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

Abstract

In this paper we study the Nirenberg problem on standard half spheres $(\mathbb {S}^n_+,g), \, n \ge 5$, 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:

$$\begin{aligned} (\mathcal {P}) \quad {\left\{ \begin{array}{ll} -\Delta _{g} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2}},\, u > 0 &{}\quad \text{ in } \mathbb {S}^n_+, \\ \frac{\partial u}{\partial \nu }\, =\, 0 &{}\quad \text{ on } \partial \mathbb {S}^n_+. \end{array}\right. } \end{aligned}$$

where $K \in C^3(\mathbb {S}^n_+)$ is a positive function. This problem has a variational structure but the related Euler–Lagrange functional $J_K$ 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 $J_K$ in their neighborhood and compute their contribution to the difference of topology between the level sets of $J_K$, 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 $(\mathcal {P})$ on half spheres of dimension $n \ge 5$.

1 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 $K \in C^{\infty }(\mathbb {S}^n)$ defined on the standard $n-$dimensional sphere $(\mathbb {S}^n, g)$ be realized as the scalar curvature of a metric $\overline{g}$ conformally equivalent to g ?

On $\mathbb {S}^2$, setting $\overline{g}=e^{2u} g$ the Nirenberg problem is equivalent to solving the following nonlinear elliptic equation

$$\begin{aligned} -\Delta _{g} u \, + \, 1 \, = \, Ke^{2u}, \, \text{ in } \, \mathbb {S}^2, \end{aligned}$$

where $\Delta _{g}$ denotes the Laplace Beltrami operator.

For spheres of dimensions $n \ge 3$ and writing the conformal metric as $\overline{g}:= u^{{4}/{(n-2)}} g$, the Nirenberg problem amounts to solve the following nonlinear elliptic equation involving the Sobolev critical exponent:

$$\begin{aligned} (\mathcal {NP}) \qquad -\Delta _{g} u \, + \frac{n(n-2)}{4} u \, = \, K u^{ \frac{n+2}{n-2}}; \quad u > 0, \, \, \text{ in } \mathbb {S}^n. \end{aligned}$$

(1)

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 (ND) (that is $\Delta K \ne 0$ at critical points of K), if the dimension $n \ge 5$. 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 $n \ge 5$, 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 $(\mathcal {NP})$:

$$\begin{aligned} (\mathcal {NP}_{\varepsilon }) \quad -\Delta _{g} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2} - \varepsilon }, \quad \, u > 0 \, \text{ in } \, \mathbb {S}^n, \end{aligned}$$

(2)

where $\varepsilon > 0$ is a small parameter. In this way one recovers the compactness and one then studies the behavior of blowing up solution $u_{\varepsilon }$ of $(\mathcal {NP}_{\varepsilon })$ as the parameter $\varepsilon$ goes to zero. Actually it can be proved that finite energy blowing up solutions of $(\mathcal {NP}_{\varepsilon })$ 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 (ND), on spheres of dimension $n \ge 5$ every $m-$tuple $(q_1, \ldots , q_m)$ of distinct critical points of K, satisfying $\Delta K(q_i) < 0$ for each $i =1, \ldots ,m$ can be realized as a concentration set of blowing up solutions of $(\mathcal {NP}_\varepsilon )$. See [34].

Regarding the high dimensional case $n \ge 5$, Malchiodi and Mayer [35] obtained recently an interesting existence criterium under some pinching condition. Their result reads as follows:

Theorem A [35] Let $n \ge 5$ and $K \in C^{\infty }(\mathbb {S}^n)$ be a positive Morse function satisfying the following conditions

(i)

$$\begin{aligned} \forall q \in \mathbb {S}^n, \quad \nabla K(q) = 0 \Rightarrow \Delta K(q) \ne 0, \end{aligned}$$

(ii)

$$\begin{aligned} {K_{max}}/{K_{min} } \, \le ({3}/{2})^{ {1}/{(n-2)}}, \, \end{aligned}$$

where $K_{max} := \max _{\mathbb {S}^n} K$ and $K_{min} := \min _{\mathbb {S}^n} K$

(iii)

$$\begin{aligned} \# \{ q \in \mathbb {S}^n; \nabla K(q) = 0; \Delta K(q) < 0 \} \ge 2, \end{aligned}$$

where $\# A$ denotes the cardinal of the set A.

Then Nirenberg Problem $(\mathcal {NP})$ has at least one solution.

In this paper we consider a version of the Nirenberg problem on standard half spheres $(\mathbb {S}^n_+,g)$. Namely we prescribe simultaneously the scalar curvature to be a positive function $0 < K \in C^3(\mathbb {S}^n_+)$ and the boundary mean curvature to be zero. This amounts to solve the following boundary value problem

$$\begin{aligned} (\mathcal {P}) \quad {\left\{ \begin{array}{ll} -\Delta _{g} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{{(n+2)}/{(n-2)}},\, u > 0 &{} \text{ in } \mathbb {S}^n_+, \\ \frac{\partial u}{\partial \nu }\, =\, 0 &{} \text{ on } \partial \mathbb {S}^n_+, \end{array}\right. } \end{aligned}$$

(3)

where $K \in C^3(\mathbb {S}^n_+)$ 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

$$\begin{aligned} (\mathcal {P}_{\varepsilon }) \quad {\left\{ \begin{array}{ll} -\Delta _{g} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2} - \varepsilon }, \, u > 0 &{} \text{ in } \mathbb {S}^n_+, \\ \frac{\partial u}{\partial \nu }\, =\, 0 &{} \text{ on } \partial \mathbb {S}^n_+. \end{array}\right. } \end{aligned}$$

(4)

Just as above, there are two alternatives for the behavior of a sequence of solutions $u_{\varepsilon }$ of $(\mathcal {P}_{\varepsilon })$. Either the $||u_{\varepsilon }||_{L^{\infty }}$ remains uniformly bounded or it blows up and if it does $u^{{2n}/{(n-2)}}_{\varepsilon } \mathcal {L}^n$ (where $\mathcal {L}^n$ 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 $\Delta K \le 0$ and the boundary points are critical points of $K_1$ the restriction of K on the boundary and satisfying that ${\partial K}/{\partial \nu } \ge 0$. See [14, 17, 25]. Furthermore a refined blow up analysis, under the non degeneracy assumption that $\Delta K \ne 0$ at interior critical points of K and that ${\partial K}/{\partial \nu } \ne 0$ 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 $(\mathcal {P}_\varepsilon )$ see [1, 2].

In this paper we study Problem $(\mathcal {P})$ 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 $\omega$-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 $K_1:= K_{\lfloor \partial \mathbb {S}^n}$, we use the following assumption:

$\mathbf (H1)$: We assume that K is a $C^3(\overline{\mathbb {S}^n_+} )$ positive function, which has only non-degenerate critical points with $\Delta K \ne 0$. (We point out that some of these points can be on the boundary.)

$\mathbf (H2)$: We assume that the restriction of K on the boundary $K_1:= K_{| \partial \mathbb {S}_+^n}$ has only non-degenerate critical points z’s. Furthermore we assume that if z is not a local maximum point of $K_1$, we have that $\partial K/\partial \nu (z) \le 0$.

$\mathbf (H3)$: If $z \in \partial \mathbb {S}^n_+$ is a critical point of $K_1$ satisfying that $\partial K/\partial \nu (z) = 0$, hence z is actually a critical point of K on $\partial \mathbb {S}^n_+$, we assume that $\Delta K(z) \ne 0$ and one of the following conditions is satisfied:

• (i) either $\partial K/\partial \nu (a) \Delta K(z) \le 0$ for each $a\in \partial \mathbb {S}^n_+$ in a small neighborhood of z,

• (ii) or $\lim _{a\in \partial \mathbb {S}^n_+; a\rightarrow z} \frac{ \partial K/\partial \nu (a) }{d( a , z ) } =0$.

Next we introduce the following subsets of critical points of K and $K_1$

$$\begin{aligned}&\mathcal {K}^{-}_{in} := \{ y\in \mathbb {S}^n_+: \nabla K (y)=0 \text{ and } \Delta K(y)<0\}, \\&\quad \mathcal {K}^+_b := \{ z\in \partial \mathbb {S}^n_+: \nabla K_1 (z)=0 \text{ and } \partial K/ \partial \nu (z ) > 0\} , \\&\quad \mathcal {K}^{0,-}_b := \{ z\in \partial \mathbb {S}^n_+: \nabla K_1 (z)=0; \, \partial K/ \partial \nu (z ) = 0 \text{ and } \Delta K(z)<0\} . \end{aligned}$$

Furthermore we define

$$\begin{aligned} \mathcal {K}^{\infty } := \, \mathcal {K}^{-}_{in} \cup \mathcal {K}^+_b \cup \mathcal {K}^{0,-}_b . \end{aligned}$$

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 $n \ge 5$ and $0 < K \in C^3(\overline{\mathbb {S}^n_+})$ satisfying the assumptions (H1), (H2) and (H3).

If the following conditions hold

(i)

$$\begin{aligned} K_{\max } / K_{\min } < (5/4)^{1/(n-2)},\end{aligned}$$

where $K_{max} := \max _{\mathbb {S}^n_+} K$ and $K_{min} := \min _{\mathbb {S}^n_+} K$.

(ii)

$$\begin{aligned} \# \mathcal {K}^{\infty } \ge 2, \end{aligned}$$

where $\# A$ denotes the cardinal of the set A. Then Problem $(\mathcal {P})$ 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 (H2), (H3) 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 $n \ge 5$ and $0 < K \in C^3(\overline{\mathbb {S}^n_+})$. Assume that the critical points of $K_1:= K_{|\partial \mathbb {S}^n_+}$ are non degenerate and that K satisfies the assumption (H3). If the following conditions hold (a)

$$\begin{aligned} K_{\max } / K_{\min } < 2^{1/(n-2)} , \end{aligned}$$

(b)

$$\begin{aligned} A_1:=\sum _{z \in \mathcal {K}^+_b\cup \mathcal {K}^{0,-}_b} (-1)^{n-1 - morse(K_1,z)} \ne 1. \end{aligned}$$

Then Problem $(\mathcal {P})$ 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 $k \in \mathbb {N}_0$.

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 $n \ge 5$ and $0 < K \in C^3(\overline{\mathbb {S}^n_+})$ satisfying the assumptions (H1), (H2) and (H3).

If the following conditions hold

(i)

$$\begin{aligned} K_{\max } / K_{\min } < (3/2)^{1/(n-2)} \text{ and } A_1 = 1, \end{aligned}$$

where $A_1$ is defined in Theorem 1.3,

(ii)

$$\begin{aligned} B_1:= \sum _{y \in \mathcal {K}^-_{in}} (-1)^{n - morse(K,y)} \ne -k, \end{aligned}$$

where $\# (\mathcal {K}^{+}_b \cup \mathcal {K}^{0,-}_b) = 2k +1, \, k \in \mathbb {N}_0$. Then Problem $(\mathcal {P})$ 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 (H2) and (H3), 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.

2 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 $H^1(\mathbb {S}^n_+)$ be the Sobolev space endowed with the norm

$$\begin{aligned} ||u||^2 := \int _{\mathbb {S}^n_+} |\nabla u|^2 \, + \, \frac{n(n-2)}{4} \int _{\mathbb {S}^n_+} u^2, \end{aligned}$$

and let $\Sigma$ denote its unit sphere.

Problem $(\mathcal {P})$ has a variational structure. Namely its solutions are in one to one correspondence with the critical points of the functional

$$\begin{aligned} J_K(u):= \frac{||u||^2}{( \int _{\mathbb {S}^n_+} K |u|^{{2n}/{(n-2)}} )^{ {(n-2)}/{n}}} \quad \text{ defined } \text{ on } \Sigma ^+:= \{ u \in \Sigma ; \, u \ge 0 \}.\end{aligned}$$

The functional $J_K$ fails to satisfy the Palais Smale condition. To describe non converging Palais-Smale sequences we introduce the following notation.

For $a \in \overline{ \mathbb {S}^n_+}$ and $\lambda > 0$ we define the standard bubble to be

$$\begin{aligned} \delta _{a,\lambda }(x) := c_0 \frac{\lambda ^{n-2/2}}{ ( \lambda ^2 + 1 + (1-\lambda ^2) \cos d(a,x))^{n-2/2}}, \end{aligned}$$

where d is the geodesic distance on $\mathbb {S}^n_+$ and $c_0$ is a constant chosen such that

$$\begin{aligned} - \Delta \delta _{a,\lambda } \, + \, \frac{n(n-2)}{4} \delta _{a,\lambda } \, = \, \delta _{a,\lambda }^{{(n+2)}/{(n-2)}} \quad \text{ in } \mathbb {S}^n_+. \end{aligned}$$

For $a \in \overline{ \mathbb {S}^n_+},$ we define projected bubble $\varphi _{a,\lambda }$ to be the unique solution of

$$\begin{aligned} - \Delta \varphi _{a,\lambda } \, + \, \frac{n(n-2)}{4} \varphi _{a,\lambda } \, = \, \delta _{a,\lambda }^{{(n+2)}/{(n-2)}} \quad \text{ in } \, \mathbb {S}^n_+; \quad \frac{\partial \varphi _{a,\lambda }}{\partial \nu } \, = 0 \text{ on } \partial \mathbb {S}^n_+. \end{aligned}$$

We point out that $\varphi _{a,\lambda } = \delta _{a,\lambda }$ if $a \in \partial \mathbb {S}^n_+$.

Next for $m \in \mathbb {N}$ and $p,q \in \mathbb {N}_0$ such that $q + 2p = m$ we define the neighborhood of potential critical points at Infinity $V(m,q,p,\varepsilon )$ as follows:

$$\begin{aligned} {V}(m,q,p, \varepsilon ) :=&\Big \{ u \in \Sigma : \, \exists \, \lambda _1, \ldots , \lambda _{p+q}> {\varepsilon ^{-1}}; \, \exists \, a_1, \ldots ,a_{q+p} \in \overline{ \mathbb {S}^n_+}, \, \text{ with } \\&\quad \lambda _i d(a_i, \partial \mathbb {S}^n_+)< \varepsilon , \, \forall \, i \le q, \text{ and } \lambda _i d(a_i,\partial \mathbb {S}^n_+)> \varepsilon ^{-1} \, \, \forall \, i > q,\\&\quad \varepsilon _{ij}< \varepsilon \text{ such } \text{ that } \Vert u - \frac{\sum _{i = 1}^{p+q} K(a_i)^{(2-n)/4}\varphi _{a_i,\lambda _i} }{ \Vert \sum _{i = 1}^{p+q} K(a_i)^{(2-n)/4}\varphi _{a_i,\lambda _i}\Vert } \Vert < \varepsilon \Big \}, \end{aligned}$$

where

$$\begin{aligned} \varepsilon _{ij}:= \Big ( \frac{\lambda _i}{\lambda _j} + \frac{\lambda _j}{\lambda _i} + 2 \lambda _i \lambda _j (1-\cos (d (a_i,a_j)))\Big )^{2-n/2}. \end{aligned}$$

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 $u_k \in \Sigma ^+$ be a sequence such that $\nabla J_K(u_k) \rightarrow 0$ and $J_K(u_k)$ is bounded. If Problem $(\mathcal {P})$ does not have a solution, then there exist $m\in \mathbb {N}$ and $p,q \in \mathbb {N}$ with $q + 2p = m$, a sequence of positive real numbers $\varepsilon _k \downarrow 0$ as well as subsequence of $u_k$, still denoted $u_k$ such that $u_k \in V(m,q,p,\varepsilon _k)$.

Following Bahri and Coron, we consider for $u \in V(m,q,p,\varepsilon )$ the following minimization problem

$$\begin{aligned} Min \left\{ \Vert u \, - \, \sum _{i=1}^{p+q} \alpha _i \varphi _{a_i,\lambda _i} \Vert ; \alpha _i> 0, \lambda _i > 0, a_i \in \partial \mathbb {S}^n_+, \forall i =1, \dots , q; \, a_i \in \mathbb {S}^n_+, \forall q+1 \le i \le q+p \right\} . \end{aligned}$$

(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 $m \in \mathbb {N}$ there exists $\varepsilon _m > 0$ such that if $\varepsilon < \varepsilon _m$ and $u \in V(m, q, p, \varepsilon )$ the minimization problem (5) has, up to permutation, a unique solution.

Hence it follows from Proposition 2.2 that every $u \in V(m, q,p, \varepsilon )$ can be written in a unique way as

$$\begin{aligned} u \, = \, \sum _{i=1}^{q} \alpha _i \delta _{a_i,\lambda _i} \, + \, \sum _{i= q+1}^{p+q} \alpha _i \varphi _{a_i,\lambda _i} \, + \, v, \end{aligned}$$

(6)

where

$$\begin{aligned} a_i \in \partial \mathbb {S}^n_+, \, i =1, \ldots ,q \text{ and } a_i \in \mathbb {S}^n_+, \, i=q+1, \ldots , p+q, \end{aligned}$$

and $v \in H^1(\mathbb {S}^n_+)$ satisfying

$$\begin{aligned} (V_0) \quad \Vert v\Vert< \varepsilon , \quad <v, \psi > = 0, \text{ for } \psi \in \bigcup _{ 1 \le i\le q; \, q + 1 \le j \le q+p } \left\{ \delta _i, \frac{\partial \delta _i}{\partial \lambda _i}, \frac{\partial \delta _i}{\partial a_i}, \varphi _j, \frac{\partial \varphi _j}{\partial \lambda _j}, \frac{\partial \varphi _j}{\partial a_j} \right\} , \end{aligned}$$

(7)

where $\delta _i := \delta _{a_i,\lambda _i}$ and $\varphi _i:= \varphi _{a_i,\lambda _i}$. In addition, the variables $\alpha _i$’s satisfy

$$\begin{aligned} | 1 - J(u)^{n/(n-2)} \alpha _i^{4/(n-2)} K(a_i) | = o_\varepsilon (1) \quad \text{ for } \text{ each } i.\end{aligned}$$

(8)

In the next lemma we deal with the v-part of $u \in V(m,q,p,\varepsilon )$ in order to prove, that its effect is negligible with the concentration phenomenon. Namely we prove:

Lemma 2.3

Let $n \ge 5$. For $\varepsilon > 0$ small, there exists a $C^1$-map which, to each $(\alpha := (\alpha _1,\ldots ,\alpha _{p+q}) , a:= (a_1,\ldots ,a_{p+q}), \lambda := (\lambda _1, \ldots ,\lambda _{p+q}) )$, such that $u \, =\, \sum _{i=1}^{p+q} \alpha _i \varphi _i \in V(m,q,p,\varepsilon )$, associates $\overline{v}=\overline{v}_{(\alpha ,a,\lambda )}$ satisfying

$$\begin{aligned} J_K\left( \sum _{i=1}^{p+q} \alpha _i \varphi _{a_i,\lambda _i} +\overline{v}\right) = \min \left\{ J_K\left( \sum _{i=1}^{p+q} \alpha _i \varphi _{a_i,\lambda _i} +v\right) , \, v \text{ satisfies } (V_0)\right\} . \end{aligned}$$

Moreover, there exists $c>0$ such that the following holds

$$\begin{aligned} \Vert \overline{v} \Vert \le c \sum _{i=1}^{q+p} \frac{| \nabla K(a_i) | }{\lambda _i} + \frac{1}{\lambda _i^2} + {\left\{ \begin{array}{ll} &{} \sum \varepsilon _{ij }^{\frac{n+2}{2(n-2)}}(\ln \varepsilon _{ij}^{-1})^{\frac{n+2}{2n}} + \sum _{i> q }\frac{\ln (\lambda _i d_i)}{(\lambda _i d_i)^{(n+2)/2}} \text{ if } n \ge 6,\\ &{} \sum \varepsilon _{ij }(\ln \varepsilon _{ij}^{-1})^{{3}/{5}} + \sum _{i > q} \frac{1}{(\lambda _i d_i)^{3}} \text{ if } n = 5. \end{array}\right. } \end{aligned}$$

Proof

The proof follows as in Proposition 3.1 in [12] (see also [9]). Indeed, easy computations imply that

$$\begin{aligned}&J_K(u+v) = J_K(u) - f(v) +(1/2) Q(v) + o( \Vert v \Vert ^2 ) \qquad \text{ where } \\&f(v):= \int _{\mathbb {S}_+^n} K u^{\frac{n+2}{n-2}} v \quad \text{ and } \quad Q(v):= \Vert v \Vert ^2 - \frac{n+2}{n-2} \sum _{i=1}^N \int _{\mathbb {S}_+^n} \delta _i ^{4/(n-2)} v^2.\end{aligned}$$

Note that Q is a positive definite quadratic form (see [5]) and we have that

$$\begin{aligned} f(v) = \sum \alpha _i^{\frac{n+2}{n-2}} \int _{\mathbb {S}_+^n} K \varphi _i ^{\frac{n+2}{n-2}} v + O \Big ( \sum _ {i\ne j } \int _{\mathbb {S}_+^n} \sup (\varphi _j, \varphi _i) ^{\frac{4}{n-2}} \inf (\varphi _j, \varphi _i) | v | \Big ).\end{aligned}$$

(9)

Observe that, for $n\ge 6$, it follows that $4/(n-2) \le 1$. Hence, using Holder’s inequality, we get

$$\begin{aligned} \int _{\mathbb {S}_+^n} \sup (\varphi _j, \varphi _i) ^{\frac{4}{n-2}}&\inf (\varphi _j, \varphi _i) | v | \le \int _{\mathbb {S}_+^n} (\varphi _j \varphi _i) ^{\frac{n+2}{2(n-2)}} | v | \nonumber \\&\le c \Vert v \Vert \Big ( \int _{ \mathbb {S}_+^n} (\delta _j \delta _i) ^{\frac{n}{n-2}} \Big )^{\frac{n+2}{2n}} \le c \Vert v \Vert \varepsilon _{ij }^{\frac{n+2}{2(n-2)}}(\ln \varepsilon _{ij}^{-1})^{\frac{n+2}{2n}} \quad \text{ if } n \ge 6, \end{aligned}$$

(10)

$$\begin{aligned} \int _{\mathbb {S}_+^5} \sup (\varphi _j, \varphi _i) ^{{4}/{3}}&\inf (\varphi _j, \varphi _i) | v | \le c \Vert v \Vert \varepsilon _{ij }(\ln \varepsilon _{ij}^{-1})^{{3}/{5}} \text{ if } n = 5. \end{aligned}$$

(11)

For the other term, for $i\le q$ (that is $a_i \in \partial \mathbb {S}_+^n$), using the fact that $\langle \delta _i, v \rangle =0$, we get

$$\begin{aligned} \int _{\mathbb {S}_+^n} K \delta _i ^{\frac{n+2}{n-2}} v= & {} O\Big ( | \nabla K(a_i) | \int _ {\mathbb {R}^n_+} | x-a_i| \delta _i ^{\frac{n+2}{n-2}} | v | + \int _ {\mathbb {R}^n_+} | x-a_i|^2 \delta _i ^{\frac{n+2}{n-2}} | v | \Big )\\= & {} O \Big (\Big ( \frac{| \nabla K(a_i) | }{\lambda _i} + \frac{1}{\lambda _i^2} \Big )\Vert v\Vert \Big ).\end{aligned}$$

For $i > q$, using Lemma 5.1, we get

$$\begin{aligned} \int _{\mathbb {S}_+^n} K \varphi _i ^{\frac{n+2}{n-2}} v&= \int _{\mathbb {S}_+^n} K \delta _i ^{\frac{n+2}{n-2}} v + O\Big (\int _{\mathbb {S}_+^n} \delta _i ^{\frac{4}{n-2}} | \varphi _i - \delta _i | | v | \Big ) \\&= O \Big (\Big ( \frac{| \nabla K(a_i) | }{\lambda _i} + \frac{1}{\lambda _i^2} \Big )\Vert v\Vert \Big ) + {\left\{ \begin{array}{ll} O \big ( \Vert v \Vert /(\lambda _i d_i ) ^{n-2} \big ) \text{ if } n = 5,\\ O \big ( \Vert v \Vert \ln (\lambda _i d_i) /(\lambda _i d_i ) ^{(n+2)/2} \big ) \text{ if } n \ge 6 \end{array}\right. } \end{aligned}$$

and the result follows. $\square$

3 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 $\bigcup _{m,q,p} V(m,q,p,\varepsilon /2)$ and satisfies the Palais-Smale condition there. Moreover in each $V(m,q,p,\varepsilon )$ it has the property to move the concentration points according to $\nabla K$ or $\nabla K_1$, the $\alpha _i$’s to their maximum values and the concentration $\lambda _i$’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 $\overline{v}$ minimizes $J_K$ in the v-space, it follows from the classical Morse Lemma that there exists a change of variable $v \rightarrow V$ such that

$$\begin{aligned} J_K\left( \sum _{i=1}^{p+q} \alpha _i \varphi _{a_i,\lambda _i} + v \right) = J_K\left( \sum _{i=1}^{p+q} \alpha _i \varphi _{a_i,\lambda _i} + \overline{v} \right) + \Vert V \Vert ^2.\end{aligned}$$

(12)

Hence, for the variable V, we will use $\dot{V} = -V$ to bring it to 0. Thus, we need to construct some vector fields by moving the variables $\alpha _i$, $a_i$ and $\lambda _i$.

3.1 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:

Proposition 3.1

Assume that K satisfies (H1) and (H3) and that the critical points of $K_1$ are non-degenerate. A pseudogradient W can be defined so that the following holds: There is a constant $c>0$ independent of $u=\alpha \varphi _{a,\lambda } \in V(2p+q, q,p,\varepsilon )$ (with $q=1$ or $p=1$) such that

$$\begin{aligned} \mathrm{(i)}\,\,&\langle -\nabla J_K(u),W\rangle \ge c {\left\{ \begin{array}{ll} &{} 1/\lambda ^2 + 1/(\lambda d)^{n-2} + | \nabla K(a) | /\lambda \text{ if } p=1; q = 0,\\ &{} 1/\mu + |1-J(u)^{\frac{n}{n-2}}\alpha ^{\frac{4}{n-2}} K(a)| \text{ if } p=0; q = 1, \end{array}\right. } \\ \mathrm{(ii)} \, \,&\langle -\nabla J_K(u+\overline{v}),W+\frac{\partial \overline{v}}{\partial (\alpha ,a,\lambda )}(W)\rangle \ge c {\left\{ \begin{array}{ll} &{} 1/\lambda ^2 + 1/(\lambda d)^{n-2} + | \nabla K(a) | /\lambda \text{ if } p=1; q = 0,\\ &{} 1/\mu + |1-J(u)^{\frac{n}{n-2}}\alpha ^{\frac{4}{n-2}} K(a)| \text{ if } p=0; q = 1, \end{array}\right. } \end{aligned}$$

where $d:= d(a,\partial \mathbb {S}^n_+)$ for $a \in \mathbb {S}^n_+$ and $\mu ^{-1}= |\nabla K(a) | /\lambda + 1/\lambda ^2$ for $a\in \partial \mathbb {S}^n_+$.

(iii) The vector field W is bounded with the property that along its flow lines, $\lambda$ increases only in the following region

• If $p=1$ then $\lambda$ increases if and only if the point a belongs to a small neighborhood of a critical point $y\in \mathbb {S}_+^n$ of K, such that $\Delta K(y) < 0$

• If $q=1$ then $\lambda$ increases if and only if the point a belongs to a small neighborhood of a critical point $z \in \partial \mathbb {S}_+^n$ of $K_1$ such that either $(\partial K/\partial \nu ) (z)>0$ or $(\partial K/\partial \nu ) (z)=0$ and $\Delta K(z) < 0$.

Proof

We start by giving the proof of Claim (i) for the case where $p=1$ and $q=0$ that is in $V(2,0,1,\varepsilon )$. First, we notice that, if a is close to a critical point y of K in $\mathbb {S}_+^n$, then $\Delta K(a) = \Delta K(y) (1+o(1))$ and therefore $\Delta K(a)$ has a constant sign.

Let M be a large constant and let $\psi _1$ be a $C^\infty$ cut off function defined by $\psi \in [0,1]$, $\psi _1(t) = 1$ if $t \ge 2$ and $\psi _1(t)=0$ if $t\le 1$. We define

$$\begin{aligned}&W := \psi _1\Big (\frac{ \lambda | \nabla K(a) | }{M} \Big ) \Big ( \frac{1}{\lambda } \frac{\partial \varphi _{a,\lambda }}{\partial a} \frac{ \nabla K(a)}{| \nabla K(a) |} - \lambda \frac{\partial \varphi _{a,\lambda }}{\partial \lambda } \Big ) \\&\quad + \Big (1 - \psi _1\Big (\frac{ \lambda | \nabla K(a) | }{M} \Big )\Big ) ( \text{ sign }(-\Delta K(a))) \lambda \frac{\partial \varphi _{a,\lambda }}{\partial \lambda } .\end{aligned}$$

We notice that, in the region where $| \nabla K(a) | \ge 2M/\lambda$, we have that $\psi _1( { \lambda | \nabla K(a) | }/{M} )=1$, therefore the Claim (i) follows from Proposition 5.7.

Next if $| \nabla K(a) | \le 2M/\lambda$ then a is very close to a critical point of K in $\overline{ \mathbb {S}^n_+}$. 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 $z\in \partial \mathbb {S}_+^n$. Since z is a non-degenerate critical point of K, we derive that $\lambda d(a,z)$ is bounded which contradicts the fact that $\lambda d(a,\partial \mathbb {S}_+^n)$ is very large. Hence our claim follows and a is close to an interior critical point y in $\mathbb {S}_+^n$.

Next using Proposition 5.7 we derive that

$$\begin{aligned} \langle -\nabla J_K(u),W\rangle \ge c \psi _1\Big (\frac{ \lambda | \nabla K(a) | }{M}\Big ) \Big ( \frac{ | \nabla K(a) | }{\lambda } + \frac{ 1 }{\lambda ^2}\Big ) + \Big (1 - \psi _1\Big (\frac{ \lambda | \nabla K(a) | }{M} \Big )\Big ) \frac{ c }{\lambda ^2}\end{aligned}$$

which implies Claim (i) in this region.

Hence Claim (i) is proved in the case where $p=1$ and $q =0$.

Concerning (ii) it follows from (i) using the estimate of $\overline{v}$ in Lemma 2.3. Finally we notice that $\lambda$ increases along the flow lines of the pseudogradient W only in the region where a is close to a critical point y with $\Delta K(y) < 0.$ 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 $a \in \partial \mathbb {S}_+^n$. In this situation we divide the set $V(1,1,0,\varepsilon )$ into 3 subsets and construct an appropriate vector field in each of these sets.

(1) Let $V_1^1:= \{ u \in V(1,1,0,\varepsilon ): | 1- J_K(u)^{\frac{n}{n-2}}\alpha ^{\frac{4}{n-2}}K(a) | \ge M /\mu \}$. In this region, we define

$$\begin{aligned} \underline{W}^1_1:= \text{ sign }( 1- J_K(u)^{\frac{n}{n-2}}\alpha ^{\frac{4}{n-2}}K(a) | )\, \, \delta _{a,\lambda }\end{aligned}$$

and using Proposition 5.6, Claim (i) follows easily (since M is chosen large).

(2) Let $V_1^2:= \{ u \in V(1,1,0,\varepsilon ): | 1- J_K(u)^{\frac{n}{n-2}}\alpha ^{\frac{4}{n-2}}K(a) | \le 2M /\mu \text{ and } | \nabla K_1(a) | \ge \eta \}$, where $\eta$ is a small fixed constant. In this region, we define

$$\begin{aligned} \underline{W}^2_1:= \frac{1}{\eta } W_a^{b} \quad \text{ where } W_a^{b}:= \frac{1}{\lambda } \frac{\partial \delta _{a,\lambda } }{\partial a} \frac{ \nabla K_1(a) }{| \nabla K_1(a)| } .\end{aligned}$$

Note that, in this region, the parameter $\mu$ is of the same order that $\lambda$. Hence, using Proposition 5.5, the proof of Claim (i) follows.

(3) Let $V_1^3:= \{ u \in V(1,1,0,\varepsilon ): | 1- J_K(u)^{\frac{n}{n-2}}\alpha ^{\frac{4}{n-2}}K(a) | \le 2M /\mu \text{ and } | \nabla K_1(a) | \le 2 \eta \}$. In this region, a is close to a critical point z of $K_1$. The pseudogradient will depend on z. We define

$$\begin{aligned}&\underline{W}^3_1:= \psi _1( {\lambda | \nabla K_1(a) | } / {M}) W_a^b + \gamma \, \, \lambda \frac{\partial \delta _{a,\lambda }}{\partial \lambda } \quad \text{ with } \gamma \in \{-1,1\} \text{ satisfying } \nonumber \\&\quad {\left\{ \begin{array}{ll} &{} \gamma = 1 \quad \text{ if } \partial K/\partial \nu (z)> 0 \text{ or } \partial K/\partial \nu (z) = 0 \text{ and } \Delta K(z)< 0, \\ &{} \gamma = - 1 \quad \text{ if } \partial K/\partial \nu (z) < 0 \text{ or } \partial K/\partial \nu (z) = 0 \text{ and } \Delta K(z) > 0.\end{array}\right. } \end{aligned}$$

(13)

Using Propositions 5.4 and 5.5, it holds

$$\begin{aligned} \langle -\nabla J_K (u), \underline{W}^3_1\rangle\ge & {} c \psi _1\Big (\frac{ \lambda | \nabla K_1(a) | }{M} \Big ) \Big ( \frac{ | \nabla K_1(a) | }{\lambda } + \frac{ 1 }{\lambda ^2}\Big )\nonumber \\&+ \gamma \Big ( \frac{c_3}{\lambda } \frac{\partial K}{\partial \nu }(a) - c \frac{\Delta K(a)}{\lambda ^2} + O(\frac{1}{\lambda ^3})\Big ). \end{aligned}$$

(14)

Observe that, if $\partial K/\partial \nu (z) \ne 0$, it follows that $\gamma \partial K/\partial \nu (a) \ge c > 0$ and therefore Claim (i) follows easily. In the other case, that is $\partial K/\partial \nu (z) = 0$, we need to make use of the assumption (H3). Indeed,

• if (i) of (H3) holds, it follows that $\gamma \partial K/\partial \nu (a) = | \partial K/\partial \nu (a) |$ and $- \gamma \Delta K(a)\ge c > 0$. Therefore, if $\lambda |\nabla K_1(a) | \ge 2 M$, in the lower bound of (14) will appear $| \nabla K_1(a) | /\lambda + | \partial K/\partial \nu (a) | / \lambda + 1 /\lambda ^2$ which is larger than $c /\mu$. Hence, Claim (i) follows in this case. However, if $\lambda |\nabla K_1(a) | \le 2 M$, it follows that $| \nabla K(a) | \le c M /\lambda$ (since we assumed that z is a non degenerate critical point). Therefore $1/\lambda ^2 \ge c (1/\lambda ^2 + |\nabla K(a)|/\lambda ) = c/\mu$. Thus Claim (i) follows in this case.

• Next we consider the case where (ii) of (H3) holds. Recall that z is a non degenerate critical point of $K_1$, thus it follows that there exists $r_1 > 0$ such that $| \nabla K_1(a) | \ge \underline{c} d(a,z)$ for each $a \in B(z,r_1)$. Let $\varrho _1 > 0$ (satisfying $\varrho _1 \max (M,1/\underline{c})$ is very small), using (ii) of (H3), there exists $r_2>0$ (with $r_2 \le r_1$) such that $| \partial K/ \partial \nu (a) | \le \varrho _1 d(a,z)$ for each $a\in B(z,r_2)$. Hence, in $B(z,r_2)$, $| \partial K/ \partial \nu (a) | = o(| \nabla K_1(a) | )$ (since $\varrho _1$ is chosen so that $\varrho _1 / \underline{c}$ is small) and therefore $| \nabla K_1(a) | = | \nabla K(a) | (1+o(1))$. Finally, as before, if $\lambda |\nabla K_1(a) | \ge 2 M$, in the lower bound of (14) will appear $| \nabla K_1(a) | /\lambda$. Furthermore, we have $- \gamma \Delta K(a)\ge c > 0$ and $| \partial K/ \partial \nu (a) | = o(| \nabla K_1(a) | )$ which imply the proof of Claim (i) in this case. In the other case, which is $\lambda |\nabla K_1(a) | \le 2 M$, it holds: $d(a,z) \le c M/\lambda$ which implies that $| \partial K/ \partial \nu (a) | \le \varrho _1 d(a,z) \le c \varrho _1 M/ \lambda ^2 = o(1/\lambda ^2)$ (by the chose of $\varrho _1$). Thus the proof of Claim (i) follows from (14).

Finally Claim (ii) follows from Claim (i) using the estimate of $\overline{v}$ in Lemma 2.3 and Claim (iii) follows immediately from the properties of the constructed vector field. $\square$

We remark that the assumption (H2) is not used in the construction of the pseudogradient in $V(1,1,0,\varepsilon )$.

3.2 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 $i =1, \ldots , m$ the scalar quantity $\mu _i$ as follows

$$\begin{aligned} \mu _i^{-1}= |\nabla K(a_i) | /\lambda _i + 1/\lambda _i^2 \text{ if } i\le q\, \, ; \quad \mu _i= \lambda _i^2 \text{ if } i\ge q+1. \end{aligned}$$

(15)

The behavior of such a quantity along the flow lines of the constructed pseudogradient plays crucial role in identifying critical points at infinity.

Proposition 3.2

Assume that K satisfies (H1), (H2) and (H3). A pseudogradient W can be defined so that the following holds: There is a constant $c>0$ independent of $u=\sum _{i=1}^q\alpha _i\delta _{a_i,\lambda _i}+\sum _{j=q+1}^{p +q} \alpha _j \varphi _{a_j,\lambda _j}\in V(m, q,p,\varepsilon )$ such that

$$\begin{aligned} \mathrm{(i)}\qquad&\langle -\nabla J_K(u),W\rangle \ge c \, \sum _{i=1}^{p+q} \frac{1}{\mu _i^{2-{1}/{(n-2)}}}+ c \sum _{i \le q} |1-J_K(u)^{\frac{n}{n-2}}\alpha _i^{\frac{4}{n-2}} K(a_i)|^{2-\frac{1}{n-2}} \\&\quad + c\, \sum _{k\ne r} \varepsilon _{kr}^{\frac{n-1}{n-2}} +c \sum _{i> q}\Big ( \frac{1}{(\lambda _id_i)^{n-1}}+\Big (\frac{|\nabla K(a_i)|}{\lambda _i}\Big )^{2-\frac{1}{n-2}} \Big )\\ \mathrm{(ii)} \qquad&\langle -\nabla J_K(u+\overline{v}),W+\frac{\partial \overline{v}}{\partial (\alpha _i,a_i,\lambda _i)}(W)\rangle \ge c\, \sum _{i = 1}^{p+q} \frac{1}{\mu _i^{2-{1}/{(n-2)}}}+ c\, \sum _{k\ne r} \varepsilon _{kr}^{\frac{n-1}{n-2}}\\&\quad +c \sum _{i\le q} |1-J_K(u)^{\frac{n}{n-2}}\alpha _i^{\frac{4}{n-2}} K(a_i)|^{2-\frac{1}{n-2}} +c \sum _{i> q}\Big ( \frac{1}{(\lambda _id_i)^{n-1}}+\Big (\frac{|\nabla K(a_i)|}{\lambda _i}\Big )^{2-\frac{1}{n-2}} \Big ) \end{aligned}$$

where $d_i:= d(a_i, \partial \mathbb {S}^n_+)$.

(iii) The vector field W is bounded with the property that along its flow lines the maximum of the $\mu _i$’s increases only if the $(q+p)-$tuple $(a_1,\ldots ,a_q, \ldots a_{q+p})$ is close to a collection of different critical points of K or $K_1$ $(z_1,\ldots ,z_q,y_{q+1}, \ldots y_{q+p})$ with the $y_i$’s are critical points of K in $\mathbb {S}^n_+$ satistying $\Delta K(y_{i}) < 0$ for each $i \ge q+1$ and the $z_i$’s are critical points of $K_1$ such that either $(\partial K/\partial \nu ) (z_{i_k})>0$ or $((\partial K/\partial \nu ) (z_{i})=0$ and $\Delta K(z_{i}) < 0)$.

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 $\alpha$-, a- and $\lambda$-component of the gradient in the neighborhood at infinity $V(m,q,p,\varepsilon )$. To perform such a construction we divide the set $V(m,q,p,\varepsilon )$ into four subsets. The first and the second ones correspond to the situation where at least one of the variables $\alpha _i$’s and $a_i$’s is not in its critical position and the $\mu _i$’s are of the same order. In the third one, the $\mu _i$’s are still of the same order but the variables $\alpha _i$’s and $a_i$’s are very close to their critical positions. Finally in the fourth one we address the case where the $\mu _i$’s are not of the same order.

To define these regions, we introduce the following notation. For $M_2$ a large constant we set:

$$\begin{aligned} \Gamma _{\alpha _k}:= & {} \frac{ | 1- J_K(u)^{\frac{n}{n-2}}\alpha _k^{\frac{4}{n-2}}K(a_k) | }{ M_2 (\sum _{r\ne k} \varepsilon _{kr} +1/\mu _k) }; \, \, \Gamma _{a_i}^b:= \frac{ | \nabla K_1 (a_i) | / \lambda _i }{ M_2 /\lambda _i^2 + (1/ M_2^2) \sum _{k\in I} \varepsilon _{ik}}\quad \text{ for } i \le q\nonumber \\ \Gamma _{a_i}:= & {} \frac{ |\nabla K(a_i)| /\lambda _i }{M_2 (\sum _{ k \ne i } \varepsilon _{ki} +{(\lambda _i d_i)^{2-n}}+ \frac{1}{\lambda _i^2})}; \, \, \Gamma _{H_i}:= H(a_i, a_i) / M_2 \lambda _i^{n-4} \quad \text{ for } i > q ,\nonumber \\ \Gamma _{\lambda _k}:= & {} \mu _k \sum _{j \ne k }\varepsilon _{jk} /M_2 \quad \text{ for } 1\le i \le q +p . \end{aligned}$$

(16)

To explain the relevance of the above quantities, we state the following Lemma

Lemma 3.3

(1) Let $a_i$ be an interior point satisfying $\Gamma _{\lambda _i} +\Gamma _{a_i} +\Gamma _{H_i} \le 8$. Then $a_i$ is close to a interior critical point y of K in $\mathbb {S}^n_+$.

(2) If $a_i, a_j$ are interior points satisfying that $\Gamma _{\lambda _k} +\Gamma _{a_k} +\Gamma _{H_k} \le 8$ for $k=i,j$ and if their corresponding concentration rates $\lambda _i$ and $\lambda _j$ are of the same order. Then $a_i$ and $a_j$ cannot be close to the same critical point.

Proof

Since i satisfies: $\Gamma _{H_i} + \Gamma _{a_i} + \Gamma _{\lambda _i} \le 8$, this implies that $| \nabla K(a_i) | \le C/\lambda _i$ 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 $z\in \partial \mathbb {S}^n_+$. Then it follows from (H1), that $\lambda _i d(a_i,z)$ is bounded, which is not allowed. Therefore, each concentration point $a_i$ is close to a critical point $y_{j_i} \in \mathbb {S}^n_+$ 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: $\lambda _k d(a_k,y)$ is bounded for $k=i,j$. Since $\lambda _i$ and $\lambda _j$ are assumed to be of the same order, it follows that $\lambda _k d(a_i , a_j )$ is bounded, which contradicts the smallness of $\varepsilon _{ij}$. $\square$

3.2.1 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 $V(m,q,p,\varepsilon )$

$$\begin{aligned} V_1(M_0):=&\{ u: \mu _{\max } \le 2M_0 \, \mu _{\min } \} \cap \{ u: \exists \, \, i> q: \Gamma _{H_i} + \Gamma _{a_i} + \Gamma _{\lambda _i} \ge 6 \} ,\\ V_2(M_0):=&\{ u: \mu _{\max } \le 2M_0\, \mu _{\min } \} \cap \{ u: \forall \, \, i> q : \Gamma _{H_i} + \Gamma _{a_i} + \Gamma _{\lambda _i} \le 8 \} \cap \Big ( \{ u: \exists \, \, i \le q : \Gamma _{\alpha _i} + \Gamma _{\lambda _i} \ge 4 \}\\&\cup \{ u: \exists \, \, i \le q: d(a_i ,\mathcal {K}^b) \ge \eta \}\Big ) \qquad \text{ where } \mathcal {K}^b:= \{ z\in \partial \mathbb {S}^n_+: \nabla K_1(z)=0\} ,\\ V_3(M_0):=&\{ u: \mu _{\max } \le 2M_0\, \mu _{\min } \} \cap \{ u: \forall \, \, i> q: \Gamma _{H_i} + \Gamma _{a_i} + \Gamma _{\lambda _i} \le 8 \} \cap \{ u: \forall \, \, i \le q: \Gamma _{\alpha _i} + \Gamma _{\lambda _i} \le 6 \}\\&\cap \{ u: \forall \, \, i \le q: d(a_i ,\mathcal {K}^b) \le 2\eta \} , \\ V_4(M_0):=&\{ u: \mu _{\max } > M_0\, \mu _{\min } \}, \end{aligned}$$

where $\mu _{\max }:= \max _j \mu _j$ and $\mu _{\min }:= \min _j \mu _j$.

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.

Remark 3.4

(1) In $V_k(M_0)$, for $k\le 3$, the variables $\mu _i$’s are of the same order. Thus, using Lemma 5.2, we derive that, for each $i\ne j \le q$, it holds

$$\begin{aligned} - \lambda _i \frac{\partial \varepsilon _{ij}}{\partial \lambda _i} \ge c \varepsilon _{ij} .\end{aligned}$$

(17)

Furthermore, for $i\ne j > q$, we deduce that $\lambda _i$ and $\lambda _j$ are of the same order and therefore (17) holds true. Now, for $i > q$ and $j \le q$, we have $\lambda _i d_i$ is very large which implies that $\lambda _i d(a_i ,a_j )$ 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 $y_{j_i} \in \mathbb {S}^n_+$ and two different points $a_i$ and $a_j$ cannot be near the same critical point y (see Lemma 3.3).

(3) In $V_3(M_0)$, for each $i \le q$, $a_i$ is close to a critical point $z_{j_i}$ of $K_1$ in $\partial \mathbb {S}^n_+$.

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 a constant $c>0$ independent of $u=\sum _{i=1}^{q}\alpha _i\delta _i + \sum _{i=q+1}^{p+q}\alpha _i\varphi _i \in V_1(M_0)$ such that

$$\begin{aligned}&\langle -\nabla J_K(u),W_1\rangle \ge \, \sum _{i=1}^{q+p} \frac{c}{\mu _i}+ c \sum _{i=1}^{q}|1-J_K(u)^{\frac{n}{n-2}}\alpha _i^{\frac{4}{n-2}} K(a_i)| \nonumber \\&\quad + c\, \sum _{k\ne r} \varepsilon _{kr} + \sum _{i=q+1}^{q+p} \frac{| \nabla K(a_i)|}{\lambda _i} . \end{aligned}$$

(18)

Furthermore, the $\lambda _i$’s are decreasing functions along the flow lines generated by this pseudogradient. In addition, the constant of $1/\mu _{\max }$ is independent of $M_0$ and $M_2$.

Proof

We start by defining the following vector fields:

$$\begin{aligned} W_{\Lambda _{in}}:= & {} - \sum _{i> q} (\psi _1(\Gamma _{\lambda _i}) +\psi _1(\Gamma _{H_i})) \lambda _i \frac{\partial \varphi _i}{\partial \lambda _i} \quad \text{ and } \quad W_a^{in}:= \sum _{i > q } \psi _1(\Gamma _{a_i}) \frac{1}{\lambda _i} \frac{\partial \varphi _i}{\partial a_i} \frac{\nabla K(a_i)}{| \nabla K(a_i)|} \end{aligned}$$

(19)

$$\begin{aligned} W_{\Lambda _{b}}:= & {} - \sum _{i \le q} \psi _1(\Gamma _{\lambda _i}) \lambda _i \frac{\partial \delta _i}{\partial \lambda _i} \quad \text{ and } \quad {W}_\alpha := - \sum _{k \le q } \psi _1 (\Gamma _{\alpha _k}) \text{ sign }(1- J_K(u)^{\frac{n}{n-2}}\alpha _k^{\frac{4}{n-2}}K(a_k)) \delta _k\nonumber \\ \end{aligned}$$

(20)

where $\psi _1$ is a $C^\infty$ function defined by $\psi _1 \in [0,1]$, $\psi _1(t) = 1$ if $t \ge 2$ and $\psi _1(t)=0$ if $t\le 1$. Observe that, using Propositions 5.4, 5.7, the estimate (17) and the definition of $\psi _1$, we derive that

$$\begin{aligned}&\langle - \nabla J_K(u), W_{\Lambda _{in}} \rangle \ge c \sum _{ i > q }(\psi _1(\Gamma _{\lambda _i})+\psi _1(\Gamma _{H_i})) \Big ( \sum _{j\ne i} \varepsilon _{ij} + \frac{H(a_i,a_i)}{\lambda _i^{n-2}} + \frac{M_2}{2}\frac{1}{\lambda _i^2} + O(R_1) \Big ) := \overline{ \Gamma }_{\Lambda _{in}} \, \end{aligned}$$

(21)

$$\begin{aligned}&\langle -\nabla J_K(u), W_{\Lambda _{b}} \rangle \ge c \sum _{i \le q}\psi _1(\Gamma _{\lambda _i}) \Big ( \sum _{ k\ne i} \varepsilon _{ik} +\frac{M_2}{2} \frac{1}{\mu _{i}} +O\Big ( \sum _{ k > q } \varepsilon _{ki} + R_1^b \Big ) \Big ) := \overline{\Gamma }_{\Lambda _{b}} . \end{aligned}$$

(22)

Moreover using Proposition 5.6, we derive that

$$\begin{aligned} \langle -\nabla J_K(u), {W}_\alpha \rangle \ge c \sum _{k \le q} \psi _1(\Gamma _{\alpha _k}) \left( | 1- J_K(u)^{\frac{n}{n-2}}\alpha _k^{\frac{4}{n-2}}K(a_k) | + \frac{M_2}{2} \left( \sum _{r\ne k} \varepsilon _{kr} +\frac{1}{\mu _k}\right) \right) := \overline{\Gamma }_\alpha .\end{aligned}$$

(23)

Such an estimate suggests to move the variable $\alpha _i$’s if $| 1- J_K(u)^{{n}/{n-2}}\alpha _i^{{4}/{n-2}}K(a_i) |$ is very large with respect to $\sum _{r\ne k} \varepsilon _{kr} +1/\mu _k$. Furthermore making use of Propositions 5.6 and 5.7, we derive that

$$\begin{aligned} \langle - \nabla J_K(u), W_a^{in} \rangle \ge c \sum _{i > q} \psi _1(\Gamma _{a_i}) \Big ( \frac{| \nabla K(a_i)|}{\lambda _i} + \frac{M_2}{2}\Big (\sum _{k\ne i} \varepsilon _{ki} +\frac{1}{(\lambda _i d_i)^{n-2}}+ \frac{1}{\lambda _i^2}\Big )\Big ) := \overline{\Gamma }_a^{in}.\end{aligned}$$

Nest we define

$$\begin{aligned} W_1:= W_{\Lambda _{in}} + W_{a}^{in} + W_{\alpha } + (1/M_2) W_{\Lambda _b}.\end{aligned}$$

Using the previous estimates, we obtain

$$\begin{aligned} \langle - \nabla J_K(u), W_{1} \rangle&\ge \overline{ \Gamma }_{\Lambda _{in}} + \overline{ \Gamma }_{a}^{in} + \overline{ \Gamma }_{\alpha } + (1/M_2) \overline{ \Gamma }_{\Lambda _b} \nonumber \\&\ge c \sum _{ i> q }(\psi _1(\Gamma _{\lambda _i})+\psi _1(\Gamma _{H_i}) + \psi _1(\Gamma _{a_i})) \Big ( \sum _{j\ne i} \varepsilon _{ij} + \frac{H(a_i,a_i)}{\lambda _i^{n-2}} + \frac{M_2}{2}\frac{1}{\lambda _i^2} \Big )\nonumber \\&\quad + c \sum _{i > q} \psi _1(\Gamma _{a_i}) \frac{| \nabla K(a_i)|}{\lambda _i} + \overline{ \Gamma }_{\alpha } + (1/M_2) \overline{ \Gamma }_{\Lambda _b} + O(R_1) . \end{aligned}$$

(24)

Regarding the above estimate, we point that we need to take care of the interaction term $O(\varepsilon _{ki})$ contained in the expression $\overline{ \Gamma }_{\Lambda _b}$. To that aim, we observe that, if $\Gamma _{H_k} + \Gamma _{a_k} + \Gamma _{\lambda _k} \ge 6$, then the $\varepsilon _{ki}$ appears in the lower bound in (24) and therefore we are able to remove the $(1/M_2) \varepsilon _{ki}$ by taking $M_2$ large. But, if $\Gamma _{H_k} + \Gamma _{a_k} + \Gamma _{\lambda _k} \le 6$, 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 $\varepsilon _{ki} = O( 1/\lambda _k^{n-2} + 1/\lambda _i ^{n-2})$ 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 $\psi _1(\Gamma _{\lambda _i})+\psi _1(\Gamma _{H_i}) + \psi _1(\Gamma _{a_i}) \ge 1$. This implies that $1/\lambda _i^2= 1/\mu _i$ appears in the lower bound of (24). Since all the $\mu _j$’s are of the same order, we are able to make appear all the $1/\mu _j$’s in this lower bound and Lemma 3.5 follows. $\square$

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 is a constant $c > 0$ independent of $u=\sum _{i=1}^{q}\alpha _i\delta _i + \sum _{i=q+1}^{p+q}\alpha _i\varphi _i \in V_2(M_0)$ such that the statement of Lemma 3.5 holds true with $W_2$ instead of $W_1$.

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 $\mathbb {S}^n_+$ and that two interior concentration points $a_i$ and $a_k$ cannot be close to the same critical point which implies that $d(a_i, a_k ) \ge c > 0$ and $\varepsilon _{ik} = O( 1/(\lambda _k\lambda _i)^{(n-2)/2})$.

Recalling that $\mathcal {K}^b:= \{ z \in \partial \mathbb {S}^n_+: \nabla K_1(z)=0\}$ we define the following pseudogradient:

$$\begin{aligned} W_2:= W_{\alpha } + W_{\Lambda _b} + \sum _{i \in D_1} \frac{1}{\lambda _i} \frac{\partial \delta _i}{\partial a_i} \frac{\nabla K_1(a_i)}{| \nabla K_1(a_i)|} \qquad \text{ where } D_1:= \{ i \le q: d(a_i, \mathcal {K}^b) \ge \eta \}.\end{aligned}$$

Using (22), (23) and Proposition 5.5, we get

$$\begin{aligned} \langle - \nabla J_K(u), W_{2} \rangle \ge \overline{ \Gamma }_{\alpha } + \overline{ \Gamma }_{\Lambda _b} + \sum _{i \in D_1} \frac{c}{\lambda _i} + O\Big ( \sum _{k \le q }\big ( \frac{1}{\lambda _i} | \frac{\partial \varepsilon _{ki}}{\partial a_i} | + \lambda _k d(a_k , a_i ) \varepsilon _{ki} ^{\frac{n+1}{n-2}} \big ) + R_1^b + \sum _{k > q } \varepsilon _{ki} \Big ). \end{aligned}$$

(25)

First, taking $i \in D_1$, for $k \le q$, two cases may occur: (i) either $d(a_i, a_k ) \le \eta /2$, and in this case we get that $|\nabla K(a_k) | \ge c$ and therefore $\mu _k$ and $\lambda _k$ are of the same order. Thus $\lambda _i$ and $\lambda _k$ are of the same order. (ii) or $d(a_i , a_k ) \ge \eta /2$. In the two cases, we deduce that

$$\begin{aligned}&\varepsilon _{ki}= \frac{1+o(1)}{(\lambda _i \lambda _k d(a_i , a_k )^2)^{\frac{n-2}{2}}}; \quad \lambda _k d(a_k , a_i ) \varepsilon _{ki} ^{\frac{n+1}{n-2}} \le \frac{c\, \varepsilon _{ki}^\frac{n-1}{n-2} }{\lambda _i |d(a_i , a_k )} \, \, \\&\quad \text{ and } \, \, \frac{1}{\lambda _i} | \frac{\partial \varepsilon _{ki}}{\partial a_i} | \le \frac{c\, \varepsilon _{ki}}{\lambda _i d( a_i , a_k )} = o( \varepsilon _{ki}) .\end{aligned}$$

Secondly, for $i \in D_1$, we have $| \nabla K_1(a_i) | \ge c(\eta )$ and therefore $\lambda _i$ and $\mu _i$ are of the same order. Since all the $\mu _j$’s are assumed to be of the same order, we are able to make appear all the $1/\mu _j$’s in the lower bound of (25). Finally, for $j\notin D_1$, (i) either $\Gamma _{\alpha _j} \ge 2$, in this case, the $|1-J_K(u)^{{n}/(n-2)}\alpha _j^{{4}/(n-2)} K(a_j)| + \sum \varepsilon _{kj}$ appears in $\overline{\Gamma }_{\alpha }$, (ii) or $\Gamma _{\alpha _j} \le 2$ and $\Gamma _{\lambda _j} \ge 2$, in this case $\sum _{kj} \varepsilon _{kj}$ appears in $\overline{\Gamma }_{\Lambda _b}$, (iii) or $\Gamma _{\alpha _j} + \Gamma _{\lambda _j} \le 4$, in this case we are able to make appear $|1-J_K(u)^{{n}/(n-2)}\alpha _j^{{4}/(n-2)} K(a_j)| + \sum \varepsilon _{kj}$ from $1/\mu _j$. Hence the lemma follows.   $\square$

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 $z \in \partial \mathbb {S}^n_+$ of $K_1$ (resp. $y\in \mathbb {S}^n_+$ of K), we denote by

$$\begin{aligned} B_z :=\{ i \le q: a_i \text{ is } \text{ close } \text{ to } z\} \quad ; \quad B_y :=\{ i > q: a_i \text{ is } \text{ close } \text{ to } y\}.\end{aligned}$$

We observe that it follows from Remark 3.4 that $\# B_y \le 1$ for each critical point y in $\mathbb {S}^n_+$. However, it is possible to have $\#B_z \ge 2$ for some critical points z’s in $\partial \mathbb {S}^n_+$.

Next we divide the set $V_3(M_0)$ into four subsets. The first three ones are defined as follows:

$$\begin{aligned} V_3^1:=&\{ u\in V_3(M_0): \exists \, z \text{ with } \partial K/\partial \nu (z) =0 \text{ and } \# B_z \ge 2\} , \\ V_3^2:=&\Big ( \{ u: \exists \, z \text{ with } \partial K/\partial \nu (z) < 0 \text{ and } B_z \ne \emptyset \} \bigcup \{ u: \exists \, y \text{ with } \Delta K> 0 \text{ and } B_y \ne \emptyset \}\\&\bigcup \{ u: \exists \, z \text{ with } \partial K/\partial \nu (z) = 0; \, \, \Delta K(z)> 0 \text{ and } \# B_z \ne 0\} \Big ) \bigcap (V_3(M_0) {\setminus } V_3^1), \\ V_3^3:=&\{ u\in V_3(M_0): \exists \, z \text{ with } \partial K/\partial \nu (z) > 0 \text{ and } \# B_z \ge 2\} \bigcap (V_3(M_0) {\setminus } (V_3^1 \cup V_3^2)) \end{aligned}$$

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:

$$ \begin{aligned} \mathcal {W}:=&\{ u \in V_3(M_0): \forall i \le q, a_i \text{ is } \text{ close } \text{ to } z_i \in \partial \mathbb {S}^n_+, \text{ with } \# B_{z_{i}} =1; ( \partial K/\partial \nu =0 \, \& \, \Delta K< 0) \nonumber \\&\quad \text{ or } \partial K/\partial \nu> 0 \} \bigcap \{ u \in V_3(M_0), \forall j > q, a_j \text{ is } \text{ close } \text{ to } y_j \in \mathbb {S}^n_+, \nonumber \\&\quad \text{ with } \# B_{y_{j}} =1 \text{ and } \Delta K(y_j) < 0 \}. \end{aligned}$$

(26)

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 $W_3^1$ such that the following holds: There is a constant $c > 0$ independent of $u=\sum _{i=1}^{q}\alpha _i\delta _i + \sum _{i=q+1}^{p+q}\alpha _i\varphi _i \in V_3^1$ such that the statement of Lemma 3.5 holds true with $W_3^1$ instead of $W_1$.

Proof

Let z be such that $\partial K/\partial \nu (z) = 0$ and $\# B_z \ge 2$. Firstly, we claim that:

$$\begin{aligned} \text{ There } \text{ exists } k \in B_z \text{ such } \text{ that: } \frac{ | \nabla K_1 (a_k) | }{\lambda _k } \ge \frac{ M_2 }{\lambda _k^2} + \frac{1}{ M_2^2} \sum _{j\ne k} \varepsilon _{jk}.\end{aligned}$$

(27)

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 $k \in B_z$,

$$\begin{aligned} c \frac{d( a_k, z )}{\lambda _k}\le & {} \frac{| \nabla K_1(a_k) |}{\lambda _k} \le \frac{ M_2 }{\lambda _k^2} + \frac{1}{ M_2^2} \sum _{j\ne k} \varepsilon _{jk} \\\le & {} \frac{ M_2 }{\lambda _k^2} + \frac{c}{ M_2} \frac{1}{\mu _k} \le \frac{c}{M_2} \frac{| \nabla K(a_k) |}{\lambda _k} + c \frac{M_2}{\lambda _k^2} \le c \frac{ d(a_k ,z ) }{M_2\lambda _k } + c \frac{M_2}{\lambda _k^2}\end{aligned}$$

which implies that $\lambda _k d(a_k , z )$ is bounded. In addition, from the definition of $\mu _k$, we get

$$\begin{aligned} \frac{1}{\lambda _k^2} \le \frac{1}{\mu _k} := \frac{| \nabla K(a_k) | }{\lambda _k} + \frac{1}{\lambda _k^2} \le c \frac{ \lambda _k d(a_k , z ) }{\lambda _k^2} + \frac{1}{\lambda _k^2} \le \frac{c}{\lambda _k^2}.\end{aligned}$$

Thus, $\mu _k$ and $\lambda _k^2$ are of the same order for each $k \in B_z$.

Next let i and j be two different indices in $B_z$. We deduce that $\lambda _j$ and $\lambda _i$ are of the same order and $\lambda _k d(a_i , a_j )$ is bounded for $k=i,j$. These give a contradiction with the fact that $\varepsilon _{ij}$ is small. Hence our claim follows.

Furthermore observe that, for k satisfying (27), it holds that $\lambda _k d(a_k, z ) \ge c M_2$.

Now, in this region, we define the following vector field:

$$\begin{aligned} W_3^1:= \sum _{i \in D_2} \frac{1}{\lambda _i} \frac{\partial \delta _i}{\partial a_i} \frac{\nabla K_1(a_i)}{| \nabla K_1(a_i)|} \qquad \text{ where } D_2:= \{ i \le q: (27) \text{ holds } \text{ with } k=i \}. \end{aligned}$$

Using Proposition 5.5, we get

$$\begin{aligned} \langle - \nabla J_K(u), W_{3}^1 \rangle \ge c \sum _{i \in D_2} \frac{| \nabla K_1(a_i) | }{\lambda _i} + O\Big ( \sum _{k \le q }\big ( \frac{1}{\lambda _i} | \frac{\partial \varepsilon _{ki}}{\partial a_i} | + \lambda _k d(a_k , a_i ) \varepsilon _{ki} ^{\frac{n+1}{n-2}} \big ) + R_1^b + \sum _{k > q } \varepsilon _{ki} \Big ). \end{aligned}$$

(28)

Recall that (see Remark 3.4), in $V_3(M_0)$, each concentration point $a_k$, for $k > q$ is close to a critical point of K in $\mathbb {S}^n_+$ which implies that $d(a_i,a_k ) \ge c > 0$ for each $i \le q$. Hence we get $\varepsilon _{ik} = O( 1/(\lambda _k\lambda _i)^{(n-2)/2})$.

Moreover for $i\in D_2$ and $k \le q$ with $k\ne i$, two cases may occur: (i) either $\lambda _k \le M_0 ^2M_2^2\lambda _i$, and in this case we get

$$\begin{aligned} \frac{1}{\lambda _i} | \frac{\partial \varepsilon _{ik}}{\partial a_i} | + \lambda _k d(a_k, a_i ) \varepsilon _{ki} ^{\frac{n+1}{n-2}} \le c\, \lambda _k d(a_i , a_k ) \varepsilon _{ik}^{\frac{n}{n-2}} \le c\, M_0 M_2\sqrt{\lambda _k \lambda _i } d( a_i, a_k ) \varepsilon _{ik}^{\frac{n}{n-2}} \le c\, M_0M_2 \varepsilon _{ik}^{\frac{n-1}{n-2}} ,\end{aligned}$$

or (ii) $\lambda _k \ge M_0 ^2 M_2^2\lambda _i$. In this case, since $\mu _k \le 2 M_0 \mu _i$ and z is a non-degenerate critical point of $K_1$, it follows that

$$\begin{aligned} c \frac{d( a_i , z )}{\lambda _i} \le \frac{| \nabla K_1 (a_i) |}{\lambda _i} \le \frac{| \nabla K(a_i) |}{\lambda _i} + \big (\frac{ 1 }{\lambda _i^2} - 2\frac{ M_0 }{\lambda _k^2}\big ) \le 2 M_0 \frac{| \nabla K(a_k) |}{\lambda _k} \le c M_0 \frac{d(a_k , z)}{\lambda _k}\end{aligned}$$

which implies that $d(a_i, z ) / d(a_k , z ) \le c M_0 \lambda _i/\lambda _k \le c/ (M_0 M_2^2)$. Thus we deduce that $d( a_i, a_k ) \ge c M_0 M_2^2 d( a_i, z )$. Therefore we obtain

$$\begin{aligned} \frac{1}{\lambda _i} | \frac{\partial \varepsilon _{ik}}{\partial a_i} | + \lambda _k d(a_k, a_i ) \varepsilon _{ki} ^{\frac{n+1}{n-2}} \le c\, \lambda _k d(a_i, a_k ) \varepsilon _{ik}^{\frac{n}{n-2}} \le \frac{c}{\lambda _i d(a_i , a_k )} \varepsilon _{ik} \le \frac{1}{M_0 M_2^2}\frac{c}{\lambda _i d(a_i, z )} \varepsilon _{ik} \le \frac{c }{ M_2^3 M_0 } \varepsilon _{ik}\end{aligned}$$

where we have used the fact that $\lambda _i d(a_i, z ) \ge c M_2$. Thus (28) becomes

$$\begin{aligned} \langle - \nabla J_K(u), W_{3}^1 \rangle \ge c \, \sum _{i \in D_2} \frac{ | \nabla K_1(a_i) | }{\lambda _i} + \frac{ M_2 }{\lambda _i^2} + \frac{1}{M_2^2} \sum _{j \ne i} \varepsilon _{ij} + O(R_1^b + \sum \frac{1}{\lambda _j ^{n-2}} ).\end{aligned}$$

(29)

Finally, we notice that $| \nabla K(a_i) | \le c d( a_i, z ) \le c | \nabla K_1(a_i) | \le c | \nabla K(a_i) |$. Thus, in (29), we can make appear $1/\mu _i$ for $i\in D_2$ and therefore all the $1/\mu _j$’s (since there are of the same order) and the proof follows as the proof of the previous lemmas. $\square$

Lemma 3.8

There exists a bounded pseudogradient $W_3^2$ such that the following holds: There is a constant $c > 0$ independent of $u=\sum _{i=1}^{q}\alpha _i\delta _i + \sum _{i=q+1}^{p+q}\alpha _i\varphi _i \in V_3^2$ such that the statement of Lemma 3.5 holds true with $W_3^2$ instead of $W_1$.

Proof

Let $D_1:= \cup _{y: \Delta K(y) > 0} B_y$, $D_2:= \cup _{z: \partial K/\partial \nu (z) < 0} B_z$ and $D_3:= \cup _{z: \partial K/\partial \nu (z) = 0;\, \Delta K(z) > 0} B_z$. We divide this region into two subsets:

1st subset: If $D_1 \cup D_2 \ne \emptyset$. In this case, we define

$$\begin{aligned} W_3^{21}:= - \sum _{i\in D_1 \cup D_2} \lambda _i \frac{\partial \varphi _i}{\partial \lambda _i} .\end{aligned}$$

By using the first assertion of Remark 3.4 and Propositions 5.4 and 5.7, it follows that

$$\begin{aligned} \langle - \nabla J_K(u), W_{3}^{21} \rangle \ge c \sum _{i \in D_1 \cup D_2} \Big ( \sum _{ j\ne i} \varepsilon _{ij} + \frac{1}{\mu _i} + O\big ( \sum \frac{1}{\lambda _j ^{n-2} }+ R_1^b + R_1\big ) \Big ).\end{aligned}$$

Hence, the proof follows.

2nd subset: $D_3 \ne \emptyset$. Note that, since we are outside of $V_3^1$, for $i \in B_z$ with $\partial K/\partial \nu (z) = 0$, it holds that $B_z =\{ i \}$, that is $d( a_i, a_j ) \ge c > 0$ for each $j \ne i$. We define

$$\begin{aligned} W_3^{22} := \sum _{i \in D_3} \psi _1(\lambda _i | \nabla K_1(a_i) | /M) \frac{1}{\lambda _i} \frac{\partial \delta _i}{\partial a_i} \frac{ \nabla K_1(a_i) }{| \nabla K_1(a_i) |} - \lambda _i \frac{\partial \delta _i}{\partial \lambda _i} \end{aligned}$$

where M is a large constant. We point out that $W_3^{22}$ is exactly the sum of of the vector fields $\underline{W}_1^{3}$ (defined in (13)) with $\gamma =-1$. Furthermore, the presence of the function $\psi _1$ implies that the point $a_i$ moves only if $| \nabla K_1(a_i) | \ge M/\lambda _i$.

Using Propositions 5.4 and 5.5, we get

$$\begin{aligned} \langle - \nabla J_K(u), W_{3}^{22} \rangle \ge&c \sum _{i \in D_3} \psi _1(\lambda _i | \nabla K_1(a_i) | /M) \Big ( \frac{ | \nabla K_1(a_i) | }{\lambda _i} + \frac{1}{\lambda _i ^2} \Big ) \nonumber \\&- \Big ( \frac{c_3}{\lambda _i} \frac{\partial K}{\partial \nu } (a_i) - c \frac{\Delta K(a_i)}{\lambda _i ^2}\Big ) + O\Big ( \sum \frac{1}{\lambda _j ^{n-2} } +R_1^b \Big ) \end{aligned}$$

(30)

which has the same form as (14). Hence, the same computations and arguments hold and the proof of the lemma follows. $\square$

3.2.2 Ruling out collapsing phenomena

We point that, the main difference between the $\mathbb {S}^n$-case (or the case of an interior blow up point for the $\mathbb {S}^n_+$-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 $\lambda$-term). Indeed when $\partial K/\partial \nu (z) \ne 0$ and $a_i$ is close to a boundary critical point $z\in \partial \mathbb {S}^n_+$, the leading term behaves like $c/\lambda _i$, while for the $\mathbb {S}^n$-case (or the case of an interior blow up point in the $\mathbb {S}^n_+$-case), for $a_i$ close to an interior critical point y with $\Delta K(y) \ne 0$, this term behaves like $c/\lambda _i^2$. 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 $\partial K/\partial \nu (z) > 0$, 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 $\partial K/\partial \nu (z) > 0$ 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 $\partial K/\partial \nu (z) > 0$. Then z is a simple blow up. More precisely if $\# B_z := \, \# \{ a_i; \text{ close } \text{ to } z\} := q_1 \ge 2$, then $J_K$ admits in the set $V(q_1,q_1,0,\varepsilon )$ a compactifying bounded pseudogradient $W(z,q_1)$. Namely there exits a constant $c>0$ independent of $u=\sum _{i=1}^{q_1}\alpha _i\delta _i$ such that

$$\begin{aligned} \langle -\nabla J_K(u),W(z,q_1)\rangle \ge c \, \sum _{i\le q_1} \Big (\frac{1}{\lambda _i^{2-{1}/{(n-2)}}}+|1-J_K(u)^{\frac{n}{n-2}}\alpha _i^{\frac{4}{n-2}} K(a_i)|^{2-\frac{1}{n-2}} \Big ) + c\, \sum _{k\ne r} \varepsilon _{kr}^{\frac{n-1}{n-2}}. \end{aligned}$$

Furthermore, the concentration rates $\lambda _i$’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 $a_i, a_j \in \partial \mathbb {S}^n_+$ be concentration points such that the corresponding rates $\lambda _i$ and $\lambda _j$ are of the same order and $d(a_k, b) \rightarrow 0$ for $k=i,j$ for some point $b\in \partial \mathbb {S}^n_+$. Then we have

$$\begin{aligned} e_{ij}:= \frac{ \partial \varepsilon _{ij} }{\partial a_i } (b- \langle a_i , b\rangle a_i) + \frac{ \partial \varepsilon _{ij} }{\partial a_j } (b- \langle a_j , b\rangle a_j) \ge c \, \varepsilon _{ij}. \end{aligned}$$

Proof

Easy computation implies that

$$\begin{aligned} \frac{ \partial \varepsilon _{ij} }{\partial a_i } = (n-2) \lambda _i \lambda _j (a_j - a_i ) \varepsilon _{ij}^{n/(n-2)}.\end{aligned}$$

Thus we get

$$\begin{aligned} e_{ij}&= (n-2) \lambda _i \lambda _j \varepsilon _{ij}^{n/(n-2)}\big ( \langle a_j - a_i , b - \langle a_i , b\rangle a_i\rangle + \langle a_i - a_j , b - \langle a_j , b\rangle a_j\rangle \big )\\&= (n-2) \lambda _i \lambda _j \varepsilon _{ij}^{n/(n-2)} \langle a_j + a_i , b \rangle (1- \langle a_i , a_j \rangle )\\&= (n-2) \lambda _i \lambda _j \varepsilon _{ij}^{n/(n-2)}| a_i - a_j |^2 (1+o(1)) \, \ge \, c \varepsilon _{ij}. \end{aligned}$$

where $| a_i - a_j |$ is the euclidian norm of $a_i-a_j$ seen as a vector in $\mathbb {R}^{n+1}$. $\square$

Lemma 3.11

Let $a, \, h \in \partial \mathbb {S}^n_+$ be close to a non degenerate local maximum z of $K_1$. Then it holds that

$$\begin{aligned} \frac{1}{K_1(a)^{n/2}} \nabla K_1(a) \big (h - \langle a, h \rangle a\big ) \ge - \frac{1}{K_1(h)^{n/2}} \nabla K_1(h) \big (a - \langle a, h \rangle h\big ) + c\, | a - h |^2.\end{aligned}$$

Proof

Let

$$\begin{aligned} \beta (t):= \frac{ h + t(a-h)}{| h + t(a-h) |} \quad , \quad g(t):= \frac{ 2/(n-2) }{ K_1(\beta (t))^{(n-2)/2}} \quad \text{ for } t\in [0,1].\end{aligned}$$

It is easy to get that

$$\begin{aligned} \beta '(t) = \frac{1}{ | h + t(a-h) |} \Big ( a - h - \langle \beta (t) , a-h \rangle \beta (t) \Big ) \quad , \quad \langle \beta (t) , a-h \rangle = O(| a - h |^2),\end{aligned}$$

and therefore it holds that $| \beta '(t) | = |a-h | (1+o(1))$ uniformly in $t\in [0,1]$. Furthermore, easy computations imply that $| \beta ''(t) | = O( |a-h | ^2)$ uniformly in $t\in [0,1]$. In another hand, we have

$$\begin{aligned} g'(t) = \frac{-1}{ K_1(\beta (t))^{n/2}} \nabla K_1(\beta (t)) \big (\beta '(t)\big ) \end{aligned}$$

and, since a and h are close to a non degenerate maximum critical point z of $K_1$, we derive that

$$\begin{aligned} g''(t)= & {} o( | \beta '(t) |^2 ) - \frac{1}{ K_1(\beta (t))^{n/2}} D^2 K_1(\beta (t)) \big (\beta '(t), \beta '(t)\big ) + o( | \beta ''(t) | )\\\ge & {} c | a- h |^2 \,\, (\text{ uniformly } \text{ in } t\in [0,1]).\end{aligned}$$

Now,

$$\begin{aligned} \frac{1}{K_1(a)^{n/2}} \nabla K_1(a) \big (h - \langle a, h \rangle a\big ) + \frac{1}{K_1(h)^{n/2}} \nabla K_1(h) \big (a - \langle a, h \rangle h\big ) = g'(1) - g'(0) = \int _0^1 g''(t) \, dt \end{aligned}$$

which implies the lemma. $\square$

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

$$\begin{aligned} \frac{ M_0}{M_4 ^2} \text{ small } \quad , \quad \max \Big ( \frac{ M_2}{M_0^{1/(q+p-1)}};\, \frac{ M_2^{(n-1)/(n-2)}}{M_0^{(1/2 + 1/(n-2))/(q+p-1)}} \Big ) \text{ small } .\end{aligned}$$

(31)

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:

$$\begin{aligned} V(z,q_1,\eta , \varepsilon ,M_0):= \{ u\in V(q_1,q_1,0, \varepsilon ): \lambda _{\max } \le M_0 \lambda _{\min }; \, d( a_i, z ) < \eta ; \Gamma _{\lambda _i} \le 2 \, \text{ and } \Gamma _{\alpha _i} \le 2 \, \, \forall \, i\}.\end{aligned}$$

Moreover, since the $\lambda _i$’s are of the same order, we have that $\varepsilon _{ij}=(1+o(1))/(\lambda _i \lambda _j d(a_i,a_j)^2)^{(n-2)/2}$ and therefore $d( a_i ,a_j ) \ge c/\lambda _1^{(n-3)/(n-2)}$ for each $i\ne j$ (since $\Gamma _{\lambda _i}$ 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 $d( a_i , a_{i_1} ) := \min d(a_r, a_\ell )$ and define $L_i:= \{i,i_1\}$. Next let $M_4$ be a large positive constant, for such an index i, we define inductively a sequence $L_{i}^s$ by setting

$$\begin{aligned}&L_i^1 := \{ j: \, \exists \, \, \ell \in L_i \, \, s.t. \, \, d(a_j , a_\ell ) \le M_4 d( a_i , a_{i_1} ) \} \quad \text{ and } \\&L_i^s := \{ j: \, \exists \, \, \ell \in L_i^{s-1} \, \, s.t. \, \, d(a_j, a_\ell ) \le M_4 \max _{r, t \in L_i^{s-1} } d( a_r , a_t ) \} .\end{aligned}$$

Observe that, since we have only $q_1$ points and $\# L_i = 2$, then there exists $m \le q_1-1$ such that $L_i^{m+1}= L_i^m$ and we set $L_i^* := L_i^m$ where m is the first index such that $L_i^{m+1}= L_i^m$. We remark that $L_i \subset L_i^*$. Next we want to move the points $a_j$’s, for $j \in L_i^*$, to their center of mass. For this aim, let $\mathbf{\overline{a}_i}$ be defined as

$$\begin{aligned} \mathbf{\overline{a}_i} := \frac{b_i}{| b_i| } \quad \text{ where } \quad b_i \in \mathbb {R}^{n+1} \text{ satisfying } \sum _{j\in L_i^*} (b_i - a_j) = 0.\end{aligned}$$

(32)

Note that, it is easy to see that $\mathbf{\overline{a}_i}$ satisfies

$$\begin{aligned} \mathbf{\overline{a}_i} \in \partial \mathbb {S}^n_+ \quad \text{ and } \quad \sum _{j\in L_i^*} a_j - \langle a_j , \mathbf{\overline{a}_i} \rangle \mathbf{\overline{a}_i} = 0.\end{aligned}$$

(33)

Now we define the following vector field:

$$\begin{aligned} W_3^i := \frac{1}{\lambda _i \gamma _i } \sum _{j\in L_i^*} \alpha _j\frac{\partial \delta _j}{\partial a_j} ( \mathbf{\overline{a}_i} - \langle a_j , \mathbf{\overline{a}_i} \rangle a_j ) \qquad \text{ where } \quad \gamma _i := \max _{j\in L_i^*} d(a_i, a_j).\end{aligned}$$

We note that $L_i^*$ has two important properties:

• If $k,\ell \in L_i^*$, we have $d( a_k ,a_\ell ) \le c M_4^m d( a_{i} , a_{i_1} )$.

• If $k \notin L_i^*$, then, for each $j \in L_i^*$, we have $d( a_j, a_k ) \ge M_4 \max _{r,\ell \in L_i^*} d(a_r, a_\ell )$. Hence, for $k \notin L_i^*$ and $j \in L_i^*$, choosing ${M_0^{(n-2)/2}}/{M_4^{n-2}}$ small, it follows that for every $\ell \in L_i^*$, we have that:

  $$\begin{aligned}&| \frac{\partial \varepsilon _{jk}}{\partial a_j} | | \mathbf{\overline{a}_i} - \langle a_j , \mathbf{\overline{a}_i} \rangle a_j | \le \frac{c d(\mathbf{\overline{a}_i}, a_j) }{(\lambda _j \lambda _k) ^{\frac{n-2}{2}} d( a_j, a_k )^{n-1}} \le \frac{ M_0^{(n-2)/2}}{M_4^{n-1}}\frac{c}{(\lambda _j \lambda _\ell ) ^{\frac{n-2}{2}} d(a_j,a_\ell )^{n-2}} = o\Big ( \varepsilon _{j\ell } \Big ) \end{aligned}$$

  (34)
  $$\begin{aligned}&\varepsilon _{jk} \le \frac{c }{(\lambda _j \lambda _k) ^{(n-2)/2} d(a_j, a_k )^{n-2}} \le \frac{c M_0^{(n-2)/2}}{M_4^{n-2}}\frac{1}{(\lambda _j \lambda _\ell ) ^{(n-2)/2} d( a_j, a_\ell )^{n-2}} = o\Big ( \varepsilon _{j\ell } \Big ) \end{aligned}$$

  (35)

  (by using (31)). We note that, in this region, we have $|1- J_K(u)^{\frac{n}{n-2}}\alpha _j^{\frac{4}{n-2}}K(a_j)| \le c M_2/{\lambda _j}$ for each j, hence Proposition 5.5 can be written as:

  $$\begin{aligned} \langle&\nabla J_K(u), \alpha _j\frac{\partial \delta _j}{\partial a_j}\rangle = \lambda _j \left[ c_4\left( 1- J_K(u)^{\frac{n}{n-2}}\alpha _i ^{\frac{4}{n-2}}K(a_i)\right) + J_K(u)^{\frac{n}{n-2}}\alpha _i ^{\frac{4}{n-2}}\frac{c_5}{\lambda _i}\frac{\partial K}{\partial \nu }(a_i)\right] e_n \nonumber \\&- J_K(u)c_2 \sum _{k\ne j} \alpha _j\alpha _k \frac{\partial \varepsilon _{kj}}{\partial a_j} - 8 c_5 \, J_K(u)^{-\frac{n-2}{2}} \frac{\nabla K_1(a_j)}{K(a_j)^{n/2}} + O\Big (\frac{1}{\lambda } + \lambda \sum \varepsilon _{kr }^{\frac{n}{n-2}}\ln (\varepsilon _{kr}^{-1})\Big ). \end{aligned}$$

  (36)

  Hence we derive that:

  $$\begin{aligned} \langle -\nabla J_K(u),&W_3 ^i \rangle = \frac{J_K(u)c_2}{\lambda _i \gamma _i} \sum _{k\ne j; j\in L_i^* } \alpha _j\alpha _k \frac{\partial \varepsilon _{kj}}{\partial a_j}( \mathbf{\overline{a}_i} - \langle a_j , \mathbf{\overline{a}_i} \rangle a_j )\nonumber \\&+ \frac{8 c_5 \, J_K(u)^{(2-n)/2} }{\lambda _i \gamma _i} \sum _{j\in L_i^*}\frac{1}{K(a_j)^{n/2}} \nabla K_1(a_j) ( \mathbf{\overline{a}_i} - \langle a_j , \mathbf{\overline{a}_i} \rangle a_j ) \nonumber \\&+ O\Big ( \frac{1}{\lambda ^2} + \sum \varepsilon _{kr }^{\frac{n}{n-2}}\ln (\varepsilon _{kr}^{-1})\Big ). \end{aligned}$$

  (37)

Next we notice that, using Lemma 3.10, il holds

$$\begin{aligned} \frac{\partial \varepsilon _{kj}}{\partial a_j}( \mathbf{\overline{a}_i} - \langle a_j , \mathbf{\overline{a}_i} \rangle a_j ) + \frac{\partial \varepsilon _{kj}}{\partial a_k} ( \mathbf{\overline{a}_i} - \langle a_k , \mathbf{\overline{a}_i} \rangle a_k )\, \ge c \, \varepsilon _{kj}, \qquad \text{ for } \text{ each } k, j \in L_i^*.\end{aligned}$$

(38)

Furthermore, using Lemma 3.11 (with $h=\mathbf{\overline{a}_i}$), it holds that

$$\begin{aligned}&\sum _{j\in L_i^*}\frac{1}{K(a_j)^{n/2}} \nabla _T K(a_j) ( \mathbf{\overline{a}_i} - \langle a_j , \mathbf{\overline{a}_i} \rangle a_j )\\&\quad \ge \sum _{j\in L_i^*}\frac{-1}{K( \mathbf{\overline{a}_i} )^{n/2}} \nabla _T K( \mathbf{\overline{a}_i} ) ( a_j - \langle a_j , \mathbf{\overline{a}_i} \rangle \mathbf{\overline{a}_i} ) + c \sum _{j\in L_i^*} | a_j - \mathbf{\overline{a}_i}| ^2 \\&\quad \ge c \sum _{j\in L_i^*} | a_j - \mathbf{\overline{a}_i}| ^2 \quad (\text{ since } \sum _{j\in L_i^*} a_j - \langle a_j , \mathbf{\overline{a}_i} \rangle \mathbf{\overline{a}_i} = 0\, (\text{ see } \text{(33) })). \end{aligned}$$

Thus we get

$$\begin{aligned} \langle -\nabla J_K(u), W_3^i \rangle \ge c \sum _{k, j \in L_i^*} \frac{\varepsilon _{kj}}{\lambda _i \gamma _i} + \sum _{j\in L_i^*} \frac{ d( a_j,\mathbf{\overline{a}_i})^2 }{\lambda _i \gamma _i} +O\Big ( \sum \varepsilon _{\ell r}^{\frac{n}{n-2}} \ln \varepsilon _{\ell r} ^{-1} + \frac{1}{\lambda _i ^2}\Big ). \end{aligned}$$

(39)

Now, since $\gamma _i:= \max _{k,r \in L_i^*} d(a_k,a_r)$ is of the same order of all the $d(a_\ell , a_j )$’s, we derive that ${\varepsilon _{kj}}/{\lambda _i \gamma _i} \ge c \varepsilon _{kj} ^{(n-1)/(n-2)}$. Furthermore, $\sum _{j\in L_i^*} d(a_j ,\mathbf{\overline{a}_i})^2 \ge c \sum _{j, r \in L_i^*} d( a_j, a_r )^2$ and therefore

$$\begin{aligned} \sum _{j\in L_i^*} d(a_j, \mathbf{\overline{a}_i})^2 / (\lambda _i \gamma _i) \ge \sum _{j, r \in L_i^*} d(a_j, a_r ) / \lambda _i \ge c /\lambda _i ^{2- 1/(n-2)}.\end{aligned}$$

Hence, in the lower bound of (39), we are able to make appear $1 /\lambda _i ^{2- 1/(n-2)}$ and therefore (since all the $\lambda _j$’s are of the same order and $\Gamma _{\alpha _k} \le 4$ for each k) we are able to make appear all the $1 /\lambda _j ^{2- 1/(n-2)}$’s and $| 1- J_K(u)^{n/(n-2)} \alpha _j^{4/(n-2)} K(a_j) |^{2- 1/(n-2)}$’s. Concerning the $\varepsilon _{kr}$, we note that the $\varepsilon _{kj}$’s which appeared in the lower bound, are only for the indices $k, j \in L_i^*$. Hence we need to make appear $\varepsilon _{jr}$ for $j\notin L_i^*$. For this aim, we remark that, for each $j, \ell$, we have $d( a_j, a_\ell ) \ge d( a_i, a_{i_1} )$ (by the definition of i and $i_1$), in addition we have that the $\lambda _k$’s are of the same order. Hence we deduce that $\varepsilon _{ii_1} \ge c \varepsilon _{j \ell }$. 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 is a constant $c > 0$ independent of $u=\sum _{i=1}^{q}\alpha _i\delta _i + \sum _{i=q+1}^{p}\alpha _i\varphi _i \in V_3^3$ such that

$$\begin{aligned} \langle -\nabla J_K(u),W_3^3\ge & {} \sum _{i=1}^{q+p} \frac{c}{\mu _i^{\frac{2n-5}{n-2}}}+ c \sum _{i=1}^{q}|1-J_K(u)^{\frac{n}{n-2}}\alpha _i^{\frac{4}{n-2}} K(a_i)|^{\frac{2n-5}{n-2}} \\+ & {} c\, \sum _{k\ne r} \varepsilon _{kr} ^\frac{n-1}{n-2} + c \sum _{i=q+1}^{q+p}\left( \frac{| \nabla K(a_i)|}{\lambda _i}\right) ^{\frac{2n-5}{n-2}} \end{aligned}$$

Furthermore, the $\lambda _i$’s do not increase along the flow lines generated by the pseudogradient $W^3_3$.

Proof

Let $z_1, \ldots ,z_\ell$ be the critical points of $K_1$ satisfying $\partial K /\partial \nu (z_j) > 0$ and $\# B_{z_j} \ge 2$. We decompose u as follows:

$$\begin{aligned} u:=\sum _{i=1}^\ell u_i + u_{\ell +1} \quad \text{ where } u_i:= \sum _{k\in B_{z_i}} \alpha _k \delta _k \text{ and } u_{\ell +1} := u- \sum _{i=1}^\ell u_i .\end{aligned}$$

From the second and the third assertions of Remark 3.4, it follows that each concentration point $a_j$ of $u_{\ell +1}$ satisfies $| a_j - a_k | \ge c$ for each $k \ne j$ and it is close to a critical point of $K_1$ with $\partial K/\partial \nu \ge 0$ or a critical point of K in $\mathbb {S}^n_+$ with $\Delta K < 0$. Furthermore, for $j \in B_{z_i}$, we have $| a_j - a_k | \ge c$ for each $k \notin B_{z_i}$. Hence the mutual interaction between two clusters $B_{z_i}$ and $B_{z_j}$ for $i \ne j$ is negligible with respect to the other terms. In this situation, we define the following vector field

$$\begin{aligned} W_3^3 := \sum _{i=1}^\ell W(z_i, \# B_{z_i}) \end{aligned}$$

where $W(z_i, \# B_{z_i})$ is defined in Lemma 3.9. Hence we obtain

$$\begin{aligned} \langle -\nabla J_K(u),W_3^3 \rangle = \sum _{i=1}^\ell \langle -\nabla J_K(u), W(z_i, \# B_{z_i}) \rangle = \sum _{i=1}^\ell \langle -\nabla J_K(u_i), W(z_i, \# B_{z_i}) \rangle + \sum _{k\in B_{z_i}; j\notin B_{z_i}} O\big ( \varepsilon _{kj}\big ) .\end{aligned}$$

(40)

We observe that, for $k\in B_{z_i}$, we have $\mu _k$ and $\lambda _k$ are of the same order. Moreover we are in the case where all the $\mu _j$’s are of the same order. Thus, using Lemma 3.9, we are able to make appear all the $1/\mu _j^{2-1/(n-2)}$’s in the lower bound of (40) (and therefore all the $|1-J_K(u)^{{n}/(n-2)}\alpha _i^{{4}/(n-2)} K(a_i)|^{2-{1}/(n-2)}$’s and the $({| \nabla K(a_i)|}/{\lambda _i})^{2-{1}/(n-2)}$’s (since the $\Gamma _{\alpha _k}$’s and the $\Gamma _{a_i}$’s are bounded). In addition, for $j\notin B_{z_i}$ and $k\in B_{z_i}$, we have

$$\begin{aligned} \varepsilon _{kj} \le \frac{c }{(\lambda _j \lambda _k)^{(n-2)/2}} \le {\left\{ \begin{array}{ll} &{} o(1/\lambda _k^2) \text{ if } n \ge 6 , \\ &{} c/\lambda _k^2 + c/\lambda _j^4 \text{ if } n =5.\end{array}\right. } \end{aligned}$$

Therefore, our lemma follows from Lemma 3.9. $\square$

Lemma 3.13

There exists a bounded pseudogradient $\mathcal {V}$ satisfying the following estimate:

There is a constant $c > 0$ independent of $u=\sum _{i=1}^{q}\alpha _i\delta _i + \sum _{i=q+1}^{p}\alpha _i\varphi _i \in \mathcal {W}$ such that (18) holds true with $\mathcal {V}$ instead of $W_1$.

Furthermore in the subset of $\mathcal {W}$ such that $\lambda _i |\nabla K_1(a_i) |$ is bounded, the $\lambda _i$’s are increasing functions along the flow lines generated by the pseudogradient $\mathcal {V}$.

Proof

Let $\psi _1$ be a $C^\infty$ cut of function defined by $\psi _1 \in [0,1]$, $\psi _1(t) = 1$ if $t \ge 2$ and $\psi _1(t)=0$ if $t\le 1$.

We define the following vector field:

$$\begin{aligned} \mathcal {V}:= W_\alpha + W_a^{in} + W_a^b + \sum _{i=1}^{p+q} \lambda _i \frac{\partial \varphi _i}{\partial \lambda _i} \end{aligned}$$

where $W_a^b := \sum _{i\in I_b} \psi _1 (\lambda _i | \nabla K_1(a_i) | / M_2) (1/\lambda _i) (\partial \delta _i/\partial a_i)(\nabla K_1(a_i) / | \nabla K_1(a_i) |)$ and $W_a^{in}$ (resp. $W_\alpha$) is defined in (19) (resp. (20)).

Observing that in $\mathcal {W}$ we have $\varepsilon _{ij} = O( 1/\lambda _i ^{n-2} + 1/\lambda _j ^{n-2})$ for each $i \ne j$ and using Propositions 5.4, 5.5, 5.7 the lemma follows. $\square$

3.2.3 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 is a constant $c > 0$ independent of $u=\sum _{i=1}^{q}\alpha _i\delta _i + \sum _{i=q+1}^{p+q}\alpha _i\varphi _i \in V_4(M_0)$ such that

$$\begin{aligned} \langle -\nabla J_K(u),W_4 \rangle\ge & {} c \, \sum _{i=1}^{q+p} \frac{1}{\mu _i^\frac{n-1}{n-2}}+ c \sum _{i=1}^{q}|1-J_K(u)^{\frac{n}{n-2}}\alpha _i^{\frac{4}{n-2}} K(a_i)|^\frac{n-1}{n-2}\\+ & {} c\, \sum _{k\ne r} \varepsilon _{kr} ^\frac{n-1}{n-2} + \sum _{i=q+1}^{q+p}\left( \frac{| \nabla K(a_i)|}{\lambda _i}\right) ^\frac{n-1}{n-2} . \end{aligned}$$

Furthermore, $\max \mu _i$ deos not increase along the flow lines generated by this pseudogradient.

Proof

For $u= \sum _{i=1}^{q}\alpha _i\delta _{a_i,\lambda _i} + \sum _{i=q+1}^{p+q}\alpha _i\varphi _{a_i,\lambda _i}$, we denote

$$ \begin{aligned} \mathcal {I}_{in} := \{ i=1, \ldots , p+q; \, a_i \in \mathbb {S}^n_+ \} \quad \& \quad \mathcal {I}_{b} := \{ i=1, \ldots , p+q; \, a_i \in \partial \mathbb {S}^n_+ \} . \end{aligned}$$

Next we reorder the parameters $\mu _i$’s as: $\mu _1 \le \cdots \le \mu _{p+q}$ and define the following subset of indices:

$$\begin{aligned}I:= \{1\} \cup \{ i \ge 2: \mu _k \le M_0^{1/(p+q -1)} \mu _{k-1} \text{ for } \text{ each } k\le i\}.\end{aligned}$$

Since we are in $V_4(M_0)$, we have $\mu _{\max } > M_0 \mu _{\min }$, it follows that $p+q \notin I$. In this region, we write u as

$$\begin{aligned} u := u_1 + u_2 \quad \text{ where } u_1 := \sum _{i\in I} \alpha _i \varphi _i \text{ and } u_2 := u - u_1.\end{aligned}$$

Let $k_0:= \max I$ (then we have $k_0 < p+q$). It follows that $\mu _{k_0} \le M_0^{(k_0 - 1 )/ (p+ q -1)} \mu _1 := \overline{ M}_0 \mu _1$, $\mu _{k_0+1} \ge M_0^{1/(p+q -1)} \mu _{k_0}$ and therefore $u_1 \in V_1(\overline{ M}_0) \cup V_2(\overline{ M}_0) \cup V_3(\overline{ M}_0)$.

Furthermore we introduce the following notation

$$ \begin{aligned} D^4_1:= \{ i \in \mathcal {I}_{in}: \Gamma _{\lambda _i} + \Gamma _{a_i} + \Gamma _{H_i} \ge 6\} \quad \& \quad D^4_2:= \{ i \in \mathcal {I}_{b}: \Gamma _{\lambda _i} + \Gamma _{\alpha _i} \ge 4\} \end{aligned}$$

and set

$$\begin{aligned} i_0:= {\left\{ \begin{array}{ll} \min D^4_1, &{} \text{ if } D^4_1 \ne \emptyset \\ p+q+1 , &{} \text{ otherwise }. \end{array}\right. } \qquad j_0:= {\left\{ \begin{array}{ll} \min D^4_2, &{} \text{ if } D^4_2 \ne \emptyset \\ p+q+1 , &{} \text{ otherwise }. \end{array}\right. } \end{aligned}$$

Next we define in case $D^4_1 \cup D^4_2 \ne \emptyset$ the following vector fields:

$$\begin{aligned} W_{i_0}:= - \sum _{i\ge i_0; i\in \mathcal {I}_{in}} 2^i \lambda _i \frac{\partial \varphi _i}{\partial \lambda _i} \quad \text{ and } \quad W_{j_0}:= - \sum _{j\ge j_0; i\in \mathcal {I}_{b}} 2^j \lambda _j \frac{\partial \delta _i}{\partial \lambda _i} \end{aligned}$$

and as in the proof of Lemma 3.5, we define

$$\begin{aligned} W_4^0:= W_{i_0} + (1/M_2) W_{j_0} + W_{\alpha } + W_a^{in}\end{aligned}$$

where $W_a^{in}$ (resp. $W_{\alpha }$) is defined in (19) (resp. (20)). Following the proof of Lemma 3.5 and using Lemma 5.2, we get

$$\begin{aligned} \langle - \nabla J_K(u), W^{0}_4 \rangle \ge&\, \, \overline{ \Gamma }_{a}^{in}+ c \sum _{i\ge i_0; i\in \mathcal {I}_{in}} \Big ( \sum _{\ell \ne i} \varepsilon _{i\ell } + \frac{1}{(\lambda _i d_i)^{n-2}} +O(R_1)\Big ) + \frac{c}{\mu _{i_0}}\nonumber \\&+ \overline{ \Gamma }_{\alpha } + \frac{c}{M_2} \sum _{ j\ge j_0; j\in \mathcal {I}_{b}} \Big ( \sum _{\ell \ne j } \varepsilon _{j\ell } +O\big (R_1^b + + c\frac{c}{\mu _{j_0}}+ \sum _{\ell \in \mathcal {I}_{in}} \varepsilon _{j\ell }\big )\Big ) := \overline{\Gamma }_4. \end{aligned}$$

(41)

Observe that, concerning the last term, for $\ell \in \mathcal {I}_{in}$, (i) either $\ell \ge i_0$, then the $\varepsilon _{j\ell }$ exists in the second term of this formula and one takes $M_2$ large to absorb the last term, or (ii) $\ell < i_0$ and in this case by Lemma 3.3, the concentration point $a_\ell$ is close to a critical point y of K in $\mathbb {S}^n_+$ and then $\varepsilon _{j\ell } \le c(1/ \lambda _j^{n-2} + 1/ \lambda _\ell ^{n-2})$. Hence, we can in either case absorb the last term.

Furthermore we notice that if $D^4_1 \cup D^4_2 \ne \emptyset$ and if $i_0 \in I$ or if $j_0 \in I$ 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:

Case 1: If $u_1 \in V_1(\overline{ M}_0) \cup V_2(\overline{ M}_0) \cup (V_3(\overline{ M}_0) {\setminus } V_3^3)$. In this region, we define the following vector field:

$$\begin{aligned} W_4^1:= W_4^0 + (1/M_2^2) \widetilde{W}(u_1), \end{aligned}$$

where $\widetilde{W}$ is the convex combination of the pseudogradients constructed in $V_1(\overline{ M}_0)$, $V_2(\overline{ M}_0)$ and $V_3(\overline{ M}_0) {\setminus } V_3^3$. It follows then that

$$\begin{aligned} \langle - \nabla J_K(u), W_4^1 \rangle \ge&\, \, \overline{\Gamma }_4 + \frac{1}{M_2^2}\left( \sum _{i \in I} \frac{c}{\mu _i}+ c\sum _{i\in I\cap \mathcal {I}_b}|1-J_K(u)^{\frac{n}{n-2}}\alpha _i^{\frac{4}{n-2}} K(a_i)| \nonumber \right. \\&\left. + c \sum _{k\ne r; k,r\in I} \varepsilon _{kr} + c \sum _{i\in I\cap \mathcal {I}_{in}} \frac{| \nabla K(a_i)|}{\lambda _i} + O\left( \sum _{j \in I; \ell \notin I} \varepsilon _{j\ell }\right) \right) . \end{aligned}$$

(42)

To complete the proof, it remains to absorb the last term. To this aim, we notice that:

(i) if "$\ell \in \mathcal {I}_{in}$ with $\ell \ge i_0$ or $\ell \in \mathcal {I}_b$ with $\ell \ge j_0$", then the term $\varepsilon _{j\ell }$ is already in $\overline{\Gamma }_4$ the lower bound of (41). Taking $M_2$ large, we will be able to absorb this term.

(ii) if "$\ell \in \mathcal {I}_{in}$ with $\ell < i_0$ or $\ell \in \mathcal {I}_b$ with $\ell < j_0$", then there holds: $\varepsilon _{j\ell } \le c \frac{M_2}{\mu _\ell } \le c (M_2/ { M}_0^{1/(q+p-1)})\frac{1}{\mu _{k_0}} = o(1/\mu _{k_0})$ by choosing $M_2/ { M}_0^{1/(q+p-1)}$ 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 $\mu _{max}$ 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 $u_1 \in V_3^3(\overline{ M}_0)$ and assume that $D_9 \cup D_8 \ne \emptyset$, where

$$\begin{aligned} D_{9}:= \{ i\in I: i \in B_z \text{ with } \# B_z =1\}; \qquad D_{8}:= I \cap \mathcal {I}_{in}. \end{aligned}$$

Here we define the following vector field:

$$\begin{aligned} W_4^2:= W_4^0 + (1/M_2^2) \sum _{i\in D_8 \cup D_9} \lambda _i \frac{\partial \varphi _i}{\partial \lambda _i} .\end{aligned}$$

We point out that, this pseudogradient increases the $\mu _i$ for $i\in D_8 \cup D_9$, but does not increase the $\mu _{\max }:= \mu _{p+q}$ since $p+q \notin I$. Furthermore observe that

$$\begin{aligned} \langle - \nabla J_K(u), \sum _{i\in D_8 \cup D_9} \lambda _i \frac{\partial \varphi _i}{\partial \lambda _i} \rangle \ge c \sum _{i\in D_8 \cup D_9} \Big ( \frac{1}{\mu _i} + O\Big ( \sum _{j =1}^{p+q} \frac{1}{\lambda _j^3} + \sum _{\ell \notin I} \varepsilon _{i\ell } \Big ) \Big ).\end{aligned}$$

Hence the result follows as the first case.

Next we set

$$\begin{aligned} D_{10}:= \left\{ i\in I: \sum _{k\in I; k\ne i} \varepsilon _{ki} \le m_1 q /\lambda _i\right\} \ne \emptyset , \quad \text{ where } m_1 \text{ is } \text{ a } \text{ small } \text{ constant }.\end{aligned}$$

Case 3: In this case we take $u_1 \in V_3^3(\overline{ M}_0)$ and assume that $D_9 \cup D_8 = \emptyset$. That is we have that $I\subset \mathcal {I}_b$ and that $\#B_z \ne 1$ for each z critical point of $K_1$. Furthermore we assume that $D_{10} \ne \emptyset$.

Next we recall that in this case, for each z such that $\#B_z \ge 2$, z has to be a local maximum point with $\partial K/\partial \nu > 0$ (which implies that $\mu _i$ and $\lambda _i$ are of the same order). Hence one can use the same pseudogradient defined in Case 2 (by replacing $D_{8} \cup D_{9}$ by $D_{10}$). Hence for $i \in D_{10}$, using Proposition 5.4, we derive that

$$\begin{aligned} \langle - \nabla J_K(u), \lambda _i \frac{\partial \delta _i}{\partial \lambda _i} \rangle \ge \frac{c}{\lambda _i} + O( \sum _{j\ne i} \varepsilon _{ij}) \ge \frac{c}{\lambda _i} + c \sum _{j\ne i; j\in I} \varepsilon _{ij} + O( \sum _{j\notin I} \varepsilon _{ij})\end{aligned}$$

and the proof follows as the previous cases.

Case 4: $u_1 \in V_3^3(\overline{ M}_0)$ and $I\subset I_b$, $\#B_z \ne 1$ for each z and $D_{10} = \emptyset$.

In this case, for each z such that $\#B_z \ge 2$, z has to be a local maximum point with $\partial K/\partial \nu > 0$ (which implies that $\mu _i$ and $\lambda _i$ are of the same order). Let $z_1,\ldots ,z_\ell$ be such that $\# B_{z_j} \ge 2$. Thus, the function u can be written as

$$\begin{aligned} u := \sum _{j=1}^\ell u_j + u_{\ell +1} \quad \text{ where } \quad u_j := \sum _{i\in B_{z_j}} \alpha _i \varphi _i \text{ for } j\le \ell \quad \text{ and } \quad u_{\ell +1} := \sum _{i\notin I} \alpha _i \varphi _i .\end{aligned}$$

Notice that, for $j \le \ell$, it follows that $u_j \in V(z_j, \#B_{z_j}, \eta , \varepsilon , \overline{M}_0)$ and in Lemma 3.9, we have constructed a pseudogradient $W(z_j, \#B_{z_j})$ in this region. Now, we define

$$\begin{aligned} W_4^4:= W_4^0 + \frac{1}{M_2^2} \sum _{j=1}^\ell W(z_j,\#B_{z_j}) (u_j).\end{aligned}$$

(43)

Observe that, by Lemma 3.9, we have

$$\begin{aligned}&\langle - \nabla J_K(u), W(z_j,\#B_{z_j}) (u_j) \rangle \\&\quad \ge c \sum _{k\in B_{z_j}} \Big ( \sum _{r\ne k; r\in B_{z_j}} \varepsilon _{kr} ^\frac{n-1}{n-2} + O\Big ( \sum _{r\notin B_{z_j}, r\in \mathcal {I}_{in}} \varepsilon _{kr} + \sum _{ r\notin B_{z_j}, r\in \mathcal {I}_{b}} \frac{1}{\lambda _k} | \frac{\partial \varepsilon _{kr}}{\partial a_k} | \Big )\Big ) .\end{aligned}$$

Furthermore we notice that, for $r\notin B_{z_j}$ and $r\in \mathcal {I}_{in}$, (i) either $r \ge i_0$ and therefore the $\varepsilon _{kr}$ exists already in $\overline{\Gamma }_4$ or (ii) $r < i_0$ and, using Lemma 3.3, it follows that $a_r$ is close to a critical point y of K in $\mathbb {S}^n_+$ which implies that $\varepsilon _{kr} \le c(1/\lambda _k^{n-2} + 1/\lambda _r^{n-2})$. Next for $r\notin B_{z_j}$ and $r\in \mathcal {I}_{b}$, three situations may occur

• (i) $r \ge j_0$ and therefore the $\varepsilon _{kr}$ exists already in $\overline{\Gamma }_4$.

• (ii) $r < j_0$ and $r \notin I$. In this case it follows that $\varepsilon _{kr } \le M_2/\lambda _r$ and thus (since $\lambda _r \ge M_0 ^{1/q+p - 1 } \lambda _k$ for each $k \in I$) we have that

  $$\begin{aligned} \frac{1}{\lambda _k} | \frac{\partial \varepsilon _{kr}}{\partial a_k} |&\le {c}{ \lambda _r d(a_r,a_k) } \varepsilon _{kr } ^{\frac{n}{n-2}} \le {c}\sqrt{\frac{ \lambda _r }{\lambda _k}} \varepsilon _{kr } ^{\frac{n-1}{n-2}} \le \frac{c \, M_2^{\frac{n-1}{n-2}}}{ \lambda _k ^{1/2}\lambda _r ^{1/2 + 1/(n-2)}} \nonumber \\&\le c \frac{M_2 ^{\frac{n-1}{n-2}} }{ M_0 ^{(^{1/2 + 1/(n-2)})/q+p - 1 }}\frac{1}{\lambda _k ^{1 + 1/(n-2)}} = o \Big (\big ( \frac{m_1}{\lambda _k}\big )^{(n-1)/(n-2)}\Big ) \end{aligned}$$

  (44)

  (by using (31)).

• (iii) $r < j_0$ and $r \in I$. In this case, it follows that $a_r \in B_{z_\ell }$ with $\ell \ne k$ and therefore we deduce that $| a_k - a_r | \ge c > 0$. Hence we get

  $$\begin{aligned} \frac{1}{\lambda _k} | \frac{\partial \varepsilon _{kr}}{\partial a_k} | = O\Big ( \frac{1}{\lambda _k ^{n-1} }+ \frac{1}{\lambda _r ^{n-1} }\Big ).\end{aligned}$$

Using (41),(43), the previous estimates and the fact that $D_{10} = \emptyset$, the lemma follows in this case. $\square$

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 $\mu _{\max }$ increases is the region $\mathcal {W}$. Finally, Claim (ii) follows from the first one and the estimate of $\Vert \overline{v} \Vert ^2$ 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.

3.2.4 Critical points at infinity and their topological contribution

For $\varepsilon _0$ a small number, we define the following neighborhood of the cone of positive solutions of the sphere in $H^1(\mathbb {S}^n_+)$:

$$\begin{aligned} V_{\varepsilon _0}(\Sigma ^+):= \{ u \in \Sigma ; \, J_K(u)^{(2n-2)/(n-2)} e^{2J(u)}|u^-|_{L^{{2n}/{(n-2)}}}^{{4}/{(n-2)}} \, < \, \varepsilon _0 \}, \quad \text{ where } u^-:= \max (0, -u). \end{aligned}$$

This set is for $\varepsilon _0$ small enough invariant under the gradient flow lines of the Euler Lagrange functional $J_K$. Namely we prove that

Lemma 3.15

For $\varepsilon _0 > 0$ small enough, the set $V_{\varepsilon _0} (\Sigma ^+)$ is invariant under the flow generated by $-\nabla J_K$.

Proof

We will write J instead of $J_K$. For $w\in L^{2n/(n+2)} (\mathbb {S}^n_+)$, we denote by $\mathcal {L}^{-1} (w)$ the solution of the following PDE:

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {L}u := -\Delta u + \frac{n(n-2)}{4} u = w \quad \text{ in } \mathbb {S}^n_+, \\ \partial u / \partial \nu = 0 \quad \text{ on } \partial \mathbb {S}^n_+ . \end{array}\right. }\end{aligned}$$

Furthermore, it holds

$$\begin{aligned}&| u | _{L^{2n/(n-2)} } \le c \Vert u \Vert _{H^1} \le c | w |_{L^{2n/(n+2)} } \nonumber \\&| \mathcal {L} ^{-1}(K|u |^{4/(n-2)}u) | _{L^{2n/(n-2)} } \le c | u |_{L^{2n/(n-2)} } ^{(n+2)/(n-2)} . \end{aligned}$$

(45)

Suppose $u_0 \in V_{\varepsilon _0} (\Sigma ^+)$ and consider

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{du(s)}{ds}= -\nabla J(u(s)) = -2J(u)\biggr (u-J(u)^{n/(n-2)} \mathcal {L} ^{-1}(K|u|^{4/(n-2)}u)\biggr )\\ u(0)=u_0. \end{array}\right. } \end{aligned}$$

Then

$$\begin{aligned}&e^{2\int _0^s J(u(t)) dt}u(s)= u_0 +2\int _0^s e^{2\int _0^tJ(u(y)) dy } J(u(t))^{\frac{2n-2}{n-2}}\mathcal {L} ^{-1}(K|u(t)|^{4/(n-2)}u(t))dt,\\&u^-(s) \le e^{-2\int _0^s J(u(t)) dt}\biggr ( u_0^- +2\int _0^se^{2\int _0^tJ(u(y)) dy} J(u(t))^{\frac{2n-2}{n-2}}\mathcal {L} ^{-1}(K(u^-(t))^{\frac{n+2}{n-2}})dt\biggr )\\&\quad := e^{-2\int _0^s J(u)}f(s).\end{aligned}$$

Setting

$$\begin{aligned} F(s)=e^{-\frac{4n}{n-2}\int _0^s J(u(t)) dt } | f (s) |_{L^{{2n}/{(n-2)}}}^{{2n}/{(n-2)}} \quad \text{ which } \text{ implies } \text{ that } \quad |u^-(s)|_{L^{{2n}/{(n-2)}}}^{{2n}/{(n-2)}} \le F(s).\end{aligned}$$

Recall that, if $u_0^{-} =0$ then u(s) is positive for all s. Hence, we can assume that $u_0^-\ne 0$ and we want to prove that F is a decreasing function. Observe that

$$\begin{aligned} F'(s)&= -\frac{4n}{n-2} J(u(s))e^{-\frac{4n}{n-2}\int _0^s J(u)} |f(s) |_{L^{{2n}/{(n-2)}}}^{{2n}/{(n-2)}}+ e^{-\frac{4n}{n-2} \int _0^s J(u)}\frac{2n}{n-2} \int _{\mathbb {S}^n_+} f' (s) f(s) ^{\frac{n+2}{n-2} } dx \\&\le \frac{2n}{n-2} e^{-\frac{4n}{n-2}\int _0^s J(u)} \left[ -2J(u(s))|u_0^-|_{L^{{2n}/{(n-2)}}}^{{2n}/{(n-2)} } + \int _{\mathbb {S}^n_+}f'(s) f(s) ^{\frac{n+2}{n-2} } dx\right] \qquad (\text{ using } f(s) \ge u_0^{-}). \end{aligned}$$

Notice that $f'(0) = u_0^{-}$ and therefore

$$\begin{aligned} \big |\int _{\mathbb {S}^n_+}f'(s)f(s)^{\frac{n+2}{n-2}}\big | dx \le c\int _{\mathbb {S}^n_+} | f'(s) | |u_0^-|^{\frac{n+2}{n-2}}+c\int _{\mathbb {S}^n_+} | f'(s) | \Big (\int _0^s |f'(t)|dt \Big )^{\frac{n+2}{n-2}} dx. \end{aligned}$$

But, we have (using (45))

$$\begin{aligned} \int _{\mathbb {S}^n_+}(u_0^-)^{\frac{n+2}{n-2}} | f'(s) | dx&= \int _{\mathbb {S}^n_+}(u_0^-)^{\frac{n+2}{n-2}}\Big ( 2 e^{2\int _0^sJ(u)} J(u(s))^{\frac{2n-2}{n-2}}\mathcal {L} ^{-1}(K(u^-(s))^{\frac{n+2}{n-2}})\Big ) dx\\&\le CJ(u(s))^{\frac{2n-2}{n-2}} e^{2\int _0^sJ(u)} |u_0^-|_{L^{{{2n}/{(n-2)}}}}^{{(n+2)}/{(n-2)}} |u^-(s)|_{L^{{{2n}/{(n-2)}}}}^{{(n+2)}/{(n-2)}}, \end{aligned}$$

and we also have (using the fact that J(u(s)) is a decreasing function)

$$\begin{aligned} \int _{ \mathbb {S}^n_+ }|f'(s)|&\big ( \int _0^s|f'(t))|dt \big )^{\frac{n+2}{n-2} }dx \le cs^{\frac{4}{n-2}} \int _{\mathbb {S}^n_+}|f'(s)|\int _0^s|f'(t)|^{\frac{n+2}{n-2}}dt dx \\&\le c s^{\frac{4}{n-2}} e^{\frac{4n}{n-2}sJ(u_0)} J(u_0)^{\frac{2n-2}{n-2}\frac{2n}{n-2}} |u^-(s)|_{L^{{{2n}/{(n-2)}}}}^{{(n+2)}/{(n-2)}} \int _0^s |u^-(t)|_{L^{{{2n}/{(n-2)}}}}^{(n+2)^2/(n-2)^2} dt. \end{aligned}$$

Hence, if $|u^-(s)|_{L^{{2n}/{(n-2)}}} \le 5|u_0^-|_{L^{{2n}/{(n-2)} }}$, for $0\le s\le 1$, we derive that

$$\begin{aligned} F'(s) \le \frac{4 n}{n-2} e^{-\frac{4n}{n-2}\int _0^s J(u)} |u_0^-|_{L^{{2n}/{(n-2)}}}^{{2n}/{(n-2)} } \Big ( - J(u(s)) + c \, J(u_0)^{\frac{2n-2}{n-2}} e^{2 J(u_0) } | u^- _0 | _{L^{{2n}/{(n-2)}}}^{{4}/{(n-2)}} \\ + c\, \Big ( J(u_0)^{\frac{2n-2}{n-2 } } e^{2 J(u_0)} | u^- _0 | _{L^{{2n}/{(n-2)}}}^{4/{(n-2)}} \Big )^{2n/(n-2)}\Big ) \end{aligned}$$

Finally, since $\inf J> c > 0$, using the fact that $u_0\in V_{\varepsilon _0}(\Sigma ^+)$, that is, $J(u_0)^{\frac{2n-2}{n-2}} e^{2J(u_0)}|u_o^-|_{L^{{2n}/{(n-2)}}}^{{4}/{(n-2)}} <\varepsilon _0$, and $\eta$ is small enough, then $F'(s) \le 0$, for $0\le s\le 1$. Therefore $J(u(s))^{\frac{2n-2}{n-2}} e^{2J(u(s))}|u(s)^-|_{L^{{2n}/{(n-2)}}}^{{4}/{(n-2)}} <\varepsilon _0$, and our result follows. $\square$

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 $- \nabla J$ outside the $V(q,p,m,\varepsilon )$’s. Let us denote the resulting global vector field by Y and define a new vector field by setting:

$$\begin{aligned} X(u):= \, Y(u) \, - \, <Y(u),u> u \qquad \text{ for } u \in V_{\varepsilon _0}(\Sigma ^+). \end{aligned}$$

We then have

Corollary 3.16

Assume that K satisfies (H1), (H2) and (H3). Then using Propositions 3.1, 3.2 and arguing as in the above Lemma, one proves that for $\varepsilon _0$ small enough, X is a pseudogradient of J which preserves $V_{\varepsilon _0}(\Sigma ^+)$. Moreover the critical points at infinity of X lie in subsets $\mathcal {W}$ (see the formula (26) for a definition)

Next we perform a Morse type reduction in the subsets $\mathcal {W}$. Namely we prove

Lemma 3.17

For $u = \sum _{i=1}^{q} \alpha _i \delta _{a_i,\lambda _i} + \sum _{q+1}^{p+q} \alpha _i \varphi _{a_i,\lambda _i}\in \mathcal {W}$, we define

$$ \begin{aligned} D_4:= \{ i \le q: a_i \text{ is } \text{ close } \text{ to } z \text{ with } \frac{\partial K}{\partial \nu }(z) = 0\} \, \, \& \, \, D_5 := \{ i \le q: a_i \text{ is } \text{ close } \text{ to } z \text{ with } \frac{\partial K}{\partial \nu }(z) > 0\}. \end{aligned}$$

Then the functional $J_K$ expands as follows

$$\begin{aligned} J_K(u)&= \frac{(\sum _{i\le q }\alpha _i^2+2 \sum _{i> q}\alpha _i^2)S_n^{2/n}}{(\sum _{i\le q }\alpha _i^{\frac{2n}{n-2}}K(a_i)+2\sum _{i> q } \alpha _i^{\frac{2n}{n-2}}K(a_i))^{\frac{n-2}{n}}}\Big ( 1- c \sum _{i> q } \frac{\Delta K(y_i)}{\lambda _i^2} + c \sum _{i \in D_5} \frac{1}{\lambda _i}\frac{\partial K}{\partial \nu }(z_i) \nonumber \\&\quad + c \sum _{i \in D_4} \Big ( \frac{c_7}{\lambda _i}\frac{\partial K}{\partial \nu }(a_i) - c_6 \frac{\Delta K(a_i)}{\lambda _i^2}\Big ) + o \Big ( \sum _{i \in D_5} \frac{1}{\lambda _i}+ \sum _{i \in D4} \frac{1}{\lambda _i^2}+ \sum _{i> q } \frac{1}{\lambda _i^2} \Big ) \Big )\nonumber \\&= S_n^{2/n} \Big ( \sum _{i \le q } \frac{1}{K(z_i)^{\frac{n-2}{2}}} + 2 \sum _{i> q } \frac{1}{K(y_i)^{\frac{n-2}{2}}} \Big )^\frac{2}{n} \Big ( 1 - \Vert \alpha \Vert ^2 + \sum _{i=1}^{p+q} \big ( | A_i^-|^2 - | A_i^+| \big ) - c \sum _{i> q } \frac{\Delta K(y_i)}{\lambda _i^2} \\&\quad + c \sum _{i \in D_5} \frac{1}{\lambda _i}\frac{\partial K}{\partial \nu }(z_i) + c \sum _{i \in D_4} \Big ( \frac{c_7}{\lambda _i}\frac{\partial K}{\partial \nu }(a_i) - c_6 \frac{\Delta K(z_i)}{\lambda _i^2}\Big ) + o \Big ( \sum _{i \in D_5} \frac{1}{\lambda _i}+ \sum _{[i > q]\cup [i \in D_4] } \frac{1}{\lambda _i^2} \Big ) \Big ), \nonumber \end{aligned}$$

(46)

where $S_n$ is defined in Proposition 5.3 (it represents the level of one boundary bubble), $\alpha \in \mathbb {R}^{q+p-1}$, $(A_i^+,A_i^-)$ are the local coordinates of the parameters $(\alpha _1,\ldots ,\alpha _{p+q})$ 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 $\alpha$ since the functional $J_K$ is homogenous with respect to this parameter.

From Propositions 3.1, 3.2 and Lemma 3.17, we derive the characterization of critical points at infinity and identify their level sets. Namely we have:

Corollary 3.18

Assume that K satisfies (H1), (H2) and (H3). Then, in $V(m,q,p,\varepsilon )$, the critical points at infinity of $J_K$ are in one to one correspondence with the collections of q critical points $z_\ell$’s of $K_1$ satisfying: either $z_\ell$ is a local maximum point with $\partial K/ \partial \nu (z_\ell ) >0$ or $\partial K/ \partial \nu (z_\ell )=0$ and $\Delta K(z_\ell ) < 0$ and p critical points $y_r$’s of K in $\mathbb {S}^n_+$ satisfying $\Delta K(y_r) < 0$. We will denote such a critical point at infinity by $(z_1, \ldots ,z_q, y_{q+1}, \ldots ,y_{q+p})_\infty$. Such a critical point at infinity is at the level (see (46))

$$\begin{aligned} C_\infty (z_1, \ldots ,z_q, y_{q+1}, \ldots ,y_{q+p}):= S_n^{2/n} \Big ( \sum _{i=1}^q \frac{1}{K(z_i)^{(n-2)/2}} + \sum _{i=q+1}^{q+p} \frac{2}{K(y_i)^{(n-2)/2}}\Big )^{2/n}. \end{aligned}$$

In particular, it holds that

$$\begin{aligned} C_{\min }^{(2p + q),\infty } :=\frac{\big ((2p + q ) S_n\big )^{2/n}}{K_{\max }^{(n-2)/n}} \le C_\infty (z_1, \ldots ,z_q, y_{q+1}, \ldots ,y_{q+p}) \le \frac{\big ( (2p + q ) S_n\big )^{2/n} }{K_{\min }^{(n-2)/n}}:= C_{\max }^{(2p + q),\infty } \end{aligned}$$

Furthermore, for such a critical point at infinity, we associate an index (which corresponds to the number of the decreasing directions for $J_K$ by using the Morse Lemma at infinity, see (46))

$$\begin{aligned} \ i_\infty (z_1, \ldots ,z_q, y_{q+1}, \ldots ,y_{q+p}):= & {} q+p-1 + \sum _{i=1}^q (n-1 - morse(K_1, z_i))\\+ & {} \sum _{i= q+1} ^{q+p} (n- morse (K,y_i)). \end{aligned}$$

Such an index will be called the $i_{\infty }$-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 $\tau _{\infty }$ be a critical point at infinity at the level $C_{\infty }(\tau _{\infty })$ with index $i_{\infty }(\tau _{\infty })$. Then for $\theta$ a small positive number and a field $\mathbb {F}$, we have that

$$\begin{aligned} H_l(J_K^{C_{\infty }(\tau _{\infty }) + \theta }, J_K^{C_{\infty }(\tau _{\infty }) - \theta }; \mathbb {F} ) = {\left\{ \begin{array}{ll} \mathbb {F} &{} \text{ if } \quad l = i_{\infty }(\tau _{\infty }), \\ 0, &{} \text{ otherwise }. \end{array}\right. } \end{aligned}$$

where $H_l$ denotes the $l-$dimensional homology group with coefficient in the field $\mathbb {F}$.

4 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:

Lemma 4.1

Let $\underline{A}$ and $\overline{A}:= ({ K_{\max }}/{ K_{\min }})^{(n-2)/n} \, \underline{A}$. Assume that $J_K$ does not have any critical point nor critical point at infinity in the set $J_K^{\overline{A}} {\setminus } J_K^{\underline{A}}$ where $J_K^A := \{ u: J_K(u) < A\}$. Then for each $c\in [ \underline{A},\overline{A}]$, the level set $J_K^c$ is contractible.

Proof

First, since we assumed that $J_K$ does not have any critical point nor critical point at infinity in $\Sigma ^+$ between the levels $\overline{A}$ and $\underline{A}$, we have that $J_K^{\overline{A}}$ retracts by deformation onto $J_K^{\underline{A}}$. Indeed such a retraction can be realized by following the flow lines of a decreasing pseudogradient $Z_K$ for $J_K$. Let $\phi _K$ denote the one parameter group corresponding to this pseudogradient. For each $u\in \Sigma ^+$, we denote by $s_K(u)$ the first time such that $\phi _K(s_K(u), u ) \in J_K^{\underline{A}}$.

Secondly we recall that, for $K\equiv 1$, 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 $J_1^A$ 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 $u\in \Sigma ^+$, will reach the bottom level $S_n$. Let us denote by $\phi _1$ the one parameter group corresponding to $Z_1$.

Next we notice that, we have

$$\begin{aligned} (1/ K_{\max }^{(n-2)/n}) J_1 (u) \le J_K (u) \le (1/ K_{\min }^{(n-2)/n}) J_1 (u) \quad \text{ for } \text{ each } u\in \Sigma , \end{aligned}$$

which implies that

$$\begin{aligned} J_K ^{\underline{A}} \subset J_1 ^{A'} \subset J_K ^{\overline{A}} \quad \text{ where } A':= K_{\max }^{(n-2)/n} \underline{A}.\end{aligned}$$

Furthermore we observe that for each $u \in \Sigma ^+$, there exists a unique $s_1(u)$ satisfying $\phi _1(s_1(u), u ) \in J_1^{A'}$.

Next we define the following map:

$$\begin{aligned} F:= [ 0,1] \times J_1^{A'} \rightarrow J_1^{A'}; \quad F(t,u) := \phi _1(s_1(\phi _K(t\, s_K(u), u )), \phi _K(t\, s_K(u), u ) ).\end{aligned}$$

We notice that F is well defined and continuous and satisfies the following properties:

• For $t=0$, we have $\phi _K(0, u )=u$. Furthermore, for each $u\in J_1^{A'}$, we have $s_1(u) = 0$. Therefore, for each $u\in J_1^{A'}$, we get $F(0,u)= \phi _1(0,u) = u$.

• For $t=1$, we have $\phi _K( s_K(u), u ) \in J_K^{\underline{A}} \subset J_1^{A'}$ (by the definition of $s_K$) which implies that $s_1(\phi _K( s_K(u), u ))= 0$ and therefore $F(1,u)= \phi _1(0,\phi _K( s_K(u), u )) = \phi _K( s_K(u), u ) \in J_K^{\underline{A}}$ for each $u \in J_1^{A'}$.

• If $u\in J_K^{\underline{A}}$, then $s_K(u)=0$ which implies that $\phi _K(t\, s_K(u), u ) = \phi _K( 0, u ) =u$. Therefore $F(t,u) = \phi _1(s_1(u),u) = \phi _1(0,u)=u$ for each $u\in J_K^{\underline{A}}$ and each $t\in [0,1]$ (we used $s_1(u)=0$ since $u\in J_K^{\underline{A}}\subset J_1^{A'}$).

Thus $J_1^{A'}$ retracts by deformation onto $J_K^{\underline{A}}$, a fact which provides the claim of the lemma since $J_1^{A'}$ itself is a contractible set. $\square$

The second deformation lemma is a consequence of the previous one, the assumptions (H1), (H2), (H3) of this paper and an appropriate pinching condition for the function K. To state it we set the following notation:

$$ \begin{aligned} \text{ for } \ell \in \mathbb {N}, \quad C^{\ell , \infty }_{\max } := (\ell S_n)^{2/n} / K_{\min } ^{(n-2)/n} \quad \& \quad C^{\ell , \infty }_{\min } := (\ell S_n)^{2/n} / K_{\max } ^{(n-2)/n}. \end{aligned}$$

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 $q + 2p= \ell$ lie between $C^{\ell , \infty }_{\min }$ and $C^{\ell , \infty }_{\max }$.

Our second deformation lemma reads as follows:

Proposition 4.2

For $k\in \mathbb {N}$ a fixed integer, let $0 < K \in C^3( \overline{\mathbb {S}^n_+})$ satisfying the conditions (H1), (H2), (H3) and the pinching condition $K_{\max } / K_{\min } < ((k+1)/k)^{1/(n-2)}$.

Assume that $J_K$ does not have any critical point under the level $C^{k+1,\infty } _{\min }$. Then, for every $1\le \ell \le k$ and every $c\in (C^{\ell ,\infty }_{\max }, C^{\ell +1,\infty }_{\min } )$, the sublevel $J_K^c$ is a contractible set.

Proof

Since we assumed that $K_{\max } / K_{\min } < ((k+1)/k)^{1/(n-2)}$, it follows that, for each $1\le \ell \le k$, we have $(k+1)/k \le (\ell +1)/\ell$ and

$$\begin{aligned} C^{\ell , \infty }_{\max }< C^{\ell , \infty }_{\max } (K_{\max } / K_{\min } )^{(n-2)/n} < C^{\ell +1 , \infty }_{\min } .\end{aligned}$$

The proof follows then from Lemma 4.1 by taking $\underline{A}= C^{\ell , \infty }_{\max } + \gamma$ with a small $\gamma >0$ so that $\overline{A} < C^{\ell +1, \infty }_{\min }$. Indeed between the levels $\underline{A}$ and $\overline{A}$ the functional $J_K$ does not have any critical point nor critical point at infinity. $\square$

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 $C_{\min }^{2,\infty }$. Hence it follows from Proposition 4.2 (with $k=1$) that under the assumption of Theorem 1.3, we have that $J^{C_{\max }^{1,\infty } +\gamma }$ is a contractible set, for $\gamma$ a small constant. Moreover it is a retract by deformation of $C_{\min }^{2,\infty }$. Furthermore follows from corollary 3.18 that critical points at infinity under the level $C_{\min }^{2,\infty }$ are in one to one correspondence with critical points of $K_1$ in $\mathcal {K}^{+}_b\cup \mathcal {K}^{0,-}_b$. Then it follows from Lemma 3.19 and the Euler–Poincaré theorem that:

$$\begin{aligned} 1 = \chi ( J^{C_{\min }^{2,\infty } +\gamma }) = \sum _{ z\in \mathcal {K}^{+}_b\cup \mathcal {K}^{0,-}_b} (-1)^{n-1- morse (K_1,z)}\end{aligned}$$

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 $C_{\min }^{3,\infty }$, we derive, using Proposition 4.2 (with $k=2$), the level sets $J_K^{C_{\max }^{1,\infty } +\gamma }$ and $J_K^{C_{\max }^{2,\infty } +\gamma }$ are contractible sets. Then it follows from the properties of the Euler-Characteristic, see Proposition 5.7, pp.105 in [26], that

$$\begin{aligned} 1 = \chi (J_K^{C^{2,\infty }_{\max }+\gamma }) \, = \, \chi (J_K^{C^{2,\infty }_{\max }+\gamma }, J_K^{C^{1,\infty }_{\max }+\gamma }) \, + \, \chi (J_K^{C^{1,\infty }_{\max }+\gamma }). \end{aligned}$$

That is $\chi (J_K^{C^{2,\infty }_{\max }+\gamma }, J_K^{C^{1,\infty }_{\max }+\gamma }) \, = \, 0.$ Moreover it follows from Corollary 3.18 that the critical points at infinity between these two levels are $(z_i,z_j)_\infty$ with $z_i \ne z_j \in \mathcal {K}^{+}_b\cup \mathcal {K}^{0,-}_b$ and $y_\infty$ with $y \in \mathcal {K}^{-}_{in}$. Thus, it follows from Lemma 3.19 and the Euler–Poincaré theorem that

$$\begin{aligned} \sum _{z_i \ne z_j \in \mathcal {K}^{+}_b\cup \mathcal {K}^{0,-}_b} (-1)^{1+ \iota (z_i) + \iota (z_j)} + \sum _{y \in \mathcal {K}^{-}_{in}} (-1)^{\iota (y)} = 0\end{aligned}$$

where $\iota (z_k):= n-1-\text{ morse }(K_1,z_k)$ and $\iota (y):= n-\text{ morse }(K,y)$.

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 $A_1 \ne 1$ or respectively $A_1 =1$ and $B_1 \ne -k$, where $\#( \mathcal {K}^{+}_b\cup \mathcal {K}^{0,-}_b)= 2k+1$, the existence of at least one solution to Problem $(\mathcal {P})$ follows from Theorem 1.4, respectively Theorem 1.3. Hence we will assume that $A_1=1$ and $B_1=-k$ and notice that

$$\begin{aligned} \# ( \mathcal {K}_{in}^{-}) \, = 2 r + k, \text{ where } r \in \mathbb {N}_0, \end{aligned}$$

and there are r even numbers $\iota (y_j)$’s and $r+k$ odd numbers $\iota (y_j)$’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 $J_K^{C_{\max }^{3,\infty } +\gamma }$ and $J_K^{C_{\max }^{4,\infty } +\gamma }$ are contractible sets. Using Corollary 3.18, we derive that the critical points at infinity whose level are lying between these values are:

• $(z_i,z_j,z_r,z_t)_\infty$ with different $z_i$’s which belong to $\mathcal {K}^{+}_b\cup \mathcal {K}^{0,-}_b$,

• $(z_i,z_j,y)_\infty$ with $y \in \mathcal {K}^{-}_{in}$ and $z_i \ne z_j \in \mathcal {K}^{+}_b\cup \mathcal {K}^{0,-}_b$,

• $(y_i,y_j)_\infty$ with $y_i \ne y_j \in \mathcal {K}^{-}_{in}$.

Hence arguing as above we derive that

$$\begin{aligned}&\sum _{z_i \ne z_j\ne z_r\ne z_t \in \mathcal {K}^{+}_b\cup \mathcal {K}^{0,-}_b} (-1)^{3+ \iota (z_i) + \iota (z_j) + \iota (z_r) + \iota (z_t)} \\&\quad + \sum _{y \in \mathcal {K}^{-}_{in};z_i \ne z_j \in \mathcal {K}^{+}_b\cup \mathcal {K}^{0,-}_b} (-1)^{2 + \iota (z_i) + \iota (z_j) +\iota (y)} + \sum _{y_i \ne y_j \in \mathcal {K}^{-}_{in}} (-1)^{1+ \iota (y_i)+\iota (y_j)} = 0.\end{aligned}$$

Observe that, the first term is exactly $-A_4$, the second one is $A_2\times B_1$ 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

$$\begin{aligned}r+k=0\end{aligned}$$

which implies that $r=k=0$. Now, from $r=k=0$, we get $\#(\mathcal {K}^{+}_b\cup \mathcal {K}^{0,-}_b) = 1$ and $\# \mathcal {K}^{-}_{in} = 0$. This leads to a contradiction with the assumption that $\#(\mathcal {K}^{+}_b\cup \mathcal {K}^{0,-}_b\cup \mathcal {K}^{-}_{in}) \ge 2$. Thereby the proof of the theorem is completed.

Appendix

1.1 Bubble estimates

Lemma 5.1

For $a\in \partial \mathbb {S}_+^n$, we have $\partial \delta _{a,\lambda } /\partial \nu = 0$ and therefore $\varphi _{a,\lambda } = \delta _{a,\lambda }$. For $a\notin \partial \mathbb {S}_+^n$, we have

$$\begin{aligned} \mathrm{(i)} \quad \delta _{a,\lambda } \le \varphi _{a,\lambda } \le 2 \delta _{a,\lambda }\, \, ; \quad | \lambda \partial \varphi _{a,\lambda } /\partial \lambda | \le c \delta _{a,\lambda }; \quad | (1/\lambda ) \partial \varphi _{a,\lambda } /\partial a^k | \le c \delta _{a,\lambda },\end{aligned}$$

where $a^k$ denotes the k-th component of a.

$$\begin{aligned}&\mathrm{(ii)} \qquad \varphi _{a,\lambda } = \delta _{a,\lambda } + c_0 \frac{H(a,.)}{\lambda ^{(n-2)/2}} + f_{a,\lambda }\qquad \text{ where } \\&\quad | f_{a,\lambda } | _{\infty } \le \frac{c}{\lambda ^{(n+2)/2} d_a^{n}}; \quad | \lambda \frac{\partial f_{a,\lambda } }{\partial \lambda } | _{\infty } \le \frac{c}{\lambda ^{(n+2)/2} d_a^{n}} \quad \text{ and } \quad | \frac{1}{\lambda } \frac{\partial f_{a,\lambda } }{\partial a^k} | _{\infty } \le \frac{c}{\lambda ^{(n+4)/2} d_a^{n+1}},\end{aligned}$$

where $d_a:= d(a,\partial \mathbb {S}^n_+$).

Proof

Using a stereographic projection, we are led to prove the corresponding estimates on $\mathbb {R}^n_+$. We still denote by G and H the Green’s function and its regular part of Laplacian on $\mathbb {R}^n_+$ under Neumann boundary conditions. In this case, we have

$$\begin{aligned} \delta _{a,\lambda }(x):= c_0 \frac{\lambda ^{(n-2)/2}}{ (1+\lambda ^2| x-a|^2)^{(n-2)/2}} \qquad \text{ and } \qquad H(a,x) := \frac{1}{ | x - \overline{a} | ^{n-2}}, \end{aligned}$$

where $\overline{a}$ denotes the symmetric point of a with respect to $\partial \mathbb {R}^n_+$. Let $\psi := \delta _{a,\lambda } + \delta _{\overline{a},\lambda }$. Easy computation implies that $\partial \psi /\partial \nu = 0$.

To prove the first inequality, let us consider $h:= \varphi _{a,\lambda } - \delta _{a,\lambda }$. Hence we get $\Delta h = 0$ and $\partial h /\partial \nu = -\partial \delta _{a,\lambda } /\partial \nu > 0$. Hence, using the Green’s representation, we derive that $h > 0$ in $\mathbb {R}^n_+$.

For the second inequality, let us consider $h := \psi - \varphi _{a,\lambda }$. Easy computations imply that $\partial h /\partial \nu = 0$ and $-\Delta h = - \Delta \delta _{\overline{a},\lambda } > 0$. Hence, $h >0$ in $\mathbb {R}^n_+$. The inequality follows from the fact that $\delta _{\overline{a},\lambda } \le \delta _{{a},\lambda }$ in $\mathbb {R}^n_+$.

For the third one, let $g:= \lambda \partial \varphi _{a,\lambda } /\partial \lambda$, observe that $\partial g /\partial \nu = 0$ and $| \Delta g | \le ((n+2)/2) \delta _{a,\lambda }^{(n+2)/(n-2)}$. Now let us consider $h:= ((n+2)/2) \psi \pm g$. It follows that $-\Delta h > 0$ and $\partial h /\partial \nu = 0$. Hence $h >0$ in $\mathbb {R}^n_+$ which gives the proof of the third inequality. The fourth one follows by the same way.

Concerning the second claim, it is easy to see that $\Delta f_{a,\lambda } = 0$ and

$$\begin{aligned} \frac{\partial f_{a,\lambda }}{\partial \nu }&= - \frac{\partial \delta _{a,\lambda }}{\partial \nu } - \frac{c_0}{\lambda ^{(n-2)/2}} \frac{\partial H(a,.)}{\partial \nu }= c_0 (n-2) \frac{\lambda ^{(n+2)/2} d_a}{(1+ \lambda ^2 | x-a|^2)^{n/2}} - \frac{c_0}{\lambda ^{(n-2)/2}} \frac{(n-2) d_a}{ | x-\overline{a} |^n}\\&= O \Big ( \frac{d_a}{ \lambda ^{(n+2)/2} | x-a|^{n+2} } \Big ). \end{aligned}$$

Now, using the Green’s representation, we get

$$\begin{aligned} | f_{a,\lambda } (x)|&\le c \int _{\partial \mathbb {R}^n_+} G(x,y) | \frac{\partial f_{a,\lambda }}{\partial \nu } (y)| dy \le \frac{c\, d_a}{ \lambda ^{(n+2)/2}} \int _{\partial \mathbb {R}^n_+} G(x,y) \frac{1}{ | y-\overline{a} |^{n+2} }dy \\&\le \frac{c}{ \lambda ^{(n+2)/2}d_a } \int _{\partial \mathbb {R}^n_+} G(x,y) \frac{1}{ | y-\overline{a} |^{n} }dy \le \frac{c}{ \lambda ^{(n+2)/2}d_a } \frac{H(a,x)}{d_a} \le \frac{c}{ \lambda ^{(n+2)/2}d_a^{n} }. \end{aligned}$$

This gives the first claim in (ii). The other ones can be done by the same way. $\square$

Lemma 5.2

1) For each $i\ne j$, we have

$$\begin{aligned} - \lambda _i \frac{\partial \varepsilon _{ij}}{\partial \lambda _i} - \lambda _j \frac{\partial \varepsilon _{ij}}{\partial \lambda _j } \ge 0\, \, \text{ and } - \lambda _i \frac{\partial \varepsilon _{ij}}{\partial \lambda _i}\ge c \varepsilon _{ij} \text{ if } \lambda _i \ge c \lambda _j \text{ or } \lambda _i d(a_i, a_j ) \ge 2. \end{aligned}$$

2) Let $i,j \in I_b:=\{k:a_k\in \partial \mathbb {S}^n_+\}$ and let $\mu _i$ and $\mu _j$ be defined by (15). Assume that $\mu _j \le c' \mu _i$ for some constant $c'$, then: (i) either there exists a constant $c''$ such that $\lambda _j \le c'' \lambda _i$, (ii) or $\lambda _i d(a_i,a_j ) \ge 2$.

Proof

The proof of the first assertion follows immediately from the definition of $\varepsilon _{ij}$. Concerning the second one, observe that, if $| \nabla K(a_i) | \ge c$ and $| \nabla K(a_j) | \ge c$, then it follows that $\mu _k$ and $\lambda _k$ are of the same order (that is: the ratio is bounded from above and below) for $k=i,j$. Hence the result follows in this case. In the other case, there exists $k\in \{i,j\}$ such that $a_k$ is close to a critical point z of K in $\partial \mathbb {S}^n_+$ (i.e. $\partial K/\partial \nu (z) =0$). Arguing by contradiction, assume that $\lambda _i d(a_i,a_j ) \le 2$ and $\lambda _j / \lambda _i$ is very large. It follows that $a_i$ and $a_j$ are close to the same critical point z. Now we claim that:

Claim 1: $\lambda _j | \nabla K(a_j) |$ is very large.

In fact, if it is not, we derive that $| \nabla K(a_j) | /\lambda _j \le c/\lambda _j^2$ which implies that $1/\mu _j \le c/\lambda _j^2$ and therefore $1/\mu _j$ is very small with respect to $1/\lambda _i^2 \le 1/\mu _i$. This gives a contradiction and therefore our claim follows.

Since z is a non degenerate critical point of $K_1$, it follows that $\lambda _j d( a_j, z )$ is very large. Moreover, Claim 1 implies that $| \nabla K(a_j) | /\lambda _j \le 1/\mu _j \le c | \nabla K(a_j) | /\lambda _j$. Now we claim that:

Claim 2: $\lambda _i | \nabla K(a_i) | \ge 1$ cannot occur.

To prove this claim, we assume that the inequality is true. Then we derive that $| \nabla K(a_i) | /\lambda _i \le 1/\mu _i \le 2 | \nabla K(a_i) | /\lambda _i$. Since $\mu _j \le c' \mu _i$, we derive that $| \nabla K(a_i) | /\lambda _i \le c | \nabla K(a_j) | /\lambda _j$ and therefore $\lambda _j d( a_i, z ) \le c \lambda _i d( a_j, z )$ which implies that $d( a_i,z )$ is very small with respect to $d(a_i,z)$ and therefore $d( a_i, z )$ is very small with respect to $d( a_i, a_j )$. Now observe that, since we assumed that $\lambda _i | \nabla K(a_i) | \ge 1$, it follows that $\lambda _i d( a_i,z ) \ge c$ and therefore $\lambda _i d( a_i, a_j )$ becomes very large which gives a contradiction. Hence Claim 2 follows.

Finally, we claim that

Claim 3: $\lambda _i | \nabla K(a_i) | \le 1$ cannot occur.

Arguing by contradiction we assume that $\lambda _i d( a_i, z ) \le c$. From $\mu _j \le c' \mu _i$, we derive that $1/\lambda _i^2 \le c| \nabla K(a_j) | /\lambda _j \le c d(a_j,z ) /\lambda _j$ and therefore $\lambda _j/\lambda _i \le c \lambda _i d(a_j, z )$, that is $\lambda _i d( a_j, z )$ is very large. But we have $\lambda _i d( a_j , a_i ) \le 2$ and $\lambda _i d( a_i, z ) \le c$ which imply that $\lambda _i d( a_j , z )$ is bounded. Hence we get a contradiction which completes the proof of Claim 3.

Hence the lemma is fully proven. $\square$

1.2 Asymptotic expansion of the functional and its gradient

Proposition 5.3

Let $n\ge 5$ and $u=\sum _{i\le q} \alpha _i \delta _i + \sum _{i > q} \alpha _i \varphi _i \in V(m,q,p,\varepsilon )$ be such that: (i) $d( a_i, a_j ) \ge c > 0$ for every $i\ne j$, (ii) for $i > q$, $a_i$ is close to a critical point $y_{j_i}$ of K in $\mathbb {S}^n_+$ and (iii) for $i \le q$, $a_i$ is close to a critical point $z_{j_i}$ of $K_1$ in $\partial \mathbb {S}^n_+$. Then the following expansion holds

$$\begin{aligned} J_K(u)= & {} \frac{(\sum _{i\le q }\alpha _i^2+2 \sum _{i> q}\alpha _i^2)S_n^{2/n}}{(\sum _{i\le q }\alpha _i^{{2n}/{n-2}}K(a_i)+2\sum _{i> q } \alpha _i^{{2n}/{n-2}}K(a_i))^{{n-2}/{n}}}\Big ( 1 - 2 c_6 J_K(u)^{\frac{n}{n-2}} \sum _{i > q } \alpha _i ^\frac{2n}{n-2} \frac{\Delta K(a_i)}{\lambda _i^2} \\&+ J_K(u)^{\frac{n}{n-2}} \sum _{i \le q} \alpha _i ^\frac{2n}{n-2} \Big ( \frac{c_7}{\lambda _i}\frac{\partial K}{\partial \nu }(a_i) - c_6 \frac{\Delta K(a_i)}{\lambda _i^2}\Big ) + \sum _{i =1} ^{p+q} O \Big ( \frac{1}{\lambda _i^3}+ \frac{ | \nabla K(a_i) |^2}{\lambda _i^2} \Big ) \Big )\end{aligned}$$

where

$$\begin{aligned} S_n:= & {} c_0^\frac{2n}{n-2} \int _{\mathbb {R}^n_+} \frac{ dx }{(1+| x| ^2)^n}; \, c_6:= \frac{n-2}{n^2} c_0^\frac{2n}{n-2} \int _{\mathbb {R}^n_+} \frac{| x |^2 dx }{(1+| x|^2)^n}; \\ c_7:= & {} 2 \frac{n-2}{n} c_0^\frac{2n}{n-2} \int _{\mathbb {R}^n_+} \frac{x_n dx }{(1+| x|^2)^n} \end{aligned}$$

Proof

From the definition of $J_K$, we need to expand (using the fact that $\overline{v} \perp \varphi _i$ for each i)

$$\begin{aligned} \Vert u\Vert ^2= & {} \sum \alpha _i^2 \Vert \varphi _i \Vert ^2 + \Vert \overline{v} \Vert ^2 + O\left( \sum \varepsilon _{ij}\right) \\= & {} S_n \left( \sum _{i\le q }\alpha _i^2+2 \sum _{i> q}\alpha _i^2\right) + \Vert \overline{v} \Vert ^2 + O\left( \sum \varepsilon _{ij} + \sum _{ i > q} \frac{1}{\lambda _i^{n-2}} \right) , \\ \int _{\mathbb {S}^n_+} K u^\frac{2n}{n-2}= & {} \sum _{i=1}^{q+p} \alpha _i ^\frac{2n}{n-2} \int _{\mathbb {S}^n_+} K \varphi _i ^\frac{2n}{n-2} + \frac{2n}{n-2} \int _{\mathbb {S}^n_+} K \big ( \sum \alpha _i \varphi _i\big )^\frac{n+2}{n-2} \overline{v} \\+ & {} O\Big ( \sum _{i\ne j} \int \varphi _i ^\frac{n+2}{n-2} \varphi _j + \Vert \overline{v} \Vert ^2 \Big ).\end{aligned}$$

The last integral is equal to $O(\varepsilon _{ij})$. The second one is presented in (9). Concerning the first one, for $i > q$, using Lemma 5.1, we get

$$\begin{aligned} \int _{\mathbb {S}^n_+} K \varphi _i ^\frac{2n}{n-2}&= \int _{\mathbb {R}^n_+} \widetilde{K} \delta _i ^\frac{2n}{n-2} + O\Big ( \frac{1}{\lambda _i ^{(n-2)/2}}\int \delta _i ^\frac{n+2}{n-2}\Big ) \\&= \int _{B(a_i,d_i)} \widetilde{K} \delta _i ^\frac{2n}{n-2} + O\Big ( \int _{\mathbb {R}^n {\setminus } B(a_i, d_i) } \delta _i^{\frac{2n}{n-2}} + \frac{1}{\lambda _i ^{(n-2)/2}}\int \delta _i ^\frac{n+2}{n-2}\Big )\\&= 2 S_n \widetilde{K}(a_i) + \frac{1}{2n} \frac{\Delta \widetilde{K}(a_i) }{\lambda _i^2} c_0^\frac{2n}{n-2} \int _{\mathbb {R}^n} \frac{| x | ^2 }{(1+| x | ^2)^n}dx + O\Big ( \frac{1}{\lambda _i^3}\Big ). \end{aligned}$$

However, for $i\le q$, we have $\varphi _i = \delta _i$ and therefore

$$\begin{aligned} \int _{\mathbb {S}^n_+} K \delta _i ^\frac{2n}{n-2}&= \int _{\mathbb {R}^n_+} \widetilde{K} \delta _i ^\frac{2n}{n-2} = S_n \widetilde{K}(a_i) + \nabla \widetilde{K}(a_i) c_0^\frac{2n}{n-2} \int _{\mathbb {R}^n_+ } \frac{ \lambda _i^n (x -a_i) }{(1+\lambda _i^2| x-a_i | ^2)^n}dx \\&\quad + \frac{1}{2} \sum \frac{\partial ^2 \widetilde{K}(a_i)}{\partial x_k\partial x_\ell } c_0^\frac{2n}{n-2} \int _{\mathbb {R}^n_+} \frac{\lambda _i^n ( x -a_i)_k (x-a_i)_\ell }{(1+ \lambda _i ^2| x -a_i | ^2)^n}dx + O\Big ( \frac{1}{\lambda _i^3}\Big )\\&= S_n \widetilde{K}(a_i) - \frac{\partial \widetilde{K}}{\partial \nu }(a_i) c_0^\frac{2n}{n-2} \int _{\mathbb {R}^n_+} \frac{x_n }{(1+| x | ^2)^n}dx \\&\quad + \frac{1}{2n} \frac{\Delta \widetilde{K}(a_i) }{\lambda _i^2} c_0^\frac{2n}{n-2} \int _{\mathbb {R}^n_+} \frac{| x | ^2 }{(1+| x | ^2)^n}dx + O\Big ( \frac{1}{\lambda _i^3}\Big ). \end{aligned}$$

Note that, since $u\in \Sigma$, we deduce that

Now, the precise expansion of $J_K$ follows from the above estimates, the estimate of $\Vert \overline{v}\Vert$ (see Lemma 2.3) and the fact that $(1+x)^{-(n-2)/n}= 1 -((n-2)/n) x + O(x^2)$. $\square$

In the following, we will present the expansion of the gradient of $J_K$ in the potential sets. We will present the results for $p+q \ge 2$. However, the results are true for $p+q =1$, it suffices to remove the terms $\varepsilon _{ij}$’s which correspond to the interaction terms of the bubbles.

Proposition 5.4

Let $n\ge 5$, for $u=\sum _{i\le q} \alpha _i \delta _i + \sum _{i > q} \alpha _j \varphi _j \in V(m,q,p,\varepsilon )$ and $i \le q$, it holds

$$\begin{aligned}&\langle \nabla J_K(u),\lambda _i \frac{\partial \delta _i }{\partial \lambda _i} \rangle = 2J_K(u)\Big [ - \frac{c_2}{2}\sum _{j\ne i; j\le q} \alpha _j \lambda _i\frac{\partial \varepsilon _{ij}}{\partial \lambda _i}(1+o(1)) \\&\quad + 2J_K(u)^\frac{n}{n-2}\alpha _i^\frac{n+2}{n-2}\Big ( - \frac{c_3}{\lambda _i}\frac{\partial K}{\partial \nu }(a_i) + c_9\frac{\Delta K(a_i)}{\lambda _i ^2}\Big )\Big ] +O\Big (\frac{1}{\lambda _i^3} + \sum _{j > q} \varepsilon _{ij} + R_1^b \Big ) \end{aligned}$$

where

$$\begin{aligned} R_1^b := \sum _{k \le q} \Big (\frac{ | \nabla K(a_k) | }{\lambda _k }\Big )^{\frac{n}{2}} + \Big ( \frac{1}{\lambda _k^2}\Big )^{\frac{n+1}{3}} + \sum _{j\ne k; j,k \le q} \varepsilon _{kj }^{\frac{n}{n-2}}\ln (\varepsilon _{kj }^{-1}) \, \, \, ; \, \, \, c_3=\frac{n-2}{2}c_0^{\frac{2n}{n-2}}\int _{\mathbb {R}_+^n}\frac{x_n(|x|^2-1)}{(1+|x|^2)^{n+1}}dx .\end{aligned}$$

Proof

$$\begin{aligned} \langle \nabla J_K(u),\lambda _i \frac{\partial \delta _i }{\partial \lambda _i} \rangle = 2J_K(u)\Big (\sum _{j\le q} \alpha _j \langle \delta _j ,\lambda _i \frac{\partial \delta _i }{\partial \lambda _i} \rangle - J(u)^{n/(n-2)} \int K \big ( \sum _{j \le q} \alpha _j \delta _j\big )^\frac{n+2}{n-2} \lambda _i \frac{\partial \delta _i }{\partial \lambda _i} + \sum _{j > q} O(\varepsilon _{ij})\Big ).\end{aligned}$$

For $j \le q$, we have $a_j \in \partial \mathbb {S}_+^n$ and therefore, using [5], we get, for $j\ne i$,

$$\begin{aligned}&\langle \delta _j ,\lambda _i \frac{\partial \delta _i }{\partial \lambda _i} \rangle = \int _{\mathbb {R}^n_+} \delta _j^\frac{n+2}{n-2} \lambda _i \frac{\partial \delta _i }{\partial \lambda _i} = \frac{1}{2}\int _{\mathbb {R}^n} \delta _j^\frac{n+2}{n-2} \lambda _i \frac{\partial \delta _i }{\partial \lambda _i}= \frac{1}{2} c_2 \varepsilon _{ij} + O\big ( \varepsilon _{ij }^{\frac{n}{n-2}}\ln (\varepsilon _{ij }^{-1})\big )\\&\langle \delta _i ,\lambda _i \frac{\partial \delta _i }{\partial \lambda _i} \rangle = \int _{\mathbb {R}^n_+} \delta _i^\frac{n+2}{n-2} \lambda _i \frac{\partial \delta _i }{\partial \lambda _i} = \frac{1}{2}\int _{\mathbb {R}^n} \delta _i^\frac{n+2}{n-2} \lambda _i \frac{\partial \delta _i }{\partial \lambda _i}=0.\end{aligned}$$

Concerning the other term, it holds

$$\begin{aligned}&\int K \big ( \sum _{j \le q} \alpha _j \delta _j\big )^\frac{n+2}{n-2} \lambda _i \frac{\partial \delta _i }{\partial \lambda _i} = \sum _{j \le q} \int K \big ( \alpha _j \delta _j\big )^\frac{n+2}{n-2} \lambda _i \frac{\partial \delta _i }{\partial \lambda _i} \\&\quad + \frac{n+2}{n-2} \int K (\alpha _i \delta _i)^\frac{4}{n-2} \big (\sum _{j \le q; j\ne i } \alpha _j \delta _j\big ) \lambda _i \frac{\partial \delta _i }{\partial \lambda _i} + O\Big ( \sum _{k\ne r} \int (\delta _k \delta _r ) ^\frac{n}{n-2} \Big ).\end{aligned}$$

Observe that, for $j\ne i$, expanding K around $a_j$, we get

$$\begin{aligned}&\int _{\mathbb {R}^n_+} K \delta _j ^\frac{n+2}{n-2} \lambda _i \frac{\partial \delta _i }{\partial \lambda _i}\\ {}&\quad = K(a_j) \int _{\mathbb {R}^n_+} \delta _j ^\frac{n+2}{n-2} \lambda _i \frac{\partial \delta _i }{\partial \lambda _i} + O\Big (| \nabla K(a_j) | \int _{\mathbb {R}^n_+} |x-a_j| \delta _j ^\frac{n+2}{n-2} \delta _i +\int _{\mathbb {R}^n_+} |x-a_j|^2 \delta _j ^\frac{n+2}{n-2} \delta _i \Big )\\&\quad = K(a_j) \frac{1}{2} c_2 \varepsilon _{ij} + O\Big ( \varepsilon _{ij }^{\frac{n}{n-2}}\ln (\varepsilon _{ij }^{-1}) + \frac{| \nabla K(a_j) | }{\lambda _j} \varepsilon _{ij }(\ln \varepsilon _{ij }^{-1} )^{\frac{n-2}{n}} + \frac{1 }{\lambda _j^2} \varepsilon _{ij }^\frac{n}{n+1} (\ln \varepsilon _{ij }^{-1} )^{\frac{n-2}{n+1}} \Big ) \\&\quad = K(a_j) \frac{1}{2} c_2 \varepsilon _{ij} + O\Big ( \varepsilon _{ij }^{\frac{n}{n-2}}\ln (\varepsilon _{ij }^{-1}) + \Big (\frac{| \nabla K(a_j) | }{\lambda _j} \Big )^{n/2} + \Big ( \frac{1}{\lambda _j^2}\Big )^{(n+1)/3}\Big ) ,\\&\int _{\mathbb {R}^n_+} K \delta _i ^\frac{n+2}{n-2} \lambda _i \frac{\partial \delta _i }{\partial \lambda _i} = \sum _{k} \frac{\partial K}{\partial x_k}(a_i) \int _{\mathbb {R}^n_+} (x-a_i)_k \delta _i ^\frac{n+2}{n-2} \lambda _i \frac{\partial \delta _i }{\partial \lambda _i} \\&\qquad + \frac{1}{2} \sum \frac{ \partial ^2 K}{\partial x_k \partial x_\ell } (a_i)\int _{\mathbb {R}^n_+} (x-a_i)_k (x-a_i)_\ell \delta _i ^\frac{n+2}{n-2} \lambda _i \frac{\partial \delta _i }{\partial \lambda _i} + O\Big ( \int _{\mathbb {R}^n_+} |x-a_i|^3 \delta _i ^\frac{2n}{n-2} \Big )\\&\quad = \frac{c_3}{\lambda _i} \frac{\partial K}{\partial \nu } (a_i) - c_9 \frac{\Delta K(a_i)}{\lambda _i^2} + O\big ( \frac{1}{\lambda _i^3}\big ). \end{aligned}$$

Finally, for $j\ne i$, it holds

$$\begin{aligned} \frac{n+2}{n-2} \int _{\mathbb {R}^n_+} K \delta _i^\frac{4}{n-2} \delta _j \lambda _i \frac{\partial \delta _i }{\partial \lambda _i}= & {} K(a_i) \langle \delta _j, \lambda _i \frac{\partial \delta _i }{\partial \lambda _i} \rangle + O \Big ( | \nabla K(a_i) | \int _{\mathbb {R}^n_+} |x-a_i | \delta _i ^\frac{n+2}{n-2} \delta _j \\+ & {} \int _{\mathbb {R}^n_+} |x-a_i |^2 \delta _i ^\frac{n+2}{n-2} \delta _j \Big ).\end{aligned}$$

Hence the proof follows. $\square$

Proposition 5.5

Let $n\ge 5$. For $u = \sum _{i\le q} \alpha _i \delta _i + \sum _{i > q} \alpha _j \varphi _j \in V (m ,q,p, \varepsilon )$ and $i\le q$, it holds:

$$\begin{aligned}&\langle \nabla J_K(u), \frac{1}{\lambda _i}\frac{\partial \delta _i}{\partial a_i}\rangle = 2J_K(u)\alpha _i e_n \left[ c_4\left( 1- J_K(u)^{\frac{n}{n-2}}\alpha _i ^{\frac{4}{n-2}}K(a_i)\right) + J_K(u)^{\frac{n}{n-2}}\alpha _i ^{\frac{4}{n-2}}\frac{c_5}{\lambda _i}\frac{\partial K}{\partial \nu }(a_i)\right] \\&\quad -J_K(u)c_2 \sum _{j\le q; j\ne i} \alpha _j \frac{1}{\lambda _i}\frac{\partial \varepsilon _{ij}}{\partial a_i}\Big (-1 + J_K(u)^{\frac{n}{n-2}}\sum _{k=i,j} \alpha _k ^{\frac{4}{n-2}}K(a_k)\Big )+ O\Big ( \frac{1}{\lambda _i^{2}} \Big )\\&\quad -4J_K(u)^{\frac{2(n-1)}{n-2}}\alpha _i ^{\frac{n+2}{n-2}}\frac{2c_5}{\lambda _i}\nabla _T K(a_i) + O\Big (R_1^b + \sum _{k\le q; k \ne i} \varepsilon _{ik}^{\frac{n+1}{n-2}}\lambda _k d(a_i,a_k) + \sum _{k > q } \varepsilon _{ik} \Big ) \end{aligned}$$

where $R_1^b$ is defined in Proposition 5.4 and

$$\begin{aligned} c_4 = (n-2)c_0^{\frac{2n}{n-2}}\int _{\mathbb {R}^n_+}\frac{x_n}{(1+|x|^2)^{n+1}}dx \quad \text{ and } c_5=\frac{n-2}{2n}c_0^{\frac{2n}{n-2}}\int _{\mathbb {R}^n}\frac{x_n^2}{(1+|x|^2)^{n+1}}dx. \end{aligned}$$

Proof

The proof can be done as the previous one. $\square$

Proposition 5.6

For $u = \sum \alpha _j \varphi _j \in V (m,q,p, \varepsilon )$ and $i \le q$, we have the following expansion:

$$\begin{aligned} \langle \nabla J_K(u),\delta _i\rangle =2J_K(u) \alpha _i {S_n} \left( 1-J_K(u)^{\frac{n}{n-2}}\alpha _i^{\frac{4}{n-2}}K(a_i)\right) + O\Big ( \frac{ | \nabla K(a_i) | }{\lambda _i }+ \frac{ 1 }{\lambda _i^2 }+ \sum _{j\ne i} \varepsilon _{ij} \Big ). \end{aligned}$$

where $S_n$ is defined in Proposition 5.3.

Proof

$$\begin{aligned} \langle \nabla J_K(u),\delta _i\rangle =2J_K(u) \alpha _i \Vert \delta _i\Vert ^2 - J_K(u)^{n/(n-2)} \int K \delta _i^\frac{2n}{n-2} +O\Big ( \sum \varepsilon _{ki}\Big ) . \end{aligned}$$

Observe that

$$\begin{aligned} \int K \delta _i^\frac{2n}{n-2} = K(a_i) \int \delta _i^\frac{2n}{n-2} + O\Big ( | \nabla K(a_i) | \int | x-a_i | \delta _i^\frac{2n}{n-2} + \int | x-a_i |^2 \delta _i^\frac{2n}{n-2} \Big )\end{aligned}$$

which gives the result. $\square$

Proposition 5.7

For $u=\sum _{j\le q} \alpha _j\delta _j + \sum _{j > q} \alpha _j\varphi _j \in V(m,q,p , \varepsilon )$ and for each $i \ge q+1$, we have:

$$\begin{aligned} \langle \nabla J_K(u),\lambda _i\frac{\partial \varphi _i}{\partial \lambda _i}\rangle&= 2J_K\biggl (-c_2 \sum _{ j\ne i}\alpha _j (1+o(1)) \lambda _i\frac{\partial \varepsilon _{ij}}{\partial \lambda _i} +c_2 \frac{n-2}{2} \sum _{j=q+1}^p\alpha _j (1+o(1)) \frac{H(a_i,a_j)}{(\lambda _i\lambda _j)^{(n-2)/2}}\\&+ c \alpha _i(1+o(1))\frac{\Delta K(a_i)}{\lambda _i^2K(a_i)}\biggr ) +O\Bigl ( \frac{1}{\lambda _i^{3}} + R_1\biggr ), \\ \langle \nabla J_K(u),\varphi _i\rangle&=2J_K(u) \alpha _i {S_n}\left( 1-J_K(u)^{\frac{n}{n-2}}\alpha _i^{\frac{4}{n-2}}K(a_i)\right) \\&+ O\Big ( \frac{ | \nabla K(a_i) | }{\lambda _i }+ \frac{ 1 }{\lambda _i^2 }+ \frac{ 1 }{(\lambda _id_i)^{n-2} } + \sum _{j\ne i} \varepsilon _{ij} \Big ), \end{aligned}$$

$$\begin{aligned} \langle \nabla J_K(u),\frac{1}{\lambda _i}\frac{\partial \varphi _i}{\partial a_i}\rangle .\nabla K(a_i)\ge c\frac{|\nabla K(a_i)|^2}{\lambda _i}+O\biggl (\Big (\frac{1}{\lambda _i^2}+ \frac{1}{(\lambda _id_i)^{n-2}} + \sum _{j\ne i}\varepsilon _{ij}\Big )|\nabla K(a_i)|\biggr ) \end{aligned}$$

where

$$\begin{aligned} R_1 := \sum _{k =1}^{q+p} \Big (\frac{ | \nabla K(a_k) | }{\lambda _k }\Big )^{\frac{n}{2}} + \Big ( \frac{1}{\lambda _k^2}\Big )^{\frac{n+1}{3}} + \sum _{j\ne k} \varepsilon _{kj }^{\frac{n}{n-2}}\ln (\varepsilon _{kj }^{-1}) + \sum _{j > q} \frac{1}{(\lambda _jd_j)^n}. \end{aligned}$$

1.3 Counting index formulae

Lemma 5.8

Let $z_1,\ldots ,z_N$ be N critical points of $K_1$ in $\partial \mathbb {S}^n_+$ and let $\iota (z_j):= n-1 - morse(K_1,z_j)$. Assume that

$$\begin{aligned} A_1:= \sum _{j=1}^N (-1)^{i(z_j)} =1.\end{aligned}$$

Then the number N has to be odd, say $N:= 2k +1$ (with $k\in \mathbb {N}_0$) and there are k odd numbers $\iota (z_j)$’s and $k+1$ even numbers $\iota (z_j)$’s. Furthermore, for each $k\ge 0$, it hold

$$\begin{aligned}&A_2 := \sum _{ j< \ell } (-1)^{\iota (z_j) + \iota (z_\ell ) } = -k \quad ; \quad A_3 := \sum _{ j< \ell< r } (-1)^{\iota (z_j) + \iota (z_\ell ) + \iota (z_r) } = -k,\\&\quad A_4 := \sum _{ j< \ell< r < t } (-1)^{\iota (z_j) + \iota (z_\ell ) + \iota (z_r) + \iota (z_t)} = \frac{1}{2} k(k-1) .\end{aligned}$$

Proof

To compute the value of $A_2$, observe that it is the sum of $+1$ and $-1$. To get $-1$, $\iota (z_j)$ and $\iota (z_k)$ have to be of different parity. However, to get $+1$, $\iota (z_j)$ and $\iota (z_k)$ have to be of the same parity. A similar argument holds for the computation of the values $A_3$ and $A_4$. Hence:

• For $k=0$, we have only one point z with an even $\iota (z)$. Thus $A_2=A_3=A_4=0$.

• For $k=1$, we have two points $z_0$ and $z_2$ with even $\iota (z_k)$ and one point $z_1$ with an odd $\iota (z_1)$. Thus, $A_4=0$, $A_3= -1$ and $A_2= 1-2=-1$.

• For $k\ge 2$, there exist $k+1$ even numbers $\iota (z_j)$ and k odd numbers $\iota (z_j)$. Thus, it holds

  $$\begin{aligned}&A_2=\begin{pmatrix} 2 \\ k+1\end{pmatrix} + \begin{pmatrix} 2 \\ k \end{pmatrix} - \begin{pmatrix} 1 \\ k+1\end{pmatrix} \begin{pmatrix} 1 \\ k \end{pmatrix} = \frac{1}{2} (k+1)k + \frac{1}{2} k(k-1) - (k+1) k = -k,\\&A_3 = \begin{pmatrix} 3 \\ 3\end{pmatrix} + \begin{pmatrix} 1 \\ 3\end{pmatrix} \begin{pmatrix} 2 \\ 2\end{pmatrix} - \begin{pmatrix} 2 \\ 3\end{pmatrix} \begin{pmatrix} 1 \\ 2\end{pmatrix} = -2\quad \text{ if } k=2\\&A_3 = \begin{pmatrix} 3 \\ k+1 \end{pmatrix} + \begin{pmatrix} 1 \\ k+1\end{pmatrix} \begin{pmatrix} 2 \\ k\end{pmatrix} - \begin{pmatrix} 2 \\ k+1\end{pmatrix} \begin{pmatrix} 1 \\ k\end{pmatrix} - \begin{pmatrix} 3 \\ k \end{pmatrix}= - k\quad \text{ if } k \ge 3\\&A_4 = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\begin{pmatrix} 2 \\ 2\end{pmatrix} - \begin{pmatrix} 3 \\ 3 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \end{pmatrix}= 1 \quad \text{ if } k = 2\\&A_4 = \begin{pmatrix} 4 \\ 4\end{pmatrix} + \begin{pmatrix} 2 \\ 4\end{pmatrix} \begin{pmatrix} 2 \\ 3\end{pmatrix} - \begin{pmatrix} 3 \\ 4\end{pmatrix} \begin{pmatrix} 1 \\ 3\end{pmatrix} - \begin{pmatrix} 1 \\ 4\end{pmatrix} \begin{pmatrix} 3 \\ 3\end{pmatrix} = 3 \quad \text{ if } k=3 \\&A_4 = \begin{pmatrix} 4 \\ k+1 \end{pmatrix} + \begin{pmatrix} 2 \\ k+1\end{pmatrix} \begin{pmatrix} 2 \\ k\end{pmatrix} + \begin{pmatrix} 4 \\ k\end{pmatrix} - \begin{pmatrix} 3 \\ k+1\end{pmatrix} \begin{pmatrix} 1 \\ k\end{pmatrix} - \begin{pmatrix} 1 \\ k+1\end{pmatrix} \begin{pmatrix} 3 \\ k \end{pmatrix}\\&\qquad = \frac{1}{2} k (k-1) \quad \text{ if } k \ge 4. \end{aligned}$$

The proof is thereby completed. $\square$

Arguing as in the above lemma, one derives the following counting formula:

Lemma 5.9

Let $y_1,\ldots ,y_L$ be L critical points of K in $\mathbb {S}^n_+$ and let $\iota (y_j):= n - morse(K,y_j)$. Assume that

$$\begin{aligned} B_1:= \sum _{j=1}^L (-1)^{\iota (y_j)} =-k \quad \text{ with } k \ge 0.\end{aligned}$$

Then the number L has to satisfy $L:= 2r +k$ (with $r \in \mathbb {N}_0$) and there are r even numbers $\iota (y_j)$’s and $r+k$ odd numbers $\iota (y_j)$’s. Furthermore, it holds

$$\begin{aligned} B_2 := \sum _{ 1\le j < \ell \le L} (-1)^{\iota (y_j) + \iota (y_\ell ) } = -r + \frac{1}{2} k(k-1); \, \, \text{ for } \text{ each } L \ge 0.\end{aligned}$$