The Nirenberg problem on high dimensional half spheres: the effect of pinching conditions

In this paper we study the Nirenberg problem on standard half spheres (S+n,g),n≥5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathbb {S}^n_+,g), \, n \ge 5$$\end{document}, which consists of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent: (P)-Δgu+n(n-2)4u=Kun+2n-2,u>0inS+n,∂u∂ν=0on∂S+n.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\mathcal {P}) \quad {\left\{ \begin{array}{ll} -\Delta _{g} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2}},\, u > 0 &{}\quad \text{ in } \mathbb {S}^n_+, \\ \frac{\partial u}{\partial \nu }\, =\, 0 &{}\quad \text{ on } \partial \mathbb {S}^n_+. \end{array}\right. } \end{aligned}$$\end{document}where K∈C3(S+n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \in C^3(\mathbb {S}^n_+)$$\end{document} is a positive function. This problem has a variational structure but the related Euler–Lagrange functional JK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_K$$\end{document} lacks compactness. Indeed it admits critical points at infinity, which are limits of non compact orbits of the (negative) gradient flow. Through the construction of an appropriate pseudogradient in the neighborhood at infinity, we characterize these critical points at infinity, associate to them an index, perform a Morse type reduction of the functional JK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_K$$\end{document} in their neighborhood and compute their contribution to the difference of topology between the level sets of JK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_K$$\end{document}, hence extending the full Morse theoretical approach to this non compact variational problem. Such an approach is used to prove, under various pinching conditions, some existence results for (P)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal {P})$$\end{document} on half spheres of dimension n≥5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 5$$\end{document}.


Introduction and statement of the results
In the early seventieth of the last century Louis Nirenberg asked the following question: Can a smooth positive function K ∈ C ∞ (S n ) defined on the standard n−dimensional sphere (S n , g) be realized as the scalar curvature of a metric g conformally equivalent to g ? On S 2 , setting g = e 2u g the Nirenberg problem is equivalent to solving the following nonlinear elliptic equation where g denotes the Laplace Beltrami operator. For spheres of dimensions n ≥ 3 and writing the conformal metric as g := u 4/(n−2) g, the Nirenberg problem amounts to solve the following nonlinear elliptic equation involving the Sobolev critical exponent: The Nirenberg problem has attracted a lot attention in the last half century. See [3,4,6,7,[10][11][12][19][20][21][22][23][24]28,31,32,36,37] and the references therein. Actually due to Kazdan-Warner obstructions, see [18,28], a positive answer to the Nirenberg's question requires imposing conditions on the function K . It turns out that finding sufficient conditions under which the Nirenberg problem is solvable depends strongly on the dimension n and the behavior of the function K near its critical points. Indeed in low dimension n < 5 index counting criteria have been obtained, see [7,20,27,31,32]. Such a counting index criterium fails, under the nondegeneracy assumption (N D) (that is K = 0 at critical points of K ), if the dimension n ≥ 5. They can be extended on high dimensional spheres in the perturbative setting (that is when K is close to a constant) see [19,24] or under some flatness assumptions see [16,22,31].
To explain the main difficulty in studying the Nirenberg problem and the differences between the low dimensional case n < 5 and the high dimensional one n ≥ 5, we point out that due to the presence of the Sobolev critical exponent, the corresponding Euler-Lagrange functional does not satisfy the Palais-Smale condition. One way to overcome such a difficulty is to consider the following subcritical approximation of the problem (N P): where ε > 0 is a small parameter. In this way one recovers the compactness and one then studies the behavior of blowing up solution u ε of (N P ε ) as the parameter ε goes to zero. Actually it can be proved that finite energy blowing up solutions of (N P ε ) can have only isolated simple blow up points which are critical points of the function K , see [23,31,32,35]. The reason of the additional difficulty in the high dimensional case lies in the complexity of the blow up phenomenon. Indeed in dimensions n = 2, 3 there are only single blow up points, see, [7,20,27,31,37] and in dimension n = 4 multiple bubbling may occur only under some extra condition, see [11,32] while, under the non degeneracy assumption (N D), on spheres of dimension n ≥ 5 every m−tuple (q 1 , . . . , q m ) of distinct critical points of K , satisfying K (q i ) < 0 for each i = 1, . . . , m can be realized as a concentration set of blowing up solutions of (N P ε ). See [34].
Regarding the high dimensional case n ≥ 5, Malchiodi and Mayer [35] obtained recently an interesting existence criterium under some pinching condition. Their result reads as follows: Theorem A [35] Let n ≥ 5 and K ∈ C ∞ (S n ) be a positive Morse function satisfying the following conditions (i) ∀q ∈ S n , ∇ K (q) = 0 ⇒ K (q) = 0, where K max := max S n K and K min := min S n K (iii) #{q ∈ S n ; ∇ K (q) = 0; K (q) < 0} ≥ 2, where # A denotes the cardinal of the set A. Then Nirenberg Problem (N P) has at least one solution.
In this paper we consider a version of the Nirenberg problem on standard half spheres (S n + , g). Namely we prescribe simultaneously the scalar curvature to be a positive function 0 < K ∈ C 3 (S n + ) and the boundary mean curvature to be zero. This amounts to solve the following boundary value problem where K ∈ C 3 (S n + ) is a positive function. This problem has been studied on half spheres of dimensions n = 2, 3, 4. See the papers [13][14][15]17,25,29,30] and the references therein. Very much like the case of spheres, to recover compactness one considers here the following subcritical approximation Just as above, there are two alternatives for the behavior of a sequence of solutions u ε of (P ε ). Either the ||u ε || L ∞ remains uniformly bounded or it blows up and if it does u 2n/(n−2) ε L n (where L n denotes the Lebesgue measure) converges to a sum of Dirac masses, some of them are sitting in the interior and the others ones are located on the boundary. The interior points are critical points of K satisfying that K ≤ 0 and the boundary points are critical points of K 1 the restriction of K on the boundary and satisfying that ∂ K /∂ν ≥ 0. See [14,17,25]. Furthermore a refined blow up analysis, under the non degeneracy assumption that K = 0 at interior critical points of K and that ∂ K /∂ν = 0 at critical points of K 1 , shows that in the dimension n = 3 multiple bubbling may occur but all blow up points are isolated simple, see [25,30]. Moreover in dimensions n = 2, 3 counting index criteria have been established, see [14,17,25,29]. Furthermore under additional condition on K 1 it has been proved in [15] that all blow up points are isolated simple, but already in dimension n = 4 counting index formulae, under the above non degeneracy conditions fail. More surprisingly and in contrast with the case of closed spheres, the Nirenberg problem on half spheres may have non simple blow up points, even for finite energy bubbling solutions of (P ε ) see [1,2]. In this paper we study Problem (P) from the viewpoint of the theory of critical points at infinity. In this approach initiated by the late Bahri, see [5][6][7][8], one studies the possible ends of non compact orbits of the (negative) gradient of the associated Euler Lagrange functional. The method consists of taking advantage of the concentration-compactness analysis of non converging Palais-Smale sequences to identify a potential neighborhood at infinity where concentration may occur. Then one constructs a global pseudogradient for which the full analysis of the ω-limit set, in this neighborhood is easier than for the genuine gradient flow and then uses it to characterize critical points at infinity. One then performs a Morse reduction near these critical points at infinity in order to compute their topological contribution to the difference of topology between the level sets of the Euler-Lagrange functional.
Before stating our main results, we set up some notation and introduce our assumptions. For the function K and its restriction on the boundary K 1 := K ∂S n , we use the following assumption: (H1): We assume that K is a C 3 (S n + ) positive function, which has only non-degenerate critical points with K = 0. (We point out that some of these points can be on the boundary.) (H2): We assume that the restriction of K on the boundary K 1 := K |∂S n + has only nondegenerate critical points z's. Furthermore we assume that if z is not a local maximum point of K 1 , we have that ∂ K /∂ν(z) ≤ 0.
(H3): If z ∈ ∂S n + is a critical point of K 1 satisfying that ∂ K /∂ν(z) = 0, hence z is actually a critical point of K on ∂S n + , we assume that K (z) = 0 and one of the following conditions is satisfied: (i) either ∂ K /∂ν(a) K (z) ≤ 0 for each a ∈ ∂S n + in a small neighborhood of z, (ii) or lim a∈∂S n d(a,z) = 0. Next we introduce the following subsets of critical points of K and K 1 K − in := {y ∈ S n + : ∇ K (y) = 0 and K (y) < 0}, Furthermore we define Our first result is an existence result under a pinching assumption, which parallels the above mentioned existence result of Malchiodi-Mayer. Namely we prove Theorem 1.1 Let n ≥ 5 and 0 < K ∈ C 3 (S n + ) satisfying the assumptions (H 1), (H 2) and (H 3).
If the following conditions hold where K max := max S n + K and K min := min S n where # A denotes the cardinal of the set A. Then Problem (P) has at least one solution. Remark 1.2 1. The above theorem is the counterpart of the existence result of Malchiodi-Mayer [35](see Theorem A quoted above). We point that the proof of Theorem 1.1, compared with the proof of Theorem A is more involved. In particular the counting index argument in our case is more subtle. Indeed due to the influence of the boundary the blow up picture is more complicated. Namely we have boundary and interior blow up as well as mixed configurations involving both of them. Such a complicated picture imposes to consider 4 critical levels instead of two critical levels needed in the case of closed spheres. Such a fact makes the index counting of the associated critical points at infinity more involved, see Lemmas 5.9, 5.8 in the "Appendix". 2. The conditions (H 2), (H 3) are used to rule out non simple blow up, see [1]. A phenomenon which does not occur in the case of closed spheres. See Sect. 3.2.2.
The above pinching condition (i) of Theorem 1.1 can be relaxed when combined with some counting index formula involving either the boundary blow up points or the interior blow points. In the next theorem we provide an existence result involving the boundary blow up points. Namely we prove: Theorem 1.3 Let n ≥ 5 and 0 < K ∈ C 3 (S n + ). Assume that the critical points of K 1 := K |∂S n + are non degenerate and that K satisfies the assumption (H 3). If the following conditions hold (a) Then Problem (P) has at least one solution.
Next we assume that the above index formula A 1 = 1, which implies, in particular that the number of boundary blow up points is an odd number, say 2k + 1, where k ∈ N 0 . The next existence result combined a pinching condition with a counting index formulae involving interior blow up points. Namely we prove: Theorem 1.4 Let n ≥ 5 and 0 < K ∈ C 3 (S n + ) satisfying the assumptions (H 1), (H 2) and (H 3).
If the following conditions hold (i) Then Problem (P) has at least one solution.
Regarding the method of proof of our main existence results, Theorems 1.1, 1.3 and 1.4 some comments are in order. Indeed although the general scheme falls in the framework of the techniques and ideas of the critical point theory at infinity , see [6,7,11], the main arguments here are of a different flavor. Indeed with respect to the case of closed spheres, treated by A.Bahri in his seminal paper [6], the case of half spheres presents new aspects: From one part the blow up picture is more complicated (interior, boundary and mixed configurations) and from another part the behavior of the self interactions of interior bubbles and boundary bubbles is drastically different. A fact which was used in [1] to construct subcritical solutions having non simple blow ups. To rule out such a possibility, under our assumption (H 2) and (H 3), we had to come up with a barycentric vector field which moves a cluster of concentration points towards their common barycenter and to prove that along the flow lines of such a vector field the functional decreases and the concentration rates of an initial value do not increase, see Lemma 3.9. Furthermore we prove that in the neighborhood of critical points at infinity, the concentration rates are comparable and the concentration points are not to close to each other. See Sects. 3.2.2 and 3.2.3.
The remainder of this paper is organized as follows: In Sect. 2 we set up the variational framework and define the neighborhood at infinity and in Sect. 3 we construct an appropriate pseudogradient in the vicinity of highly concentrated bubbles and derive from the analysis of the behavior of its flow lines the set of its critical points at infinity. Section 4 is devoted to the proof of the main existence results of this paper. Lastly we collect in the appendix some estimates of the bubble, fine asymptotic expansion of the Euler-Lagrange functional and its gradient in the neighborhood at infinity as well as useful counting index formula for the critical points of the function K and its restriction K 1 on the boundary.

Loss of compactness and neighborhood at infinity
In this section we set up the analytical framework of the variational problem associated to the Nirenberg problem and recall the description of its lack of compactness. Let H 1 (S n + ) be the Sobolev space endowed with the norm For a ∈ S n + and λ > 0 we define the standard bubble to be where d is the geodesic distance on S n + and c 0 is a constant chosen such that For a ∈ S n + , we define projected bubble ϕ a,λ to be the unique solution of We point out that ϕ a,λ = δ a,λ if a ∈ ∂S n + . Next for m ∈ N and p, q ∈ N 0 such that q + 2 p = m we define the neighborhood of potential critical points at Infinity V (m, q, p, ε) as follows: In the following we describe non converging Palais-Smale sequences. Such a description, which is by now standard, follows from concentration-compactness arguments as in [33,38] and reads as follows Proposition 2.1 Let u k ∈ + be a sequence such that ∇ J K (u k ) → 0 and J K (u k ) is bounded. If Problem (P) does not have a solution, then there exist m ∈ N and p, q ∈ N with q+2 p = m, a sequence of positive real numbers ε k ↓ 0 as well as subsequence of u k , still denoted u k such that u k ∈ V (m, q, p, ε k ).
Following Bahri and Coron, we consider for u ∈ V (m, q, p, ε) the following minimization problem (5) We then have the following proposition whose proof is identical, up to minor modification to the one of Proposition 7 in [8] Proposition 2.2 For any m ∈ N there exists ε m > 0 such that if ε < ε m and u ∈ V (m, q, p, ε) the minimization problem (5) has, up to permutation, a unique solution.
Hence it follows from Proposition 2.2 that every u ∈ V (m, q, p, ε) can be written in a unique way as where a i ∈ ∂S n + , i = 1, . . . , q and a i ∈ S n + , i = q + 1, . . . , p + q, and v ∈ H 1 (S n + ) satisfying where δ i := δ a i ,λ i and ϕ i := ϕ a i ,λ i . In addition, the variables α i 's satisfy In the next lemma we deal with the v-part of u ∈ V (m, q, p, ε) in order to prove, that its effect is negligible with the concentration phenomenon. Namely we prove:

Moreover, there exists c > 0 such that the following holds
Proof The proof follows as in Proposition 3.1 in [12] (see also [9]). Indeed, easy computations imply that Note that Q is a positive definite quadratic form (see [5]) and we have that Observe that, for n ≥ 6, it follows that 4/(n − 2) ≤ 1. Hence, using Holder's inequality, we get For the other term, for i ≤ q (that is a i ∈ ∂S n + ), using the fact that δ i , v = 0, we get For i > q, using Lemma 5.1, we get and the result follows.

Pseudogradient and Morse Lemma at infinity
This section is devoted to the construction of a pseudogradient for the functional J K , which has the property that along its flow lines there could be only finitely many isolated blow up ponits. Such a pseudogradient coincides with the gradient outside of m,q, p V (m, q, p, ε/2) and satisfies the Palais-Smale condition there. Moreover in each V (m, q, p, ε) it has the property to move the concentration points according to ∇ K or ∇ K 1 , the α i 's to their maximum values and the concentration λ i 's are moved so that the functional J K decreases along its flow lines. The global vector field is then defined by convex combining these two vector fields. Such a construction is then used to perform a Morse reduction near the singularities of the pseudogradient and to compute the difference of topology induced by the critical points at infinity between the level sets of the Euler-Lagrange functional J K . The first step in the construction of the pseudogradient is to describe the movement of the variable v. In fact, since v minimizes J K in the v-space, it follows from the classical Morse Lemma that there exists a change of variable v → V such that Hence, for the variable V , we will useV = −V to bring it to 0. Thus, we need to construct some vector fields by moving the variables α i , a i and λ i .

The case of a single concentration point
We point out that the construction of a pseudogradient satisfying the above properties becomes quite involved in the case of more than one concentration point. Indeed in the case of two bubbles sitting at different points, their mutual interaction comes into play. For this reason we start by constructing the needed pseudogradient in neighborhoods at infinity, containing one interior or one boundary point. To do so we consider two cases, the first one corresponds to p = 1 and q = 0 (case of an interior concentration point) and the second one corresponds to p = 0 and q = 1 (the case of a boundary point). Namely we prove: There is a constant c > 0 independent of u = αϕ a,λ ∈ V (2 p + q, q, p, ε) (with q = 1 or p = 1) such that Proof We start by giving the proof of Claim (i) for the case where p = 1 and (1)) and therefore K (a) has a constant sign. Let M be a large constant and let ψ 1 be a C ∞ cut off function defined by ψ ∈ [0, 1], We notice that, in the region where |∇ K (a)| ≥ 2M/λ, we have that ψ 1 (λ|∇ K (a)|/M) = 1, therefore the Claim (i) follows from Proposition 5.7. Next if |∇ K (a)| ≤ 2M/λ then a is very close to a critical point of K in S n + . We claim that this critical point cannot be on the boundary. Indeed, arguing by contradiction, we assume that a is in small neighborhood of a critical point z ∈ ∂S n + . Since z is a non-degenerate critical point of K , we derive that λd(a, z) is bounded which contradicts the fact that λd(a, ∂S n + ) is very large. Hence our claim follows and a is close to an interior critical point y in S n + . Next using Proposition 5.7 we derive that Hence Claim (i) is proved in the case where p = 1 and q = 0. Concerning (ii) it follows from (i) using the estimate of v in Lemma 2.3. Finally we notice that λ increases along the flow lines of the pseudogradient W only in the region where a is close to a critical point y with K (y) < 0. Thus the proof of the proposition follows in the case where p = 1 and q = 0.
Next we consider the case where p = 0 and q = 1, that is the case of a boundary concentration point a ∈ ∂S n + . In this situation we divide the set V (1, 1, 0, ε) into 3 subsets and construct an appropriate vector field in each of these sets.
In this region, we define and using Proposition 5.6, Claim (i) follows easily (since M is chosen large).
where η is a small fixed constant. In this region, we define Note that, in this region, the parameter μ is of the same order that λ. Hence, using Proposition 5.5, the proof of Claim (i) follows.
In this region, a is close to a critical point z of K 1 . The pseudogradient will depend on z. We define Using Propositions 5.4 and 5.5, it holds Observe that, if ∂ K /∂ν(z) = 0, it follows that γ ∂ K /∂ν(a) ≥ c > 0 and therefore Claim (i) follows easily. In the other case, that is ∂ K /∂ν(z) = 0, we need to make use of the assumption (H 3). Indeed, , in the lower bound of (14) will appear |∇ K 1 (a)|/λ + |∂ K /∂ν(a)|/λ + 1/λ 2 which is larger than c/μ. Hence, Claim (i) follows in this case. However, if λ|∇ K 1 (a)| ≤ 2M, it follows that |∇ K (a)| ≤ cM/λ (since we assumed that z is a non degenerate critical point).
which imply the proof of Claim (i) in this case. In the other case, which is λ|∇ K 1 (a)| ≤ 2M, it holds: (by the chose of 1 ). Thus the proof of Claim (i) follows from (14).
Finally Claim (ii) follows from Claim (i) using the estimate of v in Lemma 2.3 and Claim (iii) follows immediately from the properties of the constructed vector field.
We remark that the assumption (H 2) is not used in the construction of the pseudogradient in V (1, 1, 0, ε).

The case of multiple concentration points
In the next proposition we address the case where the set of the concentration points contains more than one point. Before stating our result we define for i = 1, . . . , m the scalar quantity μ i as follows The behavior of such a quantity along the flow lines of the constructed pseudogradient plays crucial role in identifying critical points at infinity.

iii) The vector field W is bounded with the property that along its flow lines the maximum of the μ i 's increases only if the
The construction of a pseudogradient satisfying (i), (ii), (iii) is quite involved and requires some preparatory Lemmas and estimates. Its construction depends on the behavior of the leading terms of the α-, a-and λ-component of the gradient in the neighborhood at infinity V (m, q, p, ε). To perform such a construction we divide the set V (m, q, p, ε) into four subsets. The first and the second ones correspond to the situation where at least one of the variables α i 's and a i 's is not in its critical position and the μ i 's are of the same order. In the third one, the μ i 's are still of the same order but the variables α i 's and a i 's are very close to their critical positions. Finally in the fourth one we address the case where the μ i 's are not of the same order.
To define these regions, we introduce the following notation. For M 2 a large constant we set: To explain the relevance of the above quantities, we state the following Lemma Then a i is close to a interior critical point y of K in S n + . (2) If a i , a j are interior points satisfying that λ k + a k + H k ≤ 8 for k = i, j and if their corresponding concentration rates λ i and λ j are of the same order. Then a i and a j cannot be close to the same critical point.
and therefore a i is close to a critical point of K . We need to exclude the case where this critical point lies on the boundary. In fact, assuming that it is the case, i.e. a i is close to z ∈ ∂S n + . Then it follows from (H 1), that λ i d(a i , z) is bounded, which is not allowed. Therefore, each concentration point a i is close to a critical point y j i ∈ S n + and the first assertion is proved. Concerning the second one, assume that two different points a i and a j are near the same critical point y. Then we have from the first assertion: λ k d(a k , y) is bounded for k = i, j. Since λ i and λ j are assumed to be of the same order, it follows that λ k d(a i , a j ) is bounded, which contradicts the smallness of ε i j .

Construction of some local pseudogradients
In this subsection we construct some local pseudogradients in some parts of the neighborhood at infinity. These vector fields will be glued together to obtain a global pseudogradient satisfying the properties required in Proposition 3.2.
For M 0 a large number we define the following subsets of V (m, q, p, ε) where μ max := max j μ j and μ min := min j μ j . Before defining a pseudogradient in each subset, we single out some of their properties that will be used in the construction of the local pseudogradients.  (1) In V k (M 0 ), for k ≤ 3, the variables μ i 's are of the same order. Thus, using Lemma 5.2, we derive that, for each i = j ≤ q, it holds Furthermore, for i = j > q, we deduce that λ i and λ j are of the same order and therefore (17) holds true. Now, for i > q and j ≤ q, we have λ i d i is very large which implies that λ i d(a i , a j ) is also very large and therefore (17) holds for these indices.
(2) In V k (M 0 ), k = 2, 3, for each i > q, the concentration point a i is close to a critical point y j i ∈ S n + and two different points a i and a j cannot be near the same critical point y (see Lemma 3.3). ( We start our construction by defining a pseudogradient in V 1 (M 0 ).

Lemma 3.5
There exists a bounded pseudogradient W 1 so that the following holds: There is Furthermore, the λ i 's are decreasing functions along the flow lines generated by this pseudogradient. In addition, the constant of 1/μ max is independent of M 0 and M 2 .
Proof We start by defining the following vector fields: Observe that, using Propositions 5.4, 5.7, the estimate (17) and the definition of ψ 1 , we derive that Moreover using Proposition 5.6, we derive that Such an estimate suggests to move the variable α i 's if |1 − J K (u) n/n−2 α 4/n−2 i K (a i )| is very large with respect to r =k ε kr + 1/μ k . Furthermore making use of Propositions 5.6 and 5.7, we derive that Nest we define Using the previous estimates, we obtain Regarding the above estimate, we point that we need to take care of the interaction term O(ε ki ) contained in the expression b . To that aim, we observe that, if H k + a k + λ k ≥ 6, then the ε ki appears in the lower bound in (24) and therefore we are able to remove the (1/M 2 )ε ki by taking M 2 large. But, if H k + a k + λ k ≤ 6, it follows that (see the second assertion of Remark 3.4) a k is close to a critical point y of K and therefore we get ) which is small with respect to our lower bound. Since we are in V 1 (M 0 ), there exists at least one index i > q such that ψ 1 ( λ i )+ψ 1 ( H i )+ ψ 1 ( a i ) ≥ 1. This implies that 1/λ 2 i = 1/μ i appears in the lower bound of (24). Since all the μ j 's are of the same order, we are able to make appear all the 1/μ j 's in this lower bound and Lemma 3.5 follows.
In the next lemma we construct a pseudogradient in the set V 2 (M 0 ). Namely we prove: Lemma 3.6 There exists a bounded pseudogradient W 2 such that the following holds: There Proof First, recall that (see Remark 3.4), in V 2 (M 0 ), each interior concentration point a k is close to a critical point of K in S n + and that two interior concentration points a i and a k cannot be close to the same critical point which implies that d(a i , a k ) ≥ c > 0 and Recalling that K b := {z ∈ ∂S n + : ∇ K 1 (z) = 0} we define the following pseudogradient: Using (22), (23) and Proposition 5.5, we get First, taking i ∈ D 1 , for k ≤ q, two cases may occur: (i) either d(a i , a k ) ≤ η/2, and in this case we get that |∇ K (a k )| ≥ c and therefore μ k and λ k are of the same order. Thus λ i and λ k are of the same order. (ii) or d(a i , a k ) ≥ η/2. In the two cases, we deduce that Secondly, for i ∈ D 1 , we have |∇ K 1 (a i )| ≥ c(η) and therefore λ i and μ i are of the same order. Since all the μ j 's are assumed to be of the same order, we are able to make appear all the 1/μ j 's in the lower bound of (25). Finally, for j / ∈ D 1 , (i) either α j ≥ 2, in this case, the |1 − J K (u) n/(n−2) α 4/(n−2) j K (a j )| + ε k j appears in α , (ii) or α j ≤ 2 and λ j ≥ 2, in this case k j ε k j appears in b , (iii) or α j + λ j ≤ 4, in this case we are able to make appear |1 − J K (u) n/(n−2) α 4/(n−2) j K (a j )| + ε k j from 1/μ j . Hence the lemma follows.
Next we consider the third set V 3 (M 0 ). We notice that in this subset each concentration point a i is close to some critical point of K or K 1 and for a critical point z ∈ ∂S n + of K 1 (resp. y ∈ S n + of K ), we denote by We observe that it follows from Remark 3.4 that # B y ≤ 1 for each critical point y in S n + . However, it is possible to have # B z ≥ 2 for some critical points z's in ∂S n + . Next we divide the set V 3 (M 0 ) into four subsets. The first three ones are defined as follows: where y is an interior critical point of K and z is a critical point of K 1 , and the last one is defined as: In the next lemma we construct a pseudogradient in the first subset. Namely we prove the following lemma: Lemma 3.7 There exists a bounded pseudogradient W 1 3 such that the following holds: There 3 such that the statement of Lemma 3.5 holds true with W 1 3 instead of W 1 . Proof Let z be such that ∂ K /∂ν(z) = 0 and #B z ≥ 2. Firstly, we claim that: There exists k ∈ B z such that: Indeed arguing by contradiction, we assume that this claim does not hold. Thus, since z is a non-degenerate critical point of K 1 , we obtain, for each k ∈ B z , z) is bounded. In addition, from the definition of μ k , we get Thus, μ k and λ 2 k are of the same order for each k ∈ B z . Next let i and j be two different indices in B z . We deduce that λ j and λ i are of the same order and λ k d(a i , a j ) is bounded for k = i, j. These give a contradiction with the fact that ε i j is small. Hence our claim follows.
Furthermore observe that, for k satisfying (27), it holds that λ k d(a k , z) ≥ cM 2 . Now, in this region, we define the following vector field: Using Proposition 5.5, we get Moreover for i ∈ D 2 and k ≤ q with k = i, two cases may occur: (i) either λ k ≤ M 2 0 M 2 2 λ i , and in this case we get In this case, since μ k ≤ 2M 0 μ i and z is a non-degenerate critical point of K 1 , it follows that z). Therefore we obtain where we have used the fact that λ i d(a i , z) ≥ cM 2 . Thus (28) becomes ). (29) Finally, we notice that |∇ K (a i )| ≤ cd(a i , z) ≤ c|∇ K 1 (a i )| ≤ c|∇ K (a i )|. Thus, in (29), we can make appear 1/μ i for i ∈ D 2 and therefore all the 1/μ j 's (since there are of the same order) and the proof follows as the proof of the previous lemmas.

Lemma 3.8 There exists a bounded pseudogradient W 2 3 such that the following holds: There is a constant c
3 such that the statement of Lemma 3.5 holds true with W 2 3 instead of W 1 .
Proof Let D 1 := ∪ y: K (y)>0 B y , D 2 := ∪ z:∂ K /∂ν(z)<0 B z and D 3 := ∪ z:∂ K /∂ν(z)=0; K (z)>0 B z . We divide this region into two subsets: 1st subset: If D 1 ∪ D 2 = ∅. In this case, we define By using the first assertion of Remark 3.4 and Propositions 5.4 and 5.7, it follows that Hence, the proof follows. 2nd subset: D 3 = ∅. Note that, since we are outside of V 1 3 where M is a large constant. We point out that W 22 3 is exactly the sum of of the vector fields W 3 1 (defined in (13)) with γ = −1. Furthermore, the presence of the function ψ 1 implies that the point a i moves only if |∇ K 1 (a i )| ≥ M/λ i . Using Propositions 5.4 and 5.5, we get which has the same form as (14). Hence, the same computations and arguments hold and the proof of the lemma follows.

Ruling out collapsing phenomena
We point that, the main difference between the S n -case (or the case of an interior blow up point for the S n + -case) and the boundary blow up point case relies essentially on the behavior of the leading term in Propositions 5.4 and 5.7 (namely the λ-term). Indeed when ∂ K /∂ν(z) = 0 and a i is close to a boundary critical point z ∈ ∂S n + , the leading term behaves like c/λ i , while for the S n -case (or the case of an interior blow up point in the S n + -case), for a i close to an interior critical point y with K (y) = 0, this term behaves like c/λ 2 i . This difference on the behavior of the leading term plays a crucial role in the nature of the critical point at infinity. Indeed in [1], for z a critical point of K 1 (which is not local maximum) satisfying ∂ K /∂ν(z) > 0, we proved that z is not a simple blow up point in the sense that B z contains more than one concentration point. In the following lemma, we consider the case of a local maximum point of K 1 satisfying ∂ K /∂ν(z) > 0 and we will prove that z is a simple blow up point. Namely we prove Lemma 3.9 Let z be a non degenerate local maximum of K 1 with ∂ K /∂ν(z) > 0. Then z is a simple blow up. More precisely if # B z := #{a i ; close to z} := q 1 ≥ 2, then J K admits in the set V (q 1 , q 1 , 0, ε) a compactifying bounded pseudogradient W (z, q 1 ). Namely there exits a constant c > 0 independent of u =

Furthermore, the concentration rates λ i 's do not increase along the flow lines generated by this pseudogradient.
For the proof of Lemma 3.9, we make use of the following technical results. Lemma 3.10 Let a i , a j ∈ ∂S n + be concentration points such that the corresponding rates λ i and λ j are of the same order and d(a k , b) → 0 for k = i, j for some point b ∈ ∂S n + . Then we have Proof Easy computation implies that Thus we get where |a i − a j | is the euclidian norm of a i − a j seen as a vector in R n+1 .

Lemma 3.11
Let a, h ∈ ∂S n + be close to a non degenerate local maximum z of K 1 . Then it holds that It is easy to get that and therefore it holds that |β (t)| = |a − h|(1 + o(1)) uniformly in t ∈ [0, 1]. Furthermore, easy computations imply that |β (t)| = O(|a − h| 2 ) uniformly in t ∈ [0, 1]. In another hand, we have and, since a and h are close to a non degenerate maximum critical point z of K 1 , we derive that which implies the lemma.

Proof of Lemma 3.9
For the construction of a suitable vector field satisfying the properties required in Lemma 3.9 as well for later purposes we will use some constants M 0 , M 2 and M 4 which are required to be large and to satisfy The first requirement is used in (34) and (35) below while the second one is used when studying a remainder term of (42) and the last one is used in (44) in the proof of Lemma 3.14.
In view of the pseudogradient constructed in Lemmas 3.6 and 3.14, it is enough to construct a pseudogradient satisfying the above estimate in the following set: Moreover, since the λ i 's are of the same order, we have that ε i j = (1 + o(1))/(λ i λ j d(a i , a j ) 2 ) (n−2)/2 and therefore d(a i , a j ) ≥ c/λ (n−3)/(n−2) 1 for each i = j (since λ i is bounded). We want to construct a pseudogradient which moves the concentration points a i to their barycenter and prove that along its flow lines the Euler-Lagrange functional decreases. To this aim, let i and i 1 be such that d(a i , a i 1 ) := min d(a r , a )  Observe that, since we have only q 1 points and #L i = 2, then there exists m ≤ q 1 − 1 such that L m+1 i = L m i and we set L * i := L m i where m is the first index such that L m+1 i = L m i . We remark that L i ⊂ L * i . Next we want to move the points a j 's, for j ∈ L * i , to their center of mass. For this aim, let a i be defined as Note that, it is easy to see that a i satisfies a i ∈ ∂S n + and j∈L * i a j − a j , a i a i = 0.
Now we define the following vector field: a j ).
We note that L * i has two important properties: small, it follows that for every ∈ L * i , we have that: (by using (31)). We note that, in this region, we have |1 − J K (u) n n−2 α 4 n−2 j K (a j )| ≤ cM 2 /λ j for each j, hence Proposition 5.5 can be written as: Hence we derive that: Next we notice that, using Lemma 3.10, il holds Furthermore, using Lemma 3.11 (with h = a i ), it holds that K (a i )(a j − a j , a i a a j , a i a i = 0 (see (33))).

Thus we get
Hence, in the lower bound of (39), we are able to make appear 1/λ 2−1/(n−2) i and therefore (since all the λ j 's are of the same order and α k ≤ 4 for each k) we are able to make appear all the 1/λ Concerning the ε kr , we note that the ε k j 's which appeared in the lower bound, are only for the indices k, j ∈ L * i . Hence we need to make appear ε jr for j / ∈ L * i . For this aim, we remark that, for each j, , we have d(a j , a ) ≥ d(a i , a i 1 ) (by the definition of i and i 1 ), in addition we have that the λ k 's are of the same order. Hence we deduce that ε ii 1 ≥ cε j . Hence the proof of the lemma follows.
In the next lemma we rule out non simple blow up for a mixed configuration involving local maxima on the boundary and other interior blow up points. Namely we prove: Lemma 3.12 There exists a bounded pseudogradient W 3 3 such that the following holds: There Furthermore, the λ i 's do not increase along the flow lines generated by the pseudogradient W 3 3 . Proof Let z 1 , . . . , z be the critical points of K 1 satisfying ∂ K /∂ν(z j ) > 0 and #B z j ≥ 2. We decompose u as follows: From the second and the third assertions of Remark 3.4, it follows that each concentration point a j of u +1 satisfies |a j − a k | ≥ c for each k = j and it is close to a critical point of K 1 with ∂ K /∂ν ≥ 0 or a critical point of K in S n + with K < 0. Furthermore, for j ∈ B z i , we have |a j − a k | ≥ c for each k / ∈ B z i . Hence the mutual interaction between two clusters B z i and B z j for i = j is negligible with respect to the other terms. In this situation, we define the following vector field Hence we obtain (40) We observe that, for k ∈ B z i , we have μ k and λ k are of the same order. Moreover we are in the case where all the μ j 's are of the same order. Thus, using Lemma 3.9, we are able to make appear all the 1/μ 2−1/(n−2) j 's in the lower bound of (40) (and therefore all the |1 − J K (u) n/(n−2) α 4/(n−2) i K (a i )| 2−1/(n−2) 's and the (|∇ K (a i )|/λ i ) 2−1/(n−2) 's (since the α k 's and the a i 's are bounded). In addition, for j / ∈ B z i and k ∈ B z i , we have Therefore, our lemma follows from Lemma 3.9.

Lemma 3.13
There exists a bounded pseudogradient V satisfying the following estimate: Furthermore in the subset of W such that λ i |∇ K 1 (a i )| is bounded, the λ i 's are increasing functions along the flow lines generated by the pseudogradient V.
Observing that in W we have ε i j = O(1/λ n−2 i + 1/λ n−2 j ) for each i = j and using Propositions 5.4, 5.5, 5.7 the lemma follows.

Ruling out bubble towers phenomena
In this subsection we prove any configuration of points of non comparable concentration rates is not critical at infinity. Indeed one can construct in the neighborhood of such points a compactifying pseudogradient. Namely we prove that: Lemma 3.14 There exists a bounded pseudogradient W 4 such that the following holds: There Furthermore, max μ i deos not increase along the flow lines generated by this pseudogradient.
. . , p + q; a i ∈ ∂S n + }. Next we reorder the parameters μ i 's as: μ 1 ≤ · · · ≤ μ p+q and define the following subset of indices: Since we are in V 4 (M 0 ), we have μ max > M 0 μ min , it follows that p + q / ∈ I . In this region, we write u as u := u 1 + u 2 where u 1 := i∈I α i ϕ i and u 2 := u − u 1 .
Furthermore we introduce the following notation Next we define in case D 4 1 ∪ D 4 2 = ∅ the following vector fields: and as in the proof of Lemma 3.5, we define where W in a (resp. W α ) is defined in (19) (resp. (20)). Following the proof of Lemma 3.5 and using Lemma 5.2, we get Observe that, concerning the last term, for ∈ I in , (i) either ≥ i 0 , then the ε j exists in the second term of this formula and one takes M 2 large to absorb the last term, or (ii) < i 0 and in this case by Lemma 3.3, the concentration point a is close to a critical point y of K in S n + and then ε j ≤ c(1/λ n−2 j + 1/λ n−2 ). Hence, we can in either case absorb the last term.
Furthermore we notice that if D 4 1 ∪ D 4 2 = ∅ and if i 0 ∈ I or if j 0 ∈ I then we can include all the indices in I in the lower bound of (41). Otherwise to make appear the terms corresponding to these indices we argue as follows: In this region, we define the following vector field: where W is the convex combination of the pseudogradients constructed in 3 3 . It follows then that To complete the proof, it remains to absorb the last term. To this aim, we notice that: (i) if " ∈ I in with ≥ i 0 or ∈ I b with ≥ j 0 ", then the term ε j is already in 4 the lower bound of (41). Taking M 2 large, we will be able to absorb this term.
(ii) if " ∈ I in with < i 0 or ∈ I b with < j 0 ", then there holds: small enough (see (31)) and where k 0 := max I . Hence, we are also able to remove this term. (Recall that, in Lemmas 3.5-3.8, 3.13, the constant over μ max is independent of M 0 and M 2 ). Hence the estimate in the first case follows as in the proof of the previous lemmas.
Case 2: In this case we take u 1 ∈ V 3 3 (M 0 ) and assume that D 9 ∪ D 8 = ∅, where Here we define the following vector field: We point out that, this pseudogradient increases the μ i for i ∈ D 8 ∪ D 9 , but does not increase the μ max := μ p+q since p + q / ∈ I . Furthermore observe that Hence the result follows as the first case. Next we set where m 1 is a small constant.
(iii) r < j 0 and r ∈ I . In this case, it follows that a r ∈ B z with = k and therefore we deduce that |a k − a r | ≥ c > 0. Hence we get Using (41),(43), the previous estimates and the fact that D 10 = ∅, the lemma follows in this case.

Proof of Proposition 3.2
The required pseudogradient will be a convex combination of the ones defined in the previous lemmas. Each one is bounded and satisfies Claim (i). Furthermore, the only case where μ max increases is the region W. Finally, Claim (ii) follows from the first one and the estimate of v 2 which is small with respect to the lower bound of Claim (i). Concerning the last claim, it follows easily from the definition of the pseudogradient. This achieves the proof of Proposition 3.2.

Critical points at infinity and their topological contribution
For ε 0 a small number, we define the following neighborhood of the cone of positive solutions of the sphere in H 1 (S n + ): This set is for ε 0 small enough invariant under the gradient flow lines of the Euler Lagrange functional J K . Namely we prove that Proof We will write J instead of J K . For w ∈ L 2n/(n+2) (S n + ), we denote by L −1 (w) the solution of the following PDE: Furthermore, it holds L 2n/(n−2) ≤ F(s). Recall that, if u − 0 = 0 then u(s) is positive for all s. Hence, we can assume that u − 0 = 0 and we want to prove that F is a decreasing function. Observe that Notice that f (0) = u − 0 and therefore But, we have (using (45)) , and we also have (using the fact that J (u(s)) is a decreasing function)  Finally, since inf J > c > 0, using the fact that L 2n/(n−2) < ε 0 , and η is small enough, then F (s) ≤ 0, for 0 ≤ s ≤ 1. Therefore J (u(s)) 2n−2 n−2 e 2J (u(s)) |u(s) − |
Next using a partition of the unity, one can define the vector field W of Proposition 3.2 globally by gluing it to the negative gradient −∇ J outside the V (q, p, m, ε)'s. Let us denote the resulting global vector field by Y and define a new vector field by setting: We then have Next we perform a Morse type reduction in the subsets W. Namely we prove Then the functional J K expands as follows where S n is defined in Proposition 5.3 (it represents the level of one boundary bubble), are the local coordinates of the parameters (α 1 , . . . , α p+q ) and a i . This expansion will be called the Morse Lemma at Infinity of J K near its critical point at infinity. Note that we loose an index for the parameter α since the functional J K is homogenous with respect to this parameter.

Such an index will be called the i ∞ -index of such a critical point at infinity.
Next as consequence of the above corollary and the Morse reduction in Lemma 3.17 we compute the topological contribution of the critical points at infinity to the difference of topology between the level sets of the functional J K . Namely we have Lemma 3. 19 Let τ ∞ be a critical point at infinity at the level C ∞ (τ ∞ ) with index i ∞ (τ ∞ ). Then for θ a small positive number and a field F, we have that where H l denotes the l−dimensional homology group with coefficient in the field F.

Proof of the main results
This section is devoted to the proof of Theorems 1.1, 1.3 and 1.4. The proof of these theorems is based on the characterization of the critical points at infinity in Corollary 3.18 and the computation of their contribution to the difference of topology in Lemma 3.19. It also uses two deformation lemmas. The first one is an abstract lemma, which is inspired by Proposition 3.1 in [35]. It reads as follows: . Indeed such a retraction can be realized by following the flow lines of a decreasing pseudogradient Z K for J K . Let φ K denote the one parameter group corresponding to this pseudogradient. For each u ∈ + , we denote by s K (u) the first time such that φ K (s K (u), u) ∈ J A K .
Secondly we recall that, for K ≡ 1, the only critical points of J 1 are minima and lie in the bottom level S n . Furthermore, for each A > S n , the set J A 1 is a contractible one. Indeed by following the flow lines of a decreasing pseudogradient Z 1 of the Yamabe functional J 1 , each flow line, starting from u ∈ + , will reach the bottom level S n . Let us denote by φ 1 the one parameter group corresponding to Z 1 .
Next we notice that, we have which implies that Furthermore we observe that for each u ∈ + , there exists a unique s 1 (u) satisfying φ 1 (s 1 (u), u) ∈ J A 1 . Next we define the following map: u)). We notice that F is well defined and continuous and satisfies the following properties: • For t = 0, we have φ K (0, u) = u. Furthermore, for each u ∈ J A 1 , we have s 1 (u) = 0. Therefore, for each u ∈ J A 1 , we get F(0, u) = φ 1 (0, u) = u.
1 (by the definition of s K ) which implies that s 1 (φ K (s K (u), u)) = 0 and therefore F (1, u . Thus J A 1 retracts by deformation onto J A K , a fact which provides the claim of the lemma since J A 1 itself is a contractible set. The second deformation lemma is a consequence of the previous one, the assumptions (H 1), (H 2), (H 3) of this paper and an appropriate pinching condition for the function K . To state it we set the following notation: We recall that it follows from Corollary 3.18 that the level of critical points at Infinity corresponding to q boundary points and p interior points such that q + 2 p = lie between C ,∞ min and C ,∞ max . Our second deformation lemma reads as follows: , the sublevel J c K is a contractible set. Proof Since we assumed that K max /K min < ((k + 1)/k) 1/(n−2) , it follows that, for each 1 ≤ ≤ k, we have (k + 1)/k ≤ ( + 1)/ and The proof follows then from Lemma 4.1 by taking A = C ,∞ max + γ with a small γ > 0 so that A < C +1,∞ min . Indeed between the levels A and A the functional J K does not have any critical point nor critical point at infinity.
Next we start the proof of our existence results by proving Theorem 1.3.

Proof of Theorem 1.3
Arguing by contradiction we assume that the functional J K does not have any critical point under the level C 2,∞ min . Hence it follows from Proposition 4.2 (with k = 1) that under the assumption of Theorem 1.3, we have that J C 1,∞ max +γ is a contractible set, for γ a small constant. Moreover it is a retract by deformation of C 2,∞ min . Furthermore follows from corollary 3.18 that critical points at infinity under the level C 2,∞ min are in one to one correspondence with critical points of Then it follows from Lemma 3.19 and the Euler-Poincaré theorem that: which contradicts the assumption (b) of Theorem 1.3. Hence the existence of at least one critical point of J K .

Proof of Theorem 1.4
Assuming that J K does not have any critical point under the level C 3,∞ min , we derive, using Proposition 4.2 (with k = 2), the level sets J C 1,∞ max +γ K and J C 2,∞ max +γ K are contractible sets. Then it follows from the properties of the Euler-Characteristic, see Proposition 5.7, pp.105 in [26], that Moreover it follows from Corollary 3.18 that the critical points at infinity between these two levels are in . Thus, it follows from Lemma 3.19 and the Euler-Poincaré theorem that where ι(z k ) := n − 1 − morse(K 1 , z k ) and ι(y) := n − morse(K , y).
Observe that, the first term is exactly −A 2 defined in Lemma 5.8. Hence, the previous equality contradicts the assumption (ii) of the theorem. The proof is thereby completed.
Proof of Theorem 1. 1 We first observe that, under the assumption of the theorem, if A 1 = 1 or respectively A 1 = 1 and B 1 = −k, where #(K + b ∪ K 0,− b ) = 2k + 1, the existence of at least one solution to Problem (P) follows from Theorem 1.4, respectively Theorem 1.3. Hence we will assume that A 1 = 1 and B 1 = −k and notice that #(K − in ) = 2r + k, where r ∈ N 0 , and there are r even numbers ι(y j )'s and r + k odd numbers ι(y j )'s.
Next arguing as in the proof of Theorem 1.4 using the assumption on K max /K min and Proposition 4.2, we deduce that J C 3,∞ max +γ K and J C 4,∞ max +γ K are contractible sets. Using Corollary 3.18, we derive that the critical points at infinity whose level are lying between these values are: • (z i , z j , z r , z t ) ∞ with different z i 's which belong to K + b ∪ K 0,− b , • (z i , z j , y) ∞ with y ∈ K − in and z i = z j ∈ K + b ∪ K 0,− b , • (y i , y j ) ∞ with y i = y j ∈ K − in . Hence arguing as above we derive that Observe that, the first term is exactly −A 4 , the second one is A 2 × B 1 and the third one is −B 2 (defined in Lemmas 5.8 and 5.9). Using the values of these terms (given in Lemmas 5.8 and 5.9), we obtain that r + k = 0 which implies that r = k = 0. Now, from r = k = 0, we get #(K + b ∪ K 0,− b ) = 1 and #K − in = 0. This leads to a contradiction with the assumption that #(K + b ∪ K 0,− b ∪ K − in ) ≥ 2. Thereby the proof of the theorem is completed.
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Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. However, for i ≤ q, we have ϕ i = δ i and therefore Note that, since u ∈ , we deduce that Now, the precise expansion of J K follows from the above estimates, the estimate of v (see Lemma 2.3) and the fact that (1 + x) −(n−2)/n = 1 − ((n − 2)/n)x + O(x 2 ).
In the following, we will present the expansion of the gradient of J K in the potential sets. We will present the results for p + q ≥ 2. However, the results are true for p + q = 1, it suffices to remove the terms ε i j 's which correspond to the interaction terms of the bubbles. Proposition 5.4 Let n ≥ 5, for u = i≤q α i δ i + i>q α j ϕ j ∈ V (m, q, p, ε) and i ≤ q, it holds For j ≤ q, we have a j ∈ ∂S n + and therefore, using [5], we get, for j = i, Observe that, for j = i, expanding K around a j , we get Finally, for j = i, it holds n + 2 n − 2 R n + K δ Hence the proof follows.