The second Yamabe invariant with singularities

Let (M,g) be a compact manifold of dimension n greater or equals to 3. We suppose that g is a given metric in a precised Sobolev space and there is a point P in M and d>o such that g is smooth on the ball B(P,d). We define the second Yamabe invariant with singularities a the minimum of the second eigenvalue of the singular Yamabe operator over a generalized class of conformal metrics to g and of volume 1. We show that this operator is attained by a generalized metric, we deduce nodal solutions to a Yamabe type equation with singularities.


Introduction
Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. The problem of finding a metric conformal to the original one with constant scalar curvature was first formulated by Yamabe ([9]) and a variational method was initiated by this latter in an attempt to solve the problem. Unfortunately or fortunately a serious gap in the Yamabe was pointed out by Trudinger who addressed the question in the case of non positive scalar curvature ( [8] ). Aubin ([2]) solved the problem for arbitrary non locally conformally flat manifolds of dimension n ≥ 6. Finally Shoen ([7]) solved completely the problem using the positive-mass theorem founded previously by Shoen himself and Yau . The method to solve the Yamabe problem could be described as follows: let u be a smooth positive function and let g = u N −2 g be a conformal metric where N = 2n/(n − 2). Up to a multiplying constant, the following equation is satisfied L g (u) = Sg|u| N −2 u where L g = 4(n − 1) n − 2 ∆ + S g and S g denotes the scalar curvature of g. L g is conformally invariant called the conformal operator. Consequently, solving the Yamabe problem is equivalent to find a smooth positive solution to the equation where k is a constant. In order to obtain solutions to this equation, Yamabe defined the quantity where Y (u)= M 4(n−1) n−2 |∇u| 2 + S g u 2 dv g ( M |u| N dv g ) 2/N . µ(M, g) is called the Yamabe invariant, and Y the Yamabe functional. In the sequel we write µ instead of µ (M, g). Writing the Euler-Lagrange equation associated to Y , we see that there exists a one to one correspondence between critical points of Y and solutions of equation (1). In particular, if u is a positive smooth function such that Y (u) = µ, then u is a solution of equation (1) and g = u (N −2) g is metric of constant scalar curvature. The key point to solve the Yamabe problem is the following fundamental results due to Aubin ( [2]). Let S n be the unit euclidean sphere. This strict inequality µ (M, g) < µ(S n ) avoids concentration phenomena. Explicitly µ(S n ) = n(n − 1)ω 2/n n where ω n stands for the volume of S n . Theorem 2. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. Then µ (M, g) ≤ µ(S n ). Moreover, the equality holds if and only if (M, g) is conformally diffeomorphic to the sphere S n .
Amman and Humbert ( [1]) defined the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the conformal class of the metric g with volume 1. Their method consists in considering the spectrum of the operator L g spec(L g ) = {λ 1,g , λ 2,g . . .} where the eigenvalues λ 1,g < λ 2,g . . . appear with their multiplicities. The variational characterization of λ 1,g is given by Then they defined the k th Yamabe invariant with k ∈ N ⋆ , by With these notations µ 1 is the Yamabe invariant. They studied the second Yamabe invariant µ 2 , they found that contrary to the Yamabe invariant, µ 2 cannot be attained by a regular metric. In other words, there does not exist g ∈ [g] , such that In order to find minimizers, they enlarged the conformal class to a larger one. A generalized metric is the one of the form g = u N −2 g , which is not necessarily positive and smooth, but only u ∈ L N (M ), u ≥ 0, u = 0 and where N = 2n/ (n − 2) . The definitions of λ 2,g and of V ol(M, g) 2/n can be extended to generalized metrics. The key points to solve this problem is the following theorems ([1]).
Theorem 3. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3, then µ 2 is attained by a generalized metrics in the following cases.
Theorem 5. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3, assume that µ 2 is attained by a generalized metricg = u N −2 g then there exist a nodal solution w ∈ C 2,α (M ) of equation In ( [5]), recently F.Madani studied the Yamabe problem with singularities when the metric g admits a finite number of points with singularities and smooth outside these points. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3, assume that g is a metric in the Sobolev space and there exist a point P ∈ M and δ > 0 such that g is smooth in the ball B p (δ), and let (H) be these assumptions. By Sobolev's embedding, we have for where [n/p] denotes the entire part of n/p. Hence the metric satisfying assumption (H) is of class C 1−[ n p ],β with β ∈ (0, 1) provided that p > n. The Christoffels symbols belong to H p 1 (M ) ( to C o (M ) in case p > n), the Riemannian curvature tensor, the Ricci tensor and scalar curvature are in L p (M ). F. Madani proved under the assumption (H) the existence of a metric g = u N −2 g conformal to g such that u ∈ H p 2 (M ), u > 0 and the scalar curvature S g of g is constant and (M, g) is not conformal to the round sphere. Madani proceeded as follows: let u ∈ H p 2 (M ), u > 0 be a function and g = u N −2 g a particular conformal metric where N = 2n/(n− 2).Then, multiplying u by a constant, the following equation is satisfied S g and the scalar curvature S g is in L p (M ). Moreover L g is weakly conformally invariant hence solving the singular Yamabe problem is equivalent to find a positive solution u ∈ H p 2 (M ) of (2) where k is a constant. In order to obtain solutions of equation (2) we define the quantity µ is called Yamabe invariant with singularities. Writing the Euler-Lagrange equation associated to Y , we see that there exists a one to one correspondence between critical points of Y and solutions of equation (2). In particular, if u ∈ H p 2 is a positive function which minimizes Y , then u is a solution of equation(2) and g = u N −2 g is a metric of constant scalar curvature and µ is attained by a particular conformal metric. The key points to solve the above problem are the following theorems ( [5]). For regularity argument we need the following results The proof is the same as in ( [5]) with some modifications. As a consequence of Lemma 7, v ∈ L s (M ), ∀s ≥ 1.
weakly. Moreover if µ > 0, then L g is coercive and invertible.
In this paper, let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. We suppose that g is a metric in the Sobolev space H p 2 (M, T * M ⊗ T * M ) with p > n/2 and there exist a point P ∈ M and δ > 0 such that g is smooth in the ball B P (δ) and we call these assumptions the condition (H).
In the smooth case the operator L g is an elliptic operator on M self-adjoint, and has a discrete spectrum Spec(L g ) = {λ 1,g , λ 2,g , . . .}, where the eigenvalues λ 1,g < λ 2,g . . . appear with their multiplicities. These properties remain valid also in the case where S g ∈ L p (M ). The variational characterization of λ 1,g is given by 2 and u > 0}, Let k ∈ N * , we define the k th Yamabe invariant with singularities µ k as λ k,g V ol(M,g) 2/n with these notations, µ 1 is the first Yamabe invariant with singularities.
In this work we are concerned with µ 2 . In order to find minimizers to µ 2 we extend the conformal class to a larger one consisting of metrics of the form g = u N −2 g where u is no longer necessarily in H p 2 (M ) and positive but u ∈ L N + (M ) = L N (M ), u ≥ 0, u = 0 such metrics will be called for brevity generalized metrics. First we are going to show that if the singular Yamabe invariant µ ≥ 0 then µ 1 it is exactly µ next we consider µ 2 , µ 2 is attained by a conformal generalized metric.
Our main results state as follows: Theorem 9. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 .We suppose that g is a metric in the Sobolev space H p 2 (M, T * M ⊗ T * M ) with p > n/2. There exist a point P ∈ M and δ > 0 such that g is smooth in the ball B P (δ), then Theorem 10. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3, we suppose that g is a metric in the Sobolev space H p 2 (M, T * M ⊗ T * M ) with p > n/2 .There exist a point P ∈ M and δ > 0 such that g is smooth in the ball B P (δ) .Assume that µ 2 is attained by a metric g = u N −2 g where u ∈ L N + (M ), then there exist a nodal solution w ∈ C 1−[n/p],β , β ∈ (0, 1), of equation Moreover there exist real numbers a, b > 0 such that Theorem 11. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3, suppose that g is a metric in the Sobolev space H p 2 (M, T * M ⊗ T * M ) with p > n/2. There exist a point P ∈ M and δ > 0 such that g is smooth in the ball B P (δ) then µ 2 is attained by a generalized metric in the following cases: If (M, g) in not locally conformally flat and, n ≥ 11 and µ > 0 If (M, g) in not locally conformally flat and, µ = 0 and n ≥ 9.

Generalized metrics and the Euler-Lagrange equation
Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. For a generalized metric g conformal to g, we define We quote the following regularity theorem and u > 0, where [n/p] denotes the integer part of n/p and β ∈ (0, 1).
Proof. The proof is the same as in ( [3]).
Moreover we can choose v and w such that It is well know that (|v m |) m is also minimizing sequence. Hence we can assume Now by the fact that L g is coercive (v m ) m is bounded in H 2 1 (M ) and after restriction to a subsequence we may assume that there exist v ∈ H 2 1 (M ), v ≥ 0 such that v m → v weakly in H 2 1 (M ), strongly in L 2 (M ) and almost everywhere in M , then v satisfies in the sense of distributions Then v is not trivial and nonnegative minimizer of λ 1,g , by Lemma 7 v is a non negative minimizer in H 2 1 of λ 1,g such that Now consider the set and define Let (w m ) be a minimizing sequence for λ ′ 2,g i.e. a sequence w m ∈ E such that The same arguments lead to a minimizer w to λ ′ 2,g with M u N −2 w 2 = 1. Hence (8) and (9) are that satisfied with λ ′ 2,g instead of λ 2,g .

Variational characterization and existence of µ 1
In this section we need the following results Theorem 13. Let (M, g) be a compact n -dimensional Riemannian manifold. For any ε > 0, there exists A(ε) > 0 such that ∀u ∈ H 2 1 (M ), where N = 2n/(n − 4) and K 2 = 4/(n(n − 2)) ω −2 n n . ω n is the volume of the round sphere S n .
and u > 0}, we define the first singular Yamabe invariant µ 1 as .
Proof. The proof will take several steps.

Step1.
We study a sequence of metrics g m = u N −2 In particular, the sequence of functions u m is bounded in L N (M ) and there exists u ∈ L N (M ), u ≥ 0 such that u m → u weakly in L N (M ). We are going to prove that the generalized metric u N −2 g minimizes µ 1 . Proposition 3 implies the existence of a sequence (v m ) of class H 2 1 (M ), v m > 0 such that now since µ > 0, by Proposition 1, L g is coercive and we infer that Together with the weak convergence of (u m ) m , we obtain in the sense of distributions Step2.

Now because of 2p/
And by the definition of µ and Lemma 2 we get And since and from inequality (10), we get and by Hölder's inequality By the strong convergence of v m in L N (M ), we get M u N −2 v 2 dv g = 1, then v and u are non trivial functions. Step4.
Let u = av ∈ L N + (M ) with a > 0 a constant such that We claim that u = v; indeed, And since the equality in Hölder's inequality holds if u = u = av then a = 1 and u = v .

Variational characterization of µ 2
Let [g] = {u N −2 g, u ∈ H p 2 (M ) and u > 0}, we define the second Yamabe invariant µ 2 as λ 2,g V ol(M, g) 2/n or more explicitly Theorem 16. [1] On compact Riemannian manifold (M, g) of dimension n ≥ 3, we have for all v ∈ H 2 1 (M ) and for all u ∈ L N + (M ) Theorem 17. [1] For any compact Riemannian manifold (M, g) of dimension n ≥ 3, there exists B 0 > 0 such that where ω n is the volume of the unit round sphere or ( 6. Properties of µ 2 We know that g is smooth in the ball B p (δ) by assumption (H), this assumption is sufficient to prove that Aubin's conjecture is valid. The case (M, g) is not conformally flat in a neighborhood of the point P and n ≥ 6, let η is a cut-off function with support in the ball B p (2ε) and η = 1 in B p (ε), where 2ε ≤ δ and v ε (q) = ( ε r 2 + ε 2 ) n−2 2 with r = d(p, q).We let u ε = ηv ε and define We obtain the following lemma where |w(P )|is the norm of the Weyl tensor at the point P and c > 0. Proof. With the same method as in [1], this lemma follows from theorem18.
7. Existence of a minimizer to µ 2 Theorem 19. If 1 − 2 − 2 n K 2 µ 2 > 0, then the generalized metric u N −2 g minimizes µ 2 Proof. Step1. We study a sequence of metrics g m = u N −2 m g with u m ∈ H p 2 (M ), u m > 0 which minimizes the infimum in the definition of µ 2 i.e. a sequence of metrics such that µ 2 = lim λ 2,m (V ol(M, g m ) 2/n . Without loss generality, we may assume that V ol(M, g m ) = 1 i.e. that M u N m dv g = 1.In particular, the sequence of functions (u m ) m is bounded in L N (M ) and there exists u ∈ L N (M ), u ≥ 0 such that u m → u weakly in L N . We are going to prove that the generalized metric u N −2 g minimizes µ 2 .Proposition 3 , implies the Together with the weak convergence of (u m ), we get in weak sense and v m → v strongly in H 2 1 (M ) .The same argument holds with (w m ), hence w m → w strongly in H 2 1 (M ) and M u N −2 w 2 dv g = 1. To show that M u N −2 vwdv g = 0, first writing and using Hôlder's inequality, we get Consequently the generalized metric u N −2 g minimizes µ 2 .
Theorem 20. If µ 2 < K −2 , then generalized metric u N −2 g minimizes µ 2 Proof. Step1. We study a sequence of metrics g m = u N −2 m g with u m ∈ H p 2 (M ), u m > 0 which attained µ 2 i.e. a sequence of metrics such that with v ≥ 0 such that v m → v, w m → w weakly in H 2 1 .Together with the weak convergence of (u m ) m , we get in the weak sense L g (v) = µ 1 u N −2 v and L g (w) = µ 2 u N −2 w where µ 1 = lim λ 1,m ≤ µ 2 .

Step2.
Now we are going to show that v m → v , w m → w strongly in H 2. 1 . By Hölder's inequality, Theorem 13, strong convergence of v m in L 2. ,we get Theorem 21. If a generalized metric u N −2 g minimizes µ 2 , then there exist a nodal solution w ∈ H p 2 ⊂ C 1−[n/p],β of equation More over there exist a, b > 0 such that u = aw + + bw − With w + = sup(w, 0) and w − = sup(−w, 0) .
Step3. The solution w of the equation (22) changes sign. Since if it does not, we may assume that w ≥ 0, by step2 the inequality in (20) is strict and by Lemma (5) we have the equality: a contradiction.