Concentration of solutions for the fractional Nirenberg problem

Abstract

The aim of this paper is to show the existence of infinitely many concentration solutions for the fractional Nirenberg problem under the condition that Qs curvature has a sequence of strictly local maximum points moving to infinity.

Multi-bump solutions for fractional Nirenberg problem

“…We note that the conditions (H 1 )-(H 3 ) and the condition (K) in [25] has some intersecion. When K satisfies both the condition (K) in [25] and our conditions (H 1 )-(H 3 ), the equation (1.4) has infinitely many 2-peak solutions according to [25].…”

“…Chen and Zheng [10] found a 2-peak solution when K(x) = 1+ε K(x) has at least two critical points and satisfies some local conditions. What is more, Liu in [25] constructed infinitely many 2-peak solutions when K has a sequence of strictly local maximum points moving to infinity. When K is a radial symmetric function, in [24] and [26] it was showed that (1.4) has infinitely many non-radial solutions.…”

On a Fractional Nirenberg problem involving the square root of the Laplacian on $\mathbb{S}^{3}$

“…As in [27,32], by the implicit function theorem, there exists a unique v τ ∈ E P,t with v σ < ν 0 satisfying (4.17) and…”

“…This lead naturally to a fractional order curvature R g σ := P g σ (1), which will be called σ-curvature in this paper. The fractional operators P g σ and their associated fractional order curvatures P g σ (1) have been the subject of many studies, for instance, see [1,2,13,12,21,22,27].…”

Fractional Yamabe solitons and fractional Nirenberg problem

“…As a generalization of the Nirenberg's problem, the fractional Nirenberg's problem is to find a conformal metric of the n-dimensional unit sphere (S n , g S n ) such that its fractional order curvature is a given function f . This has been studied in [11,12,13,14,28,29,33,34,35]. For simplicity, we write P σ = P g S n σ and R σ = R g S n σ .…”