Compactness and blow up results for doubly perturbed Yamabe problems on manifolds with umbilic boundary

Given a compact Riemannian manifold with umbilic boundary, the Yamabe boundary problem studies if there exist conformal scalar-flat metrics such that the boundary has constant mean curvature. In this paper we address to the stability of this problem with respect of perturbation of mean curvature of the boundary and scalar curvature of the manifold. In particular we prove that the Yamabe boundary problem is stable under perturbation of the mean curvature and the scalar curvature from below, while it is not stable if one of the two curvatures is perturbed from above.


Introduction
Let (M, g), a smooth, compact Riemannian manifold of dimension n ≥ 3 with boundary. In [17] Escobar asked it there exists a conformal metricg = u 4 n−2 g for which M has zero scalar curvature and constant boundary mean curvature.
This problem can be understood as a generalization of the Riemann mapping theorem and it is equivalent to finding a positive solution to the following nonlinear boundary value problem (1.1) L g u = 0 in M B g u + (n − 2)u n n−2 = 0 on ∂M .
Where L g = ∆ g − n−2 4(n−1) R g and B g = − ∂ ∂ν − n−2 2 h g are respectively the conformal Laplacian and the conformal boundary operator, R g is the scalar curvature of the manifold, h g is the mean curvature of the ∂M and ν is the outer normal with respect to ∂M . If M is of positive type, that is when In fact, if the boundary of M is umbilic, and the Weyl tensor W g never vanishes on the boundary, the full set of solution of (1.1) is compact. This is proved in [9], for dimensions n > 8, and in [12], for dimensions n = 6, 7, 8. We recall that the boundary of M is called umbilic if the trace-free second fundamental form of ∂M is zero everywhere. Also, the authors show in [10] that the problem is stable for perturbation from below of the mean curvature, while in [11] with Pistoia they prove that there is a blow up phenomenon when perturbing the mean curvature from above. This recalls a similar result from the Yamabe problems on boundaryless manifolds, in which perturbations form below of the scalar curvature preserve the compactness of the set of solutions (see [4,5]).
At this point it is interesting to study what happens when one perturbs both the scalar and the mean curvature, and to investigate compactness versus blow up of solutions in this framework. Thus, we study the linearly perturbed problem (1.2) −∆ g u + n−2 4(n−1) R g u + ε 1 αu = 0 in M ∂u ∂ν + n−2 2 h g u + ε 2 βu = (n − 2)u n n−2 on ∂M or, in a more compact form, where ε 1 , ε 2 are small positive parameters and α, β : M → R are smooth functions.
Here we choose ε 1 sufficiently such that −L g +ε 1 α is still a positive definite operator.
Our aim is to prove that, if we linearly perturb the mean curvature term h g with a negative smooth function, and jointly we perturb the scalar curvature term R g with another negative smooth function, the set of solution is still compact. On the contrary, if one between scalar and mean curvature is perturbed from above, the compactness of solutions is lost. Our main results read as Theorem 1. Let (M, g) a smooth, n-dimensional Riemannian manifold of positive type not conformally equivalent to the standard ball with regular umbilic boundary ∂M .
Let α, β : M → R smooth functions such that α, β < 0 on ∂M . Suppose that n ≥ 8 and that the Weyl tensor W g is not vanishing on ∂M . Then, there exists a positive constant C such that for any ε 1 , ε 2 ≥ 0 small enough and for any u > 0 solution of (1.2) it holds C −1 ≤ u ≤ C and u C 2,η (M) ≤ C for some 0 < η < 1. The constant C does not depend on u, ε 1 , ε 2 .
Theorem 2. Let (M, g) a smooth, n-dimensional Riemannian manifold of positive type not conformally equivalent to the standard ball with regular umbilic boundary ∂M .
Let α, β : M → R smooth functions. Suppose that n ≥ 8 and that the Weyl tensor W g is not vanishing on ∂M . If α > 0 on ∂M or β > 0 on ∂M , then there exists a sequence of solutions u ε1,ε2 of (1.2) which blows up at a point of the boundary when (ε 1 , ε 2 ) → (0, 0).
Let us shortly comment these two results.
• In a series of paper [4,5,6] Druet, Hebey and Robert studied the stability of classical Yamabe problem under perturbation of scalar curvature terms. They proved that the set of solutions of −∆ g u+ n−2 4(n−1) a(x)u = cu n+2 n−2 in M is compact if a(x) ≤ R g (x) on M , thus the problem is stable perturbing R g from below, while they found counterexamples to compactness when a(x) is greater than R g (x). In [10,11] the same problem is studied in the case of boundary Yamabe equations by perturbing the mean curvature term and a matching compactness versus blow up phenomenon appears. So there is a strong analogy between the role of R g in classical case and h g in boundary case. We continue here the same analysis, by perturbing both the curvature terms at the same time, to complete the study. It appears that the problem is stable only when perturbing both terms with non positive functions, while it is enough to perturb from above one between h g and R g to lose compactness of the solutions.
• We worked here in the framework of umbilic boundary manifolds. In a recent paper [14], we studied the case of manifold with non umbilic boundary, that is when the trace-free second fundamental form is non zero in any point of ∂M . In this case it is possible to have compactness also for positive small perturbation of the scalar curvature. We want to remind that in the case of non umbilic boundary the compactness of solution for the unperturbed case was proved by Almaraz [1] and Kim, Musso and Wei [16]. • In the unperturbed case the compactness of solutions for umbilic manifolds has been proved for dimensions n ≥ 6 (see [9,12]). It should be possible to apply the same technique also in the perturbed case to extend Theorem 1 to n = 6, 7. It is less clear to us if Theorem 2 could be extended to lower dimensions. Case n = 5 remains open also for the unperturbed problem. • Our theorems consider only perturbations that are everywhere positive or everywhere negative on M . However, in Remark 27 it is shown that it is possible to construct a sign changing α such that for any β the set of solutions in no more compact, or a sign changing β such that for any α the set of solutions in no more compact. We do not know if it would be possible to craft a sign changing perturbation for which compactness is preserved.
The paper is organized as follows. Hereafter we recall some basic definitions and all the preliminary notions useful to achieve the result. Section 2 is devoted to the proof of the compactness theorem, while in Section 3 we prove the non compactness result.

Notations and preliminary definitions.
Remark 3 (Notations). We will use the indices 1 ≤ i, j, k, m, p, r, s ≤ n − 1 and 1 ≤ a, b, c, d ≤ n. Moreover we use the Einstein convention on repeated indices. We denote by g the Riemannian metric, by R abcd the full Riemannian curvature tensor, by R ab the Ricci tensor and by R g and h g respectively the scalar curvature of (M, g) and the mean curvature of ∂M ; moreover the Weyl tensor of (M, g) will be denoted by W g . The bar over an object (e.g.W g ) will means the restriction to this object to the metric of ∂M . Finally, on the half space R n + = {y = (y 1 , . . . , y n−1 , y n ) ∈ R n , y n ≥ 0} we set B r (y 0 ) = {y ∈ R n , |y − y 0 | ≤ r} and B + r (y 0 ) = B r (y 0 ) ∩ {y n > 0}. When y 0 = 0 we will use simply B r = B r (y 0 ) and B + r = B + r (y 0 ). On the half ball B + r we set On R n + we will use the following decomposition of coordinates: (y 1 , . . . , y n−1 , y n ) = (ȳ, y n ) = (z, t) whereȳ, z ∈ R n−1 and y n , t ≥ 0.
Fixed a point q ∈ ∂M , we denote by ψ q : B + r → M the Fermi coordinates centered at q. We denote by B + g (q, r) the image of B + r . When no ambiguity is possible, we will denote B + g (q, r) simply by B + r , omitting the chart ψ q . We introduce the following notation for integral quantities which recur often in the paper By direct computation (see [1,Lemma 9.4]) it holds Also, we have the following integral identities: and, by change of variables where ω n−2 is the volume of S n−1 .
We shortly recall here the well known function U (y) := 1 which is also called the standard bubble and which is the unique solution, up to translations and rescaling, of the nonlinear critical problem .
We set . and we recall that j 1 , . . . , j n are a base of the space of the H 1 solutions of the linearized problem . Given a point q ∈ ∂M , we introduce now the function γ q which arises from the second order term of the expansion of the metric g on M (see 1.18). The choice of this function plays a twofold role in this paper. On the one hand, using the function γ q we are able to perform the estimates of Lemmas 12, 13 and Proposition 14. On the other hand, it gives the correct correction to the standard bubble in order to perform finite dimensional reduction.
For the proof of the following Lemma we refer to [11,Lemma 3] and [1, Proposition 5.1]. Lemma 4. Assume n ≥ 5. Given a point q ∈ ∂M , there exists a unique γ q : R n + → R a solution of the linear problem which is L 2 (R n + )-orthogonal to the functions j 1 , . . . , j n defined in (1.7) and (1.8). Moreover it holds (1.12) Finally the map q → γ q is C 2 (∂M ).

1.2.
Expansion of the metric. Since the boundary ∂M is umbilic, given q ∈ ∂M there exists a conformally related metricg q = Λ 4 n−2 q g such that some geometric quantities at q have a simpler form which will be summarized later in this paragraph. We have Λ q (q) = 1, ∂Λ q ∂y k (q) = 0 for all k = 1, . . . , n − 1.
Also, we have thatũ q := Λ q u, is a solution of (1. β. In the following expansion and in section 2, in order to simplify notations, we will omit the tilde symbols, since we will always work in the conformal metricg, while in Section 3 we will switch between metrics, so we will keep g andg explicitly indicated.
With this metric we have the following expansions.
Remark 5. In Fermi conformal coordinates around q ∈ ∂M , it holds (see [20]) ikjl,m y k y l y m + R ninj,k y 2 n y k + nins . All the quantities above are calculate in q ∈ ∂M , unless otherwise specified.
If we choose ε 1 sufficiently small in order to have that −L g + ε 1 α is a positive definite operator, we can define an equivalent scalar product on H 1 as (1.23) which leads to the norm · g equivalent to the usual one.
With this norm we have that Λ q is an isometry. In fact, by (1.23), for any u, v ∈ H 1 (M ), Λ q u, Λ q v g = u, v gq and, consequently, Λ q u g = u gq .
we have the well known embedding i : H 1 (M ) → L t (∂M ). We define, by the scalar product ·, · g , in the following sense: given f ∈ L 2(n−1) Notice that, if v ∈ H 1 g , then f (v) ∈ L 2(n−1) n (∂M ). Problem (1.2) has also a variational structure and a positive solution for (1.2) is a critical point for the following functional defined on H 1 (M ) We remark that, defined Given q ∈ ∂M and ψ ∂ q : R n + → M the Fermi coordinates in a neighborhood of q; we define and χ is a radial cut off function, with support in ball of radius R. In an analogous way, given γ q as in Lemma 4 we define and, given j a defined in (1.7) and (1.8) we define By means of ·, · g it is possible to decompose H 1 in the direct sum of the following two subspaces and to define the projections Notice that, since Λ q is an isometry, we have that ϕ ∈K δ,q if and only if Λ −1 q ϕ ∈ K δ,q and the same holds forK ⊥ δ,q . In Section 3, we will look for a solutionũ q = Λ q u of (1.2) which has the form whereφ ∈K ⊥ δ,q . By means of i * α this is equivalent to the following pair of equations (1.27)

The compactness result
In this section, firstly we recall a Pohozaev type identity which will gives us a fundamental sign condition to rule out the possibility of blowing up sequence (see subsection 2.4). A recall of preliminary results on blow up points is collected in subsection 2.2, while a careful analysis of blow up sequences is performed in subsection 2.3. The proof of Theorem 1 is given in subsection 3.1. Throughout this section we work ing metric. For the sake of readability we will omit the tilde symbol throughout this section.

2.1.
A Pohozaev type identity. We will use this version of a local Pohozaev type identity (see [1,9]).
Theorem 6 (Pohozaev Identity). Let u a C 2 -solution of the following problem Then P (u, r) =P (u, r).

Isolated and isolated simple blow up points.
Here we recall the definitions of some type of blow up points, and we give the basic properties about the behavior of these blow up points (see [1,7,15,21]). We will omit the proofs of the well known results. Let {u i } i be a sequence of positive solution to Shortly we say that Given x i → x 0 an isolated blow up point for {u i } i , and given ψ i : B + ρ (0) → M the Fermi coordinates centered at x i , we define the spherical average of u i as 3) We say that x i → x 0 is an isolated simple blow up point for {u i } i solutions of (2.1) if x i → x 0 is an isolated blow up point for {u i } i and there exists ρ such that w i has exactly one critical point in the interval (0, ρ).
We recall two propositions whose proofs can be found in [2] and in [7].
Then, given R i → ∞ and c i → 0, up to subsequences, we have Proof. We compute the Pohozaev identity in a ball of radius r and we set r δi =: R i → ∞. We estimate any term of P (u i , r i ) andP (u i , r i ). Set By Proposition 9 we obtain In a similar way we decomposê The terms I 3 , I 4 and I 6 are been estimated in [10]. For the sake of completeness we report here the main steps of the estimates. By Proposition 9 and by definition of v i we have Using the expansion of the metric, it easy to check that Finally, by Claim 1 of Proposition 8, by (1.5) and by (1.3) we get (1)).
In a similar way we proceed for I 5 . In fact, by (1.5), (1.3) and (1.4), we have and thus Since ε 1,i δ i , ε 2,i → 0 by Prop. 10, the proof of the next proposition is analogous to Prop. 4.3 in [1] Proposition 11. Let x i → x 0 be an isolated simple blow-up point for {u i } i and α, β < 0. Then there exist C, ρ > 0 such that  Then v i satisfies The estimates that follow are similar to the ones of [1, Lemma 6.1], [9,Section 4] and [10,Section 5], where the main differences concern the terms which contain the linear perturbations.
Lemma 12. Assume n ≥ 8. Let γ xi be defined in (1.10). There exist R, C > 0 such that Proof. Let y i such that We can assume, without loss of generality, that |y i | ≤ R 2δi . In fact, suppose that there exists c > 0 such that |y i | > c δi for all i. Then, since v i (y) ≤ CU (y), and by (1.11), we get the inequality which proves the Lemma. So, in the next we will suppose |y i | ≤ R 2δi . This will be useful later.
By contradiction, suppose that We will estimate terms b i , A i, F i obtaining that the sequence w i converges in C 2 loc (R n + ) to some w solution of (2.12) ∆w = 0 in R n + ∂ ∂ν w + nU n n−2 w = 0 on ∂R n + , then we will derive a contradiction using (2.10). Since We proceed now by estimating Q i andQ i . As in [10, Lemma 11], using the expansion of the metric and the decays properties of U and γ xi we obtain and Since |v i (y)| ≤ CU (y) from (2.15) and (2.16) we get In light of (2.10) we also have A i ∈ L p (B + R/δi ) and F i ∈ L p (∂ ′ B + R/δi ) for all p ≥ 2. Finally we remark that |w i (y)| ≤ 1, so by (2.10) (2.13), (2.14), (2.17) and by standard elliptic estimates we conclude that, up to subsequence, {w i } i converges in C 2 loc (R n + ) to some w solution of (2.12) as claimed at the beginning of the proof. The next step is to prove that |w(y)| ≤ C(1 + |y| −1 ) for y ∈ R n + . Consider G i the Green function for the conformal Laplacian Lĝ i defined on B + r/δi with boundary conditions Bĝ i G i = 0 on ∂ ′ B + r/δi and G i = 0 on ∂ + B + r/δi . It is well known that G i = O(|ξ − y| 2−n ). By the Green formula and by (2.17) we have Notice that in the third integral we used that |y| ≤ R 2δi to estimate |ξ − y| ≥ |ξ| − |y| ≥ R 2δi on ∂ + B + R/δi . Moreover, since v i (ξ) ≤ CU (ξ), we get For the other terms we use the formula where y ∈ R m+k ⊇ R m , η, l ∈ N, 0 < l < η < m (see [1, Lemma 9.2] and [3,8]) . We get By the previous estimates we infer that, for |y| ≤ R 2δi , so by assumption (2.10) we prove (2.26) |w(y)| ≤ C(1 + |y|) −1 for y ∈ R n + as claimed.
We are ready now to prove the contradiction. In fact, it is known (see [1,Lemma 2]) that any solution of (2.12) that decays as (2.26) is a linear combination of ∂U ∂y1 , . . . , ∂U ∂yn−1 , n−2 2 U + y b ∂U ∂y b . This fact, combined with (2.27), implies that w ≡ 0. Now, on the one hand |y i | ≤ R 2δi , so estimate (2.26) holds; on the other hand, since w i (y i ) = 1 and w ≡ 0, we get |y i | → ∞, obtaining which gives us the contradiction.
Lemma 13. Assume n ≥ 8 and α, β < 0. There exists R, C > 0 such that Proof. We proceed by contradiction, supposing that We define, similarly to Lemma 12, and we have that w i satisfies (2.11) where b i is as in Lemma 12 and As before, b i satisfies inequality (2.14) while so by classic elliptic estimates we can prove that the sequence w i converges in C 2 loc (R n + ) to some w. We proceed as in Lemma 12 to deduce that, by (2.28) and since Now let j n defined as in (1.8). Indeed, since w i satisfies (2.11), integrating by parts we obtain (2.31) where η i is the inward unit normal vector to ∂ + B + R δ i . One can check easily that (δ i y)β(δ i y), and β < 0, by Proposition 8, we have and thus, as in Proposition 10 (2.32) (1)).
The above lemmas are the core of the following proposition, in which we iterate the procedure of Lemma 12, to obtain better estimates of the rescaled solution v i of (2.9) around the isolated simple blow up point x i → x 0 . Proposition 14. Assume n ≥ 8 and α, β < 0. Let γ xi be defined in (1.10). There exist R, C > 0 such that Here τ = 0, 1, 2 and ∇ τ y is the differential operator of order τ with respect the first n − 1 variables.
Proof. In analogy with Lemma 12, we set and we have that w i satisfies (2.11) where b i is defined as before, As before, b i satisfies inequality (2.14) and We define the Green function G i as in the previous lemma and again, by Green's formula, by (2.42), (2.43), and Lemmas 12 and 13, we have which proves the first claim for τ = 0. The other claims follow similarly.

Sign estimates of Pohozaev identity terms.
In this section, we want to estimate P (u i , r), where {u i } i is a family of solutions of (2.1) which has an isolated simple blow up point x i → x 0 . This estimate, given in the following Proposition 16, is a crucial point for the proof of the vanishing of the Weyl tensor at an isolated simple blow up point. Since the leading term of P (u i , r) will be − B + r/δ i and we recall the following result Proof. For the proof we refer to [9] for the case n > 8 and to [12] for the case n = 8.
Proposition 16. Let x i → x 0 be an isolated simple blow-up point for u i solutions of (2.1). Let α, β < 0 and n ≥ 8. Then, fixed r, we have, for i largê Proof. We recall that the definition ofP is given in Theorem 6 and we take v i (y) as in (2.8). By Proposition 14 and by (1.11) of Lemma 4, for |y| < R/δ i we have and, recalling that α i (δ i y) → α(x 0 ) < 0 and proceeding as in Proposition 10 we get and again we get . So, for i sufficiently large we obtain Now define, in analogy with Proposition 14, and by Lemma 15 we conclude the proof. Proposition 17. Assume n ≥ 8 and α, β < 0. Let x i → x 0 be an isolated simple blow-up point for u i solutions of (2.1). Then |W (x 0 )| = 0.
Proof. By Proposition 11 and Proposition 9, and since M i = δ 2−n 2 i we have, On the other hand recalling Proposition 16 and Theorem 6 we have Recalling that when the boundary is umbilic W (q) = 0 if and only ifW (q) = 0 and R nlnj (q) = 0 (see [20, page 1618]) we conclude the proof.
Remark 18. Let x i → x 0 be an isolated blow up point for u i solutions of (2.1). We set Using Proposition 16, (2.49), and since n ≥ 8 we get where C > 0.
Proposition 19. Let x i → x 0 be an isolated blow up point for u i solutions of (2.1). Assume n ≥ 8 and |W (x 0 )| = 0. Then x 0 is isolated simple.
For the proof of this Lemma we refer to [1,9] 2.5.  [9] for the last claim when n > 8 and to [12] in the case n = 8.
Proposition 20. Given K > 0 and R > 0 there exist two constants C 0 , C 1 > 0 (depending on K, R and (M, g)) such that if u is a solution of (1) set r j := Ru(q j ) 1−p then B rj ∩ ∂M j are a disjoint collection; (2) we have u(q j ) −1 u(ψ j (y)) − U (u(q j ) p−1 y) C 2 (B + 2r j ) < K (here ψ j are the Fermi coordinates at point q j ; (3) we have In addition, if n ≥ 8 and W (x) = 0 for any x ∈ ∂M , there exists d = d(K, R) such that Hereḡ is the geodesic distance on ∂M .
We prove now the main result Proof of Theorem 1. . By contradiction, suppose that x i → x 0 is a blowup point for u i solutions of (1.2). Let q i 1 , . . . q i N (ui) the sequence of points given by Proposition 20. By Claim 3 of Proposition 20 there exists a sequence of indices k i ∈ 1, . . . N such that dḡ x i , q i ki → 0. Up to relabeling, we say k i = 1 for all i. Then also q i 1 → x 0 is a blow up point for u i . By Proposition 20 and Proposition 19 we have that q i 1 → x 0 is an isolated simple blow up point for u i . Then by Proposition 17 we deduce that W (x 0 ) = 0, contradicting the assumption of the theorem. This concludes the proof.

The non compactness result
In this section we perform the Ljapunov-Schmidt finite dimensional reduction, which relies on three steps. First, we start finding a solution of the infinite dimensional problem (1.27) with the ansatz Λ q u =W δ,q + δ 2Ṽ δ,q +φ whereφ ∈K ⊥ δ,q . This is done in subsection 3.1. Then, we study the finite dimensional reduced problem in subsection 3.2, and in the last subsection we give the proof of Theorem 2.
3.1. The finite dimensional reduction. Let us define the linear operator L : and let us define a nonlinear term N (φ) and a remainder term R as With these operators the infinite dimensional equation (1.27) becomes In this subsection we will find, for any δ, q given, a functionφ which solves equation (1.27).

Lemma 21. It holds
Proof. Several estimates for this proof has been calculated in [13], which we refer to. We report here only the main steps. Take the unique Γ such that Let us call a := n−2 4(n−1) R g . We have, by (1.23) that We have that and, by change of variables and by (1.17), we get Similarly for I 1 we have Since Rg(0) = 0 (see [20, page 1609]), we get and, using the expansion of the metricg and (1.10), one can show that thus we get For the integral I 4 we have and, since U solves (1.6), we get immediately Estimating the other terms requires more care, but, expanding (U + δ 2 γ q ) + n n−2 near U , using (1.10) and the decay estimates (1.11), one can show that (see [13] for all the details )

Thus by (3.4) and (3.5) we have
For I 2 we have Now by change of variables we have The following lemma is a standard tool in finite dimensional reduction, so we refer to [18,22] for its proof.
Proof. By Lemma 22, by (3.6) and by the properties of i α , there exists C > 0 such that Now it is easy to estimate that If n > 8, by Lemma 21 and by the previous estimates, for the map It is possible to choose ρ > 0 such that T is a contraction from the ball φ g ≤ ρ(ε 2 δ + ε 1 δ 2 + δ 3 ) in itself, so, by the fixed point Theorem, there exists a uniquẽ φ with φ g = O(ε 2 δ + ε 1 δ 2 + δ 3 ) which solves (1.27). In addition by the implicit function Theorem it is possible to prove the regularity of the map q →φ. The case n = 8 follows verbatim.
3.2. The reduced functional. Once we solved (1.27), we show that we can find a critical point of J g W δ,q + δ 2Ṽ δ,q +φ by solving a finite dimensional problem depending only on (δ, q).
HereW (q) is the Weyl tensor restricted to boundary.
3.3. Proof of Theorem 2. At first we provide a sign estimate for function ϕ(q) defined in the previous paragraph.
Lemma 26. Assume n ≥ 8 and that the Weyl tensor W g is not vanishing on ∂M . Then the function ϕ(q) defined in Lemma 25 is strictly negative on ∂M .
Proof. We can write the function ϕ(q) defined in Lemma 25 as where C 1 , C 2 are positive constants. If n > 8, since in umbilic boundary manifolds W (q) = 0 if and only ifW (q) and R ninj (q) are both zero (see [20, page 1618]), by our assumption at least one among|W (q)| and R 2 nlnj (q) is strictly positive. Since by (1.12) the term involving γ q is non positive, the lemma is proved.
When n = 8 the term involving R 2 8i8j vanishes. However in [12] a refined analysis of the term R n + γ q ∆γ q dtdz was performed, leading to the following improvement of estimate (1.12): where C 3 > 0. This was possible by a more precise description of function γ q as sum of an harmonic function with explicit rational functions, proved in Lemma 19 of the cited paper. Thus for n = 8 we have ϕ(q) ≤ −C 1 |W (q)| 2 − C 3 R 2 8i8j (q) < 0, and the proof is complete.
Proof of Theorem 2. We give a detailed proof in the case α > 0. The case β > 0 is analogous and we will emphasize the difference at the end of the proof.
To conclude the proof it lasts to find a pair (λ,q) which is a critical point for I ε1,ε2 (λ, q).
Let us call G(λ, q) := λα(q)B + λ 2 ϕ(q). We have that α(q)B is strictly positive on ∂M , by our assumptions, while by Lemma 26, ϕ is strictly negative on ∂M . At this point there exists a compact set [a, b] ⊂ R + such that the function G admits an absolute maximum in (a, b) × ∂M , which also is the absolute maximum value of G on R + × ∂M . This maximum is also C 0 -stable, in the sense that, if (λ 0 , q 0 ) is the maximum point for G, for any function f ∈ C 1 ([a, b] × ∂M ) with f C 0 sufficiently small, then the function G + f on [a, b] × ∂M admits a maximum point (λ,q) close to (λ 0 , q 0 ). By the C 0 stability of this maximum (λ 0 , q 0 ), and by Lemma 25, given ε 1 sufficiently small (and ε 2 = o(ε 2 1 )), there exists a pair (λ ε1 , q ε1 ) which is a maximum point for J g W√ λε1,q + λε 1Ṽ √ λε1,q , and, in turn, that there exists a pair λ ε1 ,q ε1 which is a maximum point for I ε1,ε2 (λ, q). This implies, in light of the above Remark, thatW √λ