Symmetric properties of positive solutions for fully nonlinear nonlocal system

Wang, Pengyan; Niu, Pengcheng

doi:10.1016/j.na.2019.04.002

Cited by 14 publications

(2 citation statements)

References 27 publications

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“…In this paper, we will develop a systematical scheme to carry on the asymptotic method of moving planes for nonlocal parabolic problems, either on bounded or unbounded domains. For nonlocal elliptic equations, these kinds of approaches were introduced in [9] and then summarized in the book [10], among which the narrow region principle and the decay at infinity have been employed extensively by many researchers to solve various problems [13,18,42]. A parallel system for the fractional parabolic equations will be established here by very elementary methods, so that it can be conveniently applied to various nonlocal parabolic problems.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic method of moving planes for fractional parabolic equations

Chen

Wang

Niu

et al. 2020

Preprint

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In this paper, we develop a systematical approach in applying an asymptotic method of moving planes to investigate qualitative properties of positive solutions for fractional parabolic equations. We first obtain a series of needed key ingredients such as narrow region principles, and various asymptotic maximum and strong maximum principles for antisymmetric functions in both bounded and unbounded domains. Then we illustrate how this new method can be employed to obtain asymptotic radial symmetry and monotonicity of positive solutions in a unit ball and on the whole space. Namely, we show that no matter what the initial data are, the solutions will eventually approach to radially symmetric functions.We firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal parabolic problems with more general operators and more general nonlinearities. CONTENTS 1. Introduction 1 1.1. Main results on qualitative properties 3 1.2. Ideas and key ingredients in the proofs 5 2. Proofs of key principles 8 3. Asymptotic symmetry of solutions in B 1 (0) 20 4. Asymptotic symmetry of solutions in R N 23 4.1. Step 1: λ sufficiently negative. 24 4.2. Step 2: Move the plane to the rightmost limiting position. 25 4.3. Step 3: All ω-limit functions are radially symmetric. 27 5. Appendix 37 References 38

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Section: Introductionmentioning

confidence: 99%

Asymptotic method of moving planes for fractional parabolic equations

Chen

Wang

Niu

et al. 2020

Preprint

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“…In 2015, Chen, Li and Li [11] developed a new technique (the direct method of moving planes) that can be applied for problems with fractional Laplace operator. It is very effective in dealing with equations involving fully nonlinear nonlocal operators or uniformly elliptic nonlocal operators, for example the results in [6,10,33,34,35] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Symmetric properties for Choquard equations involving fully nonlinear nonlocal operator

Wang,

Chen,

Niu

2019

Preprint

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In this paper, the positive solutions to Choquard equation involving fully nonlinear nonlocal operator are shown to be symmetric and monotone by using the moving plane method which has been introduced by Chen, Li and Li in 2015. The key ingredients are to obtain the "narrow region principle" and "decay at infinity" for the corresponding problems. Similar ideas can be easily applied to various nonlocal problems with more general nonlinearities.

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Symmetric Properties for System Involving Uniformly Elliptic Nonlocal Operators

Zeng

2020

Mediterr. J. Math.

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Symmetric properties of positive solutions for fully nonlinear nonlocal system

Cited by 14 publications

References 27 publications

Asymptotic method of moving planes for fractional parabolic equations

Asymptotic method of moving planes for fractional parabolic equations

Symmetric properties for Choquard equations involving fully nonlinear nonlocal operator

Symmetric Properties for System Involving Uniformly Elliptic Nonlocal Operators

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