Existence results for the fractional Q-curvature problem on three dimensional CR sphere

In this paper the fractional Q-curvature problem on three dimensional CR sphere is considered. By using the critical points theory at infinity, an existence result is obtained.


Introduction and main result
The sphere S 2n+1 is the boundary of the unit ball of C n+1 . It is a contact manifold with a standard contact form θ 1 . We denote by (S 2n+1 , θ) the contact sphere with its contact form θ. Let K : S 2n+1 → R be a C 2 positive function. The prescribed Webster scalar curvature problem on S 2n+1 is to find suitable conditions on K such that K is the Webster scalar curvature for some contact formθ on S 2n+1 , CR equivalent to θ 1 . If we setθ = u 2 n θ 1 , where u is a smooth positive function on S 2n+1 , then the above problem is equivalent to solving the following PDE: where L θ 1 is the conformal Laplacian of S 2n+1 .
In recent years, fractional calculus has attracted a lot of mathematicians' interests. The CR fractional sub-Laplacian P θ 1 γ (γ ∈ (0, 1)) is defined by Rupert L. Frank, Mari a del Mar Gonzalez, Dario D. Monticelli, and Jinggang Tan in [22]. In the paper [22], it was shown that one can treat the CR fractional sub-Laplacian as a boundary operator. In [11], the CR fractional sub-Laplacian is viewed as intertwining operator.
On the CR sphere, the general intertwining operator P θ 1 γ is defined by the following property: for each F ∈ C ∞ (S 2n+1 ).
The first author is partially supported by the NSF of China (11471170, 10621101), 973 Program of MOST (2011CB808002) and SRFDP.
If u is a critical point of the function J in Σ + , then v = (J(u)) n+1 2γ u is a solution of (1.1). However, the functional J does not satisfy the Palais-Smale condition, that is to say there exist critical points at infinity, which are the limits of noncompact orbits for the gradient flow of −J. Thinking of these sequences as critical points, a natural idea is to expand the functional J near the sets of such critical points.
We will prove the theorem by contradiction in section 5. Therefore we assume that equation (1.1) has no solutions. Our proof is based on a technical Morse Lemma at infinity; it relies the construction of a suitable pseudogradient for J. The (PS) condition is satisfied along the decreasing flow lines of this pseudogradient, as long as these flow lines stay out of the neighborhood of a finite number of critical points of K. Finally we compute the contribution of some critical points at infinity to the changes of topology for the level set of the functional, from this we can achieve a contradiction.
This paper is organized as follows. In the next section, we introduce preliminary result and the general variational framework . In section 3, we give some expansions of the functional and its gradient near the sets of its critical points at infinity. In section 4, we establish the Morse lemma at infinity, which allows us to refine the expansion of the function. In section 5, we give a proof of Theorem 1.1. In Appendices A-C, we show some useful estimates which will be used in our proof of Theorem 1.1.
Alternatively, we can use complex coordinates z = x + iy to denote elements of R × R ≃ C, so that the group law can be written as for (z ′ , t ′ ), (z, t) ∈ H 1 , and < ·, · > C is the standard Hermitian inner product in C.
We introduce Cayley transform C between the Heisenberg group and the CR sphere.
Let V (p, ε) be the subset of Σ + of the following functions: u ∈ Σ + , ∃(α, g, λ), such that The set V (p, ε) has a simple interpretation: It is a neighborhood of the critical points at infinity of the functional J on Σ + . Definition 2.1. [3] We will say that the Palais-Smale condition holds on flow-lines in the V (p, ε) if, taking an initial data u 0 in V (p, ε), with ε 0 small enough (but fixed), the solution u(s, u 0 ) of the differential equation ∂u ∂s = −∂J(u) with initial data u 0 remains outside a V (p, ε 1 ), ε 1 > 0, which depends only on u 0 .
The failure of (PS)condition is characterized as follows.
Lemma 2.1. Assume that (1.1) has no solutions. Let {u k } ⊆ Σ + be a sequence such that J ′ (u k ) → 0 and J(u k ) is bounded. Then there exists an integer p 1, a positive sequence ε k → 0 and an extracted subsequence of {u k }, such that u k ∈ V (p, ε k ).
In order to prove Lemma 2.1, we introduce Then we can follow the first part of [12] by using the functional I. The proof is by now classical, we can also see [27].
We introduce the minimization problem for ε small enough Lemma 2.2. For any p 1, there exists ε p > 0 such that for any 0 < ε < ε p and u ∈ V (p, ε), the minimization problem (2.8) has a unique solution (ᾱ,ḡ,λ). (2.9) Here ·, · denote the inner product inṠ γ (S 3 ) defined by The proof of Lemma 2.2 follows from Appendix A in [5] with some modulations.
3. The expansion of the function J Lemma 3.1. If ε > 0 small enough and u = p i=1 α i w g i ,λ i + v ∈ V (p, ε), v satisfies (2.9), we have Furthermore f is bounded by The proof of this lemma is provided in Appendix A. Now, we state the following two lemmas whose proof follow the arguments used to prove similar statements in [6] (also in [3]); see the Appendix of [26] where some necessary modifications are made.
One can follow the idea of the proof of Lemma A.2 in [6] to get a proof of this lemma. We omit the details.
For a proof of Lemma 3.3, one may follow the idea and similar estimates in the proof of Proposition 5.4 in [3](P191). We omit the details.
Since v is a minimizer, we have It yields From Lemma 3.1 and Lemma 3.3, we state the following lemma which improve the asymptotic behavior of the function J.
Lemma 3.5. Assume K be a C 2 positive function satisfying condition (1.2). For any u = p i=1 α i w g i ,λ i ∈ V (p, ε), the following estimate holds We will give the proof in Appendix B and Appendix C.

Morse lemma at infinity
This section is devoted to characterize the critical points at infinity associated to problem (1.1). The characterization is obtained through the construction of a suitable pseudogradient at infinity in the set V (p, ε), depending on a delicate expansion of the gradient of J near infinity.
There is a covering {O l } and a subset of {(α l , g l , λ l )} of the base space for the bundle V (p, ε) and a diffeomorphism ξ l : The proof of this theorem need some technical result. First we give the Morse lemma at infinity by isolating the contribution of v − v.
there is a neighborhood U of (ᾱ,ḡ,λ) such that The proof of Lemma 4.1 is similar to the proof of Lemma 3.2 in [9] for the Riemannian manifold.
, ε small enough, Then there exists a vector filed W ′ so that the following holds: there is a constant C > 0 such that where C 1 is a positive constant, W ′ is bounded.
We define a set I i for all i ∈ {1, · · · , p}. We divide it in three cases: (1) For all i ∈ {1, · · · , p}, there exists a suitable constant C > 0 such that In this case, we define I i = ∅ the empty set.
then in this case we define I i = {i}. (3) λ i satisfies (4.5), but it is not the largest concentration, i.e., we have another j such that λ j > λ i and λ j satisfies (4.5). In this case we define where c is the constant in (3.2) and c ′ > 0 is a suitable constant.
Proof of the claim. In the case (1), by (3.2), we have In the case (2), then for λ j > λ i , there holds Notice that (4.10) If λ i and λ j are comparable or λ j λ i (in this case, |λ j /λ i | ≤ C), then If they are not comparable, say λ i = o(λ j ), then Thus by (4.5), there holds Hence, by choosing a large C, it holds that where C 0 > 0 is a constant depending on γ. Then from the first inequality of (4.8) and (4.14), we have In the case (3), by a simple computation, we observe that for s > t, By using (3.1) and choosing a constant c ′ which depends on γ, there holds For a proof of (4.17), from (3.1), (4.10) and that the functional λ(u) has positive lower bound, we have where 0 < c γ < 1. In the last two estimates, we have used the inequalities (4.11) and (4.12) and choose the constant C in (4.9) large enough. λ i ∼ λ j means that λ i and λ j are comparable. It completes the the proof of claim.
Proof of Lemma 4.2. For the sake of simplicity, we assume We note that when we construct the vector field W ′ satisfying the estimate (4.2), then by the same method as in [8] and [9], we can prove (4.3). So in the following we only need to construct the vector field W ′ satisfies the inequality (4.2). We divide it into four cases.
(4. 18) In this case, by (4.7) we have Combined with (4.17), (4.19) and (3.1), for any i, we reach Here and in sequel we denote B > 0 a constant which may vary in different places, We define ν k = 2 k−1 . By simple computation, we have This can be rewritten in the following form where γ i , β i are bounded nonnegative constants depending on µ i , ν i and γ.
We now define the vector field by From the estimate (4.22), we have (4.2).
In this case, as in (4.20) we define A similar construct of W ′ can be done and the proof of (4.2) is repeated as the case 1 word by word by some mirror modifications.
, C a suitable constant. (4.26) In this case, we define so we can define vector field W ′ and give a similar proof of (4.2) as in case 1. (4.28) the above proof extends as follows. Subcase 1. Suppose in the sequence λ 1 · · · λ p , there exists i 1 such that for some 0 ≤ r ≤ p − i 1 , there holds r s=0 j i 1 +r,j =i 1 +s (4.29) We note that for a choice of (i 1 , r 0 ) satisfying (4.29), then all of (i 1 , r) with r 0 ≤ r ≤ p − i 1 satisfies (4.29). Similarly to the case 1 and case 3, we can define a vector field W (i 1 , r) in Assume i 1 is the smallest subscript satisfying (4.29). Then by choosing If i 1 = 1, we obtain the result of (4.2). Otherwise, for integer l ∈ [1, i 1 ), there holds (4.32) From (4.31) and (4.32), then (4.2) is true if Combining with (4.32), we have for j i 1 − 1, . These imply that: for j i 1 − 1, g j is close to a critical point of K which we denoted by η j , λ j d(g j , η j ) = O(1).
Combining (4.32) and (4.34), we have Since g 1 is close to a critical point of K which we denoted by η 1 and ∆ θ 1 K(η 1 ) = 0, we get ).
Now we define the vector field which satisfies (4.2).
which also satisfies (4.2). Subcase 2. Assume that indices i 1 satisfying (4.29) do not exist, i.e., for any l ∈ {1, · · · , p} p k=l i =k (4.37) By a direct argument, when i < j, η i = η j , we get Thus under the condition (4.37), for some i < j The construct of vector field is same as (4.35) or (4.38) in the previous subcase. In fact, we have reach that if two λ's for example λ i and λ j are not comparable, the vector field W ′ can be defined to satisfy (4.2). So in this subcase, we can assume that inf i =j d(g i , g j ) d 0 > 0 and all the λ's are comparable. Thus we have ).
Therefore, we have If for some η i satisfies −∆ θ i K(η i ) −c < 0, we define If for all η i , −∆ θ i K(η i ) c > 0, Now the construct of vector field is same as Since in the cases 1-4, we can adjust some constants to insure that the union of these four case is the whole discussed space, by using a partition of unity, we can define the final vector field W ′ satisfying (4.2) at all. The proof of Lemma 4.2 is complete.
and the following two statements Proof. The proof is similar to the one given in [9] and [26]. By lemma 4.2, the vector field W ′ is Lipschitz. Hence, there is a 1-parameter group h s generated by , · · · , p} and δ < 1 2 min{dist(η i , η j )}, we define V δ (η 1 , · · · , η p ) to be the set of (g, λ) satisfying g i ∈ B δ (η i ), i = 1, · · · , p. J (h s ( p i=1 α i w g i ,λ i )) and J (h s ( p i=1 α i w g i ,λ i +v(s))) are decreasing functions of s.
, there is at most one solution of the equation By using Lemma 4.2 and a similar proof as in [6], we can see that the flow line The cases in which there could be no solution of (4.42) are h s ( p i=1 α i w g i ,λ i ) exits from V (p, ε 1 ) or the decreasing flow goes to critical points at infinity. If We can choose ε > 0 sufficiently small, there is a solution of (4.42).
By continuity, (4.42) must have a solution.
Similarly, we consider the vector field −W ′ and the flow line It is easy to know that there is a unique solution for and take (g i ,λ i ) = (g i (s), λ i (s)), we have (4.39).
Following from Lemma 4.3, for any ε 1 > 0 small, there are ε > 0 and ε 2 > 0 such that From Lemma 4.1, Lemma 4.3 and this fact, we can proof Theorem 4.1.
We first define a vector field on V (p, ε) by using a partition of unity η l on the base space of (α,g,λ, V ) as X = W − m l=1 η l V l , where W (α,g,λ, V l ) = W ′ (α,g,λ). Then the vector field in the variables (α, g, λ, v) is defined as Z = X • ϕ. By direct computation, in every open set O l we have Using the estimates in Appendix A and Q(v, v) is positive definite, we obtain J ′′ (u 0 ) V l ·V l is positive definite. So in V (p, ε), if ε small enough, there holds We now suppose that the functional J has no critical point and there holds J ′ (u) ≥ C ε > 0 for u / ∈ V (p, ε 2 ). On V (p, ε 2 ), define the vector field −J ′ , and then also via a partition of unity of the two sets V (p, ε) and V (p, ε 2 ) to build a global vector fieldZ(u) on V ε 0 (Σ + ) from Z and −J ′ . It is easy to see It is important to insure that any flow line generated by the vector fieldZ with initial condition u ∈ V ε 0 (Σ + ) remains in V ε 0 (Σ + ). We have the following lemma. Proof. It is sufficient to prove that V ε 0 (Σ + ) is invariant under the negative gradient flow of J. Therefore, Therefore, |u(s) − | L 4 2−γ < ε 0 for all s > 0.
Next, we study the concentration phenomenon of the functional J.
we have the following expansion of J(u) after changing the variables: where g + i , g − i are the coordinates of g i near η β i along the stable and unstable manifold for K and h = (h 1 , · · · , h p−1 ) ∈ R p−1 with h i = h i (α 1 , · · · , α p ), i = 1, · · · , p − 1 are independent functions.
Proof. From Lemma 3.1 and Theorem 4.1, we have From the proof of Lemma 4.2, we have |∇ 1| < ε, the expansion of the functional J can be rewritten as follows: Except the term all others are positive on the right hand side of the above equality. Since g(α, g) is homogeneous in the variable α, we have a degenerated critical point (ᾱ 1 , · · · ,ᾱ p ) which satisfiesᾱ This critical point has an index equal to p − 1 (since the critical point corresponds to a maximum), On the other hand, g(α, g) has a single critical point η = (η β 1 , η β 2 , · · · , η βp ) in the g variable. Thus, using the Morse lemma, after a change of variables, we have have the following normal form, For any l-tuple τ l = (i 1 , · · · , i l ), 1 i j m 1 , j = 1, · · · , l, let c(τ l ) = l j=1 denote the associated critical value. We only consider a simple situation, where for any τ = τ ′ , c(τ ) = c(τ ′ ), and thus order as c(τ 1 ) < · · · < c(τ k 0 ). From Lemma 5.3 and a deformation lemma (see [3] and [7]) or directly the critical group theory (see [14]), we have Lemma 5.4. If c(τ l−1 ) < a < c(τ l ) < b < c(τ l+1 ), for any coefficient group G, then . If X is a topological set, then χ(X) is its Euler-Poincare characteristic with rational coefficients. Proof of the theorem. Since we assumed that (1.1) has no solution, V ε 0 (Σ + ) is retract by deformation of Σ + . Σ + is contractible, so χ(V ε 0 (Σ + )) = 1. By Lemma 5.4 and the Morse lemma, we have is a contradiction. Therefore, (1.1) has a solution u 0 ∈ V ε 0 (Σ + ).

Appendix
We first introduce some well-known inequalities which are from Taylor expansion and some computations.
In the following four lemmas, we assume α > 0.
There exists a constant M, such that for any (a, b) ∈ R 2 , a > 0, a+b > 0, There exists a constant M, such that for any (a 1 , · · · , a p ) ∈ R p , There exists a constant M, such that for any (a 1 , · · · , a p ) ∈ R p , Lemma 6.5. [3] There exists a constant M, such that for any (a 1 , · · · , a p ) ∈ R p ,

Appendix A. We set
We first expand the numerator N as follows, all the other terms are zero since v satisfies conditions. From now on, we denote a i = C −1 (g i ), ξ = C −1 (ζ).
where S is the sharp Sobolev constant given by Here O ′ ( ) means that when | | << 1, there exist two constants C 1 , C 2 > 0 such that Proof. Let ξ = (x, y, t), a i = (x i , y i , t i ) ∈ H 1 .

Thus it yields
Hence when ε ij goes to zero, we have The case µ = λ j λ i is similar to the case µ = λ i λ j . Then we consider the third case µ = λ i λ j |d ij | 2 . In this case, Without loss of generality, we assume λ i λ j , therefore By the same arguments used in the first case, we obtain We have On B 1 , we have |ξ| 9 10 λ i |d ij | , we obtain This completes the proof.
Similar to the proof of Lemma A.2, we have the following results. Lemma A.3 (1) It holds that (2) Let α, β > 1, such that α + β = 4 2−γ , θ = inf (α, β), it holds that Let us consider the denominator D of J, Proof. Using (6.1), we have In order to get more information from Lemma A.4, we now estimate the first three terms on the right hand side of (6.6). Lemma A.4.1 where c 2 = 4c

.4) and Lemma A.3, we get
Then we have By Taylor expansion, there holds Finally we estimate for i = j, by Lemma A.2, Lemma A.3 and Taylor expansion and Young inequality, Proof. By (2.2), we have where C 0 is a constant depending on γ. From (6.3) , Hölder inequality and Lemma A.3, we get Since v satisfies (2.9), we have Proof. Using (6.3), we get By Hölder inequality and Lemma A.3, we can easily get, Now we can complete the Proof of Lemma 3.1.
Therefore combining Lemma A.1, Lemma A.2 and Lemma A.3, we have This is the estimate of J(u) in Lemma 3.1.

Appendix B.
We define By direct computation, Thus we have We first take W = λ j ∂w g j ,λ j ∂λ j in (6.7), we obtain In the remainder of the part B, we will give some lemmas to complete the proof of (3.1). Lemma B.1 We have In the proof of this result, the idea is same as that in the proof of Lemma A.2. The details are omitted. Lemma B.3 There holds Proof. Using (6.5), we have Proof. There hold Proof. We have the following computations By some similar computations, we also get the following result. Lemma B.3.3 For i = j, there holds S 3 K(ζ)(w g j ,λ j ) 2γ 2−γ w g i ,λ i λ j ∂w g j ,λ j ∂λ j θ 1 ∧ dθ 1 Proof. From direct computation, we have |λ j ∂w g j ,λ j ∂λ j | w g j ,λ j . Then Similarly, we have the following result, the proof is omitted. Lemma B.3.5 By using the lemmas above, we have So we obtain the desired estimate of (3.1).

Appendix C.
In this section, we take W = 1 λ j ∂w g j ,λ j ∂g j in (6.7) and complete the proof of (3.2).
By the same reason of Lemma B.1, we have the following result.
Lemma C. 2 We have Proof. By some similar computations as in the proof of Lemma B.2, we can complete the proof of Lemma C.2. The details are omitted. Lemma C. 3 We have Proof. Using (6.5), we have 2−γ inf(α i w g i ,λ i , α k w g k ,λ k ) 1 λ j ∂w g j ,λ j ∂g j + M 5 S 3
Lemma C.3.2 For i = j, we have Proof. Similarly as in the proof of Lemma C.3.1, we have