Uniqueness and multiplicity for perturbations of the Yamabe problem on S^n

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Uniqueness and Multiplicity for Perturbations of the Yamabe problem on S n Pierpaolo Esposito - Universit`a di Roma Tre

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Introduction

In this paper we study the equation n+2

− △h u + λ(x)u = u n−2 u ∈ H12 (S n )

(1)

u>0

where (S n , h) is the n-dimensional sphere equipped with the standard metric. , the solutions are explicitely known and they are obtained by When λ(x) ≡ n(n−2) 4 the action of the conformal group on the constant solution u¯ := [

n(n − 2) n−2 ] 4 4

, as a consequence of a remarkable result by BidautIn contrast, if λ(x) ≡ λ < n(n−2) 4 Veron and Veron, see [4], (1) admits just the constant solution. The main purpose of this paper is to show that uniqueness fails for | ǫ | small if λ(x) = n(n−2) + ǫf (x) and f changes sign. 4 More precisely, we will prove Theorem 1.1 Let λǫ (x) = n(n−2) +ǫf (x)+g(ǫ, x), with k g(ǫ, x) k∞ = o(ǫ) as ǫ → 0, 4 be in C 0 (S n ). Then (1) is solvable for | ǫ | small. Furthermore, (1) has at least two R solutions if f changes sign and n ≥ 4 or if n = 3 and S n f = 0 with f 6= 0. A related problem in Rn has been considered in [1]. We wish to thank Prof. Emmanuel Hebey and his whole research group for their kind hospitality and for the helpful suggestions received in the period of study spent in the Cergy-Pontoise University. After this paper was finished, we learned from Prof. Ambrosetti that in a recent paper of Cingolani similar results are obtained.

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2

A finite dimensional reduction

Weak solutions of (1) are critical points of the energy functional Eǫ (u) := E0 (u) + G(ǫ, u) , u ∈ H12 (S n ) where 2n n(n − 2) Z n−2Z 1Z 2 2 | ∇u | dv(h) + u dv(h) − (u+ ) n−2 dv(h) E0 (u) := n n n 2 S 8 2n S S

1Z (ǫf + g(ǫ, x))u2 dv(h) G(ǫ, u) := 2 Sn The peculiar property of E0 is its conformal invariance. Let us consider, in particular, the following conformal transformations ϕσ,t (y) = πσ−1 (tπσ (y))

σ ∈ Sn , t ≥ 1

where πσ denotes the stereographic projection from σ as the north pole. They act on H12 (S n ) through the following isomorphisms Tσ,t u(y) := (u ◦ ϕσ,t )(y) | det dϕσ,t (y) |

n−2 2n

and E0 (Tσ,t u) = E0 (u)

∀(σ, t) ∈ S n × [1, +∞), ∀u ∈ H12 (S n )

¿From the conformal invariance, it easily follows that ∇E0 (Tσ,t u) = 0 if ∇E0 (u) = 0. n−2 ] 4 denotes the constant solution, In particular, if u¯ ≡ [ n(n−2) 4 Z := {Tσ,t u¯ = u¯ | det dϕσ,t (y) |

n−2 n

}

(2)

is a critical manifold for E0 . Actually, it can be shown that these are all the critical points of E0 . A proof of this claim in book form can be found in [7]. References where such constructions are used are [6] and [5]. See also [2]. The critical points of Eǫ can be found as critical points of Eǫ constrained to some manifold Zǫ close to Z, following the same perturbation technique used in [1]. Let a family of C 2 functionals {Eǫ } be defined on a Hilbert space E of the form Eǫ (u) = E0 (u) + G(ǫ, u)

(3)

We assume that the unperturbed functional E0 satisfies the following assumptions: (A1) E0 possesses a finite dimensional manifold Z of critical points at a fixed energy level b, that will be called critical manifold 2

(A2) D2 E0 (z) is a Fredholm map with index 0 ∀z ∈ Z (A3) Tz Z = Ker(D2 E0 (z)) ∀z ∈ Z (Tz Z denotes the tangent space to Z at z) (B1) there exist α > 0 and a continuous function Γ : Z → R such that uniformly on compact subsets of Z G(ǫ, z) Γ(z) = lim ǫ→0 ǫα α G ′ (ǫ, z) = o(ǫ 2 ) Under the preceding assumptions, one can use the Implicit Function Theorem to show for | ǫ | small the existence of w = w(ǫ, z) ∈ (Tz Z)⊥ such that Eǫ′ (z + w) ∈ Tz Z α

kw(ǫ, z)k = o(ǫ 2 ) uniformly on compact subsets of Z. Letting Zǫ = {z + w(ǫ, z)}, we obtain a manifold locally diffeomorphic to Z such that for | ǫ | small any critical points of Eǫ restricted to Zǫ is a stationary point of Eǫ . ¿From the development of Eǫ on Zǫ Eǫ |Zǫ (u) = Eǫ (z + w(ǫ, z)) = E0 (z) + (E0′ (z) | w(ǫ, z)) + O(kw(ǫ, z)k2 )+ +G(ǫ, z) + (G ′ (ǫ, z) | w(ǫ, z)) + O(kw(ǫ, z)k2 ) = b + ǫα Γ(z) + o(ǫα ) uniformly on compact subsets of Z, one can derive a general existence result, wich we will use in a particular case. Theorem 2.1 Let Eǫ ∈ C 2 (E, R) be of the form (3), where E0 and G(ǫ, u) satisfy A1, 2, 3 and B1 and suppose that there exists a critical point z¯ ∈ Z of Γ such that one of the following conditions holds (i) z¯ is nondegenerate (ii) z¯ is a strict local minimum or maximum (iii) z¯ is isolated and the local topological degree of Γ′ at z¯, degloc (Γ′ , 0), is different from zero. Then for | ǫ | small enough, the functional Eǫ has a critical point uǫ such that uǫ → z¯ as ǫ → 0. It is easy to check that all the assumptions are satisfied in our case; in particular, Z, given as in (2), is a smooth manifold diffeomorphic to B n+1 , the open unit ball in Rn+1 , through the map ξ = ρσ ∈ B n+1 → Tσ,(1−ρ)−1 . Furthermore, the nondegeneracy assumption A3 is satisfied, because the kernel of the linearized operator at u¯ is the eigenspace of the Laplace-Beltrami operator corresponding to the first eigenvalue λ1 = n, wich has dimension n + 1, see [3], while by conformal invariance kerD2 E0 (Tσ,t u¯) = Tσ,t kerD2 E0 (¯ u) 3

Finally, we can take α = 1 and get n−2 1 2Z f (y) | det dϕσ,(1−ρ)−1 | n dv(y) Γ(ρσ) = cn 2 Sn

] where cn = [ n(n−2) 4

3

n−2 4

(4)

.

Proof of Theorem 1.1

We begin with an expansion around boundary points of the function Γ. Lemma 3.1 If n ≥ 5, for every σ ∈ S n it results Γ(ρσ) = c2n 2n−1 (1 − ρ)2 [f (−σ)

Z

Rn

(1+ | x |2 )−(n−2) dx + o(1)]

as ρ → 1

If n = 4, for every σ ∈ S n it results Γ(ρσ) = −8ω3 c24 (1 − ρ)2 ln(1 − ρ)[f (−σ) + o(1)] as ρ → 1 If n = 3, for every σ ∈ S n , it results Γ(ρσ) =

c23 (1

Z

− ρ)[

S3

f (y) dv(y) + o(1)] as ρ → 1 1 + cos d(σ, y)

where cn = [

n(n − 2) n−2 ] 4 4

Proof. We have (see [7]) that | det dϕσ,t |

n−2 n

(y) = (t

1+ | πσ (y) |2 n−2 ) 1 + t2 | πσ (y) |2

so that, integrating in stereographic coordinates Z

Sn

f (y) | det dϕσ,t |

= 2n t−2

Z

Rn

n−2 n

n

(y)dv(y) = 2

(f ◦ πσ−1 )( xt ) (1+ | x |2 )(n−2) (1+ |

x 2 2 |) t

Z

Rn

f (πσ−1 (x))tn−2 dx = (1 + t2 | x |2 )(n−2) (1+ | x |2 )2

= 2n t−2 [f (−σ)

Z

Rn

(1+ | x |2 )−(n−2) dx+

+o(1)] as t = (1 − ρ)−1 → +∞ by dominated convergence, if n ≥ 5. If n = 4, it is enough to split the integral into two pieces: −2

I1 := 16t

Z

|x|≥1

Z (f ◦ πσ−1 )(x) (f ◦ πσ−1 )(x) −2 dx = 16t [ dx+o(1)] (t−2 + | x |2 )2 (1+ | x |2 )2 |x|≥1 | x |4 (1+ | x |2 )2

4

and I2 := 16t−2

Z t (f ◦ πσ−1 )( xt ) r3 −2 dx = 16t (f (−σ)+o(1))ω dr = 3 (1+ | x |2 )2 (1+ | xt |2 )2 0 (1 + r 2 )2

Z

|x|≤t

ln t [f (−σ) + o(1)] t2 Finally, if n = 3, exactly as for I1 , we now get = 16ω3

8

Z

R3

Z (f ◦ πσ−1 )(x)t (f ◦ πσ−1 )(x) −1 dx = 8t [ dx + o(1)] (1 + t2 | x |2 )(1+ | x |2 )2 R3 (1+ | x |2 )2 | x |2

−1 (z), after the change of variable x = Since πσ−1 ( |z|z 2 ) = π−σ

Z

R3

(f ◦ πσ−1 )(

z , |z|2

we obtain the integral

Z z 2 −1 2 −2 −3 (f ◦ π−σ )(z)( )(1+ | z | ) dz = 2 )3 (1+ | z |2 )dz = 2 |z| 1+ | z |2 R3

= 2−3

Z

S3

f (y)(1+ | π−σ (y) |2 )dv(y) = 2−2

Z

S3

f (y) dv(y) 1 + cos d(σ, y)

because 1+ | π−σ (y) |2 = 1 +

2 1 − yn+1 2 = and yn+1 = cos d(σ, y) 2 (1 + yn+1 ) 1 + yn+1

Proof. (Theorem 1.1) If n ≥ 4, we see from the Lemma that Γ vanishes at ∂B n+1 and changes sign with f around boundary points. Hence Γ has a positive maximum and a negative minimum and so it does Eǫ |Zǫ . Existence of at least one solution, wich is not necessarily a minimum, follows similarly, without sign assumption on f . Finally, if n = 3, from Z

S3

Z

dσ(

S3

Z Z f (y) dσ f (y)( dy) = )dy 3 3 1 + cos d(σ, y) S 1 + cos d(σ, y) S

and the independence on y of R

f (y) dy 1+cos d(σ,y)

R

dσ S 3 1+cos d(σ,y)

> 0 because of the symmetry, we see R

that σ → S 3 has to change sign if S 3 f = 0. So, again, we get a positive maximum and a negative minimum. Hence, from Theorem 2.1, for ǫ small, we find solutions uǫ of the equation n+2

−△h u + λǫ (x)u = (u+ ) n−2

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If ab absurdo uǫ is not nonnegative, then there exists y0 an absolute minimum point of uǫ on S n with uǫ (y0 ) < 0. So n+2 λǫ (y0 )uǫ (y0 ) = (uǫ )+ (y0 ) n−2 + △h uǫ (y0 ) ≥ 0 Then, for ǫ small, uǫ (y0 ) ≥ 0, a contradiction. The function uǫ must be nonnegative and from the maximum principle either vanishes or is positive. But uǫ cannot vanish identically for ǫ small because its energy level is near b > 0.

References n+2

[1]

Ambrosetti,A. and Azorero,J.G. and Peral,I., Perturbation of △u+u n−2 = 0, the scalar curvature problem in Rn and related topics, J. Funct. Analysis (165) , 1999 , pag. 117-149.

[2]

Aubin,T., Nonlinear analysis on manifolds - Monge-Amp`ere equations, Grundlehern der Mathematischen Wissenschaften (252) , 1982.

[3]

Berger,M. and Gauduchon,P. and Mazet,E., Le spectre d’une vari´et´e Riemannienne, Lecture Note in Mathematics (194) , 1971 , Springer-Verlag , New-York/Berlin

[4]

Bidaut-Veron,M. and Veron,L., Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Inventiones Mathematicae (106) , 1991 , pag. 489-539.

[5]

Chang,S.Y.A. and Yang,P.C., A perturbation result in prescribing scalar curvature on S n , Duke Math. Journal (64/1) , 1991, pag. 27-69.

[6]

Hebey,E., Changements de m´etriques conformes sur la sph`ere. Le probl`eme de Nirenberg, Bull. Sc. Math. (114) , 1990 , pag. 215-242.

[7]

Hebey,E., Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lecture Notes in Mathematics, CIMS, 1999.

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