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Positive solutions to semilinear elliptic equations involving a weighted fractional Lapalacian
Author(s) -
Tang De
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4184
Subject(s) - mathematics , operator (biology) , monotonic function , bounded function , fractional laplacian , antisymmetric relation , semi elliptic operator , domain (mathematical analysis) , laplace operator , antisymmetric tensor , elliptic operator , mathematical analysis , symmetry (geometry) , pure mathematics , mathematical physics , differential operator , geometry , biochemistry , chemistry , repressor , gauge theory , transcription factor , gene
In this paper, we consider a uniform elliptic nonlocal operatorA α u ( x ) = C n , α P . V . ∫R na ( x − y ) ( u ( x ) − u ( y ) ) | x − y | n + αdy , which is a weighted form of fractional Laplacian. We firstly establish three maximum principles for antisymmetric functions with respect to the nonlocal operator. Then, we obtain symmetry, monotonicity, and nonexistence of solutions to some semilinear equations involving the operatorA αon bounded domain,R + nandR n , by applying direct moving plane methods. Finally, we show the relations between the classical operator − Δ and the nonlocal operator in ( ) as α →2. Copyright © 2016 John Wiley & Sons, Ltd.