Multi-bump solutions for fractional Nirenberg problem

Abstract

We consider the multi-bump solutions of the following fractional Nirenberg problemwhere s ∈ (0, 1) and n > 2+2s. If K is a periodic function in some k variables with 1 ≤ k < n−2s 2 , we proved that (0.1) has multi-bump solutions with bumps clustered on some lattice points in R k via Lyapunov-Schmidt reduction. It is also established that the equation (0.1) has an infinitemany-bump solutions with bumps clustered on some lattice points in R n which is isomorphic to Z k + .

On the density of $σ$-curvatures and multiplicity of solutions to the fractional Nirenberg problem

“…Problem (1.1) (or (1.2)) is a focus of reserach in the recent decades, and it continues to inspire new thoughts, see for example [1,2,[20][21][22][23]36,37,40,41,[50][51][52][53]56]. Fundamental progress was made by Jin, Li and Xiong in [40,41], from which they obtained compactness and existence results by applying the blow-up analysis and the degree counting argument.…”

“…Later on, the authors in [1,2,20,22] obtained some existence criterions by establishing Euler-Hopf type index formula. Recently, there have been some works devoted to the multiplicity results, and those mainly use the Lyapunov-Schmidt reduction method (see e.g., [21,23,36,[50][51][52][53]56]).…”

“…(3) The perturbation K ε,k,m can be constructed explicitly in Section 6 and it is a way of gluing approximate solutions into genuine solutions. The method is variational, rather than through Lyapunov-Schmidt reduction method as in [21,23,36,[50][51][52][53]56], etc. The solutions we obtained have most of their mass in disjoint small balls centered at the maximum points of K ε,k,m , which are far away from each other.…”

A fractional conformal curvature flow on the unit sphere

“…When σ = 1/2 and 2, it recovers the prescribing mean curvature problem and fourth order Q-curvature problem, respectively; see [39,41] for brief reviews of the classical Nirenberg problem and its generalizations, as well as references in this area. Other studies on the fractional Nirenberg problem include Chen-Zheng [18], Abdelhedi-Chtioui-Hajaiej [1], Chen-Liu-Zheng [19], Guo-Nie-Niu-Tang [32], Liu-Ren [47], Niu-Tang-Wang [51], etc. The limiting case σ = n/2 is of particular interest; see Moser [50], Chang-Yang [14], Wei-Xu [57,58], Brendle [6,7], Da Lio-Martinazzi-Riviére [20], etc.…”

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 .…”