Existence of Conformal Metrics with Prescribed Q-Curvature

Abstract

We consider the problem of existence of conformal metrics with prescribed Q-curvature on standard sphere 𝑆^𝑛,𝑛≥5. Under the assumption that the order of flatness at critical points of prescribed Q-curvature function 𝐾(𝑥) is 𝛽∈]1,𝑛−4], we give precise estimates on the losses of the compactness, and we prove new existence and multiplicity results through an Euler-Hopf type formula.

1. Introduction and Main Result

The Paneitz operator on 𝑛-Riemannian manifold (𝑀,𝑔₀) is a fourth-order differential operator which arises in conformal geometry and satisfies a certain covariance property (see [1]). For 𝑛≥5 it is defined by 𝑃^𝑛_𝑔0𝑢=Δ²_𝑔0𝑢−div_𝑔0𝑎_𝑛𝑆_𝑔0𝑔₀+𝑏_𝑛Ric_𝑔0𝑑𝑢+𝑛−42𝑄^𝑛_𝑔0𝑢, (1) where 𝑎_𝑛=(𝑛−2)²+42(𝑛−1)(𝑛−2),𝑏_𝑛4=−,𝑄𝑛−2_𝑛𝑔01=−2()Δ𝑛−1^𝑔0𝑆_𝑔0+𝑛³−4𝑛²+16𝑛−168(𝑛−1)²(𝑛−2)²𝑆²_𝑔0−2(𝑛−2)²Ric_𝑔0². (2) Such a 𝑄^𝑛_𝑔0 is a fourth-order invariant called Q-curvature or Paneitz curvature. See [2, 3] for details about the properties of 𝑃^𝑛_𝑔0.

If 𝑔=𝑢^4/(𝑛−4)𝑔₀ is a conformal metric to 𝑔₀, where 𝑢 is a smooth positive function, then the conformal covariance property of the Paneitz operator reads as follows: ∀𝜓∈𝐶^∞(𝑀)∶𝑃^𝑛_𝑔0𝑢𝜓=𝑢^{(𝑛+4)/(𝑛−4)}𝑃^𝑛_𝑔𝜓. (3) If one prescribes the Q-curvature for the metric 𝑔 by a function 𝐾, this leads to the following equation: 𝑃^𝑛_𝑔0(𝑢)=𝑢^{(𝑛+4)/(𝑛−4)}𝑃^𝑛_𝑔(1)=𝑛−42𝐾𝑢^{(𝑛+4)/(𝑛−4)}. (4) The literature for the existence of solutions of the prescribed Q-curvature problem on compact manifolds is considerably bigger. In [4–7], existence results for the constant Q-curvature problem in 4-dimensional manifolds are given. On manifolds of dimension greater than 4, existence results were given for Einstein manifolds in [3]. On the sphere 𝑆^𝑛, we refer to results of [8–14] and the references therein.

In this paper we continue to study the problem of prescribing Q-curvature on the standard sphere (𝑆^𝑛,𝑔₀), 𝑛≥5. The Paneitz operator in this case is coercive on the Sobolev space 𝐻²₂(𝑆^𝑛) and has the following expression: 𝒫(𝑢)∶=𝑃^𝑛_𝑔0(𝑢)=Δ²_𝑔0𝑢−𝑐_𝑛Δ_𝑔0𝑢+𝑑_𝑛𝑢, (5) where 𝑐_𝑛=12𝑛²−2𝑛−4,𝑑_𝑛=𝑛−4𝑛𝑛16²−4. (6) According to problem (4), the problem can be formulated as follows. Given a smooth function 𝐾 on 𝑆^𝑛, we look for solutions of 𝒫(𝑢)=𝑛−42𝐾𝑢^{(𝑛+4)/(𝑛−4)},𝑢>0on𝑆^𝑛. (7) The special case where the manifold is a sphere endowed with its standard metric deserves particular attention. Indeed due to Kazdan-Warner type obstructions, see [3], conditions have to be imposed on the function 𝐾.

To state our result we set up the following conditions and notation.

(𝑓)_𝛽 Assume that 𝐾∶𝑆^𝑛→ℝ, 𝑛≥5 be a 𝐶¹ positive function such that for any critical point 𝑦 of 𝐾, there exists some real number 𝛽=𝛽(𝑦) such that in some geodesic normal coordinate system centered at 𝑦 (through the exponential map), we have 𝐾(𝑥)=𝐾𝑦+^𝑛_𝑘=1𝑏_𝑘𝑥−𝑦_𝑘^𝛽+𝑅𝑥−𝑦, (8) where 𝑏_𝑘=𝑏_𝑘(𝑦)≠0, forall𝑘=1,…,𝑛, ∑^𝑛_𝑘=1𝑏_𝑘≠0 and ∑^[𝛽]_𝑠=0|∇^𝑠𝑅(𝑥−𝑦)||𝑥−𝑦|^−𝛽+𝑠=𝑜(1) as 𝑥 tends to 𝑦. Here ∇^𝑠 denotes all possible derivatives of order 𝑠 and [𝛽] is the integer part of 𝛽. For each critical point 𝑦 of 𝐾, we denote 𝑖̃𝑦=∶♯𝑏_𝑘,𝑘=1,…,𝑛,suchthat𝑏_𝑘<0. (9) Let, 𝒦=𝑦∈𝑆^𝑛,∇_𝑔0𝐾𝑦𝒦=0₊=𝑦∈𝒦,−^𝑛_𝑘=1𝑏_𝑘,𝒦>0_𝑛−4=𝑦∈𝒦,𝛽=𝛽𝑦=𝑛−4. (10) For each 𝑝-tuple, 𝑝≥1 of distinct points 𝜏_𝑝∶=(𝑦_𝑙1,…,𝑦_𝑙𝑝) such that for all 𝑖=1,…,𝑝, 𝑦_𝑙𝑖∈𝒦⁺∩𝒦_𝑛−4, we define a 𝑝×𝑝 symmetric matrix 𝑀(𝜏_𝑝)=(𝑚_𝑖𝑗) by 𝑚_𝑖𝑖=𝑛−4𝑛𝑐₁−∑^𝑛_𝑘=1𝑏_𝑘𝑦_𝑙𝑖𝐾𝑦_𝑙𝑖^𝑛/4,𝑚_𝑖𝑗=2^(𝑛−4)/2𝑐₁−𝐺𝑦_𝑙𝑖,𝑦_𝑙𝑗𝐾𝑦_𝑙𝑖𝐾𝑦_𝑙𝑗^(𝑛−4)/8, (11) where 𝑐₁=_ℝ𝑛𝑑𝑥1+|𝑥|^2(𝑛+4)/2,𝑐₁=𝑐₀^{2𝑛/(𝑛−4)}_ℝ𝑛𝑥₁^𝑛−41+|𝑥|^2𝑛𝑑𝑥, (12) and 𝐺 is the Green function for the operator 𝒫(𝑢) on 𝑆^𝑛. Here 𝑥₁ is the first component of 𝑥 in some geodesic normal coordinate system.

Let 𝜌(𝜏_𝑝) be the least eigenvalue of 𝑀(𝜏_𝑝). It was first pointed out by Bahri [15] that when the interaction between the different bubbles is of the same order as the self interaction, the least eigenvalue of some matrices like (11) plays a fundamental role in the existence of solutions to problems like (7). Regarding problem (7), such kind of phenomenon appears under (𝑓)_𝛽 condition in 𝒦_𝑛−4; see [11].

(𝐴₁) Assume that 𝜌(𝜏_𝑝)≠0 for each distinct point 𝑦_𝑙1,…,𝑦_𝑙𝑝∈𝒦⁺∩𝒦_𝑛−4.

Now, we introduce the following set: 𝒞_∞∶=𝜏_𝑝=𝑦_𝑙1,…,𝑦_𝑙𝑝∈𝒦^+𝑝,𝑝≥1,s.t.𝑦_𝑖≠𝑦_𝑗𝑦∀𝑖≠𝑗,andif_𝑙1,…,𝑦_𝑙𝑝∩𝒦_𝑛−4≠∅,wedenoteby𝑦_𝑖1,…,𝑦_𝑖𝑞𝑦allelementsof_𝑙1,…,𝑦_𝑙𝑝with𝛽𝑦_𝑖𝑗𝜌𝑦=𝑛−4foreach𝑗=1,…,𝑞then_𝑖1,…,𝑦_𝑖𝑞>0. (13) The main result of this paper is the following.

Theorem 1. Assume that 𝐾 satisfies (𝐴₁) and (𝑓)_𝛽, with 1<𝛽≤𝑛−4. (14) If 𝑑=_{(𝑦1,…,𝑦𝑝)∈𝒞∞}(−1)^{𝑝−1+∑𝑝𝑘=1̃𝑛−𝑖(𝑦𝑘)}−1≠0. (15) Then (7) has at least one solution.
Moreover if (𝑛−4)/2<𝛽≤𝑛−4, for generic 𝐾 one has ♯𝑆≥|𝑑|, (16) where 𝑆 denotes the set of solutions to (7).

Our argument uses a careful analysis of the lack of compactness of the Euler Lagrange functional 𝐽 associated with problem (7). Namely, we study the noncompact orbits of the gradient flow of 𝐽 the so-called critical points at infinity following the terminology of Bahri [15]. These critical points at infinity can be treated as usual critical points once a Morse lemma at infinity is performed from which we can derive just as in the classical Morse theory the difference of topology induced by these noncompact orbits and compute their Morse index. Such a Morse lemma at infinity is obtained through the construction of suitable pseudogradient for which the Palais-Smale condition is satisfied along the decreasing flow lines, as long as these flow lines do not enter the neighborhood of a finite number 𝑦₁,…,𝑦_𝑝 of critical points of 𝐾 such that (𝑦₁,…,𝑦_𝑝)∈𝒞_∞.

Similar Morse lemma at infinity has been established for the problem (7) on the sphere 𝑆^𝑛, 𝑛≥5, under the hypothesis that the order of flatness at critical points of 𝐾 is 𝛽∈[𝑛−4,𝑛[; see [11].

The rest of this paper is organized as follows. In Section 2, we set up the variational problem and we recall the expansion of the gradient of the associated Euler-Lagrange functional near infinity. In Section 3, we characterize the critical points at infinity of the associated variational problem. Section 4 is devoted to the proof of the main result Theorem 1, while we give in Section 5 a more general statement than Theorem 1.

2. General Framework and Some Known Facts

2.1. Variational Structure and Lack of Compactness

Our problem (7) enjoys a variational structure. Indeed, solutions to (7) correspond to positive critical points of the functional 𝐽(𝑢)=∫_𝑆𝑛𝒫𝑢𝑢𝑑𝑣_𝑔0∫_𝑆𝑛𝐾|𝑢|^{2𝑛/(𝑛−4)}𝑑𝑣_𝑔0^{(𝑛−4)/2𝑛}, (17) defined on Σ=𝑢∈𝐻²₂𝑆^𝑛,‖𝑢‖²∶=_𝑆𝑛𝒫𝑢𝑢𝑑𝑣_𝑔0=1. (18)

However, it is delicate from a variational viewpoint because the functional 𝐽 does not satisfy the Palais-Smale condition. This means that there exist sequences along which 𝐽 is bounded, its gradient goes to zero and which not converge. The analysis of sequences failing Palais-Smale condition can be analyzed along the ideas introduced in [16, pages 325 and 334]. In order to describe such a characterization in our case, we need to introduce some notations. For 𝑎∈𝑆^𝑛 and 𝜆>0, let 𝛿_(𝑎,𝜆)(𝑥)=𝑐₀12^(𝑛−4)/2×𝜆^(𝑛−4)/21+𝜆²−1/2(1−cos(𝑑(𝑥,𝑎)))^(𝑛−4)/2, (19) where 𝑑 is the geodesic distance on (𝑆^𝑛,𝑔) and 𝑐₀ is chosen so that 𝛿_(𝑎,𝜆) is the family of solutions of the problem 𝒫(𝑢)=𝑢^{(𝑛+4)/(𝑛−4)},𝑢>0,on𝑆^𝑛 (20) (see [17]).

We define now the set of potential critical points at infinity associated with the function 𝐽. For 𝜀>0 and 𝑝∈ℕ^∗, let us define 𝑉=𝑝,𝜀𝑢∈Σ/∃𝑎₁,…,𝑎_𝑝∈𝑆^𝑛,∃𝜆₁,…,𝜆_𝑝>𝜀⁻¹,∃𝛼₁,…,𝛼_𝑝>0with𝑢−∑^𝑝_𝑖=1𝛼_𝑖𝛿_{(𝑎𝑖,𝜆𝑖)}𝐽(𝑢)<𝜀,_{𝑛/(𝑛−4)}𝛼_𝑖^8/(𝑛−4)𝐾𝑎_𝑖−1<𝜀∀𝑖,𝜀_𝑖𝑗<𝜀∀𝑖≠𝑗. (21) Here, 𝜀_𝑖𝑗=[(𝜆_𝑖/𝜆_𝑗)+(𝜆_𝑗/𝜆_𝑖)+(𝜆_𝑖𝜆_𝑗/2)(1−cos𝑑(𝑎_𝑖,𝑎_𝑗))]^{−(𝑛−4)/2}.

For 𝑤 a solution to (7) we also define 𝑉(𝑝,𝜀,𝑤) as 𝑉𝑝,𝜀,𝑤=𝑢∈Σ/∃𝛼₀>0s.t.𝑢−𝛼₀,𝛼𝑤∈𝑉𝑝,𝜀_{08/(𝑛−4)}𝐽(𝑢)^{𝑛/(𝑛−4)}−1<𝜀. (22) If 𝑢 is a function in 𝑉(𝑝,𝜀,𝑤), one can find an optimal representation, following the ideas introduced in Proposition 5.2 of [15] and [16, pages 348–350]. Namely, we have the following.

Proposition 2. For any 𝑝∈ℕ^∗, there is 𝜀_𝑝>0 such that if 𝜀≤𝜀_𝑝 and 𝑢∈𝑉(𝑝,𝜀,𝑤), then the following minimization problem min_{𝛼𝑖>0,𝜆𝑖>0,𝑎𝑖∈𝑆𝑛,ℎ∈𝑇𝑤(𝑊𝑢(𝑤)).}𝑢−^𝑝_𝑖=1𝛼_𝑖𝛿_{(𝑎𝑖,𝜆𝑖)}−𝛼₀(𝑤+ℎ) (23) has a unique solution (𝛼,𝜆,𝑎,ℎ), up to a permutation.

In particular, we can write 𝑢 as follows: 𝑢=^𝑝_𝑖=1𝛼_𝑖𝛿_{(𝑎𝑖,𝜆𝑖)}+𝛼₀(𝑤+ℎ)+𝑣, (24) where 𝑣 belongs to 𝐻²₂(𝑆^𝑛)∩𝑇_𝑤(𝑊_𝑠(𝑤)) and it satisfies 𝑉₀, 𝑇_𝑤(𝑊_𝑢(𝑤)) and 𝑇_𝑤(𝑊_𝑠(𝑤)) are the tangent spaces at 𝑤 of the unstable and stable manifolds of 𝑤 for a decreasing pseudogradient of 𝐽, and 𝑉₀ is the following: 𝑣,𝜓=0for𝜓∈𝛿_𝑖,𝜕𝛿_𝑖𝜕𝜆_𝑖,𝜕𝛿_𝑖𝜕𝑎_𝑖⟨⟩⟨⟩,𝑖=1,…,𝑝𝑣,𝑤=0,𝑣,ℎ=0∀ℎ∈𝑇_𝑤𝑊_𝑢(𝑤). 𝑉₀ Here, 𝛿_𝑖=𝛿_{(𝑎𝑖,𝜆𝑖)} and ⟨⋅,⋅⟩ denotes the scalar product defined on 𝐻²₂(𝑆^𝑛) by ⟨𝑢,𝑣⟩=_𝑆𝑛Δ_𝑔0𝑢Δ_𝑔0𝑣𝑑𝑣_𝑔0+𝑐_𝑛𝑆𝑛∇_𝑔0𝑢∇_𝑔0𝑣𝑑𝑣_𝑔0+𝑑_𝑛𝑆𝑛𝑢𝑣𝑑𝑣_𝑔0. (25) Notice that Proposition 2 is also true if we take 𝑤=0 and, therefore, ℎ=0 and 𝑢 in 𝑉(𝑝,𝜀).

We are ready now to state the characterization of the sequences failing the Palais-Smale condition. For technical reasons, we introduce the following subsets of Σ. Let Σ⁺={𝑢∈Σ,𝑢≥0} and let for 𝜂 positive very small, 𝑉_𝜂Σ⁺=𝑢∈Σ,𝐽(𝑢)^{(2𝑛−4)/(𝑛−4)}𝑒^2𝐽(𝑢)‖𝑢⁻‖_𝐿^8/(𝑛−4)_{2𝑛/(𝑛−4)}<𝜂, (26) where 𝑢⁻=max(0,−𝑢). Using the idea introduced in [16, 18], see also [19], we have the following proposition.

Proposition 3. Let (𝑢_𝑘) be a sequence in 𝑉_𝜂(Σ⁺) such that 𝐽(𝑢_𝑘) is bounded and 𝜕𝐽(𝑢_𝑘) goes to zero. Then there exists an integer 𝑝∈ℕ^∗, a sequence (𝜀_𝐾)>0, 𝜀_𝑘 tends to zero, and an extracted subsequence of 𝑢_𝑘's, again denoted (𝑢_𝑘), such that 𝑢_𝑘∈𝑉(𝑝,𝜀_𝑘,𝑤) where 𝑤 is zero or a solution to (7).

Now arguing as in [16] (pages 326, 327, and 334), we have the following Morse lemma which completely gets rid of the 𝑣-contributions and shows that it can be neglected with respect to the concentration phenomenon.

Proposition 4. There is a 𝒞¹-map which to each (𝛼_𝑖,𝑎_𝑖,𝜆_𝑖,ℎ) such that ∑^𝑝_𝑖=1𝛼_𝑖𝛿_{(𝑎𝑖,𝜆𝑖)}+𝛼₀(𝑤+ℎ) belongs to 𝑉(𝑝,𝜀,𝑤) associates 𝑣=𝑣(𝛼,𝑎,𝜆,ℎ) such that 𝑣 is unique and satisfies 𝐽^𝑝_𝑖=1𝛼_𝑖𝛿_{(𝑎𝑖,𝜆𝑖)}+𝛼₀(𝑤+ℎ)+𝑣=min_𝑉𝑣∈0𝐽^𝑝_𝑖=1𝛼_𝑖𝛿_{(𝑎𝑖,𝜆𝑖)}+𝛼₀(𝑤+ℎ)+𝑣. (27) Moreover, there exists a change of variables 𝑣−𝑣→𝑉 such that 𝐽^𝑝_𝑖=1𝛼_𝑖𝛿_{(𝑎𝑖,𝜆𝑖)}+𝛼₀(𝑤+ℎ)+𝑣=𝐽^𝑝_𝑖=1𝛼_𝑖𝛿_{(𝑎𝑖,𝜆𝑖)}+𝛼₀(𝑤+ℎ)+𝑣+‖𝑉‖². (28)

The following proposition gives precise estimates of 𝑣.

Proposition 5 (see [11, Lemma 3.1]). Let 𝑢=∑^𝑝_𝑖=1𝛼_𝑖𝛿_𝑖+𝛼₀(𝑤+ℎ)∈𝑉(𝑝,𝜀,𝑤) and let 𝑣 be defined in Proposition 4. One has the following estimates: there exists 𝑐>0 independent of 𝑢 such that the following holds: ‖𝑣‖≤𝑐^𝑝_𝑖=11𝜆_𝑖^𝑛/2+1𝜆^𝛽_𝑖+∇𝐾𝑎_𝑖𝜆_𝑖+log𝜆_𝑖^{(𝑛+4)/2𝑛}𝜆_𝑖^(𝑛+4)/2∑+𝑐^𝑘≠𝑟𝜀^{(𝑛+4)/2(𝑛−4)}_𝑘𝑟log𝜀⁻¹_𝑘𝑟^{(𝑛+4)/2𝑛}∑,𝑖𝑓𝑛≥12_𝑘≠𝑟𝜀_𝑘𝑟log𝜀⁻¹_𝑘𝑟^{(𝑛−4)/𝑛},𝑖𝑓𝑛<12. (29)

At the end of this subsection, we give the following definition.

Definition 6 (see [16, 18]). A critical point at infinity of 𝐽 on 𝑉_𝜂(Σ⁺) is a limit of a flow line 𝑢(𝑠) of the equation 𝜕𝑢(𝑢(𝑠,𝑢(0)𝜕𝑠=−𝜕𝐽))=𝑢₀, (30) such that 𝑢(𝑠) remains in 𝑉(𝑝,𝜀(𝑠),𝑤) for 𝑠≥𝑠₀. Here 𝑤 is either zero or a solution to (7) and 𝜀(𝑠) is some positive function tending to zero when 𝑠→+∞. Using Proposition 2, 𝑢(𝑠) can be written as 𝑢(𝑠)=^𝑝_𝑖=1𝛼_𝑖(𝑠)𝛿_{(𝑎𝑖(𝑠),𝜆𝑖(𝑠))}+𝛼₀(𝑠)(𝑤+ℎ(𝑠))+𝑣(𝑠). (31) Denoting 𝛼_𝑖∶=lim_𝑠→+∞𝛼_𝑖(𝑠),𝑦_𝑖∶=lim_𝑠→+∞𝑎_𝑖(𝑠), we denote by ^𝑝_𝑖=1𝛼_𝑖𝛿_{(𝑦𝑖,∞)}+𝛼₀𝑤or𝑦₁,...,𝑦_𝑝,𝑤_∞ (32) such a critical point at infinity. If 𝑤≠0, it is called of 𝑤-type or mixed type.

Notice that 𝑉_𝜂(Σ⁺) remains invariant under the flow generated by (−𝜕𝐽) (see Lemma 5.1 of [20]; see also Lemma 4.1 of [18]).

2.2. Expansion of the Gradient of the Functional

In this subsection, we recall the expansion of the gradient of the functional 𝐽 in 𝑉(𝑝,𝜀), 𝑝≥1.

Proposition 7 (see [11]). For any 𝑢=∑^𝑝_𝑗=1𝛼_𝑗𝛿_𝑗 in 𝑉(𝑝,𝜀), the following expansion holds
(i) 𝜕𝐽(𝑢),𝜆_𝑖𝜕𝛿_𝑖𝜕𝜆_𝑖=−2𝑐₂𝐽(𝑢)_𝑖≠𝑗𝛼_𝑗𝜆_𝑖𝜕𝜀_𝑖𝑗𝜕𝜆_𝑖+𝑜_𝑖≠𝑗𝜀_𝑖𝑗1+𝑜𝜆_𝑖, (33) where 𝑐₂=𝑐₀^{2𝑛/(𝑛−4)}∫_ℝ𝑛𝑑𝑦/(1+|𝑦|²)^(𝑛+4)/2.
(ii) If 𝑎_𝑖∈𝐵(𝑦_𝑗𝑖,𝜌), 𝑦_𝑗𝑖∈𝒦 and 𝜌 is a positive constant small enough, one has 𝜕𝐽(𝑢),𝜆_𝑖𝜕𝛿_𝑖𝜕𝜆_𝑖=2𝐽(𝑢)−𝑐_2𝑗≠𝑖𝛼_𝑗𝜆_𝑖𝜕𝜀_𝑖𝑗𝜕𝜆_𝑖+𝑛−4𝑐2𝑛^{02𝑛/(𝑛−4)}𝛽𝛼_𝑖𝐾𝑎_𝑖×∑^𝑛_𝑘=1𝑏_𝑘𝜆^𝛽_𝑖^ℝ𝑛signe𝑥_𝑘+𝜆_𝑖𝑎_𝑖−𝑦_𝑗𝑖𝑘×𝑥_𝑘+𝜆_𝑖𝑎_𝑖−𝑦_𝑗𝑖𝑘^𝛽−1×𝑥_𝑘1+|𝑥|^2𝑛𝑑𝑥+𝑜_𝑗≠𝑖𝜀_𝑖𝑗+^𝑝_𝑗=11𝜆^𝛽_𝑗. (34)
(iii) Furthermore if 𝜆_𝑖|𝑎_𝑖−𝑦_𝑗𝑖|<𝛿, for 𝛿 very small, one then has 𝜕𝐽(𝑢),𝜆_𝑖𝜕𝛿_𝑖𝜕𝜆_𝑖=2𝐽(𝑢)×𝑛−42𝑛𝛽𝑐₃𝛼_𝑖𝐾𝑎_𝑖∑^𝑛_𝑘=1𝑏_𝑘𝜆^𝛽_𝑖−𝑐_2𝑗≠𝑖𝛼_𝑗𝜆_𝑖𝜕𝜀_𝑖𝑗𝜕𝜆_𝑖+𝑜_𝑗≠𝑖𝜀_𝑖𝑗+^𝑝_𝑗=11𝜆^𝛽_𝑗, (35) where 𝑐₃=𝑐₀^{2𝑛/(𝑛−4)}∫_𝑆𝑛|𝑥₁|^𝛽/(1+|𝑥|²)^𝑛𝑑𝑥.

Proposition 8 (see [11]). Letting 𝑢=∑^𝑝_𝑗=1𝛼_𝑗𝛿_𝑗∈𝑉(𝑝,𝜀), one has
(i) (𝑢),1𝜕𝐽𝜆_𝑖𝜕𝛿_𝑖𝜕𝑎_𝑖=−𝑐₅(𝐽(𝑢))^{2(𝑛−2)/(𝑛−4)}𝛼_𝑖^{(𝑛+4)/(𝑛−4)}∇𝐾𝑎_𝑖𝜆_𝑖+𝑂_𝑖≠𝑗1𝜆_𝑖𝜕𝜀_𝑖𝑗𝜕𝑎_𝑖+𝑜_𝑖≠𝑗𝜀_𝑖𝑗+1𝜆_𝑖, (36) where 𝑐₅=∫_ℝ𝑛𝑑𝑦/(1+|𝑦|²)^𝑛.
(ii) If 𝑎_𝑖∈𝐵(𝑦_𝑗𝑖,𝜌), 𝑦_𝑗𝑖∈𝒦, one has (𝑢),1𝜕𝐽𝜆_𝑖𝜕𝛿_𝑖𝜕𝑎_𝑖𝑘=−2(𝑛−4)𝑐₀^{2𝑛/(𝑛−4)}𝛼_𝑖^{(𝑛+4)/(𝑛−4)}(𝐽(𝑢))^{2(𝑛−2)/(𝑛−4)}1𝜆^𝛽_𝑖×_ℝ𝑛𝑏_𝑘𝑥_𝑘+𝜆_𝑖𝑎_𝑖−𝑦_𝑗𝑖𝑘^𝛽𝑥_𝑘1+|𝑥|^2𝑛+1𝑑𝑦+𝑜_𝑖≠𝑗𝜀_𝑖𝑗+𝑜^𝑝_𝑖=11𝜆^𝛽_𝑖+𝑂_𝑖≠𝑗1𝜆_𝑖𝜕𝜀_𝑖𝑗𝜕𝑎_𝑖, (37) where 𝑘=1,…,𝑛 and (𝑎_𝑖)_𝑘 is the 𝑘th component of 𝑎_𝑖 in some geodesic normal coordinates system.

3. Characterization of the Critical Points at Infinity

This section is devoted to the characterization of the critical points at infinity in 𝑉(𝑝,𝜀),𝑝≥1, under 𝛽-flatness condition with 1<𝛽≤𝑛−4. This characterization is obtained through the construction of a suitable pseudogradient at infinity for which the Palais-Smale condition is satisfied along the decreasing flow-lines as long as these flow lines do not enter in the neighborhood of finite number of critical points 𝑦_𝑖,𝑖=1,…,𝑝 of 𝐾 such that (𝑦₁,…,𝑦_𝑝)∈𝒞_∞.

Now we introduce the following main result.

Theorem 9. Assume that 𝐾 satisfies (𝐴₁) and (𝑓)_𝛽, 1<𝛽≤𝑛−4.
Let 𝛽∶=max{𝛽(𝑦)/𝑦∈𝒦}. For 𝑝≥1, there exists a pseudogradient 𝑊 in 𝑉(𝑝,𝜀) so that the following holds.
There exists a constant 𝑐>0 independent of 𝑢=∑^𝑝_𝑖=1𝛼_𝑖𝛿_{(𝑎𝑖𝜆𝑖)}∈𝑉(𝑝,𝜀) such that (𝑖)⟨𝜕𝐽(𝑢),𝑊(𝑢)⟩≤−𝑐^𝑝_𝑖=11𝜆^𝛽_𝑖+^𝑝_𝑖=1∇𝐾𝑎_𝑖𝜆_𝑖+_𝑗≠𝑖𝜀_𝑖𝑗.(𝑖𝑖)𝜕𝐽(𝑢+𝑣)(𝑢)+𝜕,𝑊𝑣𝜕𝛼_𝑖,𝑎_𝑖,𝜆_𝑖(𝑊(𝑢))≤−𝑐^𝑝_𝑖=11𝜆^𝛽_𝑖+^𝑝_𝑖=1∇𝐾𝑎_𝑖𝜆_𝑖+_𝑗≠𝑖𝜀_𝑖𝑗. (38) Furthermore |𝑊| is bounded and the only case where the maximum of the 𝜆_𝑖's is not bounded is when 𝑎_𝑖∈𝐵(𝑦_𝑙𝑖,𝜌) with 𝑦_𝑙𝑖∈𝒦, forall𝑖=1,…,𝑝, (𝑦_𝑙1,…,𝑦_𝑙𝑝)∈𝒞_∞.

We will prove Theorem 9 at the end of the section. Now we state two results which deal with two specific cases of Theorem 9. Let 𝑉₁𝑝,𝜀=𝑢=^𝑝_𝑖=1𝛼_𝑖𝛿_𝑖∈𝑉𝑝,𝜀s.t.,𝑎_𝑖∈𝐵𝑦_𝑙𝑖,𝑦,𝜌_𝑙𝑖∈𝒦⧵𝒦_𝑛−4,𝑉∀𝑖=1,…,𝑝₂𝑝,𝜀=𝑢=^𝑝_𝑖=1𝛼_𝑖𝛿_𝑖∈𝑉𝑝,𝜀s.t.,𝑎_𝑖∈𝐵𝑦_𝑙𝑖,𝑦,𝜌_𝑙𝑖∈𝒦_𝑛−4∀𝑖=1,…,𝑝. (39) We then have the following.

Proposition 10 (see [11], Proposition 3.7). For 𝑝≥1 there exists a pseudogradient 𝑊₂ in 𝑉₂(𝑝,𝜀) such that for all 𝑢=∑^𝑝_𝑖=1𝛼_𝑖𝛿_𝑖∈𝑉₂(𝑝,𝜀), one has 𝜕𝐽(𝑢),𝑊₂(𝑢)≤−𝑐^𝑝_𝑖=11𝜆_𝑖^𝑛−4+_𝑖≠𝑗𝜀_𝑖𝑗+^𝑝_𝑖=1∇𝐾𝑎_𝑖𝜆_𝑖, (40)  where 𝑐 is a positive constant independent of 𝑢. Furthermore, one has |𝑊₂| which is bounded and the only case where the maximum of 𝜆_𝑖’s is not bounded is when 𝑎_𝑖∈𝐵(𝑦_𝑙𝑖,𝜌),𝑦_𝑙𝑖∈𝒦⁺, for all 𝑖=1,…,𝑝, with 𝜌(𝑦_𝑙1,…,𝑦_𝑙𝑝)>0.

Proposition 11. For 𝑝≥1, there exists a pseudogradient 𝑊₁ in 𝑉₁(𝑝,𝜀) so that the following holds.
There exists 𝑐>0 independent of 𝑢=∑^𝑝_𝑖=1𝛼_𝑖𝛿_𝑖∈𝑉₁(𝑝,𝜀) such that 𝜕𝐽(𝑢),𝑊₁(𝑢)≤−𝑐^𝑝_𝑖=11𝜆^𝛽_𝑖+_𝑖≠𝑗𝜀_𝑖𝑗+^𝑝_𝑖=1∇𝐾𝑎_𝑖𝜆_𝑖. (41) Furthermore |𝑊₁| is bounded and the only case where the maximum of the 𝜆_𝑖's is not bounded is when 𝑎_𝑖∈𝐵(𝑦_𝑙𝑖,𝜌) with 𝑦_𝑙𝑖∈𝒦⁺, for all 𝑖=1,…,𝑝, and 𝑦_𝑖≠𝑦_𝑗 for all 𝑖≠𝑗.

Observe that in 𝑉₁(𝑝,𝜀) the interaction of two bubbles is negligible with respect to the self-interaction. Similar phenomena occur for the scalar curvature problem, see [21], so the proof of Proposition 11 is similar to the corresponding statement in [21].

Before giving the proof of Theorem 9, we state the following notations extracted from [11].

Let 𝑀₁ be a fixed positive constant large enough and let 𝑢=∑^𝑝_𝑖=1𝛼_𝑖𝛿_{(𝑎𝑖,𝜆𝑖)}∈𝑉(𝑝,𝜀) such that 𝑎_𝑖∈𝐵(𝑎_𝑖,𝜌), 𝑦_𝑙𝑖∈𝒦, for all 𝑖=1,…,𝑝. For any index 𝑖, 1≤𝑖≤𝑝, we define the following vector fields: 𝑍_𝑖(𝑢)=𝛼_𝑖𝜆_𝑖𝜕𝛿_{(𝑎𝑖,𝜆𝑖)}𝜕𝜆_𝑖,𝑋_𝑖=𝛼_𝑖^𝑛_𝑘=11𝜆_𝑖𝜕𝛿_{(𝑎𝑖,𝜆𝑖)}𝜕𝑎_𝑖𝑘ℝ^𝑛𝑏_𝑘𝑥_𝑘+𝜆_𝑖𝑎_𝑖−𝑦_𝑙𝑖𝑘^𝛽𝑖1+𝜆_𝑖𝑎_𝑖−𝑦_𝑙𝑖𝑘^𝛽𝑖−1×𝑥_𝑘1+|𝑥|^2𝑛+1𝑑𝑥. (42) We claim that 𝑋_𝑖 is bounded. Indeed, the claim is trivial if 𝜆_𝑖|𝑎_𝑖−𝑦_𝑙𝑖|≤𝑀₁. If 𝜆_𝑖|𝑎_𝑖−𝑦_𝑙𝑖|>𝑀₁, for any 𝑘, 1≤𝑘≤𝑛, such that 𝜆_𝑖|(𝑎_𝑖−𝑦_𝑙𝑖)_𝑘|>𝑀₁/√𝑛, we have the following estimate: _ℝ𝑛𝑥_𝑘+𝜆_𝑖𝑎_𝑖−𝑦_𝑙𝑖𝑘^𝛽𝑖𝑥_𝑘1+|𝑥|^2𝑛+1𝑑𝑥=𝑐signe𝜆_𝑖𝑎_𝑖−𝑦_𝑙𝑖𝑘𝜆_𝑖𝑎_𝑖−𝑦_𝑙𝑖𝑘^𝛽𝑖−1×(1+𝑜(1)). (43) Hence our claim follows. Next, we will say that 𝑖∈𝐿₁if𝜆_𝑖𝑎_𝑖−𝑦_𝑙𝑖≤𝑀₁,𝑖∈𝐿₂if𝜆_𝑖𝑎_𝑖−𝑦_𝑙𝑖>𝑀₁, (44) and we will denote by 𝑘_𝑖 the index satisfying 𝑎_𝑖−𝑦_{𝑙𝑖𝑘𝑖}=max_{1≤𝑘≤𝑛}𝑎_𝑖−𝑦_𝑙𝑖𝑘. (45) It easy to see that if 𝑖∈𝐿₂, then 𝜆_𝑖|(𝑎_𝑖−𝑦_𝑙𝑖)_𝑘𝑖|>𝑀₁/√𝑛.

Now, we introduce the following two lemmas.

Lemma 12. Let 𝑢=∑^𝑝_𝑖=1𝛼_𝑖𝛿_{(𝑎𝑖,𝜆𝑖)}∈𝑉(𝑝,𝜀), such that 𝑎_𝑖∈𝐵(𝑦_𝑙𝑖,𝜌), 𝑦_𝑙𝑖∈𝒦, for all 𝑖=1,…,𝑝. One then has 𝜕𝐽(𝑢),𝑍_𝑖(𝑢)=−2𝑐₂𝐽(𝑢)_𝑗≠𝑖𝛼_𝑖𝛼_𝑗𝜕𝜀_𝑖𝑗𝜕𝜆_𝑖1+𝑂𝜆^𝛽𝑖_𝑖+𝑂𝑎_𝑖−𝑦_{𝑙𝑖𝑘𝑖}^𝛽𝑖−2𝜆²_𝑖,𝑖𝑓𝑖∈𝐿₂+𝑜_𝑗≠𝑖𝜀_𝑖𝑗+𝑜^𝑝_𝑗=11𝜆^𝛽𝑗_𝑗, (46) where 𝑘_𝑖 is defined in (45).

Proof. Using Proposition 7, we have 𝜕𝐽(𝑢),𝑍_𝑖(𝑢)=−2𝑐₂𝐽(𝑢)_𝑗≠𝑖𝛼_𝑖𝛼_𝑗𝜆_𝑖𝜕𝜀_𝑖𝑗𝜕𝜆_𝑖+𝑛−4𝑐2𝑛^{02𝑛/(𝑛−4)}𝛽𝛼²_𝑖𝐾𝑎_𝑖×_ℝ𝑛signe𝑥_𝑘+𝜆_𝑖𝑎_𝑖−𝑦_𝑗𝑖𝑘𝑥_𝑘+𝜆_𝑖𝑎_𝑖−𝑦_𝑗𝑖𝑘^𝛽−1×𝑥_𝑘1+|𝑥|^2𝑛𝑑𝑥+𝑜_𝑗≠𝑖𝜀_𝑖𝑗+𝑜^𝑝_𝑗=11𝜆^𝛽𝑗_𝑗. (47) Observe that for 𝑘∈{1,…,𝑛}, if 𝜆_𝑖|(𝑎_𝑖−𝑦_𝑙𝑖)_𝑘|>𝑀₁/√𝑛, we have signe𝑥_ℝ𝑛𝑘+𝜆_𝑖𝑎_𝑖−𝑦_𝑗𝑖𝑘×𝑥_𝑘+𝜆_𝑖𝑎_𝑖−𝑦_𝑙𝑖𝑘^𝛽𝑖−1𝑥_𝑘1+|𝑥|^2𝑛𝜆𝑑𝑥=𝑐signe_𝑖𝑎_𝑖−𝑦_𝑗𝑖𝑘×𝜆_𝑖𝑎_𝑖−𝑦_𝑙𝑖𝑘^𝛽𝑖−2(1+𝑜(1)), (48) taking 𝑀₁ large enough. If not, we have _ℝ𝑛𝑥_𝑘+𝜆_𝑖𝑎_𝑖−𝑦_𝑙𝑖𝑘^𝛽𝑖−1𝑥_𝑘1+|𝑥|^2𝑛𝑑𝑥=𝑂(1). (49) Using the fact that 𝑘_𝑖 defined in (45) satisfies 𝜆_𝑖|(𝑎_𝑖−𝑦_𝑙𝑖)_𝑘𝑖|>𝑀₁/√𝑛, if 𝑖∈𝐿₂, Lemma 12 follows.

Lemma 13. For 𝑢=∑^𝑝_𝑖=1𝛼_𝑖𝛿_{(𝑎𝑖,𝜆𝑖)}∈𝑉(𝑝,𝜀), such that 𝑎_𝑖∈𝐵(𝑦_𝑙𝑖,𝜌), 𝑦_𝑙𝑖∈𝒦, for all 𝑖=1,…,𝑝, one has 𝜕𝐽(𝑢),𝑋_𝑖(𝑢)≤𝑂_𝑗≠𝑖1𝜆_𝑖𝜕𝜀_𝑖𝑗𝜕𝑎_𝑖1+𝑂𝜆^𝛽𝑖_𝑖,𝑖𝑓𝑖∈𝐿₁+1−𝑐𝜆^𝛽𝑖_𝑖+𝑎_𝑖−𝑦_{𝑙𝑖𝑘𝑖}^𝛽𝑖−1𝜆_𝑖,𝑖𝑓𝑖∈𝐿₂+𝑜^𝑝_𝑗=11𝜆^𝛽𝑗_𝑗, (50) where 𝑘_𝑖 is defined in (45).

Proof. Using Proposition 8, we have 𝜕𝐽(𝑢),𝑋_𝑖(𝑢()𝑐)=−2𝑛−4₀^{2𝑛/(𝑛−4)}𝛼_𝑖^{2𝑛/(𝑛−4)}1𝜆_𝑖^𝛽𝑖×^𝑛_{𝑘=1ℝ𝑛}𝑏_𝑘𝑥_𝑘+𝜆_𝑖𝑎_𝑖−𝑦_𝑙𝑖𝑘^𝛽𝑖1+𝜆_𝑖𝑎_𝑖−𝑦_𝑙𝑖𝑘^{(𝛽𝑖−1)/2}×𝑥_𝑘1+|𝑥|^2𝑛+1𝑑𝑥²+𝑂_𝑗≠𝑖1𝜆_𝑖𝜕𝜀_𝑖𝑗𝜕𝑎_𝑖+𝑜^𝑝_𝑗=11𝜆^𝛽𝑗_𝑗1≤−𝑐𝜆_𝑖^{𝛽𝑖ℝ𝑛}𝑏_𝑘𝑖𝑥_𝑘+𝜆_𝑖𝑎_𝑖−𝑦_{𝑙𝑖𝑘𝑖}^𝛽𝑖1+𝜆_𝑖𝑎_𝑖−𝑦_{𝑙𝑖𝑘𝑖}^{(𝛽𝑖−1)/2}×𝑥_𝑘𝑖1+|𝑥|^2𝑛+1𝑑𝑥²+𝑂_𝑗≠𝑖1𝜆_𝑖𝜕𝜀_𝑖𝑗𝜕𝑎_𝑖+𝑜^𝑝_𝑗=11𝜆^𝛽𝑗_𝑗. (51) Using (43) and the fact that 𝜆_𝑖|(𝑎_𝑖−𝑦_𝑙𝑖)_𝑘𝑖|>𝑀₁/√𝑛, if 𝑖∈𝐿₂, Lemma 13 follows.

Proof of Theorem 9. In order to complete the construction of the pseudogradient 𝑊 suggested in Theorem 9, it only remains (using Propositions 10 and 11) to focus attention on the two following subsets of 𝑉(𝑝,𝜀).
Subset 1. We consider here the case of 𝑢=∑^𝑝_𝑖=1𝛼_𝑖𝛿_{(𝑎𝑖𝜆𝑖)}=∑_𝑖∈𝐼1𝛼_𝑖𝛿_{(𝑎𝑖𝜆𝑖)}+∑_𝑖∈𝐼2𝛼_𝑖𝛿_{(𝑎𝑖𝜆𝑖)} such that 𝐼₁≠∅,𝐼₂≠∅,_𝑖∈𝐼1𝛼_𝑖𝛿_{(𝑎𝑖𝜆𝑖)}∈𝑉₁♯𝐼₁,𝜀_𝑖∈𝐼2,𝛼_𝑖𝛿_{(𝑎𝑖𝜆𝑖)}∈𝑉₂♯𝐼₂,𝜀. (52) Without loss of generality, we can assume that 𝜆₁≤⋯≤𝜆_𝑝. (53) We distinguish three cases.
Case  1. 𝑢₁∶=∑_𝑖∈𝐼1𝛼_𝑖𝛿_{(𝑎𝑖𝜆𝑖)}∉𝑉¹₁(♯𝐼₁,𝜀) = {𝑢=∑^♯𝐼1_𝑗=1𝛼_𝑗𝛿_{(𝑎𝑗,𝜆𝑗)}, 𝑎_𝑗∈𝐵(𝑦_𝑙𝑗,𝜌), 𝑦_𝑙𝑗∈𝒦⁺ for all 𝑗=1,…,♯𝐼₁ and 𝑦_𝑙𝑗≠𝑦_𝑙𝑘 for all 𝑗≠𝑘}.
Let 𝑊₁ be the pseudogradient on 𝑉(𝑝,𝜀) defined by 𝑊₁(𝑢)=𝑊₁(𝑢₁), where 𝑊₁ is the vector filed defined by Proposition 11 in 𝑉₁(♯𝐼₁,𝜀). Note that if 𝑢₁∉𝑉¹₁(♯𝐼₁,𝜀), then the pseudo-gradient 𝑊₁(𝑢₁) does not increase the maximum of the 𝜆_𝑖's, 𝑖∈𝐼₁. Using Proposition 11, we have 𝜕𝐽(𝑢),𝑊₁(𝑢)≤−𝑐_𝑖∈𝐼11𝜆^𝛽𝑖_𝑖+_{𝑗≠𝑖,𝑖,𝑗∈𝐼1}𝜀_𝑖𝑗+_𝑖∈𝐼1∇𝐾𝑎_𝑖𝜆_𝑖+𝑂_{𝑖∈𝐼1,𝑗∈𝐼2}𝜀_𝑖𝑗. (54) An easy calculation yields 𝜀_𝑖𝑗1=𝑂𝜆_𝑖𝜆_𝑗^(𝑛−4)/21=𝑜𝜆^𝛽𝑖_𝑖1+𝑜𝜆^𝛽𝑗_𝑗, (55)
Fix 𝑖₀∈𝐼₁, we denote by 𝐽₁=𝑖∈𝐼₂,s.t,𝜆_𝑖^𝑛−4≥12𝜆^𝛽𝑖0_𝑖0,𝐽₂=𝐼₂⧵𝐽₁. (56) Using (54) and (55), we find that 𝜕𝐽(𝑢),𝑊₁(𝑢)≤−𝑐_{𝑖∈𝐼1∪𝐽1}1𝜆^𝛽𝑖_𝑖+_𝑖∈𝐼1∇𝐾𝑎_𝑖𝜆_𝑖+_{𝑗≠𝑖∈𝐼1}𝜀_𝑖𝑗+𝑜^𝑝_𝑖=11𝜆^𝛽𝑖_𝑖. (57) From another part, by Lemma 12 we have 𝜕𝐽(𝑢),_𝑖∈𝐽1−2^𝑖𝑍_𝑖(𝑢)≤𝑐_{𝑗≠𝑖,𝑖∈𝐽1}2^𝑖𝜆_𝑖𝜕𝜀_𝑖𝑗𝜕𝜆_𝑖+𝑂_𝑖∈𝐽11𝜆^𝛽𝑖_𝑖+𝑂_{𝑖∈𝐽1∩𝐿2}𝑎_𝑖−𝑦_{𝑙𝑖𝑘𝑖}^𝛽𝑖−2𝜆²_𝑖. (58) Observe that using a direct calculation, we have 𝜆_𝑖𝜕𝜀_𝑖𝑗𝜕𝜆_𝑖≤−𝑐𝜀_𝑖𝑗,if𝜆_𝑖≥𝜆_𝑗or𝜆_𝑖∼𝜆_𝑗or𝑎_𝑖−𝑎_𝑗≥𝛿₀>0. (59) Since for 𝑖<𝑗, we have 2^𝑖𝜆_𝑖𝜕𝜀_𝑖𝑗𝜕𝜆_𝑖+2^𝑗𝜆_𝑗𝜕𝜀_𝑖𝑗𝜕𝜆_𝑗≤−𝑐𝜀_𝑖𝑗, (60) and for 𝑖∈𝐽₁ and 𝑗∈𝐽₂ we have 𝜆_𝑗≤𝜆_𝑖, we obtain 𝜆_𝑖(𝜕𝜀_𝑖𝑗/𝜕𝜆_𝑖)≤−𝑐𝜀_𝑖𝑗. These estimates yield 𝜕𝐽(𝑢),_𝑖∈𝐽1−2^𝑖𝑍_𝑖(𝑢)≤−𝑐_{𝑗≠𝑖,𝑖∈𝐽1,𝑗∈𝐽1∪𝐽2}𝜀_𝑖𝑗+𝑂_𝑖∈𝐽11𝜆^𝛽𝑖_𝑖+𝑂_{𝑖∈𝐽1∩𝐿2}𝑎_𝑖−𝑦_{𝑙𝑖𝑘𝑖}^𝛽𝑖−2𝜆²_𝑖+𝑂_{𝑖∈𝐽1,𝑗∈𝐼1}𝜀_𝑖𝑗. (61) It is easy to see that for any index 𝑖∈𝐿₂, we have 𝑎_𝑖−𝑦_{𝑙𝑖𝑘𝑖}^𝛽𝑖−2𝜆²_𝑖≤√𝑛𝑀₁𝑎_𝑖−𝑦_{𝑙𝑖𝑘𝑖}^𝛽𝑖−1𝜆_𝑖, (62) where 𝑘_𝑖 is defined in (45) and 𝑀₁ is large enough. Thus, we derive that 𝑎_𝑖−𝑦_{𝑙𝑖𝑘𝑖}^𝛽𝑖−2𝜆²_𝑖𝑎=𝑜^𝑖−𝑦_{𝑙𝑖𝑘𝑖}^𝛽𝑖−1𝜆_𝑖,forany𝑖∈𝐿₂. (63)
Let 𝑚₁>0 be small enough; using Lemma 13, (63), and (55), we get 𝜕𝐽(𝑢),_𝑖∈𝐽1−2^𝑖𝑍_𝑖(𝑢)+𝑚_{1𝑖∈𝐽1∩𝐿2}𝑋_𝑖(𝑢)≤−𝑐_{𝑗≠𝑖,𝑖∈𝐽1,𝑗∈𝐽1∪𝐽2}𝜀_𝑖𝑗+_𝑖∈𝐽1∇𝐾𝑎_𝑖𝜆_𝑖+𝑂_𝑖∈𝐽11𝜆^𝛽𝑖_𝑖+𝑜^𝑝_𝑖=11𝜆^𝛽𝑖_𝑖, (64) and by (57) we obtain 𝜕𝐽(𝑢),𝑊₁(𝑢)+𝑚_{1𝑖∈𝐽1}−2^𝑖𝑍_𝑖(𝑢)+𝑚_{1𝑖∈𝐽1∩𝐿2}𝑋_𝑖(𝑢)≤−𝑐_{𝑖∈𝐼1∪𝐽1}1𝜆^𝛽𝑖_𝑖+_{𝑖≠𝑗∈𝐼1}𝜀_𝑖𝑗+_{𝑗≠𝑖,𝑖∈𝐽1,𝑗∈𝐽1∪𝐽2}𝜀_𝑖𝑗+_{𝑖∈𝐼1∪𝐽1}∇𝐾𝑎_𝑖𝜆_𝑖+𝑜^𝑝_𝑖=11𝜆^𝛽𝑖_𝑖. (65) We need to add the remainder indices 𝑖∈𝐽₂. Note that 𝑢∶=∑_𝑗∈𝐽2𝛼_𝑗𝛿_𝑗∈𝑉₂(♯𝐽₂,𝜀). Thus using Proposition 10, we apply the associated vector field which we will denote by 𝑊₂. We then have the following estimate: 𝜕𝐽(𝑢),𝑊₂(𝑢)≤−𝑐_𝑗∈𝐽21𝜆^𝛽𝑗_𝑗+_{𝑖≠𝑗,𝑖,𝑗∈𝐽2}𝜀_𝑖𝑗+_𝑗∈𝐽2∇𝐾𝑎_𝑗𝜆_𝑗+𝑂_{𝑗∈𝐽2,𝑖∈𝐽1}𝜀_𝑖𝑗+𝑜^𝑝_𝑖=11𝜆^𝛽𝑖_𝑖, (66) since |𝑎_𝑖−𝑎_𝑗|≥𝜌 for 𝑗∈𝐽₂ and 𝑖∈𝐼₁.
Let in this case 𝑊=𝑊₁+𝑚₁(𝑊₂+∑_𝑖∈𝐽1−2^𝑖𝑍_𝑖+𝑚₁∑_{𝑖∈𝐽1∩𝐿2}𝑋_𝑖).
From (65) and (66) we obtain ⟨𝜕𝐽(𝑢),𝑊(𝑢)⟩≤−𝑐^𝑝_𝑖=11𝜆^𝛽𝑖_𝑖+^𝑝_𝑖=1∇𝐾𝑎_𝑖𝜆_𝑖+_𝑖≠𝑗𝜀_𝑖𝑗. (67)
Case  2. 𝑢₁∶=∑_𝑖∈𝐼1𝛼_𝑖𝛿_{(𝑎𝑖𝜆𝑖)}∈𝑉¹₁(♯𝐼₁,𝜀) and 𝑢₂∶=∑_𝑖∉𝐼2𝛼_𝑖𝛿_{(𝑎𝑖𝜆𝑖)}∉𝑉¹₂(♯𝐼₂,𝜀)∶={𝑢=∑^♯𝐼2_𝑗=1𝛼_𝑗𝛿_{(𝑎𝑗,𝜆𝑗)}, 𝑎_𝑗∈𝐵(𝑦_𝑙𝑗,𝜌), 𝑦_𝑙𝑗∈𝒦⁺, for all 𝑗=1,…,♯𝐼₂ and 𝜌(𝑦_𝑙1,…,𝑦_♯𝐼2)>0}.
Since 𝑢₂∈𝑉₂(♯𝐼₂,𝜀), by Proposition 10, we can apply the associated vector field which we will denote by 𝑉₁. We get 𝜕𝐽(𝑢),𝑉₁(𝑢)≤−𝑐_𝑖∈𝐼21𝜆^𝛽𝑖_𝑖+_𝑖∈𝐼2∇𝐾𝑎_𝑖𝜆_𝑖+_{𝑖≠𝑗,𝑖,𝑗∈𝐼2}𝜀_𝑖𝑗+𝑂_{𝑖∈𝐼2,𝑗∈𝐼1}𝜀_𝑖𝑗. (68) Observe that 𝑉₁(𝑢) does not increase the maximum of the 𝜆_𝑖's, 𝑖∈𝐼₂, since 𝑢₂∉𝑉¹₂(♯𝐼₂,𝜀). Fix 𝑖₀∈𝐼₂ and let 𝐽₁=𝑖∈𝐼₁,s.t,𝜆^𝛽𝑖_𝑖≥12𝜆_𝑖^𝑛−4₀,𝐽₂=𝐼₁⧵𝐽₁. (69) Using (68) and (55), we get 𝜕𝐽(𝑢),𝑉₁(𝑢)≤−𝑐_{𝑖∈𝐼2∪𝐽1}1𝜆^𝛽𝑖_𝑖+_𝑖∈𝐼2∇𝐾𝑎_𝑖𝜆_𝑖+_{𝑖≠𝑗,𝑖,𝑗∈𝐼2}𝜀_𝑖𝑗+𝑜^𝑝_𝑖=11𝜆^𝛽𝑖_𝑖. (70) We need to add the indices 𝑖, 𝑖∈𝐽₂. Letting 𝑢∶=∑_𝐽𝑗∈2𝛼_𝑗𝛿_{(𝑎𝑗𝜆𝑗)}, since 𝑢∈𝑉₁(♯𝐽₂,𝜀), we can apply the associated vector field given by Proposition 10. Let 𝑉₂ be this vector field. By Proposition 11 we have 𝜕𝐽(𝑢),𝑉₂(𝑢)≤−𝑐_𝐽𝑗∈21𝜆^𝛽𝑗_𝑗+_𝐽𝑗∈2∇𝐾𝑎_𝑗𝜆_𝑗+_{𝐽𝑖≠𝑗,𝑖,𝑗∈2}𝜀_𝑖𝑗+𝑂_{𝐽𝑗∈2𝐽,𝑖∉2}𝜀_𝑖𝑗. (71) Observe that 𝐼₁=𝐽₁∪𝐽₂ and we are in the case where for all 𝑖≠𝑗∈𝐼₁, we have |𝑎_𝑖−𝑎_𝑗|≥𝜌. Thus by (55), we get 𝑂_{𝐽𝑗∈2𝐽,𝑖∉2}𝜀_𝑖𝑗=𝑜^𝑝_𝑖=11𝜆^𝛽𝑖_𝑖, (72) and hence 𝜕𝐽(𝑢),𝑉₁(𝑢)+𝑉₂(𝑢)≤−𝑐^𝑝_𝑖=11𝜆^𝛽𝑖_𝑖+_{𝑖∈𝐼2∪𝐽2}∇𝐾𝑎_𝑖𝜆_𝑖+_𝑖≠𝑗𝜀_𝑖𝑗. (73) Let in this case 𝑊=𝑉₁+𝑉₂+𝑚₁∑_𝐽𝑖∈1𝑋_𝑖(𝑢), 𝑚₁ small enough.
Using the previous estimate and Lemma 13, we find that ⟨𝜕𝐽(𝑢),𝑊(𝑢)⟩≤−𝑐^𝑝_𝑖=11𝜆^𝛽𝑖_𝑖+^𝑝_𝑖=1∇𝐾𝑎_𝑖𝜆_𝑖+_𝑖≠𝑗𝜀_𝑖𝑗. (74)
Case  3. 𝑢₁∈𝑉¹₁(♯𝐼₁,𝜀) and 𝑢₂∈𝑉¹₂(♯𝐼₂,𝜀).
Let 𝑉₁(resp., 𝑉₂) be the pseudo-gradient in 𝑉(𝑝,𝜀) defined by 𝑉₁(𝑢)=𝑊₁(𝑢₁) (resp., 𝑉₂(𝑢)=𝑊₂(𝑢₂)) where 𝑊₁ (resp., 𝑊₂) is the vector field defined by Proposition 11 (resp., Proposition 10) in 𝑉¹₁(♯𝐼₁,𝜀) (resp., 𝑉¹₂(♯𝐼₂,𝜀)) and let in this case 𝑊=𝑉₁+𝑉₂. (75) Using Proposition 10, Proposition 11, and (55) we get ⟨𝜕𝐽(𝑢),𝑊(𝑢)⟩≤−𝑐^𝑝_𝑖=11𝜆^𝛽𝑖_𝑖+^𝑝_𝑖=1∇𝐾𝑎_𝑖𝜆_𝑖+_𝑖≠𝑗𝜀_𝑖𝑗. (76)
Notice that in the first and second cases, the maximum of the 𝜆_𝑖's, 1≤𝑖≤𝑝, is a bounded function and hence the Palais-Smale condition is satisfied along the flow lines of 𝑊. However in the third case all the 𝜆_𝑖's, 1≤𝑖≤𝑝, will increase and go to +∞ along the flow lines generated by 𝑊.
Subset 2. We consider the case of 𝑢=∑^𝑝_𝑖=1𝛼_𝑖𝛿_𝑖∈𝑉(𝑝,𝜀), such that there exist 𝑎_𝑖 satisfying 𝑎_𝑖∉∪_𝑦∈𝒦𝐵(𝑦,𝜌).
In this region, the construction of the pseudo-gradient 𝑊 proceeds exactly as the proof of Theorem 3.2, of subset 2, of [11].
Finally, observe that our pseudogradient 𝑊 in 𝑉(𝑝,𝜀) satisfies claim (i) of Theorem 9 and it is bounded, since ‖𝜆_𝑖(𝜕𝛿_{(𝑎𝑖𝑖,𝜆)}/𝜕𝜆_𝑖)‖ and ‖(1/𝜆_𝑖)(𝜕𝛿_{(𝑎𝑖𝑖,𝜆)}/𝜕𝑎_𝑖)‖ are bounded. From the definition of 𝑊, the 𝜆_𝑖's, 1≤𝑖≤𝑝, decrease along the flow lines of 𝑊 as long as these flow lines do not enter in the neighborhood of finite number of critical points 𝑦_𝑙𝑖, 𝑖=1,…,𝑝, of 𝒦 such that (𝑦_𝑙1,…,𝑦_𝑙𝑝)∈𝒞_∞.
Now, arguing as in Appendix 2 of [16], see also Appendix B of [18], claim (ii) of Theorem 9 follows from (i) and Proposition 5. This completes the proof of Theorem 9.

Corollary 14. Let 𝑝≥1. The critical points at infinity of 𝐽 in 𝑉(𝑝,𝜀) correspond to 𝑦_𝑙1,…,𝑦_𝑙𝑝∞∶=^𝑝_𝑖=11𝐾𝑦_𝑙𝑖^(𝑛−4)/2𝛿_{(𝑦𝑙𝑖,∞)}, (77) where (𝑦_𝑙1,…,𝑦_𝑙𝑝)∈𝒞_∞. Moreover, such a critical point at infinity has an index equal to 𝑖(𝑦_𝑙1,…,𝑦_𝑙𝑝)_∞=𝑝−1+∑^𝑝_𝑖=1𝑛−𝑖(̃𝑦).

4. Proof of Theorem 1

We prove the existence result by contradiction. Assume that 𝐽 has no critical point in 𝑉_𝜂(Σ⁺). It follows from Corollary 14 that the critical points at infinity of the associated variational problem are in one to one correspondence with the elements of 𝒞_∞ defined in (13).

Notice that, just like for usual critical points, it is associated with each critical point at infinity 𝑤_∞ of 𝐽 stable and unstable manifolds 𝑊^∞_𝑠(𝑤_∞) and 𝑊^∞_𝑢(𝑤_∞) (see [16, pages 356-357]). These manifolds can be easily described once a finite-dimensional reduction like the one we performed in Section 3 is established.

For any 𝑤_∞=(𝑦_𝑖1,…,𝑦_𝑖𝑝)∈𝒞_∞, let 𝑐(𝑤)_∞=𝑆_𝑛(∑^𝑝_𝑗=11/𝐾(𝑦_𝑖𝑗)^(𝑛−4)/2)^4/𝑛 denote the associated critical value. Here we choose to consider a simplified situation, where for any 𝑤_∞≠𝑤^′_∞, 𝑐(𝑤)_∞≠𝑐(𝑤^′)_∞, and thus order the 𝑐(𝑤)_∞'s, 𝑤_∞∈𝒞_∞ as 𝑐𝑤_1∞<⋯<𝑐𝑤_𝑘0∞. (78) By using a deformation lemma (see [22, proposition 7.24 and Theorem 8.2]), we know that if 𝑐(𝑤_𝑘−1)_∞<𝑎<𝑐(𝑤_𝑘)_∞<𝑏<𝑐(𝑤_𝑘+1)_∞, then 𝐽_𝑏≃𝐽_𝑎∪𝑊^∞_𝑢𝑤_𝑘∞, (79) where 𝐽_𝑏={𝑢∈𝑉_𝜂(Σ⁺),𝐽(𝑢)≤𝑏} and ≃ denotes retracts by deformation.

We apply the Euler-Poincaré characteristic of both sides of (79); we find that 𝜒𝐽_𝑏=𝜒𝐽_𝑎+(−1)^{𝑖(𝑤𝑘)∞}, (80) where 𝑖(𝑤_𝑘)_∞ denotes the index of the critical point at infinity (𝑤_𝑘)_∞. Let 𝑏₁<𝑐𝑤_1∞=min_{𝑢∈𝑉𝜂(Σ+)}𝐽(𝑢)<𝑏₂<𝑐𝑤_2∞<⋯<𝑏_𝑘0<𝑐𝑤_𝑘0∞<𝑏_𝑘0+1. (81) Since we have assumed that (4) has no solution, 𝐽_𝑏𝑘0+1 is a retard by deformation of 𝑉_𝜂(Σ⁺). Therefore 𝜒(𝐽_𝑏𝑘0+1)=1, since 𝑉_𝜂(Σ⁺) is a contractible set. Now using (80), we derive after recalling that 𝜒(𝐽_𝑏1)=𝜒(∅)=01=^𝑘0_𝑗=1(−1)^{𝑖(𝑤𝑗)∞}. (82) Hence if (82) is violated, 𝐽 has a critical point in 𝑉_𝜂(Σ⁺).

Now, arguing as in the proof of theorem of [18, pages 659-660], we prove that such a critical point is positive.

To prove the multiplicity part of the statement, we observe that it follows from Sard-Smale theorem that for generic 𝐾's, the solutions to (7) are all nondegenerate, in the sense that the associated linearized operator does not admit zero as eigenvalue. We need to introduce the following lemma extracted from [11].

Lemma 15 (see [11, Section 3.2]). Let 𝑤 be a solution to (7). Assume that the function 𝐾 satisfies condition (𝑓)_𝛽, with (𝑛−4)/2<𝛽≤𝑛−4; then for each 𝑝∈ℕ^∗, there are neither critical points nor critical points at infinity in 𝑉(𝑝,𝜀,𝑤).

Once the existence of mixed critical points at infinity is ruled out, it follows from the previous arguments that 𝑉_𝜂Σ⁺≃^𝑘0_𝑗=1𝑊^∞_𝑢𝑤_𝑗∞∪_{(𝑤)𝑤,𝜕𝐽=0}𝑊_𝑢(𝑤). (83) Now using the Euler-Poincaré theorem, we derive that 1=^𝑘0_𝑗=1(−1)^{𝑖(𝑤𝑗)∞}+_{(𝑤)𝑤,𝜕𝐽=0}(−1)^morse(𝑤). (84) Hence our theorem follows.

5. A General Existence Result

In this last section of this paper, we give a generalization of Theorem 1. Namely, in view of the result of Theorem 1, one may think about the situation where the degree 𝑑 given in Theorem 1 is equal to zero; that is, the total sum in 𝑑 is equal to 1, but a partial one is not equal to 1. A natural question arises: is it possible in this case to use such information to derive an existence result? In the following theorem we give a partial answer to this question.

Theorem 16. Let 𝐾 be a function satisfying (𝐴₁) and (𝑓)_𝛽, with 1<𝛽≤𝑛−4. If there exists 𝑘∈ℕ such that (1)∀𝑤_∞∈𝒞_∞,𝑖(𝑤)_∞(2)≠𝑘+1_{𝑤∞∈𝒞∞,𝑖(𝑤∞)≤𝑘}(−1)^{𝑖(𝑤∞)}≠1, (85) then the problem (7) has at least one solution.
Moreover for generic 𝐾's, if (𝑛−4)/2<𝛽≤𝑛−4, then the number of solutions is lower bounded by 1−_{𝑤∞∈𝒞∞𝑤,𝑖∞≤𝑘}(−1)^{𝑖(𝑤∞)}. (86)

Let 𝑙_♯ be the maximal index over all elements of 𝒞_∞. Please observe that the integer 𝑘=𝑙_♯ satisfies (1) of Theorem 16; it follows that Theorem 1 is a corollary of Theorem 16.

Proof of Theorem 16. We set 𝑋^∞_𝑘=_{𝑥∞∈𝒞∞(𝑥),𝑖∞≤𝑘}𝑊^∞_𝑢𝑥_∞, (87) where 𝑊^∞_𝑢(𝑥_∞) is the closure of the unstable manifolds of 𝑥_∞. Observe that 𝑋^∞_𝑘 is a stratified set of top dimension 𝑘, which is contractible set in 𝑉_𝜂(Σ⁺), since 𝑉_𝜂(Σ⁺) is a contractible set. Let 𝑈 denote the image of such a contraction. To prove the first part of Theorem 16, arguing by contradiction, we assume that (7) has no solution. Using the pseudo-gradient constructed in Theorem 9, we can deform 𝑈. By transversality arguments, we can assume that such a deformation avoids all critical points at infinity of index greater or equal to 𝑘+1. Note that, from assumption (1) of Theorem 16, there is no critical point at infinity with index 𝑘+1. It follows then from a theorem of Bahri and Rabinowitz [22] that 𝑈≃𝑋^∞_𝑘. (88) Hence from the Euler-Poincaré theorem, we get 1=𝜒𝑋^∞_𝑘=_{𝑤∞∈𝐶∞𝑤,𝑖∞≤𝑘}(−1)^{𝑖(𝑤∞)}, (89) which is a contradiction with assumption (2) of Theorem 16.
Regarding the multiplicity result, we observe that for generic 𝐾's the functional 𝐽 admits only nondegenerate critical points. Hence by Lemma 15, the set 𝑈 will be deformed into 𝑈≃𝑋^∞_𝑘∪_{𝑤∈𝐶𝑘}𝑊_𝑢(𝑤), (90) where 𝐶_𝑘 denotes the set of the critical points of 𝐽 of Morse index less than or equal to 𝑘, which are dominated by 𝑈. Finally by using the Euler-Poincaré theorem, the proof of Theorem 16 follows.