Vol. 9, No. 6, 2016

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A complete study of the lack of compactness and existence results of a fractional Nirenberg equation via a flatness hypothesis, I

Wael Abdelhedi, Hichem Chtioui and Hichem Hajaiej

Vol. 9 (2016), No. 6, 1285–1315
Abstract

We consider a nonlinear critical problem involving the fractional Laplacian operator arising in conformal geometry, namely the prescribed σ-curvature problem on the standard n-sphere, n 2. Under the assumption that the prescribed function is flat near its critical points, we give precise estimates on the losses of the compactness and we provide existence results. In this first part, we will focus on the case 1 < β n 2σ, which is not covered by the method of Jin, Li, and Xiong (2014, 2015).

Keywords
fractional Laplacian, critical exponent, $\sigma$-curvature, critical points at infinity
Mathematical Subject Classification 2010
Primary: 35J60, 35B33, 35B99, 35R11, 58E30
Milestones
Received: 17 March 2015
Revised: 26 November 2015
Accepted: 12 April 2016
Published: 3 October 2016
Authors
Wael Abdelhedi
Department of Mathematics
Faculty of Sciences of Sfax
3018 Sfax
Tunisia
Hichem Chtioui
Department of Mathematics
Faculty of Sciences of Sfax
3018 Sfax
Tunisia
Hichem Hajaiej
New York University Shanghai
1555 Century Avenue, Pudong
200122 Shanghai
China