The second Yamabe invariant with singularities

BLAISE PASCAL

Mohammed Benalili & Hichem Boughazi

The second Yamabe invariant with singularities Volume 19, n^o1 (2012), p. 147-176.

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The second Yamabe invariant with singularities

Mohammed Benalili Hichem Boughazi

Abstract

Let (M, g) be a compact Riemannian manifold of dimensionn≥3.We suppose that g is a metric in the Sobolev space H₂^p(M, T^∗M ⊗T^∗M) with p > ⁿ₂ and there exist a point P ∈M andδ > 0 such thatg is smooth in the ball Bp(δ).

We define the second Yamabe invariant with singularities as the infimum of the second eigenvalue of the singular Yamabe operator over a generalized class of conformal metrics togand 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.

Dedicated to the memory of T. Aubin.

1. 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 problem was pointed out by Trudinger who addressed the question in the case of non positive scalar curvature ([9]). Aubin ([2]) solved the problem for arbitrary non locally conformally flat manifolds of dimension n≥6. Finally Shoen ([8]) solved completely the problem using the positive-mass theorem found previously by Shoen himself and Yau. The method to solve the Yamabe problem could be described as follows: letube a smooth positive function and let g=u^N−2g be a conformal metric whereN = 2n/(n−2). Up to a multiplying constant, the following equation is satisfied

Lg(u) =Sg˜|u|^N−2u

Keywords:Second Yamabe invariant, singularities, Critical Sobolev growth.

Math. classification:58J05.

where

Lg= 4(n−1)

n−2 ∆ +Sg

and S_g denotes the scalar curvature of g. L_g is conformally invariant called the conformal operator. Consequently, solving the Yamabe prob- lem is equivalent to finding a smooth positive solution to the equation

Lg(u) =ku^N−1 (1)

where kis a constant.

In order to obtain solutions to this equation, Yamabe defined the quan- tity

µ(M, g) = inf

u∈C^∞(M), u>0Y(u) where

Y(u) = R

M

_4(n−1)

n−2 |∇u|²+S_gu²dv_g R

M|u|^Ndvg

2/N .

µ(M, g) is called the Yamabe invariant, andY the Yamabe functional. In the sequel we writeµinstead ofµ(M, g). Writing the Euler-Lagrange equa- tion associated toY, we see that there exists a one to one correspondence between critical points of Y and solutions of equation (1). In particular, ifuis a positive smooth function such that Y(u) =µ, then uis 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 Sn be the unit euclidean sphere.

Theorem 1.1. Let (M, g) be a compact Riemannian manifold of dimen- sion n ≥ 3. If µ(M, g) < µ(Sn), then there exists a positive smooth solution u such thatY(u) =µ(M, g).

This strict inequality µ(M, g) < µ(Sn) avoids concentration phenom- ena. Explicitly µ(Sn) =n(n−1)ωn^2/n where ωn stands for the volume of S_n.

Theorem 1.2. Let (M, g) be a compact Riemannian manifold of dimen- sion n≥3. Then

µ(M, g)≤µ(Sn).

Moreover, the equality holds if and only if (M, g) is conformally diffeo- morphic 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(Lg) ={λ_1,g, λ2,g. . .}

where the eigenvaluesλ1,g < λ2,g. . . appear with their multiplicities. The variational characterization of λ1,g is given by

λ_1,g= inf

u∈C^∞(M), u>0

R

M

_4(n−1)

n−2 |∇u|²+Sgu²dvg

R

Mu²dvg

. Then they defined the k^th Yamabe invariant withk∈N^?, by

µk= inf

g∈[g]λk,gV ol(M,˜g)^2/n where

[g] ={u^N−2g, u∈C^∞(M), u >0}.

With these notations µ₁is the Yamabe invariant. They studied the second Yamabe invariant µ2, they found that contrary to the Yamabe invariant, µ₂ cannot be attained by a regular metric. In other words, there does not exist g∈[g] , such that

µ₂ =λ_2,gV ol(M,˜g)^2/n.

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−2g, which is not necessarily positive and smooth, but only u ∈L^N(M),u ≥0, u6= 0 and where N = 2n/(n−2). The definitions ofλ_2,g and ofV ol(M, g)^2/ncan be extended to generalized metrics. The key points to solve this problem is the following theorems ([1]).

Theorem 1.3. Let (M, g) be a compact Riemannian manifold of dimen- sion n ≥3, then µ₂ is attained by a generalized metric in the following cases.

µ >0, µ₂ <^h(µ^n/2+ (µ(S_n))^n/2i2/n

and

µ= 0, µ2 < µ(Sn)

Theorem 1.4. The assumptions of the last theorem are satisfied 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.

Theorem 1.5. Let (M, g) be a compact Riemannian manifold of dimen- sion n≥3, assume that µ₂ is attained by a generalized metricg˜=u^N−2g then there exists a nodal solution w∈C^2,α(M) of equation

L_g(w) =µ₂|u|^N−2w such that

|w|=u where α≤N −2.

Recently F.Madani studied (see [6]) the Yamabe problem with singu- larities when the metricg admits a finite number of points with singular- ities and is smooth outside these points. Let (M, g) be a compact Rie- mannian manifold of dimension n ≥ 3, assume that g is a metric in the Sobolev space H₂^p(M, T^∗M ⊗ T^∗M) with p > ⁿ₂ 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 p > ⁿ₂, H₂^p(M, T^∗M⊗T^∗M)⊂C^1−[n/p],β(M, T^∗M⊗T^∗M), where [n/p] denotes the entire part of n/p. Hence the metric satisfying assumption (H) is of class C¹⁻

_n

p

,β

with β ∈ (0,1) provided that p > n. The Christoffels symbols belong to H₁^p(M) ( to C^o(M) in case p > n), the Riemann- ian 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−2g conformal to g such that u ∈ H₂^p(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∈H₂^p(M),u >0 be a function and g = u^N−2g a particular conformal metric where N = 2n/(n−2).

Then, multiplying u by a constant, the following equation is satisfied L_gu= n−2

4(n−1)S_g˜|u|^N−2u

where

L_g = ∆_g+ n−2 4(n−1)S_g

and the scalar curvature Sg is in L^p(M). Moreover Lg is weakly confor- mally invariant hence solving the singular Yamabe problem is equivalent to finding a positive solution u∈H₂^p(M) of

L_gu=k|u|^N−2u (2)

wherekis a constant. In order to obtain solutions of equation (2) we define the quantity

µ= inf

u∈H₂^p(M),u>0

Y(u) where

Y(u) = R

M

|∇u|²+_4(n−1)⁽ⁿ⁻²⁾ S_gu²dv_g (^R_M|u|^Ndv_g)^2/N .

µ is called the 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 ofY and solutions of equation (2).

In particular, if u ∈ H₂^p(M) is a positive function which minimizes Y, thenu is a solution of equation (2) andg=u^N−2gis 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 ([6]).

Theorem 1.6. Ifp > n/2and µ < K⁻²then equation 2 admits a positive solution u ∈ H₂^p(M) ⊂ C^1−[n/p],β(M) ; [n/p] is the integer part of n/p, β ∈(0,1) which minimizes Y, where K² = _n(n−1)⁴ ω^−2/nn with ω_n denotes the volume of S_n. If p > n , then u∈H₂^p(M)⊂C¹(M).

Theorem 1.7. Let (M, g) be a compact Riemannian manifold of dimen- sion n≥3. g is a metric which satisfies the assumption (H). If (M, g) is not conformal to the sphere Sn with the standard Riemannian structure then

µ < K⁻²

Theorem 1.8. Let(M, g)be an-dimensional compact Riemannian man- ifold. If u ≥ 0 is a non trivial weak solution in H₁²(M) of equation

∆u+hu = 0, with h ∈ L^p(M) and p > n/2, then u ∈ C^1−[n/p],β and u >0; [n/p]is the integer part of n/pand β ∈(0,1).

Denote by

L^N₊(M) =ⁿu∈L^N(M) :u≥0,u6= 0^o. For regularity argument we need the following results

Lemma 1.9. Letu∈L^N₊(M)andv ∈H₁²(M)a weak solution toL_g(v) = u^N−2v, then

v∈L^N+(M) for some ε >0.

The proof is the same as in ([6]) with some modifications. As a conse- quence of Lemma 7, v∈L^s(M),∀s≥1.

Proposition 1.10. If g ∈H₂^p(M, T^∗M⊗T^∗M) is a Riemannian metric on M with p > n/2. If g = u^N−2g is a conformal metric to g such that u∈H₂^p(M), u >0thenL_g is weakly conformally invariant, which means that ∀v ∈H₁²(M), |u|^N−1Lg(v) =Lg(uv)weakly. Moreover if µ > 0, then L_g is coercive and invertible.

In this paper, let (M, g) be a compact Riemannian manifold of di- mension n ≥ 3. We suppose that g is a metric in the Sobolev space H₂^p(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 BP(δ) and we call these assump- tions the condition (H).

In the smooth case the operator Lg is an elliptic operator on M self- adjoint, and has a discrete spectrum Spec(L_g) = {λ_1,g , λ_2,g, . . .}, where the eigenvaluesλ_1,g< λ_2,g. . .appear with their multiplicities. These prop- erties remain valid also in the case where Sg ∈ L^p(M). The variational characterization of λ_1,g is given by

λ1,g= inf

u∈H₁²,u>0

R

M

|∇u|²+_4(n−1)⁽ⁿ⁻²⁾S_gu²dv_g R

Mu²dvg

Let [g] ={u^N−2g : u ∈H₂^p and u > 0}, Let k∈N^∗, we define the k^th Yamabe invariant with singularities µk as

µ_k= inf

g∈[g]λ_k,gV ol(M,˜g)^2/n

with these notations, µ₁ is the first Yamabe invariant with singularities.

In this work we are concerned withµ₂. In order to find minimizers toµ₂ we extend the conformal class to a larger one consisting of metrics of the form g =u^N−2g where u is no longer necessarily inH₂^p(M) and positive butu ∈L^N₊(M) =ⁿL^N(M), u≥0, u6= 0^o such metrics will be called for brevity generalized metrics. First we are going to show that if the singular Yamabe invariant µ ≥0 then µ₁ it is exactlyµ next we consider µ₂ and show thatµ2 is attained by a conformal generalized metric.

Our main results state as follows:

Theorem 1.11. Let (M, g) be a compact Riemannian manifold of di- mension n ≥ 3. We suppose that g is a metric in the Sobolev space H₂^p(M, T^∗M ⊗T^∗M) with p > n/2. If there exist a pointP ∈ M and δ >0 such that g is smooth in the ball B_P(δ), then

µ1 =µ.

Theorem 1.12. Let(M, g)be a compact Riemannian manifold of dimen- sion n≥3, we suppose that g is a metric in the Sobolev space

H₂^p(M, T^∗M⊗T^∗M) withp > n/2.

There exist a point P ∈ M and δ > 0 such that g is smooth in the ball B_P(δ). Assume that µ₂ is attained by a metric g = u^N−2g where u ∈ L^N₊(M), then there exist a nodal solution w∈C^1−[n/p],β(M), β ∈(0,1), of equation

Lgw=µ2u^N−2w.

Moreover there exist real numbers a, b >0 such that u=aw₊+bw−

with w₊= sup(w,0)and w−= sup(−w,0) .

Theorem 1.13. Let(M, g)be a compact Riemannian manifold of dimen- sion n≥3, suppose thatg is a metric in the Sobolev space H₂^p(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 µ₂ is attained by a generalized metric in the following cases:

If (M, g) is not locally conformally flat and,n≥11 and µ >0 If (M, g) is not locally conformally flat and,µ= 0 andn≥9.

2. Generalized metrics and the Euler-Lagrange equation

Let

L^N₊(M) =ⁿu∈L^N₊(M):u≥0, u6= 0^o where N = _n−2²ⁿ .

As in ([1])

Definition 2.1. For all u ∈ L^N₊(M), we define Gr_k^u(H₁²(M)) to be the set of all k−dimensional subspaces of H₁²(M) with span(v₁, v₂, ..., v_k) ∈ Gr^u_k(H₁²(M)) if and only if v1, v2, ..., v_k are linearly independent onM − u⁻¹(0).

Let (M, g) be a compact Riemannian manifold of dimensionn≥3. For a generalized metricg conformal to g, we define

λ_k,g= inf

V∈Gr^u_k(H₁²(M))

sup

v∈V

R

MvLg(v)dvg

R

M|u|^N−2v²dv_g. We quote the following regularity theorem

Theorem 2.2. [7] On a n -dimensional compact Riemannian manifold (M, g), ifu≥0 is a non trivial weak solution in H₁²(M) of the equation

∆u+hu=cu^N−1 with h∈L^p(M) and p > n/2, then

u∈H₂^p(M)⊂C^1−[n/p],β(M)

and u >0, where [n/p]denotes the integer part of n/p andβ ∈(0,1).

Proposition 2.3. Let (vm) be a sequence in H₁²(M) such that vm → v strongly in L²(M), then for all any u∈L^N₊(M)

Z

M

u^N−2(v²−v_m²)dv_g →0.

Proof. The proof is the same as in ([3]).

Proposition 2.4. If µ > 0, then for all u ∈ L^N₊(M), there exist two functions v, w in H₁²(M) with v ≥ 0 satisfying in the weak sense the equations

L_gv=λ_1,gu^N−2v (7) and

Lgw=λ2,gu^N−2w (8) Moreover we can choose v and w such that

Z

M

u^N−2w²dvg = Z

M

u^N−2v²dvg = 1 and Z

M

u^N−2wvdvg = 0. (9)

Proof. Let (v_m)_m be a minimizing sequence for λ_1,˜g i.e. a sequence v_m ∈ H₁²(M) such that

limm

R

Mv_mL_g(v_m)dv_g R

M|u|^N−2v_m²dvg

=λ_1,˜g

It is well know that (|v_m|)_m is also minimizing sequence. Hence we can assume thatv_m ≥0. We normalize (v_m)_m by

Z

M

|u|^N−2v²_mdv_g = 1.

Now by the fact that L_g is coercive ckv_mk_H2

1 ≤

Z

M

vmLg(vm)dvg ≤λ1,˜g+ 1.

(v_m)_mis bounded inH₁²(M) and after restriction to a subsequence we may assume that there exist v ∈H₁²(M), v ≥ 0 such that v_m → v weakly in H₁²(M), strongly inL²(M) and almost everywhere inM, then v satisfies in the sense of distributions

L_gv=λ_1,gu^N−2v.

If u∈H₂^p(M) ⊂C¹⁻

_n

p

,β

(M) then Z

M

u^N−2(v²−v²_m)dv_g→0

and Z

M

u^N−2v²dvg= 1.

Thenvis not trivial and is a nonnegative minimizer ofλ_1,g, by Lemma7 h=Sg−λ1,gu^N−2 ∈L^p(M)

and by Theorem 1.8

v∈C¹⁻

_n

p

,β

(M) and

v >0.

If u∈L^N₊(M), by Proposition 2.3 , we get Z

M

u^N−2(v²−v²_m)dv_g→0 so

Z

M

u^N−2v²dv_g= 1.

v is a non negative minimizer inH₁²(M) of λ_1,g such that Z

M

u^N−2v²dv_g= 1.

Now consider the set

E={w∈H₁²(M): such that u^N−22 w6= 0and Z

M

u^N−2wvdv_g = 0}.

Obviously E is not empty and define λ⁰_2,g = inf

w∈E

R

MwL_g(w)dv_g R

M|u|^N−2w²dvg

.

Let (wm) be a minimizing sequence for λ⁰_2,g i.e. a sequencewm ∈E such that

limm

R

Mw_mL_g(w_m)dv_g R

M|u|^N−2w²_mdvg

=λ⁰_2,g.

The same arguments lead to a minimizerwtoλ⁰_2,gwith^R_Mu^N−2w²= 1.

Now writing Z

M

u^N−2wvdv_g = Z

M

u^N−2v(w−w_m)dv_g+ Z

M

u^N−2w_mvdv_g and taking account of ^R_Mu^N−2w_mvdv_g = 0 and the fact thatw_m→w weakly in L^N(M) and sinceu^N−2v∈L^N−1N (M), we infer that

Z

M

u^N−2wvdvg = 0.

Hence (8) and (9) are satisfied with λ⁰_2,g instead ofλ_2,g.

Proposition 2.5. We have

λ⁰_2,g=λ2,g. .

Proof. The proof is the same as in ([3]) so we omit it.

Remark 2.6. If p > n then u ∈H₂^p(M) ⊂C¹(M), by Theorem 9, v and w∈C¹(M) withv >0.

Remark 2.7. If p > n then u ∈ H₂^p(M) ⊂ C¹(M) and λ_2,g = λ_1,g, we see that |w|is a minimizer for the functional associated to λ1,g, then |w|

satisfies the same equation as v and by Theorem 9 we get |w| > 0, this contradicts relation (9), necessarily

λ_2,g> λ_1,g.

3. Variational characterization and existence of µ₁ In this section we need the following Sobolev’s inequality (see [5])

Theorem 3.1. Let(M, g) be a compact n -dimensional Riemannian man- ifold. For any ε >0, there existsA(ε)>0 such that∀u∈H₁²(M),

kuk²_N ≤(K²+ε)k∇uk²₂+A(ε)kuk²₂ where N = 2n/(n−4) and K² = 4/(n(n−2)) ω

−2

nn . ωn is the volume of the round sphere S_n.

Let [g] ={u^N−2g:u∈H₂^p(M) andu >0}, we define the first singular Yamabe invariant µ1 as

µ1 = inf

g∈[g]λ1,gV ol(M,g)˜ ^2/n then we get

µ₁ = inf

u∈H₂^p,V∈Gr^u₁(H₁²)

sup

v∈V

R

MvL_g(v)dv_g R

M|u|^N−2v²dv_g( Z

M

u^Ndv_g)ⁿ². Lemma 3.2. We have

µ₁ ≤µ < K⁻². .

Proof. Ifp≥2n/(n+ 2), the embedding H₂^p(M)⊂H₁² (M) is true, so µ₁ = inf

u∈H₂^p,V∈Gr₁^u(H²₁(M))

sup

v∈V

R

MvL_g(v)dv_g R

M|u|^N−2v²dv_g( Z

M

u^Ndv_g)ⁿ²

≤ inf

u∈H₂^p,V∈Gr₁^u(H₂^p(M))

sup

v∈V

R

MvLg(v)dvg

R

M|u|^N−2v²dv_g( Z

M

u^Ndvg)²ⁿ. in particular for p > ⁿ₂ and u=v we get

µ1≤ inf

v∈H₂^P,V∈Gr^u₁(H₂^P(M))

sup

v∈V

R

MvLg(v)dvg

R

M|v|^N−2v²dvg

( Z

M

v^Ndvg)ⁿ² =µ i.e

µ₁ ≤µ < K⁻².

Theorem 3.3. If µ > 0, there exits conform metric g = u^N−2g which minimizes µ₁.

Proof. The proof will take several steps.

Step 1: We study a sequence of metrics gm = u^N−2_m g with um ∈ H₂^p(M), u_m > 0 which minimize µ₁ i.e. a sequence of metrics such that

µ1= lim

m λ1,m(V ol(M, gm)^2/n.

Without loss of generality, we may assume that V ol(M, g_m) = 1

i.e. Z

M

u^N_mdvg= 1.

In particular, the sequence of functionsu_m is bounded inL^N(M) and there exists u ∈L^N(M),u ≥0 such that um →u weakly in L^N(M). We are going to prove that the generalized metricu^N−2g minimizes µ₁. Proposition 2.4 implies the existence of a sequence (v_m) in H₁²(M),v_m >0 such that

L_g(v_m) =λ_1,mu^N−2_m v_m

and Z

M

u^N−2_m v_m²dv_g = 1.

now since µ >0, by Proposition 1.10, Lg is coercive and we infer that

ckv_mk_H2

1 ≤

Z

M

vmLg(vm)dvg =λ1,m≤µ1+ 1.

The sequence (vm)_m is bounded in H₁²(M), we can find v ∈ H₁²(M), v ≥ 0 such that vm → v weakly in H₁²(M). Together with the weak convergence of (u_m)_m, we obtain in the sense of distributions

L_g(v) =µ₁u^N−2v.

Step 2: Now we are going to show thatv_m→vstrongly inH₁²(M).

We put

z_m=v_m−v

then z_m → 0 weakly in H₁²(M) and strongly inL^q(M) with q <

N, and writing Z

M

|∇v_m|²dvg = Z

M

|∇z_m|²dvg+ Z

M

|∇v|²dvg+ 2 Z

M

∇z_m∇vdv_g we see that

Z

M

|∇v_m|²dvg = Z

M

|∇z_m|²dvg+ Z

M

|∇v|²dvg+o(1).

Now because of 2p/(p−1)< N , we have Z

M

n−2

4(n−1)Sg(vm−v)²dvg ≤ n−2

4(n−1)kS_gk_pkv_m−vk²2p p−1

→0 so

Z

M

n−2

4(n−1)S_gv_m²dv_g = Z

M

n−2

4(n−1)S_gv²dv_g+o(1) and

Z

M

|∇v_m|²dvg+ Z

M

n−2

4(n−1)Sg(vm)²dvg

= Z

M

|∇z_m|²dv_g+ Z

M

|∇v|²dv_g+ Z

M

n−2

4(n−1)S_g(v)²dv_g+o(1).

Then Z

M

v_mL_gv_mdv_g

= Z

M

|∇z_m|²dv_g+ Z

M

|∇v|²dv_g+ Z

M

n−2

4(n−1)S_gv²dv_g+o(1) And by the definition ofµand Lemma 3.2 we get

Z

M

|∇v|²dv_g+ Z

M

n−2

4(n−1)S_g(v)²dv_g ≥µ(

Z

M

v^Ndv_g)^N2 ≥µ₁( Z

M

v^Ndv_g)^N2 then

Z

M

vmLg(vm)dvg ≥ Z

M

|∇z_m|²dvg+µ1( Z

M

v^Ndvg)^N2 +o(1).

And since Z

M

vmLg(vm)dvg =λ1,m≤µ1+o(1) and

Z

M

|∇z_m|²dvg+µ1( Z

M

v^Ndvg)^N2 ≤µ1+o(1) i.e

µ1kvk²_N+k∇z_mk²₂ ≤µ1+o(1) (10) Now by Brézis-Lieb lemma ([4]) , we get

limm

Z

M

v_m^N+z_m^Ndv_g = Z

M

v^Ndv_g i.e.

limm kv_mk^N_N − kz_mk^N_N =kvk^N_N. Hence

kv_mk^N_N +o(1) =kz_mk^N_N+kvk^N_N.

By Hölder’s inequality and ^R_Mu^N−2_m v_m²dv_g= 1, we get kv_mk^N_N ≥1

i.e.

Z

M

(v^N +z^N_m )dvg= Z

M

v^N_mdvg+o(1)≥1 +o(1).

Then Z

M

v^Ndvg

_N² +

Z

M

z^N_mdvg

_N²

≥1 +o(1) i.e.

kz_mk²_N+kvk²_N ≥1 +o(1).

Now by Theorem 3.1 and the fact zm →0 strongly inL²(M), we get

kz_mk²_N ≤(K²+ε)k∇z_mk²₂+o(1)

1 +o(1)≤ kz_mk²_N +kvk²_N ≤ kvk²_N + (K²+ε)k∇z_mk²₂+o(1).

So we deduce

1 +o(1)≤ kvk²_N + (K²+ε)k∇z_mk²₂+o(1) and from inequality (10), we get

k∇z_mk²₂+µ1kvk²_N ≤µ1((K²+ε)k∇z_mk²₂+kvk²_N) +o(1).

So if µ1K²<1, we get

(1−µ1(K²+ε))k∇z_mk²₂)≤o(1) i.e.vm →v strongly inH₁²(M).

Step 3: We have

limm

Z

M

u^N−2_m v_m² −u^N−2v²+u^N−2_m v²−u^N−2_m v²dvg

= lim

m

Z

M

u^N−2_m (v²_m−v²) + (u^N_m⁻²−u^N−2)v²dv_g.

Now sinceu_m →ua.e. so doesu^N−2_m →u^N−2and^R_Mu^N−2_m dv_g ≤c, henceu^N−2_m is bounded inL^N/(N−2)(M) and up to a subsequence u^N_m⁻² → u ^N−2 weakly in L^N/(N−2)(M). Since v² ∈L^N2 (M), we have

limm

Z

M

(u^N_m⁻²−u^N−2)v²dvg = 0 and by Hölder’s inequality

limm

Z

M

u^N_m⁻²(v_m−v)²dv_g

≤( Z

M

u^N_mdvg)^(N−2)/N( Z

M

|v_m−v|^Ndvg)^N2 ≤0.

By the strong convergence of (vm) in L^N(M), we get Z

M

u^N−2v²dv_g= 1, thenv and u are non trivial functions.

Step 4: Let u = av ∈ L^N₊(M) with a > 0 a constant such that R

Mu^Ndvg = 1 withv a solution of L_g(v) =µ₁u^N−2v with the constraint

Z

M

u^N−2v²dv_g= 1.

We claim thatu=v; indeed,

µ1 ≤ R

MvLg(v)dvg

R

Mu^N−2v²dv_g

≤ R

MvLg(v)dvg

R

M(av)^N−2v²dv_g = a²µ1R

Mu^N−2v²dvg

R

Mu^N−2(av)²dv_g and Hölder’s inequality lead

≤µ₁ Z

M

(u)^N−2(av)²dv_g

≤µ₁( Z

M

(u)^N−2N−2N )^N−2N ( Z

M

(av)^2N2 dv_g)^N2 ≤µ₁. And since the equality in Hölder’s inequality holds if

u=u=av thena= 1 and

u=v.

Then v satisfiesLgv =µ1v^N−1 , by Theorem 2.2 we get v =u ∈ H₂^p(M)⊂C¹⁻

_n

p

,β

(M) withβ ∈(0,1) and v=u >0 , Resuming, we have

Lg(v) =µ1v^N−1, Z

M

v^Ndvg = 1 andv =u∈H₂^p(M)⊂C¹⁻

_n

p

,β

(M) so the metric ˜g=u^N−2g minimizesµ₁.

4. Yamabe conformal invariant with singularities Theorem 4.1. If µ≥0, then µ₁ =µ

Proof. Step 1: If µ > 0. Let v such that Lg(v) = µ1v^N−1 and R

Mv^Ndv_g = 1 then µ₁ =

Z

M

vL_g(v)dv_g≥ckvk_H2 1

and v in non trivial function then µ1 >0. On the other hand

µ= inf R

MvL_g(v)dv_g (^R_Mv^Ndv_g)^N2

≤ Z

M

vL_g(v)dv_g=µ₁ and by Lemma 3.2 , we get

µ1=µ

Step 2: If µ= 0, Lemma 3.2 implies thatµ₁≤0 , hence µ₁ = 0.

5. Variational characterization of µ₂

Let [g] ={u^N−2g,u∈H₂^p(M) and u >0}, we define the second Yamabe invariant µ₂ as

µ₂ = inf

g∈[g]λ_2,gV ol(M, g)^2/n or more explicitly

µ₂= inf

u∈H₂^P,V∈Gr^u₂(H₁²(M))

sup

v∈V

R

MvL_g(v)dv_g R

M|u|^N−2v²dv_g( Z

M

u^Ndv_g)²ⁿ

Theorem 5.1 ([1]). On a compact Riemannian manifold (M, g) of di- mension n≥3, we have for allv∈ H₁²(M)and for all u∈L^N₊(M)

2²ⁿ Z

M

|u|^N−2v²dv_g ≤(K² Z

M

|∇v|²dv_g+ Z

M

B₀v²dv_g)(

Z

M

u^Ndv_g)ⁿ² Or

2ⁿ² Z

M

|u|^N−2v²dv_g≤µ₁(S_n)(

Z

M

C_n|∇v|²+B₀v²dv_g)(

Z

M

u^Ndv_g)ⁿ²