scholarly journals A direct method of moving planes for a fully nonlinear nonlocal system

2017 ◽  
Vol 16 (5) ◽  
pp. 1707-1718 ◽  
Author(s):  
Pengyan Wang ◽  
◽  
Pengcheng Niu
Keyword(s):  
Direct Method ◽  
Fully Nonlinear ◽  
Method Of Moving Planes ◽  
Moving Planes ◽  
A Direct Method

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 Monotonicity results for the fractional p-Laplacian in unbounded domains

2021 ◽  
pp. 1-29
Author(s):  
Leyun Wu ◽  
Mei Yu ◽  
Binlin Zhang
Keyword(s):  
Singular Integrals ◽  
Direct Method ◽  
Unbounded Domains ◽  
Method Of Moving Planes ◽  
Monotonicity Result ◽  
New Ideas ◽  
Moving Planes ◽  
De Giorgi Conjecture ◽  
A Direct Method ◽  
Conditions At Infinity

In this paper, we develop a direct method of moving planes in unbounded domains for the fractional p-Laplacians, and illustrate how this new method to work for the fractional p-Laplacians. We first proved a monotonicity result for nonlinear equations involving the fractional p-Laplacian in [Formula: see text] without any decay conditions at infinity. Second, we prove De Giorgi conjecture corresponding to the fractional p-Laplacian under some conditions. During these processes, we introduce some new ideas: (i) estimating the singular integrals defining the fractional p-Laplacian along a sequence of approximate maxima; (ii) estimating the lower bound of the solutions by constructing sub-solutions.

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