A Direct Method of Moving Planes to Fractional Power SubLaplace Equations on the Heisenberg Group

2021 ◽  
Vol 37 (2) ◽  
pp. 364-379
Author(s):  
Xin-jing Wang ◽  
Peng-cheng Niu
Keyword(s):  
Heisenberg Group ◽  
Direct Method ◽  
Fractional Power ◽  
Method Of Moving Planes ◽  
Moving Planes ◽  
A Direct Method

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.

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.

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