Fractional Choquard equation with critical nonlinearities

• Published in: Nonlinear differential equations and applications Vol. 24; no. 6
• Main Authors: Mukherjee, T., Sreenadh, K.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2017; Springer Nature B.V
• Subjects: Analysis; Mathematics; Mathematics and Statistics; Variational methods; 35R09; 35A15; Choquard equation; 35R11; Brezis–Nirenberg problem; Critical exponent; Fractional Laplacian
• ISSN: 1021-9722; 1420-9004
• DOI: 10.1007/s00030-017-0487-1

In this article, we study the Brezis–Nirenberg type problem of nonlinear Choquard equation involving the fractional Laplacian ( - Δ ) s u = ∫ Ω | u | 2 μ , s ∗ | x - y | μ d y | u | 2 μ , s ∗ - 2 u + λ u in Ω , u = 0 in R n \ Ω , where Ω is a bounded domain in R n with Lipschitz boundary, λ is a real parameter, s ∈ ( 0 , 1 ) , n > 2 s , 0 < μ < n and 2 μ , s ∗ = ( 2 n - μ ) / ( n - 2 s ) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. We obtain some existence, nonexistence and regularity results for weak solution of the above problem using variational methods.