonexistence of Positive Solutions for Weighted HLS System of Integral Equations on a Half Space

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Linfen Cao ◽  
Zhaohui Dai
Keyword(s):  
Integral Equations ◽  
Half Space ◽  
Positive Solutions ◽  
Type Theorem ◽  
Integral Form ◽  
System Of Integral Equations ◽  
Method Of Moving Planes ◽  
Rotationally Symmetric ◽  
Moving Planes ◽  
Pohozaev Type Identity

We consider system of integral equations related to the weighted Hardy-Littlewood-Sobolev (HLS) inequality in a half space. By the Pohozaev type identity in integral form, we present a Liouville type theorem when the system is in both supercritical and subcritical cases under some integrability conditions. Ruling out these nonexistence results, we also discuss the positive solutions of the integral system in critical case. By the method of moving planes, we show that a pair of positive solutions to such system is rotationally symmetric aboutxn-axis, which is much more general than the main result of Zhuo and Li, 2011.

Monotonicity and symmetry of positive solutions to fractional p-Laplacian equation

2021 ◽  
pp. 2150005
Author(s):  
Wei Dai ◽  
Zhao Liu ◽  
Pengyan Wang
Keyword(s):  
Dirichlet Problem ◽  
Nonlinear Equations ◽  
Positive Solutions ◽  
Unbounded Domain ◽  
Fractional Laplacian ◽  
Unbounded Domains ◽  
Laplacian Operator ◽  
Maximum Principles ◽  
Laplacian Equation ◽  
Laplacian Equations

In this paper, we are concerned with the following Dirichlet problem for nonlinear equations involving the fractional [Formula: see text]-Laplacian: [Formula: see text] where [Formula: see text] is a bounded or an unbounded domain which is convex in [Formula: see text]-direction, and [Formula: see text] is the fractional [Formula: see text]-Laplacian operator defined by [Formula: see text] Under some mild assumptions on the nonlinearity [Formula: see text], we establish the monotonicity and symmetry of positive solutions to the nonlinear equations involving the fractional [Formula: see text]-Laplacian in both bounded and unbounded domains. Our results are extensions of Chen and Li [Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math. 335 (2018) 735–758] and Cheng et al. [The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math. 19(6) (2017) 1750018].

scholarly journals RADIAL SYMMETRY OF POSITIVE SOLUTIONS TO EQUATIONS INVOLVING THE FRACTIONAL LAPLACIAN

2014 ◽  
Vol 16 (01) ◽  
pp. 1350023 ◽  
Author(s):  
PATRICIO FELMER ◽  
YING WANG
Keyword(s):  
Positive Solutions ◽  
Fractional Laplacian ◽  
Radial Symmetry ◽  
Open Unit ◽  
Open Unit Ball ◽  
Lipschitz Continuous ◽  
Local Character ◽  
Locally Lipschitz ◽  
Moving Planes ◽  
Locally Lipschitz Continuous

The aim of this paper is to study radial symmetry and monotonicity properties for positive solution of elliptic equations involving the fractional Laplacian. We first consider the semi-linear Dirichlet problem [Formula: see text] where (-Δ)αdenotes the fractional Laplacian, α ∈ (0, 1), and B1denotes the open unit ball centered at the origin in ℝNwith N ≥ 2. The function f : [0, ∞) → ℝ is assumed to be locally Lipschitz continuous and g : B1→ ℝ is radially symmetric and decreasing in |x|. In the second place we consider radial symmetry of positive solutions for the equation [Formula: see text] with u decaying at infinity and f satisfying some extra hypothesis, but possibly being non-increasing.Our third goal is to consider radial symmetry of positive solutions for system of the form [Formula: see text] where α1, α2∈(0, 1), the functions f1and f2are locally Lipschitz continuous and increasing in [0, ∞), and the functions g1and g2are radially symmetric and decreasing. We prove our results through the method of moving planes, using the recently proved ABP estimates for the fractional Laplacian. We use a truncation technique to overcome the difficulty introduced by the non-local character of the differential operator in the application of the moving planes.

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