Prescribing the scalar curvature problem on the four-dimensional half sphere

In this paper, we consider the problem of prescribing scalar curvature under minimal boundary conditions on the standard four-dimensional half sphere. We describe the lack of compactness of the associated variational problem and we give new existence and multiplicity results.


Introduction and main results
Let (M n , g) be an n-dimensional Riemannian manifold with boundary, n ≥ 3, and letg = u 4/(n−2) g be a conformal metric to g, where u is a smooth positive function. Then, the scalar curvatures R g and Rg and the mean curvatures of the boundary h g and hg, with respect to g andg respectively, are related by the following equations: (1.2) When K and H are constants, this problem is called "The Yamabe Problem on Manifolds with boundary". It has also been studied through the works [4,18,26,27,31,32]. When K = 0, the problem is called Boundary Mean curvature problem which has been studied by Escobar (see [28]) on manifolds which are not equivalent to the standard ball. On the ball, sufficient conditions in dimensions 3 and 4 are given in [1,2,25,29]. When H = 0, the problem is called scalar curvature under minimal boundary condition and has been studied in [14][15][16][17]23]. Previously, Cherrier [22] studied the regularity question for this equation. He showed that solutions of (1.2) which are of class H 1 are also smooth. We observe that the above problem is a natural generalization of the well-known "Scalar Curvature Problems on Closed manifolds": to find a positive smooth solution to the following equation: to which much work has been devoted (see [3,[5][6][7][8]10,11,13,20,21,30,34,35,37]).
In this paper, we consider the case where H = 0, on the standard four-dimensional half sphere under minimal boundary conditions. More precisely, let K be a C 2 positive Morse function on S 4 + , we look for conditions on K to ensure the existence of a positive solution of the problem ⎧ ⎨ ⎩ L g u = − g u + 2u = K u 3 where g is the standard metric of S 4 + = {x ∈ R 5 /|x| = 1, x 5 > 0}. The main analytic difficulty of this problem comes from the presence of the critical Sobolev exponent on the right hand side of our equation, which generates blow up and lack of compactness. Indeed, due to the fact that the embedding H 1 (S 4 + ) → L 4 (S 4 + ) is not compact, the Euler-Lagrange functional J associated with our problem fails to satisfy the Palais Smale condition. That is there exist noncompact sequences along which the functional is bounded and its gradient goes to zero. Therefore, it is not possible to apply the standard variational methods to prove the existence of solution. There are also topological obstructions of Kazdan-Warner type to solve (1.4) [similar to the one associated to (1.3)], and so a natural question arises: under which conditions on K , (1.4) has a positive solution?
This problem has been studied by Li [33], and Djadli-Malchiodi-Ould Ahmedou [24], on the threedimensional standard half sphere, using the blow-up analysis of some subcritical approximations and the use of the topological degree tools. In [16,17], the authors gave some topological conditions on K to prescribe the scalar curvature under minimal boundary conditions on half spheres of dimension bigger than or equal to 4 using the method of "critical points at infinity" due to Bahri [9] and Bahri-Coron [11]. In particular, they obtained an Euler-Hopf-type criterium reminiscent to the formula obtained by Bahri-Coron [11] for the scalar curvature problem on S 3 , see also Chang-Gursky-Yang [21].
In this paper, we give new existence as well as multiplicity results, extending the previous all known ones. To state our results, we need to introduce some notations and assumptions. We denote by G the Green's function of the conformal Laplacian L g on S 4 + and H its regular part defined by (1.5) Let 0 < K ∈ C 2 (S 4 + ) be a positive Morse function. We say that the function K satisfies the condition (H 0 ): • If y is a critical point of Denoting K the set of critical point of K , we set To each p-tuple τ p := (y 1 , . . . , y p ) ∈ K + , we associate a matrix M(τ p ) = (M i j ) defined by, We denote by ρ(τ p ) the least eigenvalue of M(τ p ), and we say that a function K satisfies the condition (H 1 ) if for every τ p ∈ (K + ) p , we have ρ(τ p ) = 0. We set and we define an index i : where ind(K , y i ) denotes the Morse index of K at its critical point y i . Now, we state our main result.
Then, there exists a solution to the problem (1.4) of Morse index less or equal than k + 1.
Moreover, for generic K , it holds where N k+1 denotes the set of solutions of (1.4) having their Morse indices less than or equal to k + 1. Please observe that, taking in the above k to be , where is the maximal index over all elements of F ∞ , the second assumption is trivially satisfied. Therefore, in this case, we have the following corollary, which recovers the previous existence result of Ben Ayed et al. [17].
then there exists at least one solution to (1.4).

Moreover, for generic K , it holds
where S denotes the set of solutions of (1.4).
We point out the main new contribution of Theorem 1.1 is that we address here the case where the total sum in the above corollary equals 1, but a partial one is not equal 1. The main issue being the possibility to use such an information to prove the existence of solution to the problem (1.4). Moreover, our result does not only give existence results, but also, under generic conditions, gives a lower bound on the number of solutions of (1.4). Such a result is reminiscent to the celebrated Morse Theorem, which states that, the number of critical points of a Morse function defined on a compact manifold, is lower bounded in terms of the topology of the underlying manifold. Our result can be seen as some sort of Morse Inequality at Infinity. Indeed, it gives a lower bound on the number of metrics with prescribed curvature in terms of the topology at infinity. The remainder of this paper is organized as follows. In Sect. 2, we set up the variational structure and the lack of compactness of Problem (1.4). In Sect. 3, we characterize the critical points at infinity associated with our problem. The last section is devoted to the proof of the main result.

Variational structure and lack of compactness
In this section, we recall the functional setting and the variational problem and its main features. Problem (1.4) has a variational structure, the Euler-Lagrange functional is We denote by the unit sphere of H 1 (S 4 + ), and we set + = {u ∈ , u > 0}. Problem (1.4) is equivalent to finding the critical points of J subjected to the constraint u ∈ + . The Palais-Smale condition fails to be satisfied for J on + . To describe the sequences failing the Palais-Smale condition, we need to introduce some notations. For a ∈ S 4 + and λ > 0, let Let Pδ a,λ be the unique solution of We define now the set of potential critical points at infinity associated with the function J . Let, for ε > 0, p ∈ N * and w either a solution of (1.4) or zero, The failure of Palais-Smale condition can be described, following the idea of [19,36,38] as follows: Proposition 2.1 Let (u k ) be a sequence in + , such that J (u k ) is bounded and ∂ J (u k ) goes to zero. Then, there exists an integer p ∈ N * , a sequence (ε k ) > 0, ε k tends to zero, and an extracted subsequence of u k 's, again denoted (u k ), such that u k ∈ V ( p, ε k , w), where w is zero or a solution of (1.4).
If u is a function in V ( p, ε, w), one can find an optimal representation, following the ideas introduced in Proposition 5.2 of [9] (see also pages 348-350 of [10]). Namely, we have has a unique solution (α, λ, a, h), up to a permutation.
In particular, we can write u as follows: ) and it satisfies (V 0 ), and T w (W u (w)) and T w (W s (w)) are the tangent spaces at w of the unstable and stable manifolds of w for a decreasing pseudo-gradient of J and (V 0 ) is the following: Here, Pδ i = Pδ (a i ,λ i ) and < ., . > denotes the scalar product defined on H 1 (S 4 + ) by Notice that Proposition 2.2 is also true if we take w = 0, and therefore, h = 0. In the next, we will say that v ∈ (V 0 ) if v satisfies (V 0 ). Now, arguing as in [10, pages 326, 327 and 334], we have the following Morse lemma which completely gets rid of the v contributions and shows that it can be neglected with respect to the concentration phenomenon. a, λ, h), such that v is unique and satisfies:

Moreover,there exists a change of variables
We notice that in the V variable, we define a pseudo-gradient by setting where μ is a very large constant. Then, at s = 1, V (s) = e −μs V (0), will be very small as we wish. This shows that, to define our deformation, we can work as if V was zero. The deformation will extend immediately with the same properties to a neighborhood of zero in the V variable.

Characterization of critical points at infinity
Following Bahri [9], we introduce the following definition. Definition 3.1 A critical point at infinity of J in + is a limit of a flow line u(s) of the following equation: Here, w is either zero or a solution of (1.4), and ε(s) is some function tending to zero when s → +∞. Using Proposition 2.2, u(s) can be written as: Denoting by a i = lim a i (s) and α i = lim α i (s) , we denote by such a critical point at infinity. If w = 0, it is called w-type. , w), we have the following expansion:

Proposition 3.2 For each u
Thus, Since the function h belongs to T w (W u (w)), which has a finite dimension equal to the index of w. Thus, Therefore, Using the fact that α 2 0 J (u) 2 = 1 + o(1), we get, Observe that arguing as in [10] (page 354), the quadratic form negative definite. Hence, our proof follows.

Proposition 3.3 For each u
, we have the following expansion:

7)
where c 1 and c 2 are some positive constants.
Proof We have Using [16], for u = p j=1 α j Pδ j ∈ V ( p, ε), we have (3.8) The stereographic projection and a direct calculation show that (3.9) Similarly, we have (3.10) From another part, we have (3.11) A straight for word computation yields: (3.12) (3.13) Using the above estimates and the fact that J 2 (u)α 2 i K (a i ) = 1 + o(1), Proposition 3.3 follows using similar argument as in [9]. , w), we have the following expansion:

Proposition 3.4 For each u
(3.14) Proof First observe that arguing as in [9], easy computations show the following estimates: (3.21) Using the above estimates and the fact that J 2 (u)α 2 i K (a i ) = 1 + o(1), Proposition 3.4 follows using similar argument as in [9].
3.2 Ruling out the existence of critical point at infinity in V ( p, ε, w) for w = 0 The aim of this section is to prove that, for K , a C 2 positive function satisfying the condition of theorem and w a solution of (1.4), then for each p ∈ N, there is no critical point or critical point at infinity of J in the set V ( p, ε, w).

Proposition 3.5 For p ≥ 1, there exists a pseudo-gradient W , so that the following holds: There is a constant
This pseudo-gradient satisfies the PS condition and it increases the least distance to the boundary along any flow line.
Proof Observe first, from Proposition 3.2, we have . . , p}, we introduce the following condition: Let d 0 > 0 be a fixed small enough constant. We divide the set {1, . . . , p} into the following: In T 2 ∪ T 3 , we order the λ i 's: λ i 1 ≤ · · · ≤ λ i s . Let c > 0, a fixed constant small enough, we define For a fixed constant c > 0 small enough, we also define From Proposition 3.3, we have Observe that if i ∈ T 2 ∪ T 3 , we have w(a i ) > cd i , and therefore, 1 (3.25) We need to add some more terms in our upper-bound. For this, let (3.26) Since i s ∈ I , we can make appearing the term 1 (λ i 1 d i 1 ) −3 in the last upper bound, and so all the terms 1 (λ i d i ) −3 . Observe that, if k ∈ T 1 and k = j, we have (3.27) Thus, we can make appearing k∈T 1 , j =k ε 3 2 i j in the last upper bound. From another part, for k, j ∈ I λ is , we have |ν k − ν j | = O(|a k − a j |). Thus, It remains to estimate the case where k ∈ I λ is and j ∈ I λ is . If k ∈ T 2 ∪ T 3 or j ∈ T 2 ∪ T 3 , we have If j, k ∈ T 1 , we claim that: and the claim follows. Using now (3.30), we deduce (3.39) Observe that 1 (3.40) For two fixed large enough constants m 3 > m 2 > 0, one has (3.41) The pseudo-gradient W will be defined by W = m 4 (Y + m 2 X + m 3 Z 1 ) + h, where m 4 > 0 is a large enough fixed constant. Thus, the first claim of the proposition follows. The second claim can be obtained once we have (i) arguing as in [10].  V ( p, ε, w). Now once mixed critical points at infinity are ruled out, it follows from [17], that the critical points at infinity are in one-to-one correspondence with the elements of the set F ∞ defined in (1.7), that is, a critical point at infinity corresponds to τ p := (y 1 , . . . , y p ) ∈ (K + ) p , such that the related matrix M(τ p ) defined in (1.6) is positive definite. Such a critical point at infinity will be denoted by τ ∞ p . Like a usual critical point, it is associated with a critical point at infinity x ∞ of the problem (1.4), which are combination of classical critical points with a one-dimensional asymptote, stable and unstable manifolds, W ∞ s (x ∞ ) and W ∞ u (x ∞ ). These manifolds can be easily described once a Morse-type reduction is performed, see [10]. In the following definition, we extend the notation of domination of critical points to critical points at infinity. Recall that i(x ∞ ), the Morse index, of such critical point at infinity is equal to the dimension of W ∞ u (y) ∞ . Definition 3.7 x ∞ is said to be dominated by another critical point at infinity x ∞ if W ∞ u (x ∞ )∩ W ∞ s (x ∞ ) = ∅. If we assume that the intersection is transverse, then we obtain i(x ∞ ) ≥ (x ∞ ) + 1.

Proof of Theorem 1.1
Setting l := max i(τ p ), τ p ∈ F ∞ . (4.1) It follows then that where N k+1 denotes the set of solutions of (1.4) having their Morse indices ≤ k + 1. This conclude the proof of Theorem 1.1.
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