Constant q-curvature metrics in arbitrary dimension

Constant Q-curvature metrics in arbitrary dimension Cheikh Birahim NDIAYE SISSA, via Beirut 2-4, 34014 Trieste, Italy.

Abstract: Working in a given conformal class, we prove existence of constant Q-curvature metrics on compact manifolds of arbitrary dimension under generic assumptions. The problem is equivalent to solving a nth-order nonlinear elliptic differential (or integral) equation with variational structure, where n is the dimension of the manifold. Since the corresponding Euler functional is in general unbounded from above and below, we use critical point theory, jointly with a compactness result for the above equation.

Key Words: Geometric PDEs, Conformally invariant integral equations, Pseudodifferential operators, Blow-up analysis, Variational methods, Min-max Schemes. AMS subject classification: 35B33, 35J35, 53A30, 53C21

1

Introduction

The study of the relationships between conformally covariant operators, their associated conformal invariants and the related partial differential equations has received much attention in the last decades. A model example is the Laplace-Beltrami operator on compact surfaces (Σ, g), which governs the transformation laws of the Gauss curvature. In fact under the conformal change of metric g˜ = e2u g, we have (1)

∆g˜ = e−2u ∆g ;

−∆g u + Kg = Kg˜ e2u ,

where ∆g and Kg (resp. ∆g˜ and Kg˜ ) are the Laplace-Beltrami operator and the Gauss curvature of (Σ, g) (resp. of (Σ, g˜)). R Moreover we have the Gauss-Bonnet formula which relates Σ Kg dVg and the topology of Σ : Z Kg dVg = 2πχ(Σ); Σ

R where χ(Σ) is the Euler-Poincar´e characteristic of Σ. From this we deduce that Σ Kg dVg a topological invariant (hence also a conformal one). There exists also an other example of conformally covariant differential operator on four dimensional manifolds called the Paneitz operator and to which is associated a natural concept of curvature. This operator discovered by Paneitz in 1983 ( see [46]) and the corresponding Q-curvature introduced by Branson ( see [9]) are defined in terms of Ricci tensor Ricg and scalar curvature Rg of the manifold (M, g) as follows (2) 1 E-mail

2 Pg ϕ = ∆2g ϕ + divg ( Rg g − 2Ricg )dϕ; 3 addresses: [email protected]

1

1 (∆g Rg − Rg2 + 3|Ricg |2 ), 12 where ϕ is any smooth function on M . As the Laplace-Beltrami operator governs the transformation laws of the Gauss curvature, we have also that the Paneitz operator does the same for the Q-curvature. Indeed under a conformal change of metric g˜ = e2u g we have Qg = −

Pg˜ = e−4u Pg ;

Pg u + 2Qg = 2Qg˜ e4u .

Apart from this analogy, we have also an extension of the Gauss-Bonnet formula which is the ChernGauss-Bonnet formula Z |Wg |2 (Qg + )dVg = 4π 2 χ(M ), 8 M where Wg denotes the Weyl tensor of (M, g), see [26]. Hence from the pointwise conformal invariance of |Wg |2 dVg , it follows that the integral of Qg over M is also a conformal invariant. On the other hand there are high-order analogues to the Laplace-Beltrami operator and to the Paneitz operator for high dimensional compact manifolds and also to the associated curvatures. More precisely given a compact n-dimensional manifold (M, g), in [34] it was introduced a family of conformally covarin ant differential operators P2m ( for every positive integer m if n is odd and for every positive integer m such that 2m ≤ n if n is even ) whose leading term is (−∆g )m . These operators are usually referred to n as the GJMS operators. Moreover after passing to stereographic projection P2m coincide with (−∆g )m if M is the sphere and g its standard metric. In [7], some curvature invariants Qn2m was defined, natn . urally associated to P2m Now for n even let us set Pgn = Pnn ; Qng = Qnn . Then in low dimensions we have the following relations

and

Pg2 = ∆g ;

Q2g = Kg ,

Pg4 = Pg ;

Q4g = 2Qg .

It turns out that Pgn is self-adjoint and annihilates constants. Furthermore as for the Laplace-Beltrami operator on compact surfaces and the Paneitz operator on compact four dimensional manifold, for every compact n-dimensional manifolds with n even, we have that after a conformal change of metric g˜ = e2u g Pgn = e−nu Pgn ;

Pgn u + Qng = Qng˜ enu .

To give some geometric applications we discuss for the four dimensional case three results proven by Gursky, [36] , and by Chang, Gursky and Yang,R [14] , [15] . If a manifold which has a conformal metric of positive constant scalar curvature satisfies M Qg dVg > 0, then its first Betti number vanishes. Moreover up to a conformal metric it has positive Ricci R tensor, andR hence M has a finite fundamental group. Furthermore, if the quantitative assumption M Qg dVg > 81 M |Wg |2 dVg holds then M must be diffeomorphic to the four-sphere or to the projective space. In particular the last result is an improvement of a theorem by Margerin, [45] with a conformally invariant assumption, while the one of Margerin assumes pointwise pinching conditions on the Ricci tensor in terms of Wg . Finally, we also point out that the Paneitz operator, the Q-curvature and their high-dimensional analogues, see [7] , [8] appear in the study of Moser-Trudinger type inequalities, log-determinant formulas and the compactification of locally conformally flat manifolds, see [10], [15], [16], [17]. As for the uniformization theorem for compact surfaces, one can ask whether every compact ndimensional manifold (M, g) ( n even) carries a conformal metric g˜ for which the corresponding Qcurvature Qng˜ is a constant. Writting g˜ = e2u g, the problem is equivalent to finding a solution of the equation (3)

¯ nu Pgn u + Qng = Qe 2

in M ;

¯ is a real constant. where Q

For the four-dimensional case, problem (3) has been solved in [18] under the assumption that Pg is R a non-negative operator and M Qg dVg < 8π 2 ( 8π 2 is the integral of the Q-curvature on the standard sphere). Under these assumptions by the Adams inequality (see [17]) Z 1 log e4(u−¯u) dVg ≤ hPg u, ui + C, u ∈ H 2 (M ), 8π 2 M where u ¯ is the average of u and where C depends only on M , the functional II is bounded from below, coercive and lower semicontinuous, hence solutions can be found as global minima using the Direct Methods of the Calculus of Variations. A first sufficient condition to ensure these R hypotheses was given by Gursky in [36]. He proved that if the Yamabe invariant is positive and if Qg dVg > 0, then Pg M R is positive and moreover M Qg dVg ≤ 8π 2 , with the equality holding if and only if M is conformally equivalent to S 4 . More recently Djadli and Malchiodi (see [26]) proved existence of solutions for (3) still in the four-dimensional case under generic assumptions. More precisely they have proved existence R of solutions when Pg has no kernel and M Qg dVg ∈ / R8π 2 N. These conditions include manifolds with negative curvature or negative Yamabe class, for which M Qg dVg can be bigger than 8π 2 . For the n-dimensional case with n even, setting Z κPgn = Qng dVg ; M

problem (3) has been solved under the condition that Pgn is a non-negative operator and κPgn < (n−1)!ωn ( (n − 1)!ωn is the value of κPgn on the standard sphere) using a geometric flow (see [11]). On the other hand since under these assumptions by a Moser-Trudinger type inequality, see Section 4, the functional is bounded from below, coercive and lower semicontinuous, then solutions can be found also by Minimization via Weierstrass theorem in the Calculus of Variations, as for the case of [18]. We are interested here in generalizing the result of Djadli and Malchiodi to every compact n-dimensional manifold under similar generic assumptions. More precisely we prove the following result. Theorem 1.1 Suppose n is even, KerPgn = {constants}, and assume that κPgn 6= k(n − 1)!ωn for k = 1, 2, ..... Then M admits a conformal metric with constant Q-curvature. Remark 1.2 (a) Our assumptions are conformally invariant and generic, so the result applies to a large class of compact n-dimensional manifolds. (b) Under these assumptions, by Corollary 1.4 below, we have that blow ups of sequences of solutions to (3) is not possible. Indeed, these turn out to be bounded in C m (M ) for every integer m. Our assumptions include those made in [11] and (one) of the following two possibilities (or both) (4)

κPgn ∈ (k(n − 1)!ωn , (k + 1)(n − 1)!ωn ) , for some k ∈ N

(5)

Pgn possesses k¯ negative eigenvalues (counted with multiplicity).

To prove Theorem 1.1 we exploit the fact that the problem has a variational structure. Hence in view of standard elliptic regularity theory, solutions can be found as critical points of the following functional Z Z ­ ® n (6) II(u) = n Pgn u, u + 2n Qng udVg − 2κPgn log enu dVg ; u ∈ H 2 (M ). M

M

3

Since this Euler-Lagrange functional is in general unbounded from above and below, it is necessary to find extrema which are possibly saddle points. Hence we will use a min-max scheme following the method of Djadli and Malchiodi in [26]. By classical arguments, the scheme yields a Palais-Smale sequence, namely n a sequence (ul )l ∈ H 2 (M ) satisfying the following properties (7)

II(ul ) → c ∈ R;

0

II (ul ) → 0 as l → +∞.

Then, to recover existence, one should prove for example that (ul )l is bounded, or a similar compactness criterion. But since we do not know if the Palais-Smale condition holds or even if Palais-Smale sequences are bounded, we will employ a monotonicity argument due to Struwe , see [49]. Hence it is useful to study compactness of solutions to perturbations of (3), like ¯ l enul Pgn ul + Ql = Q

(8)

in M ;

where (9)

¯ l −→ Q ¯0 Q

in C 1 (M );

(10)

Ql −→ Q0

in C 1 (M );

¯ 0 > 0. Q

(11)

Following standard terminology, we say that a sequence (ul ) of solution to (8) blows up if the following holds: (12)

there exist xl ∈ M such that ul (xl ) → +∞ as l → +∞,

and we prove the following compactness result. Theorem 1.3 Suppose n is even, KerPgn = {constants} and that (ul ) is a sequence of solutions of (8) with ¯ l satisfying (9), Ql satisfying (10),and Q ¯ 0 satisfying (11). Assuming that (ul )l blows up, there exists Q ∗ N ∈ N such that Z (13) Q0 dVg = N (n − 1)!ωn . M

From this we derive a corollary which will be used in order to recover existence of solutions to (3) via perturbation, by ensuring compactness of the solutions to the perturbed equation. Corollary 1.4 Suppose n is even, KerPgn = {constants}. ¯ l satisfying (9), Ql satisfying (10) and Q ¯ 0 satisfying (11). a) Let (ul ) be a sequence of solutions of (8) with Q Assume also that Z (14) k0 = Q0 dVg 6= k(n − 1)!ωn k = 1, 2, 3, . . . . M α

Then (ul )l is bounded in C (M ) for any α ∈ (0, 1). ¯ Then (ul )l is bounded in b) Let (ul ) be a sequence of solutions to (3) for a fixed value of the constant Q. m C (M ) for every positive integer m. Remark 1.5 a) For the seek of simplicity of the exposition, we will give the proof of Theorem 1.1 in the case where Pgn is non-negative. At the end of Section 4 a discussion to settle the general case is made. b) The proof of Corollary 1.4 is a trivial application of Theorem 1.3 and standard elliptic regularity theory. 4

We are going to describe our approach to prove the above results. Since the proof of Theorem 1.1 is based on the compactness result of Theorem 1.3 ( see also Corollary 1.4), it is convenient to discuss first the latter. Our method follows up to some extent [27] and [42]. However some new ideas are needed since some of the arguments in [27] and [42] rely on the fact of being in low dimensions (more precise comments are given below). We study equation (8) as an integral one. This is possible since one can show that Pgn admits a Green’s function G(x, y) ( see Section 2, Lemma 2.1) which is symmetric and for 1 for x ∼ y (for the value of cn , see Section 2, Lemma 2.1). Hence from the which G(x, y) ∼ c1n log dg (x,y) existence of the Green’s function, we have that equation (8) can be written as Z ¯ l enul (y) − Ql (y))dVg (y) x ∈ M. (15) ul (x) = G(x, y)(Q M

As a first issue in the proof of Theorem 1.3 we determine the profile of solutions near blow-up points. For doing this in [27] and [42] it is used scaling argument and a classification result by C.S Lin [40]. n Unfortunately this classification result for entire solutions of (−∆) 2 u = enu (without growth condition at infinity) is available only in dimension 2 and 4. In higher dimension it is convenient instead to use the full strength of (15) and still after a scaling argument to arrive to the following integral equation on Rn Z |y| 1 (16) u(x) = σn log( )enu dy − log(kn ). |x − y| n n R ( for the definition ofR σn and kn , see Section 3). Assuming only that Rn enu dx < ∞, solutions of (16) have been classified by X.Xu in [52] as standard bubbles and this allows us to deduce the profile of blow ups of (3). Moreover using a generalized Pohozaev equality proven by X.Xu in [52] we derive a volume quantization near the blow ups points. At this stage the analysis is only local, and the next issue is to obtain a global volume quantization as in the statement of Theorem 1.3. After proving a Harnack type inequality one is reduced to study the R 1 behavior of the radial average u ¯l (r) = V olg (∂B u dσg . For doing this in [42] this function ∂Bxl (r) l xl (r)) was studied by an ODE analysis while in [27] it was mainly done using a classification results of some singular solutions to a PDE in R4 . On the other hand one can still exploit entirely the properties of (15). Here indeed we can radialize also (15) (see Section 3, Step 4) and study the radial function u ¯l (r) as solution of a suitable integral inequality in one variable. This approach seems rather natural (we refer the reader also to Remark 1.6). For convenience we divide the proof of Theorem 1.3 into five Steps. Now, having this compactness result we can describe the proof of Theorem 1.1 assuming for simplicity that Pgn is non-negative. A first ingredient is a Moser-Trudinger type inequality and its improvement. To obtain the Moser-Trudinger type inequality we follow an argument of S. Y. A. Chang, P. Yang, see [18], and Fontana [31]. Concerning the improvement of the inequality it was noticed in [26] that if the conformal volume enu is spread into (k + 1) distincts sets where k is as in (4), then the functional II stays bounded from below. As a consequence we derive that if k is given as in (4) and if II(ul ) → nul −∞ along a sequence, has to concentrate R then e Pk if we assume Pknear at most k points of M . Hence, the normalization M enul dVg = 1, then enul ' i=1 ti δxi , where ti ≥ 0, xi ∈ M, i=1 ti = 1 for II(ul ) → −∞. Therefore, as in [26] we can map enul onto Mk for l large, where Mk is the formal set of barycenters of M of order k, see Section 2. Precisely for L À 1 we can define a continuous projection Ψ : {II ≤ −L} → Mk which is homotopically non-trivial. The non-triviality of this map comes from the fact that Mk is non-contractible and from the existence of another map Φλ¯ such that Φλ¯ ◦ Ψ is homotopic to the identity on Mk . Furthermore the map Φλ¯ is such that II(Φλ¯ (Mk )) can become arbitrary large negative, so that Ψ is well-defined on its image. Some comments in the construction of the map Φλ¯ are in order. We use basically the same function as in [26], however we point out that in [26] the estimates of II(ϕλ¯ ) were done by explicit calculations which was possible since the dimension was fixed and low. Here instead, since we want to let n being arbitrary , we need a more systematic approach , which both simplifies and extends that in [26], see Lemma 4.5 and its proof. 5

At this point we are in position to run a min-max scheme similar to the one in [26] based on the use of a topological cone over Mk . The scheme yields a Palais-Smale sequence for II, but unfortunately we cannot ensure convergence directly. As anticipated, following an argument of Struwe , see [49], we introduce an auxiliary functional IIρ , where ρ belongs to a neighborhood of 1. Running the same scheme on the functional IIρ , via some monotonicity argument, we get existence of critical points for almost every ρ, and in particular along a sequence ρk → 1. To conclude, it is sufficient to apply the compactness result in Corollary 1.4. Some comments are in order. The method used here to prove Theorem 1.1 was for the first time used by Djadli and Malchiodi in [26]. Recently it has been used by Djadli in [24] to study a mean field equation depending on a real parameter λ, and prove existence of solutions when λ 6= k8π without any assumption on the topology on the surface. It has also been employed by Malchiodi and the author [43] for the study of the 2 × 2 Toda system.

Remark 1.6 In the paper of Fefferman and Graham, see [29], it was developed a tool which is referred to as FG ambient metric construction and allows them to show existence of scalar conformal invariants. Latter the same tool was used to derive the GJMS operators. On the other hand Branson [9] defines the Q-curvature in the even dimensional case via a continuation argument in the dimension, while in the paper of Graham an Zworsky, see [35], Qng was derived by an analytic continuation in a spectral parameter. Furthermore, inspired by this work, Fefferman and Graham derive the Q-curvature by solving some Lapace problem associated to the formal Poincar´e metric in the ambient space and considering formal asymptotics of the solutions. Moreover this new approach of Fefferman and Graham to derive the Qcurvature allows them to defines analogues of Pgn and Qng also when n is odd. In this case Pgn and Qng enjoy several properties similar to the counterparts in even dimension. More precisely Pgn is self-adjoint and also annihilates constants. Moreover Pgn governs the transformation laws of Qng . On the other hand there is a difference because in the odd case Pgn turns out to be a pseudodifferential operator. In the context of conformal geometry the role of Pgn and Qng in addition is not clear yet since the definition of Pgn and Qng does not depend only on the conformal class of the boundary of the ambient space but also on the extension of the formal Poincar´e metric to a metric in the interior. On the other hand our analysis ( which is mostly based on the Green’s representation formula) could still be applied. For this reason we added a brief fifth Section to discuss also the case of pseudodifferential operators, namely the odd-dimensional case. Now for n odd we set as in the even case Z n (17) κ Pg = Qng dVg ; M

and we have that the result of Brendle [11] in the even case extends to the odd, namely if Pgn is nonnegative and κPgn < (n − 1)!ωn then (3) is solvable. Indeed by the fact that the existence of the Green function for Pgn in the even case extends to the odd case we derive easily a Moser-Trudinger type inequality (see Section 5 for more comments). So from this inequality we have that the associated Euler-Lagrange functional is bounded from below and coercive. Hence from the lower semicontinuity of the Euler-Lagrange functional, we derive existence of solutions of (3) via minimization by the Weierstrass theorem in the Calculus of Variations. For the seek of simplicity again, in Section 5 the discussion will be done assuming that Pgn is non-negative, since when it has some negative eigenvalues we can argue as in the even case The structure of the paper is the following. In Section 2 we collect some notations and give some preliminaries like the existence of the Green function for Pgn and a Moser-Trudinger type inequality. In Section 3 we give the proof of Theorem 1.3 (from which the proof of Corollary (1.4) becomes a trivial application). In Section 4 we finally prove Theorem 1.1. The latter Section is divided into three 6

subsections. The first one is concerned about an improvement of Moser-Trudinger type inequality and applications. The second one deals with estimates of test functions and applications. And in the last one we perform the min-max scheme. The last Section is devoted to the odd-dimensional case. Acknowledgements: I would like to thank Professor Andrea Malchiodi for having proposed to me the study of this problem and for many helpful and interesting suggestions. I am also grateful to Professor Sagun Chanillo for his useful comments about pseudodifferential calculus and to Professor Sun Yang Alice Chang for useful informations about the topic of the paper. The author have been supported by M.U.R.S.T within the PRIN 2004 Variational methods and nonlinear differential equations.

2

Notation and Preliminaries

In this brief section we collect some useful notations, state a lemma giving the existence of the Green function of the operator Pgn with its asymptotics near the singularity and a analogue of the well-known Moser-Trudinger inequality for the operator Pgn when it is non-negative. In the following Bp (r) stands for the metric ball of radius r and center p, B 0 (r) stands for the euclidean ball of center 0 and radius r. We also denote by dg (x, y) the metric distance between two points x and y of M . H s (M ), for s ∈ R, stands for the usual Sobolev space of functions on M which are of class H s in each coordinate system. Large positive constants are always denoted by C, and the value of C is allowed to vary from formula to formula and also within the same line. M 2 stands for the cartesian product M × M , while Diag(M ) is the Rdiagonal of M 2 . Given a functionR u ∈ L1 (M ), u ¯ −1 denotes its average on M , that is u ¯ = (V olg (M )) u(x)dV (x) where V ol (M ) = dV . g g g M M ωn stands for the volume of the unit sphere in Rn+1 ωn−1 stands for the volume of the unit sphere in Rn N denotes the set of non-negative integers. N∗ stands for the set of positive integers. Al = ol (1) means that Al −→ 0 as the integer l −→ +∞. A² = o² (1) means that A² −→ 0 as the real number ² −→ 0. Aδ = oδ (1) means that Aδ −→ 0 as the real number δ −→ 0. Al = O(Bl ) means that Al ≤ CBl for some fixed constant C.. injg (M ) stands for the injectivity radius of M . dVg denotes the Riemannian measure associated to the metric g. dσg stands for the surface measure associated to g. Given an operator P acting on functions u(x, y) defined on M 2 , Py means the action of P with respect to the variable y ∈ M . Given a metric g on M , |g(x)|, x ∈ M , stands for determinant of the matrix with entries gi,j (x) where gi,j (x) are the components of g(x) in some system of coordinates. We let Mk denote the family of formal sums (18)

k k X X Mk = { ti δxi , ti ≥ 0, ti = 1; xi ∈ M }, i=1

i=1

endowed with the weak topology of distributions. This is known in literature as the formal set of barycenters of M (of order k), see [1], [2], [11], [26]. From now up to Section 4 (included), the integer n (namely the dimension of M ) is supposed to be even. Some comments regarding the case of n odd will begin in Section 5. As mentioned before we begin by stating a lemma giving the existence of the Green function of Pgn and its asymptotics near the singularities. Estimates in this spirit can be found in some unpublished work by

7

Caitlin Wang. Lemma 2.1 Suppose KerPgn = {constants}. Then the Green function G(x, y) of Pgn exists in the following sense : a) For all functions u ∈ C n (M ), we have Z (19) u(x) − u ¯= G(x, y)Pgn u(y)dVg (y) x 6= y ∈ M M

b) (20)

G(x, y) = H(x, y) + K(x, y)

is smooth on M 2 \ Diag(M 2 ), K extends to a C 2+α function on M 2 and µ ¶ 1 1 (21) H(x, y) = log f (r) cn r where 2cn = (n − 1)!ωn , r = dg (x, y) is the geodesic distance from x to y; f (r) is a C ∞ positive decreasing function , f (r) = 1 in a neighborhood of r = 0 and f (r) = 0 for r ≥ injg (M ). Proof. Let x ∈ M be fixed and m be a positive integer large enough. By [37] Theorem 5.1, there exists a metric g˜ conformal to g such that |˜ g (x, y)| = 1 + O(rm )

(22)

for y close to x.

Now in coordinates ∆g˜,y has the following expression, ´ ³ p 1 ∆g˜,y v = p ∂i g˜i,j |˜ g |∂j v |˜ g| On the other hand in conformal normal coordinates we have that g˜i,j = δi,j + O(rm ), and

∂i g˜i,j = O(rm−1 ).

Hence working in this coordinate system we have by easy calculations that the following holds: (23)

|Pgn H(x, y)| ≤ C|Pg˜n H(x, y)| ≤ Cr2−n f or r ≤ C −1 injg (M ).

On the other hand by considering the expression, Z Z H(x, y)Pg˜n u(y)dVg˜ (y) − M \Bx (²)

we have by integration by parts that, (24) Z Z n H(x, y)Pg˜ u(y)dVg˜ (y)− M \Bx (²)

M \Bx (²)

M \Bx (²)

u(y)Pg˜n H(x, y)u(y)dVg˜ (y);

Z u(y)Pg˜n H(x, y)dVg˜ (y)

n

∂ν (−∆) 2 −1 H(x, y)u(y)dVg˜ (y)+o² (1).

= ∂Bx (²)

Now by using the fact that close to x in conformal normal coordinate ∆g is close to ∆Rn , we obtain by letting ² go to 0 Z Z (25) u(x) = H(x, y)Pg˜n u(y)dVg˜ (y) − Pg˜n H(x, y)u(y)dVg˜ (y). M

M

8

Hence using the conformal invariance property of Pgn , for every x ∈ M we obtain Z Z (26) u(x) = H(x, y)Pgn u(y)dVg (y) − Pgn H(x, y)u(y)dVg (y). M

M

Now we can apply the same method as in [6] (Theorem 4.13) to construct parametrix for the Green’s function. We set q X G(x, y) = H(x, y) + Zi (x, y) + F (x, y); i=1

where q >

n 2,

(27)

Z Zi (x, y) =

Γi (x, ζ)H(ζ, y)dVg (ζ); M

and Γi are defined inductively as follows,

Z Γi (x, ζ)Γ(ζ, y)dVg (ζ);

Γi+1 (x, y) = M

with

n Γ1 (x, y) = Γ(x, y) = −Pg,y H(x, y);

and F being the solution of the equation n Pg,y F (x, y) = Γk+1 (x, y) −

(28)

1 . V olg (M )

Now from (26) we have that Zi satisfies n Pg,y Zi (x, y) = Γi (x, y) − Γi+1 (x, y).

(29)

We observe that the following estimate holds for Γ(x, y), |Γ(x, y)| ≤ Cr2−n ;

(30)

hence by using the results in [6] ( Proposition 4.12), we obtain the following estimate for Γi (x, y), |Γi (x, y)| ≤ Cr2i−n

(31)

So arriving at this stage by still the same result in [6] (Proposition 4.12), we have that Γq (x, y) and Γq+1 (x, y) are continuous hence using elliptic regularity we get Zq (x, y) and F (x, y) are in C n−1+α (M 2 ). The regularity in both the variables x and y can be deduced by the symmetry of G, which follows from the self adjointness of Pgn and reasoning as in [6] (Proposition 4.13). Further from (31) we deduce n < p < n−2 for all i = 1, . . . , q − 1. Hence by using standard elliptic regthat Γi ∈ Lp with n−2 n ularity we infer that Zi (x, y) ∈ H n,p . So from the Sobolev embedding theorem and the fact that n−2 n < p < n−2 we get Zi (x, y) ∈ C 2+α (M 2 ) for all i = 1, . . . , q − 1 for some α. Hence setting n Pq K(x, y) = i=1 Zi (x, y) + F (x, y), the Lemma is proved. Next we state a Moser-Trudinger type inequality in the case where Pgn is a nonnegative operator. Proposition 2.2 Assume Pgn is a non-negative operator with KerPgn = {constants}. Then there exists a constant C = C(M, g) so that Z (32)

ncn (u−u) ¯ 2 n u,u Pg

e h

i dV ≤ C, g

M n

for all u ∈ H 2 (M ), and hence Z (33)

en(u−¯u) ≤ C +

log M

9

® n ­ n Pg u, u . 4cn

p n Proof. Since Pgn is a nonnegative operator with KerPgn = {constants} then Pg is well defined see [18] (in that case the authors are concerned with the four dimensional case but the same construction remains true for all n). Moreover from the point a) of the Lemma (2.1) and the self adjointness of Pgn we obtain, Z q q (34) u(x) − u ¯= Pgn G(x, y) Pgn u(y)dVg (y); ∀ u C n (M ). M

p p n ˜ y) = Pgn G(x, y) is the Green function of Hence G(x, Pg G(x, y)(see [11] ( Section 3 in the proof n 2 of the boundedness p nof ω in H ). Moreover it is a well knownn fact in the theory of pseudodifferential operator that Pg is a pseudodifferential operator of order 2 and whose leading order symbol is as n ˜ y) the one of (−∆) 4 (see [32]). Hence, the leading term in the asymptotic expansion of its kernel G(x, n coincide with that of the Green’s function for the operator (−∆) 4 in Rn . So by a well know formula for n Fourier transform of radial functions (see [48], Theorem 3.3) we infer that the leading term is an r− 4 where an is a dimensional constant . Hence arriving at this step we can follow the same proof as in [31] (Propostion 2.2) to conclude the first inequality. Moreover from the basic inequality nab ≤ a2 cn +

(35)

nb2 4cn

∀a, b ∈ R;

­ ® setting a = u − u ¯ and b = Pgn u, u , taking the exponential and integrating we obtain the last one.

3

Proof of Theorem 1.3

First integrating (8) we get Z (36)

Z ¯ l enul dVg Q

Q0 dVg + ol (1) = M

M

We recall now the following result of X.Xu ( Theorem 1.2 in [52]). Theorem 3.1 ([52]) There exists a dimensional constant σn > 0 such that, if u ∈ C 1 (Rn ) is solution of the integral equation ¶ µ Z |y| enu(y) dy + c0 , u(x) = σn log |x − y| n R where c0 is a real number, then eu ∈ Ln (Rn ) implies, there exists λ > 0 and x0 ∈ Rn such that µ ¶ 2λ u(x) = log . λ2 + |x − x0 |2 Now, if cn is given in Lemma 2.1 and σn in theorem 3.1 we set kn = σn cn and γn = 2(kn )n The proof is divided into five steps. Step 1 There exists N ∈ N∗ , N converging points (xi,l ) i = 1, ..., N , N sequences (µi,l ) i = 1; ...; N ; of positive real numbers converging to 0 such that the following hold:

10

a) dg (xi,l , xj,l ) −→ +∞ µi,l

i 6= j i, j = 1, .., N

and

¯ l (xi,l )µni.l enul (xi,l ) = 1; Q

b) vi,l (x) = ul (expxi,l (µi,l x)) − ul (xi,l ) −

4γn 1 log(kn ) −→ V0 (x) := log( 2 ) n 4γn + |x|2

1 in Cloc (Rn );

c) Z ∀i = 1, .., N we have,

lim

¯ l (y)enul (y) dVg (y) = (n − 1)!ωn ; Q

lim

R→+∞ l→+∞

Bxi,l (Rµi,l )

d) T here exists C > 0 such that

inf

i=1,...,N

dg (xi,l , x)n enul (x) ≤ C

∀x ∈ M, ∀l ∈ N.

Proof of Step 1 Let xl ∈ M be such that ul (xl ) = maxx∈M ul (x), then we have that ul (xl ) −→ +∞. ¯ l (xl )µn enul (xl ) = 1. Since Q ¯ l −→ Q ¯ 0 C 1 (M ), Q ¯ 0 > 0 and ul (xl ) −→ +∞, Let µl > 0 be such that Q l we have that µl −→ 0. −1 Now let B 0 (δµ−1 l ) be the euclidean ball of center 0 and radius δµl , with δ > 0 small fixed . For −1 x ∈ B 0 (δµl ), we set (37)

vl (x) = ul (expxl (µl x)) − ul (xl ) −

1 log(kn ); n

(38)

˜ l (x) = Ql (expx (µl x)); Q l

(39)

¯ l (expx (µl x)); Q˜¯l (x) = Q l

(40)

¡ ¢ gl (x) = exp∗xl g (µl x).

2 We have that gl −→ dx2 Cloc (Rn ) as l −→ +∞. Now from the Green representation formula we have, Z (41) ul (x) − u ¯l = G(x, y)Pgn ul (y)dVg (y)

∀x ∈ M,

M

where G is the Green function of Pgn (see Lemma 2.1). Now using equation (8) and differentiating (41) with respect to x we obtain that for k = 1, 2 Z k ¯ l (y)enul (y) − Ql (y)|dVg |∇ ul |g (x) ≤ |∇k G(x, y)|g |Q M Z ¯ l (y)enul (y) dVg + O(1), ≤ |∇k G(x, y)|g Q M

(42) since Ql −→ Q0 in C 1 (M ). Now for yl ∈ Bxl (Rµl ), R > 0 fixed we write that, ! Ã Ã Z Z Z −k k nul (y) nul |∇ G(yl , y)|g e dVg (y) = O µl e dVg + O enul (xl ) M

M \Byl (µl )

! −k

dg (yl , y)

dvg (y)

Byl (µl )

= O(µ−k l )

(43) 11

¯ l (xl )µn enul (xl ) = 1 to (36) and Lemma (2.1). thanks to the fact that ul ≤ ul (xl ), to the relation Q l Together with the definition of vl (see (37)) and the fact the fact that vl (x) ≤ vl (0) = − n1 log(kn ) ∀x ∈ Rn , we obtain (vl )l is uniformly bounded in C 2 (K) for all compact subsets K of Rn . Hence by Arzel`a-Ascoli theorem we infer that (44)

vl −→ V0

1 in Cloc (Rn ),

hence we have that V0 (x) ≤ V0 (0) = − n1 log(kn ) ∀x ∈ Rn . Clearly V0 is a Lipschitz function since the constant which bounds the gradient of vl is independent of the compact set K. On the other hand from the Green’s representation formula we have for x ∈ Rn fixed and for R big enough such that x ∈ B 0 (R) Z (45) ul (expxl (µl x)) − u ¯l = G(expxl (µl x), y)Pgn ul (y)dVg (y). M

Now remarking that ul (expxl (µl x)) − ul (xl ) = ul (expxl (µl x)) − ul (expxl (0)), we have the following relation ul (expxl (µl x)) − ul (xl ) = ((ul (expxl (µl x)) − u ¯l ) − (ul (expxl (0)) − u ¯l ) . Hence (45) gives Z ul (expxl (µl x)) − ul (xl ) =

M

(G(expxl (µl x), y) − G(expxl (0), y)) Pgn ul (y)dVg (y).

Moreover using (8) we obtain Z (46)

¯ l (y)enul (y) dVg (y) (G(expxl (µl x), y) − G(expxl (0), y)) Q M Z − (G(expxl (µl x), y) − G(expxl (0), y)) Ql (y)dVg (y).

ul (expxl (µl x)) − ul (xl ) =

(47)

M

Now setting Z (48)

Il (x) = Bxl (Rµl )

¯ l (y)enul (y) dVg (y); (G(expxl (µl x), y) − G(expxl (0), y)) Q

Z (49)

IIl (x) = M \Bxl (Rµl )

¯ l (y)enul (y) dVg (y); (G(expxl (µl x), y) − G(expxl (0), y)) Q

Z (50)

IIIl (x) = M

(G(expxl (µl x), y) − G(expxl (0), y)) Ql (y)dVg (y);

we find (51)

ul (expxl (µl x)) − ul (xl ) = Il (x) + IIl (x) + IIIl (x).

So using the definition of the vl ’s we arrive to (52)

vl (x) = Il (x) + II l (x) + IIIl (x) − 12

1 log(kn ). n

Now let study each of the terms Il (x), IIl (x), IIIl (x) separately. Using the change of variables y = expxl (µl z) we have (53) Z Il (x) = B0 (R)

¯ l (expx (µl z))enul (expxl (µl z)) µn dVg (z). (G(expxl (µl x), expxl (µl z)) − G(expxl (0), expxl (µl z)) Q l l l

¯ l )µn enul (xl ) = 1 and (37)-(40), we obtain Now using the relation Q(x l Z (54)

Q˜¯l (z) nvl (z) e dVgl (z). kn (G(expxl (µl x), expxl (µl z)) − G(expxl (0), expxl (µl z)) ¯ Q(xl ) B0 (R)

Il (x) =

Now from the asymptotics of the Green’s function (see Lemma (2.1)) we have, µ

Z (55)

kn

Il (x) = B0 (R)

1 log cn

µ

|z| |x − z|

¶

¶ ˜¯ Ql (z) nvl (z) + Kl (x, y) ¯ e dVgl (z) for l large enough, Q(xl )

with (56)

Kl (x, z) = (K(expxl (µl x), expxl (µl z)) − K(expxl (0), expxl (µl z))) .

2 Hence since K is of class C 1 on M 2 and gl −→ dx2 in Cloc (Rn ), then letting l −→ +∞ we derive the following equality µ ¶ Z |z| (57) lim Il (x) = σn log enV0 (z) dz. l |x − z| 0 B (R)

Now to estimate IIl (x) we write for l large µ ¶ Z 1 dg (expxl (0), y) ¯ l (y)enul (y) dVg (y) (58) Q IIl (x) = log c d (exp (µ x), y) n g x l M \Bxl (Rµl ) l Z ¯ l (x, y)Q ¯ l (y)enul (y) dVg (y), + K M \Bxl (Rµl )

where ¯ l (x, y) = (K(expx (µl x), y) − K(expx (0), y)) . K l l

(59)

Taking the absolute value in both sides of the equality (58) and using the change of variable y = expxl (µl z) and the fact that K ∈ C 1 we obtain, Z (60) |IIl (x)| ≤

Rn \B0 (R)

¯ µ ¶¯ ˜¯ Z ¯ Ql (z) nv (z) 1 ¯¯ |z| l ¯ l (y)enul (y) dVg (z). ¯ e dV (z) + Rµ Q log gl l ¯ l) cn ¯ |x − z| ¯ Q(x M \Bxl (Rµl )

Hence letting l −→ +∞ we deduce by (36) that (61)

lim sup IIl (x) = oR (1). l

Now using the same method one proves that (62)

IIIl (x) −→ 0 as l −→ +∞.

So we have that µ

Z (63)

V0 (x) =

B 0 (R)

σn log

|z| |x − z|

¶ enV0 (z) dz −

13

1 log(kn ) + lim IIl (x). l n

Hence letting R −→ +∞ we obtain that V0 solve the following conformally invariant integral equation ¶ µ Z 1 |z| enV0 (z) dz − log(kn ). (64) V0 (x) = σn log |x − z| n Rn Now since V0 is Lipschitz then the theory of singular integral operator gives that V0 ∈ C 1 (Rn ). 2 Moreover by using a change of variables and the fact that gl converges to the Euclidean metric in Cloc (Rn ) we obtain, Z Z ¯ l enul dVg = kn (65) Q enV0 dx; lim l−→+∞

Bxl (Rµl )

B0 (R)

hence (36) implies that eV0 ∈ Ln (Rn ). So by a classification result by X.Xu for the solutions of (64) (see Theorem 3.1) we get that, ¶ µ 2λ (66) V0 (x) = log λ2 + |x − x0 |2 for some λ > 0 x0 ∈ Rn . On the other hand from V0 (x) ≤ V0 (0) = − n1 log(kn ) ∀x ∈ Rn , we have that λ = 2kn and x0 = 0 namely, µ ¶ 4γn V0 (x) = log (67) . 4γn2 + |x|2 It is then easily checked that, Z

Z (68)

lim

¯ l (y)enul (y) dVg (y) = kn Q

lim

R→+∞ l→+∞

Bxl (Rµl )

enV0 dx. Rn

Furthermore from a generalized Pohozaev identity by X.Xu (see Theorem 1.1) in [52] for the conformally invariant integral equation (64) we obtain that Z σn enV0 (y) dy = 2, Rn

hence we derives that (69)

lim

Z ¯ l (y)enul (y) dVg (y) = 2cn = (n − 1)!ωn . Q

lim

R→+∞ l→+∞

Bxl (Rµl )

Now for k ≥ 1 we say that (Hk ) holds if there exists k converging points (xi,l )l i = 1, ..., k, k sequences (µi,l ) i = 1, ¡ ..., ¢ k of positive real numbers converging to 0 such that the following hold A1k dg (xi,l , xj,l ) −→ +∞ µi,l ¡ 2¢ Ak

i 6= j

i, j = 1, .., k and

vi,l (x) = ul (expxi,l (µi,l x))−ul (xi,l )−

¯ l (xi,l )µni.l enul (xi,l ) = 1; Q

1 log(kn ) −→ V0 (x) = log n

µ

4γn 4γn2 + |x|2

¶

¡ 3¢ Ak Z ∀ i = 1, .., k, one has

lim

¯ l (y)enul (y) = (n − 1)!ωn . Q

lim

R→+∞ l→+∞

Bxi,l (Rµi,l )

14

in

1 Cloc (Rn )

∀i = 1, .., k;

Clearly, by the above arguments (H1 ) holds. We let now k ≥ 1 and assume that (Hk ) holds. We also assume that (70)

sup Rk,l (x)n enul (x) −→ +∞ as l −→ +∞, M

where Rk,l (x) = min dg (xi,l , x). i=1;..;k

We prove in the following that in this situation (Hk+1 ) holds. For this purpose we let xk+1,l ∈ M be such that (71)

Rk,l (xk+1,l )n enul (xk+1,l ) = sup Rk,l (x)n enul (x) , M

and we set µ µk+1,l =

(72)

1 ¯ Q(xk+1,l )enul (xk+1,l )

¶ n1 .

Since M is compact then (70), (71) and (72) imply that (73) (74)

µk+1,l −→ +∞ as l −→ +∞; dg (xi,l , xk+1,l ) −→ +∞ as l −→ +∞ ∀i = 1, ..., k. µk+1,l

Indeed from (70) we have that Rk,l (xk+1,l )n enul (xk+1,l ) −→ +∞, and since Rk,l (xk+1,l ) is bounded because M compact then we obtain that, enul (xk+1,l ) −→ +∞. ¯ l −→ Q ¯ 0 in C 0 (M ) and Q ¯ 0 > 0 we infer that Now from (72), Q µk+1,l −→ 0. On the other hand we have that Rk,l (xk+1,l ) dg (xi,l , xk+1,l ) ¯ k+1,l )enul (xk+1,l ) ) n1 , ≥ = (Rk,l (xk+1,l )n Q(x µk+1,l µk+1,l hence (70) and (71) give that dg (xi,l , xk+1,l ) −→ +∞. µk+1,l Now thanks to (A2k ), we can prove that (75)

dg (xi,l , xk+1,l ) −→ +∞ as l −→ +∞ ∀i = 1, ..., k. µi,l

Indeed if dg (xi,l , xk+1,l ) stays away from 0 then since µi,l −→ 0, dg (xi,l , xk+1,l ) ≤ ², ² small enough and set, x ˜k+1,l =

exp−1 xi,l (xk+1,l ) µi,l 15

.

we are done. So suppose that

We have that,

dg (xi,l , xk+1,l ) dg (xi,l , xk+1,l ) µk+1,l = . µi,l µk+1,l µi,l

On the other hand we have also that (

¯ k,l ) ¯ k,l ) Q(x µk+1,l n Q(x ) = ¯ = ¯ . nu (x )−u (x ) µi,l Q(xk+1,l )e k k+1,l k k,l Q(xk+1,l )envi,l (˜xk+1,l ) µ

k+1,l Hence if (˜ xk+1,l )l is bounded in Rn we have thanks to (A2k ) that converges to a positive µi,l number hence we are done. If (˜ xk+1,l )l were not bounded, then the relation

dg (xi,l , xk+1,l ) = µi,l k˜ xk+1,l k shows that

dg (xi,l , xk+1,l ) −→ +∞ as l −→ +∞; µi,l

¢ ¡ hence A1k+1 holds. Moreover it follows from (71) and (A1k+1 ) that (76)

lim

l−→+∞ z∈Bx

sup

k+1,l

(ul (z) − ul (xk+1,l )) = 0.

(Rµk+1,l )

Indeed from (71) we have that , Rk,l (xk+1,l )n enul (xk+1,l ) ≥ Rk,l (x)n enul (x) ∀x ∈ M ; hence the following holds Rk,l (xk+1,l )n enul (xk+1,l ) ≥ Rk,l (z)n enul (z) ∀z ∈ Bxk+1,l (Rµk+1,l ). So taking the n-th root in both sides of the inequality we obtain that 0

Rk,l (xk+1,l )eul (xk+1,l ) ≥ Rk,l (z)eul (z) ∀z ∈ Bxk+1,l (Rµk+1,l ); hence dividing by eul (xk+1,l ) Rk,l (z) in both sides we get eul (z)−ul (xk+1,l ) ≤

Rk,l (xk+1,l ) . Rk,l (z)

0

Now let zk+1,l ∈ Bxk+1,l (Rµk+1,l ) be such that ul (zk+1,l ) − ul (xk+1,l ) =

sup

(ul (z) − ul (xk+1,l )) ;

z∈Bxk+1,l (Rµk+1,l )

so we have eul (zk+1,l )−ul (xk+1,l ) ≤

Rk,l (xk+1,l ) , Rk,l (zk+1,l )

and let ik+1,l ∈ {1, ..., k} be such that, Rk,l (zk+1,l ) = dg (xik+1,l ,l , zk+1,l ); so we have that (77)

eul (zk+1,l )−ul (xk+1,l ) ≤

dg (xik+1,l ,l , xk+1,l ) Rk,l (xk+1,l ) ≤ dg (xik+1,l ,l , zk+1,l ) dg (xik+1,l ,l , zk+1,l ) 16

eul (zk+1,l )−ul (xk+1,l ) ≤ 1 +

(78)

dg (z, xk+1,l ) Rµk+1,l ≤1+ . dg (xik+1,l ,l , zk+1,l ) dg (xik+1,l ,l , zk+1,l )

On the other hand the following chain of inequality holds dg (xik+1,l ,l , zk+1,l ) dg (xik+1,l ,l , xk+1,l ) dg (xk+1,l , zk+1,l ) dg (xik+1,l ,l , xk+1,l ) ≥ − ≥ − R; µk+1,l µk+1,l µk+1,l µk+1,l but from (A1k+1 ) we deduce that,

dg (xik+1,l ,l , xk+1,l ) −→ +∞; µk+1,l

hence

dg (xik+1,l ,l , zk+1,l ) −→ +∞; Rµk+1,l

which imply with (78) that lim sup (ul (zk+1,l ) − uk (xk+1,l )) ≤ 0; l

and since (ul (zk+1,l ) − uk (xk+1,l )) ≥ 0; then we have proved that, lim

l−→+∞ z∈Bx

sup

k+1,l

(ul (z) − ul (xk+1,l )) = 0.

(Rµk+1,l )

Now mimicking what we did above thanks to the Green’s representation formula (see in particular formula (43)) and using (76) then one proves that up to a subsequence, µ ¶ 1 4γn 1 vk+1,l (x) = ul (expxk+1,l (µk+1,l x)) − ul (xk+1,l ) − log(kn ) −→ V0 (x) = log (Rn ), in Cloc n 4γn2 + |x|2 and

Z lim

¯ l (y)enul (y) dVg (y) = (n − 1)!ωn . Q

lim

R→+∞ l→+∞

Bxk+1,l (Rµk+1,l )

Hence recollecting the informations above, one gets that (Hk+1 ) holds. Moreover since of Hk imply that Z nul (y) ¯ Q(y)e dVg (y) ≥ (n − 1)!ωn k + ol (1), M

¡ 1¢ ¡ ¢ Ak and A3k

R 1 then we easily get thanks to (36) that there exists a maximal k , 1 ≤ k ≤ (n−1)!ω Q0 (y)dVg (y) , M n such that (Hk ) holds. Arriving to this maximal k, we get that (70) cannot hold. Hence setting N = k the proof of Step 1 is done.

Step 2 There exists a constant C > 0 such that (79)

Rl (x)|∇ul |g (x) ≤ C

∀x ∈ M and ∀l ∈ N ;

where Rl (x) = min dg (xi,l , x); i=1,..,N

17

and the xi,l ’s are as in Step 1. Proof of Step 2 We use again the Green’s representation formula for ul , that we differentiate. We let xl ∈ M be such that xl 6= xi,l for all i =, .., N . Note that, for xl = xi,l , the estimates of the proposition are obvious. We write thanks to the asymptotics of the Green function of Pgn see (Lemma 2.1) that µZ (80)

|∇ul |g (xl ) = O M

¶ 1 enul (y) dVg (y) + O(1). dg (xl , y)

Now for i = 1, ...N , we set (81)

Ωi,l = {y ∈ M, Rl (y) = dg (xi,l , y)};

and we write that Z (82) Ωi,l

1 enul (y) dVg (y) = Ii,l + IIi,l + IIIi,l ; (dg (xl , y))

with Z (83)

Ii,l =

(84)

IIi,l =

dg (xl ,xi,l ) Ωi,l ∩Bxi,l ( ) 2

Z Ωi,l \Bxl (5dg (xl ,xi,l ))

1 enul (y) dVg (y); (dg (xl , y)) 1 enul (y) dVg (y); (dg (xl , y))

and Z (85)

IIIi,l =

dg (xl ,xi,l ) Ωi,l ∩Bxl (5dg (xl ,xi,l ))\Bxi,l ( ) 2

d (x ,x

1 enul (y) dVg (y). (dg (xl , y))

)

To estimate Ii,l we use the fact that y ∈ Bxi,l ( g l2 i,l ), the triangle inequality and equation (3) to find that µ ¶ 1 (86) Ii,l = O . (dg (xl , xi,l )) On the other hand using the fact that y ∈ / Bxl (5dg (xi,l , xl )), and equation (2.1) we have that ¶ µ 1 (87) IIi,l = O . (dg (xl , xi,l )) Moreover using the fact that we are in Ωi,l , assumption d) of Step 1 implies that ÃZ ! 1 (88) IIIi,l = O ; n dg (xl ,xi,l ) Bx (5dg (xl ,xi,l ))\Bx ) (dg (xl , y)dg (xi,l , y) ) 2 l

i,l

d (xl ,xi,l ) ), 2

hence using the fact that y ∈ / Bxi,l ( g à (89)

IIIi,l = O

we obtain

1 (dg (xl , xi,l ))n

Z Bxl (5dg (xl ,xi,l ))

18

1 (dg (xl , y)

! .

Now working in geodesic polar coordinatesat xl we have that Z ¡ ¢ 1 (90) = O (dg (xi,l , xl ))n−1 ; Bxl (5dg (xl ,xi,l )) dg (xl , y) hence we derive µ (91)

IIIi,l = O

1 dg (xl , xi,l )

¶ .

So we have Z (92) Ωi,l

1 enul (y) dVg (y) = O (dg (xl , y))

µ

1 (dg (xl , xi,l ))

¶ ;

hence Step 2 clearly follows. Step 3 Set (93)

Ri,l = min dg (xi,l , xj,l ); i6=j

we have that 1) There exists a constant C > 0 such that ∀ r ∈ (0, Ri,l ] ∀ s ∈ ( 4r , r] (94)

¡ ¢ ¡ ¢ |ul expxi,l (rx) − ul expxi,l (sy) | ≤ C

2) If di,l is such that 0 < di,l ≤

Ri,l 2

and

di,l µi,l

f or all x, y ∈ Rn such that |x|, |y| ≤

3 . 2

−→ +∞ then we have that,if

Z ¯ l (y)enul (y) dVg (y) = (n − 1)!ωn + ol (1); Q

(95) Bxi,l (di,l )

then

Z ¯ l (y)enul (y) dVg (y) = (n − 1)!ωn + ol (1). Q Bxi,l (2di,l )

3) Let R be large and fixed. If di,l > 0 is such that di,l −→ 0,

di,l µi,l

−→ +∞, di,l
Rη , we have that n(ν−1)

dg (x, xi,l )nν enul ≤ ηµi,l

(129) r

Since µi,l −→ +∞ see (128) and i,l implies that Z

Ri,l 2

Bxi,l (

∀x ∈ Bxi,l (ri,l ) \ Bxi,l (Rµi,l ).

≥ ri,l see (127), we have

Ri,l C

Ri,l Cµi,l

−→ +∞, hence point c) of Step 1

¯ l enul = (n − 1)!ωn + ol (1). Q )

On the other hand, by continuity and by the definition of ri,l it follows that 0

(130)

ϕi,l (ri,l ) = 0. R

0

i,l −→ +∞ . We will show next that ϕi,l (ri,l ) < 0 for l large Let us assume by contradiction that ri,l contradicting the above equality (130). To do so we will study u ¯i,l (·).

23

R

i,l First let us remark that since M is compact then ri,l −→ +∞ implies that ri,l −→ 0. From the Green’s representation formula for ul we have the following equation, Z Z Z n nul (y) ¯ ul (x) = G(x, y)Pg ul (y)dVg (y) + u ¯l = G(x, y)Ql (y)e dVg (y) + u ¯l − G(x, y)Ql (y)dVg (y).

M

M

M

Hence Z

Z ¯ l (y)enul (y) dVg (y)dσg (x) + u G(x, y)Q ¯l

u ¯i,l (r) = (V olg (∂Bxi,l (r)))−1

(131)

∂Bxi,l (r)

M

Z

−(V olg (∂Bxi,l (r)))−1

(132) Setting

Z G(x, y)Ql (y)dVg (y)dσg (x).

∂Bxi,l (r)

Z

Z

−1

Fi,l (r) = (V olg (∂Bxi,l (r)))

M

G(x, y)Ql (y)dVg (y)dσg (x); ∂Bxi,l (r)

M

we obtain Z

Z ¯ l (y)enul (y) dVg (y)dσg (x) + u G(x, y)Q ¯l − Fi,l (r).

u ¯i,l = (V olg (∂Bxi,l (r)))−1

∂Bxi,l (r)

M

Since Ql −→ Q0 in C 1 (M ) we have that Fi,l is of class C 1 for all i, l and moreover, 0

|Fi,l (r)| ≤ C;

(133) Now let (134) Z

injg (M ) 4

0, there exists a constant C = C(¯ ², δ0 , γ0 ) such that Z ­ n ® n (153) log Pg u, u en(u−¯u) ≤ C + 4cn (l + 1) − ²¯ M n

for all the functions u ∈ H 2 (M ) satisfying R nu e dV g R Si ≥ γ0 , i ∈ {1, .., l + 1}. (154) enu dV g M Proof. The proof follows that of Lemma 2.2 in [26]. The argument is based on constructing some cutoff functions gi which are identically 1 on Si ; and which have disjoint support. Then ∀i by (154) we have that Z Z Z 1 CM enu dVg ≤ (155) enu dVg ≤ engi u dVg γ γ 0 Si 0 M M On the other hand using the Leibniz rule and interpolation inequalities we obtain Z Z ­ n ® ­ ® (156) Pg gi v, gi v ≤ gi2 (Pgn v, v)dVg + ² Pgn v, v + C²,δ0 v 2 dVg . M

M

R Applying Moser-Trudinger inequality (see (2.2) ) to ugj , choosing i such that M gi2 (Pgn v, v)dVg = R minj M gj2 (Pgn v, v)dVg , and by using interpolation inequalities we obtain the required statement. In the next Lemma we show a criterion which implies the situation described in the first condition in (154). The result is proven in [26] Lemma 2.3. Lemma 4.2 Let l be a given positive integer, and suppose that ² and r are positive numbers. Suppose that for a non-negative function f ∈ L1 (M ) with kf kL1 (M ) = 1 there holds Z f dVg < 1 − ε for every `-tuples p1 , . . . , p` ∈ M ∪`i=1 Br (pi )

Then there exist ε > 0 and r > 0, depending only on ε, r, ` and M (but not on f ), and ` + 1 points p1 , . . . , p`+1 ∈ M (which depend on f ) satisfying Z Z f dVg > ε, . . . , f dVg > ε; B2r (pi ) ∩ B2r (pj ) = ∅ for i 6= j. Br (p1 )

Br (p`+1 )

An interesting consequence of Lemma 4.1 is the following one. It characterize some functions in n H 2 (M ) for which the value of II is large negative. Lemma 4.3 Under the assumptions of Theorem (1.1), and for k ≥ 1 given by (4), the following property holds. For any ² > 0 and Rany r > 0 there exists large positive L = L(², r) such that for any u ∈ n H 2 (M ) with II(u) ≤ −L, M enu dVg = 1 there exists k points p1,u , . . . , pk,u ∈ M such that Z (157) enu dVg < ² M \∪k i=1 Bpi,u (r)

Proof. Suppose by contradiction that the statement is not true. Then we can apply Lemma 4.2 with l = k, f = enu , and in turn Lemma 4.1 with δ0 = 2¯ r, S1 = Bp¯1 (¯ r), . . . , Sk+1 = Bp¯k+1 (¯ r). This implies Z κPgn n ® ­ n ® n­ n II(u) ≥ Pg u, u + n Qn udVg − CκPgn − Pg u, u − nκPgn u ¯. 2 4cn (k + 1) − ²¯ M 29

κP n n

g Since κPgn < 2cn (k + 1), we can choose ²¯ > 0 so small that n2 − 4cn (k+1)−¯ ² > δ > 0. Hence using also the Poincar´e inequality we deduce Z ­ ® II(u) ≥ δ Pgn u, u + n Qn (u − u ¯)dVg − CκPgn

M

­ ® ­ ®1 ≥ δ Pgn u, u − nC Pgn u, u 2 − CκPgn ≥ −C.

(158) This concludes the proof.

R Pk By Lemma 4.3 one finds that, if the normalization M enu dVg = 1 holds true then enu ' i=1 ti δxi , where δxi stands for the Dirac delta at xi ∈ M , and the ti ’s are nonnegative numbers such that Pk i=1 ti = 1. The set of formal convex combinations of Dirac deltas at points of M (with a fixed number k of sumands) is known in the literature as the set of formal barycenters of M (see [26], [42] ) and recalling its properties we mention that this is a non-contractible set. From Lemma 4.3 , applying the arguments of Proposition 3.1 in [26], one can find the folowing result Proposition 4.4 For k ≥ 1 given as in (4), there exists a large L > 0 and a continuous map Ψ from the sublevel {II < −L} into Mk which is topologically non-trivial. By the non-contractibility of Mk , the non-triviality of the map is apparent from b) of Propostion 4.8 below.

4.2

Technical estimates for Mapping Mk into large negative sublevels of II

In this subsection we will define some test functions depending on a parameter λ and give estimate of the quadratic part of the functional II on those functions as λ tends to infinity. And as a corollary we define a continuous map from Mk into large negative sublevels of II. For δ > 0 small, consider a smooth non-decreasing cut-off function χδ : R+ → R satisfying the following properties (see [26]):   χδ (t) = t, f or t ∈ [0, δ]; χδ (t) = 2δ, f or t ≥ 2δ; (159)  χδ (t) ∈ [δ, 2δ], f or t ∈ [δ, 2δ]. Pk Then, given σ ∈ Mk , σ = i=1 ti δxi and λ > 0, we define the function ϕλ,σ : M → R by (160)

µ ¶n k X 1 2λ ϕλ,σ (y) = log ti n 1 + λ2 χ2δ (di (y)) i=1

where we have set di (y) = dist(y, xi ),

xi , y ∈ M,

with dist(·, ·) denoting the distance function on M . We define also (161)

dmin (y) = min di (y). i

When n = 4m we set (162)

Tn ϕλ,σ = (−∆)m ϕλ,σ ,

and when n = 4m + 2 we set (163)

Tn ϕλ,σ = ∇((−∆)m ϕλ,σ )

Now we state a Lemma giving an estimate (uniform in σ ∈ Mk ) of 30

­

® Pgn ϕλ,σ , ϕλ,σ as λ → +∞.

Lemma 4.5 Suppose ϕλ,σ as in (160) and let ² > 0 small enough. Then as λ → +∞ one has ­ n ® (164) Pg ϕλ,σ , ϕλ,σ ≤ (4kcn + ² + oδ (1)) log λ + C²,δ R 2 Proof. We first give an estimate of M (Tn ϕλ,σ ) dVg and after use interpolation inequalities to conclude. Let Θ be large and fixed, then by induction in the degree of differentiation we have that the following pointwise estimates holds in ∪ki=1 Bxi ( Θ λ ): n

(165)

|Tn ϕλ,σ | ≤ Cλ 2 ,

hence we obtain Z 2

(166) Θ ∪k i=1 Bxi ( λ )

(Tn ϕλ,σ ) dVg ≤ CΘn

Now to have a further simplification of the expression of ϕσ,λ , it is convenient to get rid of the cutoff functions χδ . In order to do this, we divide the set of points {x1 , . . . , xk } in a suitable way. Since the number k is fixed, there exists δˆ and sets B1 , . . . , Bj , j ≤ k with the following properties

(167)

 −1 C δ ≤ δˆ ≤ δ ;    Bk ∪ · · · ∪ B 16= {x , . . . , x }; 1 j 1 k ˆ dist(x , x ) ≤ δ if x , x  i s i s ∈ Ba ;   ˆ dist(xi , xs ) ≥ 4δ if xi ∈ Ba , xs ∈ Bb , a 6= b,

where Ck is a positive constant depending only on k. Now we define ˆ ˆa = {y ∈ M : dist(y, Ba ) ≤ 2δ}. B

(168) By definition of δˆ it follows that (169)

χδ (di (y)) = di (y),

ˆa , for xi ∈ Ba , y ∈ B

and (170)

ˆ χδ (di (y)) ≥ 2δ,

ˆa . for xi ∈ Ba , y ∈ /B

Furthermore one has (171)

ˆa ∩ B ˆb = ∅ for a 6= b B

On the other hand it is also easy to see that the following holds, ˆa . |Tn ϕλ,σ | ≤ Cδˆ in M \ ∪ja=1 B ´ ¢ ³ j ¡ ) ∪ ∪ B . Since we are taking Θ large, then in the set Now set MΘ,σ,λ.δˆ = M \ ∪ki=1 Bxi ( Θ a a=1 λ MΘ,σ,λδˆ the following estimates hold: (173) n (1+λ2 d2i ) = (1+oδ,Θ (1))λ2 d2i , ∂ β (1+λ2 d2i ) = (1+oδ,Θ (1))λ2 ∂ β d2i ; for all muti-indices β : |β| ≤ 2 (172)

First let suppose k = 1 and after we treat the case k > 1. In the case k = 1 we have ϕσ,λ takes the simple form (174)

ϕσ,λ (x) = log

2λ in MΘ,σ,λδˆ. 1 + λ2 d21 (x)

31

Hence from (173) we obtain ∂ β ϕσ,λ = 2∂ β log

(175)

1 1 + oδ,Θ (1) |β| d1 d

for all multi-indices β : |β| ≤

1

n 2

So we have that (Tn ϕσ,λ )2 = 4(Tn log

(176)

1 2 1 ) + oδ,Θ (1)( n ). d1 d1

On the other hand we have that in geodesic coordinates around x1 (177)

(Tn log

1 2 cn ) ≤ (1 + or (1)); d1 ωn−1 rn

Hence working now in geodesic polar coordinates we obtain Z (178) (Tn ϕσ,λ )2 ≤ 4cn log λ(1 + oδ,Θ (1)) + Cδ,Θ . MΘ,σ,λ.δˆ

So with what is said above we have that by fixing Θ large we arrive to Z (179) (Tn ϕσ,λ )2 ≤ 4cn log λ(1 + oδ (1)) + Cδ . M

³ Now let treat the case k > 1. For this let C large and let aj,σ,Θ,λ (x) = tj i ∈ {1, . . . , k} define the set Aλ,σ,i,C by the following formula. (180)

2λ

1+λ2 χ2δ (di (x))

´n

. Next for

Aλ,σ,i,Θ,C = {x ∈ MΘ,σ,λ,hatδ / ai,σ,λ (x) > Caj,σ,λ (x) for all j 6= i}.

By definition of ϕσ,λ (x) and aj,σ,λ (x) we have that, (181)

ϕλ,σ (x) =





k X

1 1 1 log( aj,σ,λ (x)) = log(ai,σ,λ (x)) + log 1 + n n n j=1

X aj,σ,λ (x) j6=i

ai,σ,λ (x)

 in A λ,σ,i,Θ,C .

Moreover the following holds : (182)

X aj,σ,λ (x) j6=i

ai,σ,λ (x)

=

X tj 1 + λ2 χδ (di )2 ( )n . ti 1 + λ2 χδ (dj )2 j6=i

So By the above arguments we have that the following holds in Aλ,σ,i,Θ,C (183)

X aj,σ,λ (x) j6=i

ai,σ,λ (x)

=

X tj 1 + λ2 d2 i n ( ) = oΘ,C (1), ti 1 + λ2 d2j j6=i

hence from (173) we deduce that X tj d2n j = oδ,Θ,C (1). ti d2n i

(184)

j6=i

By differentiation and reasoning as in (173) we obtain (185) X t j 1 + λ2 d 2 1 1 n i n ∂ β ϕλ,σ (x) = 2∂ β log( ) + ∂ β ( ) + oΘ,C (1)( |β| ) for all multi-indices β : |β| ≤ ; 2 2 di ti 1 + λ d j 2 d j6=i=k

i

32

where oΘ,C (1) → 0 as Θ, C → +∞. Hence using again (173) we obtain (186) X tj 1 d2 1 n ∂ β ϕλ,σ (x) = 2∂ β log( ) + (1 + oδ,Θ (1)) ∂ β ( i2 )n + oΘ,C (1)( |β| ) for all multi-indices β : |β| ≤ di ti dj 2 d j6=i

i

Moreover by easy calculations we have that the folowing holds, ∂β (

(187)

d2i n d2i n 1 n ) = O(( ) |β| ) for all multi-index β such that |β| ≤ . 2 2 dj dj d 2 min

Hence we infer that X tj

(188)

j6=i

ti

∂β (

d2i n X tj d2n 1 j ) = O( |β| ), 2 2n dj ti di d j6=i

min

so from (4.2) we obtain X tj

(189)

ti

j6=i

∂β (

d2i n 1 ) = oδ,Θ,C (1)( |β| ). 2 dj d min

Hence we have that, ∂ β ϕλ,σ (x) = 2∂ β log(

(190)

1 1 n ) + oδ,Θ,C (1)( |β| ) for all multi-index β such that |β| ≤ . di 2 d min

k f Now define the set M Θ,σ,λ,δˆ = ∪i=1 Ai,σ,λ,Θ,C . Since (Ai,σ,λ,Θ,C )i=1,...,k are disjoint, then we have that

Z (191)

2

f M ˆ Θ,σ,λ,δ

(Tn ϕλ,σ ) dVg =

k Z X i=1

2

(Tn ϕλ,σ ) dVg =

Aλ,σ,i,Θ,C

k Z X i=1

Aλ,σ,i,Θ,C ∩({di ≥ Θ λ })

2

(Tn ϕλ,σ ) dVg .

From (190) we have that, Z (192)

2

f M ˆ Θ,σ,λ,δ

(Tn ϕλ,σ ) dVg =

¶ µ 1 1 4(Tn log( ))2 + oδ,Θ,C (1)( n ) dVg . di dmin Aλ,σ,i,Θ,C ∩({di ≥ Θ λ })

k Z X i=1

On the other hand working in polar coordinates we have that (193)

(Tn log(

1 2 cn )) ≤ (1 + or (1)); di ωn−1 rn

hence we obtain Z (194)

· ¸ 1 2 1 4(Tn log( )) + oδ,Θ,C (1)( n ) dVg ≤ (4cn + oδ,Θ,C (1)) log λ + Cδ,δ,Θ,C , ˆ di dmin Aλ,σ,i,θ,C ∩({di ≥ Θ λ })

hence we have that

Z 2

(195) Now let us estimate

f M ˆ Θ,σ,λ,δ

(Tn ϕλ,σ ) dVg ≤ (4kcn + oδ,Θ,C (1)) log λ + Cδ,Θ,C .

R f MΘ,σ,λ,δˆ\M ˆ Θ,σ,λ,δ

2

(Tn ϕλ,σ ) dVg

fΘ,σ,λ . We have that the following holds First of all we give a characterization of the set MΘ,σ,λ \ M f MΘ,σ,λ,δˆ \ M Θ,σ,λ,δˆ = {x ∈ MΘ,σ,λ : ∀i there exists an index j 6= i such that ai,σ,λ (x) ≤ Caj,σ,λ (x)}. 33

f Hence we have that x ∈ MΘ,σ,λ,δˆ \ M Θ,σ,λ,δˆ is equivalent also to the fact that 1

∀i there exists an index j 6= i such that

d2i (x)

tin

≥

1 n

(1 + oδ,Θ (1))d2j (x).

1 n

C tj

f So from this fact an using an iterative argument we have that if x ∈ MΘ,σ,λ,δˆ \ M Θ,σ,λ,δˆ then 1 n

(196)

∃j 6= i such that C

t −1 j

1

tin

1

d2i (x)(1

+ oΘ,C (1)) ≤

d2j (x)

≤C

tjn 1

tin

d2i (x)(1 + oΘ,C (1)).

Hence the following holds : k f there exists l = l(k) ∈ N such that MΘ,σ,λ,δˆ \ M Θ,σ,λ,δˆ ⊂ ∪i=1 Ai ,

where Ai is the annulus Ai = Byi (bi ) \ Byi (ai ), with yi ∈ {x1 , . . . , xk } and abii ≤ CΘ,C,k . On the other hand reasoning as in (187) we have that (197)

|Tn ϕλ,σ | = O(

1 n 2

dmin

fΘ,σ,λ . in MΘ,σ,λ \ M

)

Hence working again on polar coordinates as for (198) we find that Z 2 (198) (Tn ϕλ,σ ) dVg ≤ CC,Θ . fΘ,σ,λ MΘ,σ,λ \M

So from (166), (195) and (198), by fixing C and Θ large enough we obtain we obtain Z 2 (199) (Tn ϕλ,σ ) dVg ≤ (4kcn + oδ (1)) log λ + Cδ . M

Hence we obtain for every k ≥ 1 Z 2 (200) (Tn ϕλ,σ ) dVg ≤ (4kcn + oδ (1)) log λ + Cδ . M

® ­ Now let us estimate Pgn ϕλ,σ , ϕλ,σ . We have from the self-adjointness of Pgn and the fact that it annihilates constants that the following holds, ­ n ® ­ ® Pg ϕλ,σ , ϕλ,σ = Pgn (ϕλ,σ − ϕλ,σ ), ϕλ,σ − ϕλ,σ . Hence using interpolation inequalities (see [41]) we have that Z Z ­ n ® 2 (201) Pg ϕλ,σ , ϕλ,σ ≤ (1 + ²) (Tn ϕλ,σ ) dVg + C² M

|ϕλ,σ − ϕλ,σ |2 dVg .

M

We notice first that the following fact holds true as one can check easily, (202)

ϕλ,σ (x) = log

(203)

log

2λ , 1 + 4λ2 δ 2

for y ∈ M \ ∪ki=1 Bxi (2δ);

2λ ≤ ϕλ,σ (x) ≤ log 2λ in ∪ki=1 Bxi (2δ); 1 + 4λ2 δ 2 34

and (204)

log

Next let us estimate (205)

2λ 2λ ≤ ϕλ,σ (x) ≤ log . 1 + 4λ2 δ 2 1 + χ2δ (dmin (x))

R

|ϕλ,σ − ϕλ,σ |2 dVg . By remarking the trivial identity Z 2λ 1 2λ ϕσ,λ − log = (ϕσ,λ − log )dVg 2 2 1 + 4λ δ V olg (M ) M 1 + 4λ2 δ 2 M

we have, by the bilinearity of the inner product that the following holds Z

Z (ϕσ,λ − ϕσ,λ )2 dVg =

M

(ϕσ,λ − log M

¶2 2λ dV g 1 + 4λ2 δ 2 M 2λ +V olg (M )| log − ϕσ,λ |2 , 1 + 4λ2 δ 2

2 2λ )2 dVg − 1 + 4λ2 δ 2 V olg (M )

(206) hence we find Z Z 2 (207) (ϕσ,λ − ϕσ,λ ) dVg ≤ (ϕσ,λ − log M

M

R

2

µZ

ϕσ,λ − log

2λ 2λ )2 dVg + V olg (M )| log − ϕσ,λ |2 . 1 + 4λ2 δ 2 1 + 4λ2 δ 2 R 2λ 2 it suffices to do it for M (ϕσ,λ − log 1+4λ 2 δ 2 ) dVg and

So in order to estimate M (ϕσ,λ − ϕσ,λ ) dVg R 2λ for M (ϕσ,λ − log 1+4λ )dVg . R2 δ2 2λ Let us first estimate M (ϕσ,λ − log 1+4λ 2 δ 2 )dVg . From (202) the following holds Z (208)

(ϕσ,λ − log M

2λ )dVg = 1 + 4λ2 δ 2

Z ∪k i=1 Bxi (2δ)

(ϕσ,λ − log

2λ )dVg . 1 + 4λ2 δ 2

Using (204) we have that the following holds Z

k

X 2λ )dV ≤ (ϕσ,λ − log g 1 + 4λ2 δ 2 M i=1

(209)

Z (log Bxi (2δ)

2λ 2λ − log )dVg . 1 + 4λ2 χ2δ (di ) 1 + 4λ2 δ 2

Now working in geodesic normal coordinates around the points xi we find (210) ÃZ ¶ Z 2δ µ ¶! µ Z δ 2 2 2 2 2λ 2λ 1 + 4λ δ 1 + 4λ δ . (log −log )dVg ≤ C sn−1 log ds + log ds 1 + 4λ2 χ2δ (di ) 1 + 4λ2 δ 2 1 + λ2 s 2 1 + λ2 χ2δ (s) Bxi (2δ) 0 δ Now recalling that χδ is non-decreasing we have that Z (211)

(log Bxi (2δ)

2λ 2λ − log )dVg ≤ C 1 + 4λ2 χ2δ (di ) 1 + 4λ2 δ 2

Z

δ 0

µ ¶ 1 + 4λ2 δ 2 sn−1 log ds + O(δ n ). 1 + λ2 s 2

On the other hand by performing the change of variables λs = z we obtain Z

δ

(212)

s 0

n−1

µ ¶ Z δ√λ Z δλ 1 + 4λ2 δ 2 1 + 4λ2 δ 2 1 1 + 4λ2 δ 2 1 n−1 n−1 log log ds ≤ z log dz + dz √ z 2 2 n 2 n 1+λ s λ 0 1+z λ δ λ 1 + z2

It is easy to see that the following holds (213)

1 λn

Z 0

√ δ λ

z n−1 log

n 1 + 4λ2 δ 2 dz = O(δ n λ− 2 log λ), 2 1+z

35

and 1 λn

(214)

Z

δλ √

z n−1 log

δ λ

1 + 4λ2 δ 2 dz = O(δ n ). 1 + z2

Therefore we obtain Z (215)

(ϕσ,λ − log M

n 2λ )dVg ≤ O(δ n λ− 2 log λ) + O(δ n ). 1 + 4λ2 δ 2

Furthemore using the same procedure one finds Z n 2λ (216) (ϕσ,λ − log )2 dVg ≤ O(δ n λ− 2 (log λ)2 ) + O(δ n ). 2 δ2 1 + 4λ M Hence using (215), (216) and (205) we obtain Z (217) (ϕσ,λ − ϕσ,λ )2 dVg ≤ oδ (1) log λ + Cδ . M

From (217), (201) and (199), the Lemma is proved. Next we state a lemma giving estimates of the remainder part of the functional II along ϕσ,λ . The proof is the same as formula 40 and formula 41 in the proof of Lemma 4.3 in [26]. Lemma 4.6 Soppose ϕσ,λ as in (160). Then as λ → +∞ one has Z (218) Qng ϕσ,λ = −κPgn log λ + O(δ n log λ) + O(log δ) + O(1); M

Z (219)

enϕσ,λ = O(1).

log M

n

Now for λ > 0 we define the map Φλ : Mk → H 2 (M ) by the following formula (220)

∀ σ ∈ Mk Φλ (σ) = ϕσ,λ .

We have the following Lemma which is a trivial application of Lemmas 4.5 and 4.6. ¯ such Lemma 4.7 For k ≥ 1 (given as in (4) ), given any L > 0 , there exists a small δ and a large λ that II(Φλ¯ (σ)) ≤ −L for every σ ∈ Mk . Next we state a proposition giving the existence of the projection from Mk into large negative sublevels of II, and the non-triviality of the map Ψ of the proposition (4.4). Proposition 4.8 Let Ψ be the map defined in proposition 4.4 . Then assuming k ≥ 1 (given as in (4)), for every L > 0 sufficiently large (such that proposition 4.4 applies), there exists a map (221)

n

Φλ¯ : Mk −→ H 2 (M )

with the following properties a) (222)

II(Φλ¯ (z)) ≤ −L for any z ∈ Mk ;

b) Ψ ◦ Φλ¯ is homotopic to the identity on Mk . 36

Proof. The statement (a) follows from Lemma 4.7. To prove (b) it is sufficient to consider the family of maps Tλ : Mk → Mk defined by Tλ (σ) = Ψ(Φ(σ)),

σ ∈ Mk nϕσ,λ

We recall that when λ is sufficiently large this composition is well defined. Therefore , since R eenϕσ,λ dVg * M σ in the weak sens of distributions, letting λ → +∞ we obtain an homotopy between Ψ ◦ Φ and IdMk . This conclude the proof.

4.3

Min-max Scheme

In this subsection we describe the min-max scheme based on the set Mk in order to prove Theorem 1.1. As anticipated in the introduction, we define a modified functional IIρ for which we can prove existence of solutions in a dense set of the values of ρ. Following a idea of Struwe ( see [49]), this is done proving the a.e differentiability of the map ρ → II ρ ( where II ρ is the minimax value for the functional IIρ ). dk denote the We now introduce the minimax scheme which provides existence of solutions for (8). Let M dk = (Mk × [0, 1]) with Mk × 0 collapsed to a (contractible) cone over Mk , which can be represented as M ¯ be so large ( single point. Let first L be so large that Proposition 4.4 applies with L4 , and then let λ ¯ we define the following class. that Proposition 4.8 applies for this value of L. Fixing λ, (223)

dk → H n2 (M ) : π is continuous and π(· × 1) = Φλ¯ (·)}. IIλ¯ = {π : M

Then we have the following properties. Lemma 4.9 The set IIλ¯ is non-empty and moreover, letting II λ¯ = inf

π∈IIλ ¯

sup II(π(m)), d m∈M k,

L there holds II λ¯ > − . 2

Proof. The proof is the same as the one of Lemma 5.1 in [26]. But we will repeat it for the reader’s convenience. To prove that II λ¯ is non-empty, we just notice that the following map π ¯ (·, t) = tΦλ¯ (·)

(223)

belongs to IIλ¯ . Now to prove that II λ¯ > − L2 , let us argue by contradiction. Suppose that II λ¯ ≤ − L2 : II(π(m)) ≤ − 38 L. Hence since Proposition 4.4 then there exists a map π ∈ II λ¯ such that supm∈M d k L applies with 4 , writing m = (z, t) with z ∈ Mk we have that the map t → Ψ ◦ π(·, t) is an homotopy in Mk between Ψ ◦ Φλ¯ and a constant map. But this is impossible since Mk is noncontractible and Ψ ◦ Φλ¯ is homotopic to the identity by Proposition 4.8. Next we introduce a variant of the above minimax scheme, following [49] and [26]. For ρ in a small n neighborhood of 1, [1 − ρ0 , 1 + ρ0 ], we define the modified functional IIρ : H 2 (M ) → R Z Z ­ ® n (224) IIρ (u) = n Pgn u, u + 2nρ Qng udVg − 2ρκPgn log enu dVg ; u ∈ H 2 (M ). M

M

37

Following the estimates of the previous section, one easily checks that the above minimax scheme applies ¯ sufficiently large. More precisely, given any large number uniformly for ρ ∈ [1 − ρ0 , 1 + ρ0 ] and for λ ¯ L > 0, there exist λ sufficiently large and ρ0 sufficiently small such that (225)

sup

sup II(π(m)) < −2L;

π∈IIλ d ¯ m∈∂ M k

II ρ inf

π∈IIλ ¯

L sup IIρ (π(m)) > − ; 2 d m∈Mk

ρ ∈ [1 − ρ0 , 1 + ρ0 ],

where IIλ¯ is defined as in (223). Moreover, using for example the test map, one shows that for ρ0 ¯ such that sufficiently small there exists a large constant L ¯ II ρ ≤ L,

(226)

for every ρ ∈ [1 − ρ0 , 1 + ρ0 ].

We have the following result regarding the dependence in ρ of the minimax value II ρ . ¯ and ρ0 such that (225) holds. Then the function Lemma 4.10 Let λ ρ→

II ρ ρ

is non-increasing in [1 − ρ0 , 1 + 1 − ρ0 ]

0

Proof. For ρ ≥ ρ , there holds IIρ (u) IIρ0 (u) n − = 0 ρ ρ 2

(227)

µ

1 1 − 0 ρ ρ

¶

­ n ® Pg u, u

Therefore it follows easily that also II ρ0 II ρ − 0 ≤ 0, ρ ρ

(228) hence the Lemma is proved.

From this Lemma it follows that the function ρ → obtain the following corollary.

II ρ ρ

is a.e. differentiable in [1 − ρ0 , 1 + ρ0 ], and we

¯ and ρ0 be as in Lemma 4.10, and let Λ ⊂ [1 − ρ0 , 1 + ρ0 ] be the (dense) set of Corollary 4.11 Let λ II

ρ for which the function ρρ is differentiable. Then for ρ ∈ Λ the functional IIρ possesses a bounded Palais-Smale sequence (ul )l at level II ρ . Proof. The existence of Palais-Smale sequence (ul )l at level II ρ follows from (225) and the bounded is proved exactly as in [23], Lemma 3.2. Next we state a Proposition saying that bounded Palais-Smale sequence of IIρ converges weakly (up to a subsequence) to a solution of the perturbed problem. The proof is the same as the one of Proposition 5.5 in [26]. n

Proposition 4.12 Suppose (ul )l ⊂ H 2 (M ) is a sequence for which Z 0 IIρ (ul ) → c ∈ R; IIρ [ul ] → 0; enul dVg = 1 kul kH n2 (M ) ≤ C. M

Then (ul ) has a weak limit u0 (up to a subsequence) which satisfies the following equation: Pgn u + ρQng = ρκPgn enu

38

in M.

Now we are ready to make the proof of Theorem 1.1. Proofof Theorem 1.1 By (4.11) and (4.12) there exists a sequence ρl → 1 and ul such that the following holds : Pgn u + ρl Qng = ρl κPgn enul

R

in M.

¯ l = ρl κP n we have that Now since κPgn = M Qng dVg then applying corollary 1.4 with Ql = ρl Qng and Q g α ul is bounded in C for every α ∈ (0, 1). Hence up to a subsequence it converges uniformly to a solution of (3). Hence Theorem 1.1 is proved. Remark 4.1 As said in the introduction, the condition Pgn non-negative is only required to make the exposition clear. Indeed if Pgn has some negative eigenvalues the arguments change in the following way. To obtain Moser-Trudinger type inequality and its improvement we impose the additional condition kˆ uk ≤ C where u ˆ is the component of u in the direct sum of the negative eigenspaces. Furthemore another aspect has to be considered that is not only enu can concentrate but also kˆ uk can also tend to infinity. And to deal with this we have to substitute the set Mk with an other one, Ak,k¯ which is defined in terms of the integer k (given in (4)) and the number k¯ of negative eigenvalues of Pgn , as done in [26]. This also requires suitable adaptation of the min-max scheme and of the monotonicity formula in Lemma 4.10, which in general becomes ρ→

II ρ − Cρ is non-increasing in [1 − ρ0 , 1 + ρ0 ]; ρ

for a fixed constant C > 0.

5

The odd-dimensional case

In this section we show how the proofs of Theorem 1.1 and Theorem 1.3 could be extended to the odddimensional case as well. We first remark that the proof of the Theorem 1.3 can be reproduced as soon as Lemma 2.1 is valid, since we use only the Green’s representation formula. Hence we will prove the latter fact to conclude for the proof of Theorem 1.3 following the above steps. On the other hand we remark that for Theorem 1.1 the same proof is valid if we have the analogues of Lemma 4.1 and Lemma 4.5. Hence we need only to prove their counterparts. The Section is organized as follows. We first give a counterpart of Lemma 2.1, then Lemma 4.1 and finally Lemma 4.5 Lemma 5.1 Suppose KerPgn = {constants}. Then the Green function G(x, y) of Pgn exists in the following sense : a) For all functions u ∈ C n (M ) Z (229) u(x) − u ¯= G(x, y)Pgn u(y)dVg (y) M

b) (230)

G(x, y) = H(x, y) + K(x, y), x 6= y ∈ M 2

2

is smooth on M \ Diag(M ), K extends to a C 2+α function on M 2 and µ ¶ 1 1 (231) H(x, y) = log f (r) cn r n

where 2cn = 2n π 2 Γ n2 = (n − 1)!ωn , r = dg (x, y) is the geodesic distance from x to y; f (r) is a C ∞ positive decreasing function , f (r) = 1 in a neighborhood of r = 0 and f (r) = 0 for r ≥ injg (M ). 39

Proof. We remark that if the following properties are valid (232)

|Pgn H(x, y)| ≤ C|Pg˜n H(x, y)| ≤ Cr2−n f or r ≤ C −1 injg (M );

(233)

u(x) =

Z

Z M

H(x, y)Pg˜n u(y)dVg˜ (y) −

M

Pg˜n H(x, y)u(y)dVg˜ (y),

then the proof for the even case can be easily adapted. Hence to conclude we need only to prove (232)(233). We first start by the second one which is less technical. Using the self adjointness of Pgn we have Z Z Z (234) Pg˜n H(x, y)u(y)dVg˜ (y) = H(x, y)Pg˜n u(y)dVg˜ (y) − Pg˜n H(x, y)u(y)dVg˜ (y). M \Bx (²)

M

Bx (²)

Letting ² → 0 we are done. Now let us prove the first one. Writting n = 2k + 1 and recalling we are working in conformal normal coordinates around x, up to errors terms we can suppose we are on flat 1 space and that we have to compute (−∆) 2 (−∆)k H. First, reasoning as in the even case we have the following estimate for (−∆)k H(r) (235) Now we recall a well known that we will use to continue transform that we denote by (236)

(−∆)k H(r) = O(r2−2k ). formula for Fourier transform of radial functions, see [48] (Theorem 3.3) our analysis. Given f ∈ L1 (Rn ) radial, it is well known that its Fourier fˆ is still radial and verifies the following formula Z ∞ n − n−2 ˆ 2 f (s)J n−2 (2πrs)s 2 ds, f (r) = 2πr 2

0

where J n−2 is the Bessel function of first kind and of order 2 following asymptotics at 0 (237)

J n−2 (t) = t

n−2 2

2

n−2 2 .

On the other hand J n−2 has the 2

(bn + ot (1)),

where ot (1) → 0 as t → 0 and bn is a dimensional constant. Furthermore it has also the following asymptotics at infinity (238)

1

J n−2 (t) = O(t− 2 ). 2

For reference about the asymptotics at zero and infinity of J n−2 , see [48] ( Lemma 3.11). 2 Now using (235)-(238), by easily calculations we obtain (239)

\ k H(r) = O(r 2k−2−n ); (−∆)

\ k H(r) stands for the Fourier transform of (−∆)k H(r). where (−∆) 1 On the other hand using the definition of (−∆) 2 , again (236)-(238) and (239) we have that (240)

1

(−∆) 2 (−∆)k H = O(r1−2k ).

Hence from the trivial identity 2k − 1 = n − 2, we are done. So this conclude also the proof of the Lemma.

The next discussion is concerned about extending the improvement of the Moser-Trudinger inequality (see Lemma 4.1). We first remark that the proof in the even case relies on two ingredients . The first one is the Moser-Trudinger inequality whose proof is based only on the existence of the Green’s representation, hence from the previous lemma it holds also for the odd case. The second ingredient is the use of the Leibniz rule to obtain formula (187). Hence as soon as we have a counterpart of this formula the same proof holds. In order to recover this we will use the notion of commutator in the theory of pseudodifferential operators and an interpolation inequality involving it. So we state the Lemma and give a proof of the adaptation. 40

1 Lemma 5.2 let S1 · · · Sl+1 be subsets of M satisfying dist(Si , Sj ) ≥ δ0 for i 6= j, and let γ0 ∈ (0, l+1 ). Then, for any ²¯ > 0, there exists a constant C = C(¯ ², δ0 , γ0 ) such that Z ­ n ® n (241) log en(u−¯u) ≤ C + Pg u, u 4cn (l + 1) − ²¯ M n

for all the functions u ∈ H 2 (M ) satisfying R nu e dV g R Si (242) ≥ γ0 , i ∈ {1, .., l + 1}. enu dV g M n

Proof. For every v ∈ H 2 (M ) we have that Z ­ n ® ­ ® (243) Pg gi v, gi v = gi2 (Pgn v, v)dVg + Pgn gi v − gi Pgn v, gi v . M

On the other hand by H¨older’s inequality we obtain ­ n ® (244) Pg gi v − gi Pgn v, gi v ≤ ||Pgn gi v − gi Pgn v||L2 ||gi v||L2 . Now using the property of commutators, (see [50] Corollary 4.2) we have that ||Pgn gi v − gi Pgn v||L2 ≤ C||v||H n−1 ;

(245)

so using interpolations as in the even case we obtain Z Z ­ n ® ­ ® (246) Pg gi v, gi v ≤ gi2 (Pgn v, v)dVg + ² Pgn v, v + C²,δ0 M

v 2 dVg . M

The rest of the proof of the Lemma follows that of Lemma 4.1. Next we state the counterpart of Lemma 4.5 and give a sketch of the proof. Lemma 5.3 Suppose ϕλ,σ as in (160) and let ² > 0 small enough. Then as λ → +∞ one has ­ n ® (247) Pg ϕλ,σ , ϕλ,σ ≤ (4kcn + ² + oδ (1)) log λ + C²,δ . Proof. We first remark that as soon we have the formula (200) in the even case the same proof holds. So we show how to obtain it in the odd case. We first introduce some notations. If n = 4k + 1 we set 1

Tn ϕσ,λ = (−∆) 4 (−∆)k ϕσ,λ ;

(248) and if n = 4k + 3 we set

3

Tn ϕσ,λ = (−∆) 4 (−∆)k ϕσ,λ .

(249)

Now following the proof of the even case in Lemma 4.5 everything remain true up formula (110), that is (250)

∂ β ϕσ,λ = 2∂ β log

1 1 + oδ,Θ (1) |β| d1 d

for all multi-indices β : |β| ≤

1

Hence we obtain (251)

(Tn ϕσ,λ )2 = 4(Tn log

1 1 2 ) + oδ,Θ (1)(Tn )2 . d1 d1

41

n . 2

On the other hand working in geodesic polar coordinates and reasoning as in the proof of Lemma 5.1 we obtain (252)

(Tn log

1 2 cn ) ≤ (1 + or (1)). d1 ωn−1 rn

1

3

Now by using the definition of (−∆) 4 or (−∆) 4 and still by reasoning as in the proof of Lemma 5.1 we find by easy calculations (253)

Tn

1 1 = o( n ). d1 r2

Hence at this step we can continue the proof of the even case to get the estimate for the case k = 1. Now let show the adaptations to do to get the case k > 1. Focusing on two steps, we follow the proof in the even case up to formula 190 that we recall (254)

∂ β ϕλ,σ (x) = 2∂ β log(

1 1 n ) + oδ,Θ,C (1)( |β| ) for all multi-index β such that |β| ≤ . di 2 d min

Hence from this we obtain (255)

(Tn ϕλ,σ )2 = 4(Tn log(

1 2 1 2 )) + oδ,Θ,C (1)((Tn ) ). di dmin

So reasoning as in the case k = 1 we can continue the proof in the even case up to arriving to the formula (195). Moreover to continue the proof of the even case we need only one more adaptation to obtain our result which is the formula (197). To do this we still argue as in the case k = 1. Hence continuing to adapt the proof for the even case we get our Lemma.

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