Positive solutions to semilinear elliptic equations involving a weighted fractional Lapalacian. (15th September 2016)

• Record Type: Journal Article
• Authors: Tang, De
• Is Part Of: Mathematical methods in the applied sciences. Volume 40:Number 7(2017)
• Page Start: 2596
• Page End: 2609
• Publication Date: 2016-09-15
• Subjects: uniform elliptic nonlocal operator -- direct moving plane methods -- radial symmetry -- nonexistence -- monotonicity -- Liouville‐type theorem; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.4184
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : In this paper, we consider a uniform elliptic nonlocal operator 1 A α u ( x ) = C n, α P . V . ∫ R n a ( x − y ) ( u ( x ) − u ( y ) ) | x − y | n + α dy, 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 operator A α on bounded domain, R + n and R n, by applying direct moving plane methods. Finally, we show the relations between the classical operator − Δ and the nonlocal operator in (1 ) as α →2. Copyright © 2016 John Wiley & Sons, Ltd.