CONCENTRATION OF SOLUTIONS FOR THE FRACTIONAL NIRENBERG PROBLEM

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

1. Introduction and main results. The classical N irenberg problem is to ask whether, considering the standard sphere (S N , g S N ), N ≥ 2, one can deform conformally the metric in such a way that the scalar curvature (Gauss curvature in N = 2) becomes a prescribed functionK on S N . The equivalent problem is to consider the following equations − ∆ g S N w + 1 =Ke 2w on S 2 (1) and where C(N ) = N −2 4(N −1) , R 0 = N (N − 1) is the scalar curvature of (S N , g S N ) and v = e N −2 4 w. It is well known that the linear operators in (1) and (2) are conformal Laplacians associated to the metric g S N and denoted by P . This definition can also be extended to a general compact Riemannian manifold (M, g) of dimension N ≥ 2. There is another conformally covariant operator discovered by Paneitz defined as follows N is the Q-curvature and Ric g is the Ricci curvature of g, A N , B N are suitable constants depending on N . See [16,31] for more details. Graham, Jenne, Mason and Sparling [20] constructed a sequence of conformally covariant elliptic operators {P g k } on Riemannian manifolds for all positive integers k if N is odd, and for k ∈ {1, 2, · · · , N/2} if N is even. Moreover, P g 1 is the conformal Laplacian Lg := −∆ g + c(N )R g and P g 2 is the Paneitz operator. Up to positive constants, P g 1 (1) is the scalar curvature of g and P g 2 (1) is the Q-curvature. In [26,27], Li

ZHONGYUAN LIU
Yanyan et al gave a complete characterization for fully nonlinear conformally covariant differential operators of any integer order on R N . Later on, Peterson [33] constructed an intrinsically defined conformally covariant pseudo-differential operator of arbitrary real number order. Using a generalized Dirichlet to Neumann map, Graham and Zworski [21] introduced a meromorphic family of conformally invariant operators on the conformal infinity of asymptotically hyperbolic manifolds. Recently, Chang and González [11] took the way of Graham and Zworski to define conformally invariant operators P g s of noninteger order s ∈ (0, N/2) and the localization method of Caffarelli and Silvestre [6] for the fractional Laplacian (−∆) s on the Euclidean space R N . These lead naturally to a fractional order curvature problem R g s := P g s (1). The problem of fractional order curvature was studied extensively: see, e.g., [1,13,14,17,18] and the references therein.
The operator P g s of noninteger order s ∈ (0, N/2) is conformally covariant in the sense that if f is any smooth function and g = v Similar to the formula for scalar curvature and the Paneitz-Branson Q-curvature, the Q-curvature for g of order 2s can be defined as Q g s = P g s (1). Thus, this raises a natural question: is there a metric {g} in the conformal class [g] such that Q g s equals to a prescribed function K on a smooth compact Riemannian manifold (M, g) of dimension N ≥ 2? By (3), one needs to solve the following semilinear equation, If (M, g) is the standard sphere (S N , g S N ), (4) can be viewed as the fractional Nirenberg problem which has been studied in [1,14,22,23,24]. The operator P g S N s is the 2s order conformal Laplacian on S N and can be uniquely expressed as following where Γ is the Gamma function and ∆ g S N is the Laplace-Beltrami operator on (S N , g S N ). The operator P s can be seen more concretely on R N by using stereographic projection. Let N be the north pole of S N and be the inverse of stereographic projection operator from S N \ {N } to R N . Then, by the conformal invariance of P s , one has the following relation where |J Φ | = 2 1+|x| 2 N . Then, for a solution v of (5),

CONCENTRATION OF SOLUTIONS 565
In this paper, we study the fractional Nirenberg problem (6) with s ∈ (0, 1). We focus on concentration of solutions for (6). Actually, we consider the following problem where s ∈ (0, 1), 2s < N , For s ∈ (0, 1), (−∆) s is the fractional Laplacian operator with the following representation where d s,N is a positive constant depending only on s, N , P.V. is in the sense of the principal value. Caffarelli and Silvestre [6] found another local representation , where d s > 0 is a constant depending on s andũ =ũ(x, z) is the solution of the following boundary value problem in the half space R N +1 The homogeneous fractional Sobolev spaceḢ s (R N ) is defined by the completion of 1 2 andû(ξ) is the Fourier transformation of u defined bŷ It follows from [19] that the above norm is equivalent to the Gagliardo seminorm [u] s of u, Consider the equation It has been proved in [12,25] that the following function, for x ∈ R N and λ > 0,

ZHONGYUAN LIU
We assume that K(y) satisfies the following condition: (K). K(y) > 0 has a sequence of strictly local maximum point z j ∈ R N such that |z j | → +∞ and there are constants K j > 0 and β j ∈ (N − 2s, N ) such that for some constants a 1 ≥ a 0 > 0 independent of j and R j (y) satisfies R j (y) = O |y| βj +σ for some σ > 0 independent of j.
We now state our main result of the paper: holds, then for µ > 0 small enough and each strictly local maximum point z i1 of K(y), there exists another strictly local maximum point z i2 such that (7) has a solution of the form The proof of our results is inspired by the methods of [10,32,35]. More precisely, we will use a reduction argument similar to [32,35] to prove Theorem 1.1, see also [7,8,9,29,30,34]. It is worthwhile to point out that the solutions constructed in this paper have exactly two maximum points, the distance between which is very large. From the proof of the Theorem 1.1, we find that the interaction between two approximated solutions concentrated at two different strictly local maximum points of K(y), plays the important role in our construction of the solutions. This paper is organized as follows. In Section 2, we give some basic estimates. In Scetion 3, we carry out the finite dimensional reduction procedure. The main results will be proved in Section 4.

2.
Preliminaries: Basic estimates. In this section, we will give some basic estimates used in the later sections. Since the estimates are very similar to those in [2], we only give a sketch here. Set The proof of this lemma can be found in [14]. Define To solve the finite dimensional problem in Section 4, we need to expand the functional I(u) explicitly as follows.
Using the following inequality, Combining the above estimates, we can obtain the desired expansion.
3. Finite Dimensional Reduction. In this section, we are ready to start the finite dimensional reduction. Define for µ > 0 small, The solutions we want to construct will be critical points of I of the form where x j is close to one of the local maximum points of K(y), v ∈Ḣ s (R N ) is sufficiently small. Consider the following 2(N + 1)-codimensional submanifold It is well-known that for µ > 0 small enough, ∂J(x, λ, v) The following proposition is to reduce the problem of seeking a solution for (8) to that of finding a critical point for a function defined in a finite dimensional domain. Proposition 1. Assume (K) holds. There exists µ 0 > 0, such that for µ ∈ (0, µ 0 ) and (x, λ) ∈ D µ , there exists a unique C 1 -map: (x, λ) ∈ D µ → v(x, λ) ∈ E x,λ such that v(x, λ) satisfies (9) for some A j , B ji , (i = 1, 2, · · · , N, j = 1, 2). Moreover, v(x, λ) satisfies the following estimate where τ > 0 is some constant.
Proof. We first expand J(x, λ, v) near v = 0 as follows where f ∈ E x,λ is the linear form given by Qv, v is the quadratic form on E x,λ given by and R(v) is the higher order term satifying where θ > 0 is some constant. By Proposition 2, we know that Q is invertible and Q −1 ≤ C for some C > 0 independent of x and λ. Now, following the arguments in [34], we find There exists an equivalence between the existence of v such that (9) holds for (x, λ, v) and As in [34], by the implicit function theorem, there exists µ 0 > 0 and a C 1 -map v : (x, λ) ∈ D µ → E x,λ for µ ∈ (0, µ 0 ) satisfying (15) and v ≤ C f .
Thus, we only need to estimate f . By Hölder inequality and Lemma 2.1, we find Therefore, we complete the proof.
Proposition 2. Assume (K) holds. Let (x, λ) ∈ D µ , then for µ > 0 small enough, there exists a > 0 such that Proof. We argue by contradiction. Suppose that there are µ n → 0, (x n , λ n ) ∈ D µ and v n ∈ E x n ,λn such that where o(1) → 0 as n → ∞. Without loss of generality, we may assume v n = 1.
Letṽ j,n (y) = λ 2s−N 2 j,n v n (λ −1 j,n y + x j,n ), j = 1, 2. Then,ṽ j,n (y) is bounded iṅ H s (R N ). Up to a subsequence, we may assume that there is v j ∈Ḣ s (R N ) such thatṽ j,n (y) v j weakly inḢ s (R N ). We will show that v j ≡ 0, j = 1, 2. DefineŨ j,n = λ 2s−N 2 j,n U x j,n ,λj,n (λ −1 j,n y + x j,n ), Since v n ∈ E x n ,λn , it is easy to see that v j,n (y) ∈Ẽ n Now we claim that v j satisfies For any φ ∈ C ∞ 0 (R N ), define φ n ∈Ẽ n as follows c j,n W j,n .

CONCENTRATION OF SOLUTIONS 571
where c = lim n→∞ c j,n and c i = lim n→∞ c j,i,n . Note that Thus, we get By the fact in [15] that U 0,1 is nondegenerate, we find v j = 0. Therefore, where o R (1) → 0 as R → ∞. It follows from (17), we obtain that v n → 0 as n → ∞. But this contradicts with v n = 1. So we have completed the proof of the proposition.

4.
Proof of the main results. In this section, we use a minimization procedure to prove Theorem 1.1.
Suppose thatλ 1 = γ 2 L 1 = γ 2 d So if we take γ 2 > 0 large enough depending on γ 1 , we can also obtain a contradiction. Since the similar argument can also be applied toλ 2 , the claim follows. As a result, the minimizer (x,λ) of J is an interior point of D µ,2 . Thus, u = ij Uxj ,λj +v(x,λ) is a critical point of I. By Lemma A.6 in [14], we see that u > 0. Therefore, we finish our proof.