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

Harnack‐type inequality for fractional elliptic equations with critical exponent

“…and  3R (x) is the ball in R N+1 with radius 3R and its center at the x;  + 3R =  3R ∩ R N+1 + is the upper half ball; and ′  + 3R is the flat part of  + 3R , which is the ball B 3R in R N . For other results of fractional Laplacian equations, please see some works [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] and reference therein.…”

Multi-bump solutions for fractional Nirenberg problem

“…The classic Nirenberg problem asks that on the standard sphere (S n , g S n ) with n ≥ 2, whether there exists a function w such that the scalar curvature( Gauss curvature in the dimension 2) of the conformal metric g = e w g S n equals to a prescribed functionK. This probelm is equivalent to solving the following equations − ∆ g S n w + 1 =Ke 2w on S 2 (1.1) and − ∆ g S n v + n − 2 4(n − 1) R g S n v = n − 2 4(n − 1)K v n+2 n−2 on S n for n ≥ 3, (1.2) where R g S n = n(n − 1) is the scalar curvature of (S n , g S n ) and v = e n−2 4 w . The linear operators defined on left-hand side of the equation (1.1) and (1.2) are called the conformal Laplacian on S n .…”

A non-compactness result on the fractional Yamabe problem in large dimensions

“…weakly (in the sense of the adequate modification of Definition 3.3). Assume also that there exists a function W ∈ D 1,2…”

“…We now choose appropriate coefficients a 0 and a 1 of the polynomial f (t) = a 0 + a 1 t in (2.2) so that the function J γ 1 + J γ 2 introduced in (4.4) and (4.5) has a strict local minimum at (1,0), provided that the dimension n is sufficiently large. By (4.12) and (4.32), it suffices to confirm three conditions…”

“…We refer the reader for developments on this issue to [63,64,3,33,14] and reference therein. The problem of compactness of the solution set for the 1 2 -fractional Yamabe problem is studied in the conformally flat case with umbilic boundary in [36], and in the case of dimension 2 in [37]. Related results on compactness were obtained by Almaraz in [5] and by Han-Li [44].…”