A Morse lemma at infinity for Yamabe type problems on domains

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A MORSE LEMMA AT INFINITY FOR YAMABE TYPE PROBLEMS ON DOMAINS

Mohamed BEN AYED^∗, Hichem CHTIOUI, Mokhless HAMMAMI

Département de mathématiques, faculté des sciences de Sfax, route Sokra 3018, km 3,5, Sfax, Tunisia Received 5 November 2001, revised 19 March 2002

ABSTRACT. – In this paper we consider the following nonlinear elliptic problem (P):−u= u^p, u >0 in , u=0 on ∂, where is a bounded and smooth domain in Rⁿ, n4, p+1=2n/(n−2)is the critical Sobolev exponent. We prove a version of Morse lemmas at infinity for this problem. As application of these lemmas we will give a characterization of the critical points at infinity of the functional corresponding to (P).

2003 Éditions scientifiques et médicales Elsevier SAS MSC: 35J65

Keywords: Morse lemma; Elliptic problems with critical Sobolev exponent

RÉSUMÉ. – Dans cet article nous considérons le problème elliptique non linéaire (P) :

−u=u^p,u >0 dans,u=0 sur∂, oùest un domaine borné et régulier deRⁿ,n4 et p+1=2n/(n−2)est l’exposent critique de Sobolev. Nous prouvons une version des lemmes de Morse à l’infini pour le problème (P) et nous appliquerons ces lemmes afin de donner une caractérisation des points critiques à l’infini de la fonctionnelle correspondante au problème (P).

2003 Éditions scientifiques et médicales Elsevier SAS

Mots Clés: Lemme de Morse; Problèmes elliptiques avec exposant critique de Sobolev

1. Introduction and the main results

We prove in this paper a version of a Morse Lemma at infinity for Yamabe-type problems on domains in Rⁿ, n4. These problems are related to the study of the equation

−u=uⁿⁿ⁺⁻²², u >0 in,

u=0 on∂, (P)

∗Corresponding author.

E-mail addresses: [email protected] (M. Ben Ayed), [email protected] (M. Hammami).

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where ⊂Rⁿ, n4, is a bounded and regular set. This problem has a variational structure. The related functional is

J (u)=

|u|ⁿ²ⁿ⁻² −ⁿ⁻_n²

defined on

=

u∈H₀¹()|

|∇u|²=1

.

The problem (P) is delicate from a variational viewpoint because the functional J does not satisfy the Palais–Smale condition (P.S. for short). This means that there exist sequences along whichJ is bounded, its gradient goes to zero and which do not converge. The P.S. condition fails forJ on

⁺= {u∈s.t.u0}.

Its failure has been analyzed throughout the works of [13,14,16]. The analysis carried out in [11] and [15] comes out here virtually without any change. These various studies have led to the characterization of the sequences failing the Palais–Smale condition. In order to describe this characterization, we need to introduce some notations.

Let, fora∈andλ >0 given

δ(a,λ)(x)=c0

λ 1+λ²|x−a|²

⁽ⁿ⁻²⁾₂

(1.1) c0is chosen so thatδ(a,λ)is the family of solutions of the Yamabe problem onRⁿ. LetP be the projection fromH¹()ontoH₀¹()i.e.u=Pf is the solution of

u=f in, u=0 on∂.

Let, forε >0, p∈N^∗andweither a solution of (P) or zero V (p, ε, w)=

u∈s.t.∃(a1, . . . , ap)∈^p, ∃(λ1, . . . , λp)∈(]ε⁻¹,+∞[)^p,

∃(α0, α1, . . . , αp)∈(]0,+∞[)^p⁺¹s.t.λid(ai, ∂) > ε⁻¹,

u−α0w− p

i=1

αiP δ(a_i,λ_i)

H₀¹

< ε, αi

αj

−1< ε, εij< ε (1.2) where, fori=j,

ε⁻_ij¹= λi

λj

+λj

λi

+λiλj|ai−aj|² ⁿ⁻₂²

. The failure of Palais–Smale condition can be described as follows.

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PROPOSITION 1.1. – Let(uk)∈⁺ be a sequence such that (∂J (uk))tends to zero and (J (uk)) is bounded. Then, after possibly having extracted a subsequence, there exists p∈N^∗, a sequence (εk), εk tends to zero, and w (either a solution of (P) or zero) such thatuk∈V (p, εk, w).

Thinking of these sequences which do not satisfy the (P.S.) condition as “critical points”, a natural idea is to try to find suitable parameters in order to complete a Morse Lemma at infinity. For manifolds without boundary, this program has been completed in [4] and [9]. We would here extend the proof of the existence of a Morse Lemma at infinity to the case of Dirichlet boundary conditions. We introduce the following parameterization of the set V (p, ε, w) (where w is a solution of (P)). We denote by Wu(w)the unstable manifold ofwfor a decreasing pseudogradient ofJ. If a functionu belongs toV (p, ε, w)then, forε >0 small enough, the minimization problem

min

u−α0(w+h)− p i=1

αiP δ(ai, λi) H₀¹

, αi>0, ai ∈, λi>0, h∈Tw

Wu(w) (1.3)

has an unique solution, up to permutation (see [4–6]).

Therefore, forε >0 sufficiently small, anyuinV (p, ε, w)can be uniquely written as u=α0(w+h)+

p i=1

αiP δ(ai, λi)+v wherevsatisfies the following conditions

(V0)

(v, P δ(ai, λi))_H1

0()=(v,_∂λ^∂

iP δ(ai, λi))_H1

0()=0,

(v,_∂a^∂

iP δ(ai, λi))=(v, w)_H¹

0()=(v, h)_H¹

0()=0, (1.4) and theαi’s satisfy

αi

αj

∈ [1−ε,1+ε] ∀i, ∀j.

We denote byGthe Green’s function of the Laplacian with Dirichlet boundary condition onand byH its regular part i.e.

G(x, y) = |x−y|²⁻ⁿ−H (x, y) for(x, y)∈², xH = 0 in², G=0 on∂(²).

For q ∈ N^∗, and x =(x1, . . . , xq) ∈ ^q, such that xi = xj ∀i = j, we denote by M(x)=(mij)1i,jqthe matrix defined by

mii=H (xi, xi), mij= −G(xi, xj) fori=j, (1.5) by ρ(x) its least eigenvalue and bye(x)the eigenvector corresponding to ρ(x)whose norm is 1 and whose components are strictly positive (see [8] and [9]).

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In this paper we assume that zero is a regular value of ρ for each q p (such assumption is true if, for example, is a thin or a large annuli (see [1,2])). Our main results are the following Morse lemmas at infinity.

THEOREM 1.2. – For ε > 0 sufficiently small given, there exists a change of variables, such that for anyu=^pi=1αiP δ(a_i,λ_i)+v belongs toV (p, ε),(ai, λi, v)→ (a_i, λ_i, V )whereV belongs to a neighborhood of zero in a fixed Hilbert space so that

J _p

i=1

αiP δ(ai,λi)+v

=J _p

i=1

αiP δ(a_i,λ_i)

+ |V|².

Furthermore, if each ai belongs to a neighborhood of xi such that ρ(X) > 0 and ρ(X)=0, where X =(x₁, . . . , xp), then we can find another change of variables (ai, λi)→(a_i, λ_i)such that, forηa fixed small constant

J _p

i=1

αiP δ(ai,λi)

=/(α, a, λ)

:= (Sn)^2/n^p_i₌₁α²_i (^p_i₌₁α^2n/(n_i ⁻²⁾)⁽ⁿ⁻^2)/n

1+

c₁ pSn

−η

ρ(a) p i=1

1 λ_iⁿ⁻²

where α = (α1, . . . , αp), a = (a₁, . . . , a_p), λ = (λ₁, . . . , λ_p) and c₁ is a positive constant and whereSn=_Rⁿδ^2n/(n⁻²⁾.

THEOREM 1.3. – For ε >0 sufficiently small given, there exists a change of vari- ables, such that for any u=^pi=1αiP δ(a_i,λ_i)+α0(w+h)+v belongs to V (p, ε, w), (ai, λi, h, v)→(a_i, λ_i, H, V ) where each of H and V belongs to a neighborhood of zero in a fixed Hilbert space so that

J _p

i=1

αiP δ(ai,λi)+α0(w+h)+v

=J _p

i=1

αiP δ(a_i,λ_i)+α0w

+ |V|²− |H|². The proof of these theorems is of course quite difficult and extremely technical. In principle, it relies on the construction of a suitable pseudogradient Z at infinity, as in [4,5,9], which in turn relies on very delicate expansion ofJ and∂J near infinity.

This construction is even more difficult when there is a boundary, because the distance to the boundary appears (in the denominator) in the expansion (see Section 2). We need to complete much more delicate and careful expansion.

However, these Theorems are useful. They should be useful for the study of the existence of multiple solutions to (P). At this point, we will illustrate their usefulness through the following three results.

PROPOSITION 1.4. – Assume that zero is a regular value ofρ. There are no critical points at infinity and no critical points in a neighborhood ofV (p, ε, w) forw=0 i.e.

we can define a pseudogradient which satisfies the (P.S.) condition on decreasing flow lines and has no asymptotes in this neighborhood.

PROPOSITION 1.5. – Assume that zero is a regular value ofρ. Then we have:

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(i) Forε small enough,J does not have any critical point inV (p, ε).

(ii) The only critical points at infinity ofJ correspond to^p_i₌₁δ(x_i,∞)wherep∈N^∗ and thexi’s satisfy

ρ(x1, . . . , xp) >0 and ρ(x1, . . . , xp)=0.

(iii) There isp0∈N^∗such that, above(p0Sn)^2/n,J does not have any critical point at infinity.

We also give a new proof, based on these theorems, of the formula for the difference of topology between the setsWpandWp−1where

Wp=u∈⁺s.t.J (u) <(p+1)Sn

2/n

due toV (p, ε). This formula was stated in [6] and a proof was provided in [8]. Namely, we prove

PROPOSITION 1.6. – We assume thatJ has no critical point in⁺ H_∗(Wp, Wp−1)=H_∗^p×σpp−1, ^p×∂p−1∪σpIp×p−1

whereIp= {x∈^ps.t.ρ(x)0},p−1= {(α1, . . . , αp)s.t.αi0, αi=1}andσp

is the permutation group.

The remainder of the present paper is organized as follow. Section 2 will be devoted to the expansion of J and its gradient. In Section 3 we will study the v-part of u. In Section 4 we will construct a suitable pseudogradient and then we will prove Theorem 1.2, Proposition 1.5 and Proposition 1.6. The proof of Theorem 1.3 and Proposition 1.4 are given in Section 5. Lastly, the proofs require some technical estimates which may be found in appendices.

2. The expansion of the functional and its gradient

In this section, we will give the expansion ofJ (^p_i₌₁αiP δi+v),(∂J (^p_i₌₁αiP δi),

1 λi

∂P δ_i

∂ai ),(∂J (^p_i₌₁αiP δi), λi∂P δ_i

∂λi ).

PROPOSITION 2.1. – Forp∈N^∗, ε >0, small and u=^pi=1αiP δi+v∈V (p, ε), we have the following expansion

J (u)= (^p_i₌₁α_i²)S_n^2/n p

i=1α^2n/(n_i ⁻²⁾⁽ⁿ^−2)/n

1+ c₁ Sn

p i=1

H (ai, ai) λⁿ_i⁻²

2α_i^2n/(n⁻²⁾

α_j^2n/(n⁻²⁾ − α_i² α_j²

− c₁ Sn

i=j

εij− H (ai, aj) (λiλj)⁽ⁿ⁻^2)/2

2α⁽ⁿ_i ⁺^2)/(n⁻²⁾αj

α_j^2n/(n⁻²⁾ − αiαj

α_k²

+ 1 α_j²Sn

Q(v, v)

−f, v+O p

k=1

log(λkdk) (λkdk)ⁿ +

i=j

ε

n−2n

ij logε_ij⁻¹+O|v|^inf(ⁿ²ⁿ⁻²^,3)

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wherec₁is a positive constant, Q(v, v)= |v|²_H1

0 −n+2

n−2

α_k²

α^2n/(n_k ⁻²⁾ αiP δi

_n−2⁴ v² and

(f, v)=2 α

n−22n

j Sn

−1 αiP δi

ⁿ_n⁺₋²₂ v.

Proof. –

J (u)= |αiP δi +v|²

αiP δi+v^2n/(n⁻²⁾⁽ⁿ⁻^2)/n = N D⁽ⁿ⁻^2)/n, N =

αiP δi+v²=α_i²|P δi|²+ |v|²+

i=j

αiαj

P δi, P δj

.

From Lemmas A.1 and A.3 (in Appendix A), we obtain N=α_i²

Sn−c₁H (ai, ai) λⁿ_i⁻²

+

i=j

αiαjc₁

+ |v|²+R2, (2.1)

whereR2satisfies

R2=O

k

i=j

ε

n n−2

ij logε_ij⁻¹. (2.2) We are left forD. Observe that, forq=2n/(n−2), we have

D= αiP δi

q

+q αiP δi

q−1

v +q(q−1)

2 αiP δi

q−2

v²+O|v|^inf(3,q). We have also

p i=1

αiP δi

q

= p i=1

α_i^q

P δ_i^q +q

i=j

α_i^q⁻¹αj

P δ_i^q⁻¹P δj

+O

i=j

sup(αiP δi, αjP δj)ⁿ⁻⁴² inf(αiP δi, αjP δj)²

. Using Lemmas A.2, A.3 and A.4, and the fact that 4/(n−2)2, we get

D=α_i^q

Sn−c₁qH (ai, ai) λⁿ_i⁻²

+qc₁

i=j

α_i^q⁻¹αj

+q αiP δi

q−1

v+q(q−1)

2 αiP δi

q−2

v²

+R2+O|v|^inf(3,q). (2.3)

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Combining (2.1) and (2.3), the proof of Proposition 2.1 follows. ✷

PROPOSITION 2.2. – Let n 4. For u = ^pi=1αiP δi ∈ V (p, ε), we have the following expansion

∂J (u), λi

∂P δi

∂λi

=2J (u)c₁

−n−2 2 αi

1+o(1)

−

j=i

αj

λi

∂εij

∂λi +n−2 2

H (ai, aj) (λiλj)⁽ⁿ⁻^2)/2

1+o(1)+R2

, whereR2is defined in (2.2).

Proof. – We have

∂J (u)=2J (u)u+J (u)ⁿ⁻²ⁿ ⁻¹uⁿⁿ⁻²⁺². (2.4)

Thus

∂J (u), λi

∂P δi

∂λi

=2J (u)αj

P δj, λi

∂P δi

∂λi

−J (u)ⁿ⁻ⁿ² αjP δj

ⁿ⁺²_n₋₂ λi

∂P δi

∂λi

. (2.5)

Observe that, sincen4 αjP δj

ⁿ_n−2⁺²

=(αjP δj)ⁿⁿ⁺²⁻² +n+2 n−2

i=j

(αiP δi)ⁿ⁻⁴²αjP δj

+O

i=j

(αiP δi)ⁿ⁻²²(αjP δj)ⁿ⁻ⁿ²

+O

k=j;k,j=i

(αjP δj)ⁿ⁻⁴²(αkP δk)

. (2.6)

Combining (2.6), (2.5), Lemmas A.5–A.9 and the following facts: |λi∂P δi/∂λi| cδi, P δk δk and J (u)^n/(n⁻²⁾α_j^4/(n⁻²⁾ = 1 +o(1), ∀j = 1, . . . , p, Proposition 2.5 follows. ✷

PROPOSITION 2.3. – Letn4. Foru=^pi=1αiP δi belonging toV (p, ε), we have the following expansion

∂J (u), 1 λi

∂P δi

∂ai

=J (u)c₁ αi

λⁿ_i⁻¹

∂H (ai, ai)

∂ai

1+o(1)

+2

j=i

αj

1 λi

∂εij

∂ai

−

∂H (a_i,a_j)

∂ai

λi(λiλj)⁽ⁿ⁻^2)/2

×1−J (u)ⁿ⁻ⁿ²α

4 n−2

j +α

4 n−2

i

+R3, whereR3satisfies

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R3=O

k

k=j

ε

n n−2

kj logε_kj⁻¹+

j=i

λj|ai −aj|ε

n+1n−2

ij

. (2.7) As in the proof of Proposition 2.2, we get (2.5) but with λi∂P δi/∂λi changed by λ⁻_i ¹∂P δi/∂ai. Thus, using Lemmas A.10–A.14, the proposition follows.

PROPOSITION 2.4. – Letn4, foru=^pi=1αiP δi+α0(w+h)inV (p, ε, w), we

have

−∂J (u), hc|h|²+O λ²_i⁻ⁿ

.

Proof. – Letu=αiP δi +α0(w+h), using (2.4) and the fact that αiP δi+α0(w+h)

ⁿ_n⁺₋²₂

=α

n+2n−2

0

wⁿⁿ⁻²⁺² +n+2 n−2wⁿ⁻²⁴ h

+O|h|ⁿⁿ⁻²⁺² (2.8) +O

|h|^inf(ⁿⁿ⁺²⁻²^,2)+wⁿ⁻⁴² + |h|ⁿ⁻⁴²P δi+P δ

n+2n−2

i

we need to estimate

δ

n+2 n−2

i |h||h|L^∞

δ

n+2 n−2

i =Oλ⁻

n−2 2

i |h|L^∞

=o|h|²L^∞

+O 1

λⁿ_i⁻²

, (2.9)

δi|h||h|L^∞

B

δi=O

|h|L^∞λ

2−n 2

i

=o|h|²_L∞

+O 1

λⁿ_i⁻²

, (2.10) where⊂B=B(ai, R).

The functionhbelongs toTw(Wu(w))which has a dimension equal to the index ofw.

Thus

|h|L^∞=O|h|H₀¹

;

wⁿⁿ⁻²⁺²h=(w, h)=0. (2.11) Therefore

∂J (u), h=2J (u)

α0|h|²_H¹

0 −n+2

n−2J (u)ⁿ⁻²ⁿ α

n+2n−2

0

wⁿ⁻²⁴ h² +o|h|²_H¹

0

+O 1 λⁿ_i⁻²

. Sinceu=αiP δi+α0(w+h)∈V (p, ε, w), we have

J (u)ⁿ⁻²ⁿ α

n−24

0 =1+o(1), |h|²−n+2 n−2

wⁿ⁻²⁴ h²−β0|h|². Thus the result follows. ✷

PROPOSITION 2.5. – Let n4, foru=^pi=1αiP δi +α0(w+h)∈V (p, ε, w) we

have

∂J (u), λi

∂P δi

∂λi

=2J (u)c₁ n−2

2 α0

w(ai) λ⁽ⁿ_i ⁻^2)/2

1+o(1)−n−2 2 αi

1+o(1)

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−

j=i

αj

λi

∂εij

∂λi

+n−2 2

H (ai, aj) (λiλj)⁽ⁿ⁻^2)/2

1+o(1)

+o

|h|²+ 1

λ^n/2_k +ε

n−1n−2

kr

+R2+O

log(λidi) λ^n/2_i

, whereR₂is defined in (2.2).

Proof. – We have

αjP δj+α0(w+h)

n+2 n−2

λi

∂P δi

∂λi

= p k=1

α

n+2 n−2

k

P δ

n+2 n−2

k λi

∂P δi

∂λi +α

n+2 n−2

0

wⁿⁿ⁺²⁻²λi

∂P δi

∂λi

+n+2 n−2α

4 n−2

i P δ

4 n−2

i λi

∂P δi

∂λi j=i

αjP δj+α0w

+O _p

j=1

δ

n−24

j δi|h| + δi|h|ⁿⁿ⁺⁻²² + wⁿ⁻⁴²|h|δi

+

j=i

δjw

wⁿ⁻⁴²δjδi +

wδj

δ

n−24

j wδi+

B(ai,di)

δ

n−2n

i

+

B^c(ai,di)

δ

n+2 n−2

i +

B^c(ai,di)

δ_i²+ε

n n−2

kr logε_kr⁻¹

. Observe that

B(a_i,d_i)

δ

n n−2

i =O

log(λidi) λ^n/2_i

,

B^c(ai,di)

δ

n+2 n−2

i c

λ⁽ⁿ_i ⁻^2)/2 1 (λidi)²=o

1 λ^n/2_i

+O

1 (λidi)ⁿ

,

B^c(a_i,d_i)

δ²_i =(ifn5)

o 1

λ^n/2_i

+O 1

(λidi)ⁿ

+(ifn=4)o di

λi + 1 (λidi)²

. We also have, forn5, using Holder’s inequality

δjδiε

n−1 n−2

ij

logε_ij⁻¹ⁿ⁻¹ⁿ ⁿ_n−1⁻⁴

log(λj)

λ^n/2_j +log(λi) λ^n/2_i

⁴_n

=o

ε

n−1n−2

ij +λ⁻

n 2

j +λ⁻_i ^n/2

(2.12) (since(n−4)/(n−1)+4/n >1). Forn=4,

δiδj 1 λi

(logλi)^1/2 1 λj

logλj

1/2

=o di

λi

+ 1

(λidi)²+ 1 λ²_j

.

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Observe also, using Holder’s inequality, we obtain

wδ_j

δ

4 n−2

j wδi c

δ

n n−2

j δi ε

n−1n−2

ij

logε⁻_ij¹

n−1 n

ⁿ_n⁻₋²₁ log(λj)

λ^n/2_j ²_n

=oε

n−1n−2

ij +λ⁻

n 2

j

(since(n−2)/(n−1)+2/n >1 forn4). Using the Lemmas A.5–A.9, A.15, A.16, and the fact thatJ (u)^n/4αi=1+o(1)for eachi=0, . . . , p, the result follows. ✷

PROPOSITION 2.6. – For n 4 and u=^pi=1αiP δi +α0(w+h) belonging to V (p, ε, w), we have

∂J (u), 1 λi

∂P δi

∂ai

=J (u)c₁ αi

λⁿ_i⁻¹

∂H (ai, ai)

∂ai

− α0

λ^n/2_i Dw(ai)1+o(1)

+o 1

λ^n/2_k

+2

j=i

αj

1 λi

∂εij

∂ai

−

∂H (a_i,a_j)

∂ai

λi(λiλj)⁽ⁿ⁻^2)/2

×1−J (u)ⁿ⁻ⁿ²α

4 n−2

j +α

4 n−2

i

+R3+o|h|², whereR3is defined in Proposition 2.3.

Proof. –

∂J (u), 1 λi

∂P δi

∂ai

=2J (u) _p

j=1

αj

P δj, 1

λi

∂P δi

∂ai

+α0

w+h, 1 λi

∂P δi

∂ai

−J (u)ⁿ⁻²ⁿ

^p

j=1

αjP δj+α0w ⁿ_n⁺₋²₂

1 λi

∂P δi

∂ai

+(ifn5)O 1 λⁿ_k⁻²

+o|h|² +(ifn=4)o

1

λ²_i + 1

(λidi)³ +

j=i

ε_ij^3/2

. We consider now the term

p j=1

αjP δj+α0w ⁿ_n⁺²₋₂

1 λi

∂P δi

∂ai

= p j=1

α

n+2n−2

j

P δ

n+2n−2

j

1 λi

∂P δi

∂ai

+α

n+2n−2

0

wⁿⁿ⁻²⁺² 1 λi

∂P δi

∂ai

+n+2 n−2α

n−24

i

P δ

n−24

i

1 λi

∂P δi

∂ai j=i

αjP δj+α0w

+O

k=r

ε

n−2n

kr logε⁻_kr¹+

P δ_iw

wⁿ⁻²⁴ δi

1 λi

∂P δi

∂ai

+

wδ_j

wδ

n−24

j δi