Blow-up behavior of a fractional Adams–Moser–Trudinger-type inequality in odd dimension. Issue 10 (2nd October 2016)

• Record Type: Journal Article
• Authors: Maalaoui, Ali; Martinazzi, Luca; Schikorra, Armin
• Is Part Of: Communications in partial differential equations. Volume 41:Issue 10(2016)
• Page Start: 1593
• Page End: 1618
• Publication Date: 2016-10-02
• Subjects: Fractional Laplacian -- Moser-Trudinger inequality -- Q-curvature -- quantization; 35J30 -- 35J61 -- 35S05; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2016.1222544
• Languages: English
• ISSNs: 0360-5302

Abstract

ABSTRACT: Given a smoothly bounded domain Ω ⊂ ℝ n with n ≥ 1 odd, we study the blowup of bounded sequences of solutions to the nonlocal equation where λ k → λ∞ ∈ [0, ∞), and denotes the Lions–Magenes spaces of functions u ∈ L 2 (ℝ n ) which are supported in Ω and with . Extending previous works of Druet, Robert-Struwe, and Martinazzi, we show that if the sequence ( u k ) is not bounded in L ∞ (Ω), a suitably rescaled subsequence η k converges to the function, which solves the prescribed nonlocal Q -curvature equation recently studied by Da Lio–Martinazzi–Rivière when n = 1, Jin–Maalaoui–Martinazzi–Xiong when n = 3, and Hyder when n ≥ 5 is odd. We infer that blowup can occur only if .