Harnack‐type inequality for fractional elliptic equations with critical exponent. (15th February 2020)

• Record Type: Journal Article
• Authors: Huang, Shuibo; Tian, Qiaoyu
• Is Part Of: Mathematical methods in the applied sciences. Volume 43:Number 8(2020)
• Page Start: 5380
• Page End: 5397
• Publication Date: 2020-02-15
• Subjects: fractional Laplacian equation -- Harnack‐type inequality -- method of moving planes; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.6280
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : In this paper, we use the method of moving planes to derive the Harnack‐type inequality max B R u · min B 2 R u ≤ C R N − 2 α, for nonnegative solutions to fractional semilinear elliptic equations ( − Δ ) α u = f ( u ), x ∈ B 3 R ( 0 ), u ( x ) > 0, x ∈ B 3 R ( 0 ), u ( x ) = 0, x ∈ R N ∖ B 3 R ( 0 ), with 0 < R < 1, 0 < α < 1, and 2 α < N . We obtain an asymptotic blow‐up rate estimate, as an application, for solutions to the above equation with a singular set.