The Bahri–Coron Theorem for Fractional Yamabe-Type Problems

Wael Abdelhedi; Hichem Chtioui; Hichem Hajaiej

doi:10.1515/ans-2017-6035

Article Open Access

The Bahri–Coron Theorem for Fractional Yamabe-Type Problems

Wael Abdelhedi , Hichem Chtioui and Hichem Hajaiej

Published/Copyright: November 4, 2017

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From the journal Advanced Nonlinear Studies Volume 18 Issue 2

Abstract

We study the following fractional Yamabe-type equation:

{ A s ⁢ u = u n + 2 ⁢ s n - 2 ⁢ s , u > 0 in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω ,

Here Ω is a regular bounded domain of ℝn, n≥2, and As, s∈(0,1), represents the fractional Laplacian operator (-Δ)s in Ω with zero Dirichlet boundary condition. We investigate the effect of the topology of Ω on the existence of solutions. Our result can be seen as the fractional counterpart of the Bahri–Coron theorem [3].

Keywords: Fractional PDE; Variational Method; Critical Exponent; Loss of Compactness

MSC 2010: 35J65; 35R11; 58J20; 58C30

1 Introduction

In this work, we consider the following fractional Yamabe-type equation:

(1.1) { A s ⁢ u = u n + 2 ⁢ s n - 2 ⁢ s , u > 0 in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω ,

where Ω⊂ℝn, n≥2, is a regular bounded domain and As, s∈(0,1), represents the fractional Dirichlet Laplacian operator (-Δ)s in Ω defined by using the spectrum of the Laplacian -Δ in Ω with zero Dirichlet boundary condition. It can be viewed as the nonlocal version of the Yamabe-type equation

(1.2) { - Δ ⁢ u = u n + 2 n - 2 , u > 0 in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω .

The fractional Laplacian has attracted the attention of a lot of researchers in the last years since it appears in numerous applications in diverse domains including medicine, biology, physics, modeling populations, mathematical finance and nonlocal diffusion; see [10] and the references [1, 7, 19, 38, 58] therein. The nonlocal character of the fractional Laplacian makes it difficult to handle. After the breakthrough work of Caffarelli–Silvestre [6] who provided to the fractional Laplacian a local interpretation in one more dimension, a large amount of studies were developed on problems involving the fractional Laplacian. Here we point out only some results related to equation (1.1). In [5], Cabré and Tan studied the subcritical cases, that is, equation (1.1) with subcritical nonlinearities in the particular case s=12. They transformed the equation in a local form as the Caffarelli–Silvestre extension and established the existence of positive solutions. For similar extensions, we refer to [4, 7, 18]. For more results concerning the subcritical cases, we refer to [14].

Motivated by the work of Pohozaev [13] on equation (1.2), Tan [17] proved that if Ω is a starshaped domain, equation (1.1) has no solutions in the case s=12. For such a non-existence result, the author used a Pohozaev-type identity.

The resemblance between (1.1) and (1.2) leads us to investigate the effect of the topology of Ω on the existence of solutions of (1.1). By assuming that Ω admits a non-trivial homological group with ℤ2 coefficients Hk⁢(Ω,ℤ2) of an order k∈ℕ*, we get a fractional analog of the Bahri–Coron existence result [3] in form of the following theorem.

Theorem 1.1.

If there exists k∈N* such that Hk⁢(Ω,Z2)≠0, then equation (1.1) admits a solution.

The proof of Theorem 1.1 hinges on the “critical points at infinity” method and the algebraic-topological tools of [3]. Nevertheless, the nonlocal properties of the fractional Laplacian involve many additional difficulties and require some novelties in the proofs.

In Section 2, we recall some known notations related to the operator As, and state the associated variational formulations including the local equivalent extension of problem (1.1). Section 3 will be devoted to an asymptotic expansion of the associated Euler–Lagrange functional J. In Section 4, we will apply Bahri–Coron topological tools and prove Theorem 1.1.

2 Local Equivalent Problem and Variational Structure

First, we recall some preliminaries related to the fractional Laplacian. Let (ek)k∈ℕ be the basis of L2⁢(Ω) such that for any k∈ℕ one has ∥ek∥L2⁢(Ω)=1, 〈ek,eℓ〉=0 for all k≠ℓ, and

{ - Δ ⁢ e k = λ k ⁢ e k in ⁢ Ω , e k = 0 on ⁢ ∂ ⁡ Ω .

So λk>0 for any k∈ℕ.

The fractional Laplacian As, s∈(0,1), is defined by

H 0 s ⁢ ( Ω ) → H 0 - s ⁢ ( Ω ) ≃ H 0 s ⁢ ( Ω ) ,

u = ∑ k = 1 ∞ b k ⁢ e k ↦ A s ⁢ ( u ) = ∑ k = 1 ∞ b k ⁢ λ k s ⁢ e k ,

where

H 0 s ⁢ ( Ω ) := { u = ∑ k = 1 ∞ b k ⁢ e k ∈ L 2 ⁢ ( Ω ) : ∑ k = 1 ∞ b k 2 ⁢ λ k s < ∞ }

and H0-s⁢(Ω) is the dual space of the Hilbert fractional Sobolev space H0s⁢(Ω). Concerning the local equivalent problem to (1.1), we follow the results of [6] for Ω=ℝn, and [5] for a bounded domain Ω; see also [4, 7, 15, 18]. Therefore, we consider the associated local problem on the half cylinder with base Ω:

C = Ω × [ 0 , ∞ ) = { ( x , t ) : x ∈ Ω ⁢ and ⁢ t ∈ [ 0 , ∞ ) } .

Let

C 0 ⁢ L ∞ ⁢ ( C ) = { v ∈ C ∞ ⁢ ( C ¯ ) : v = 0 ⁢ on ⁢ ∂ L ⁡ C } ,

where ∂L⁡C denotes the lateral boundary of C, which is defined by ∂⁡Ω×[0,∞). Let H0⁢Ls⁢(C) be the Hilbert Sobolev space defined as the closure of C0⁢L∞⁢(C) with respect to

| v | = ( ∫ C t 1 - 2 ⁢ s ⁢ | ∇ ⁡ v | 2 ) 1 2 ,

and equipped by the following inner product:

〈 v , w 〉 H 0 ⁢ L s ⁢ ( C ) = ∫ C t 1 - 2 ⁢ s ⁢ ∇ ⁡ v ⁢ ∇ ⁡ w for all ⁢ v , w ∈ H 0 ⁢ L s ⁢ ( C ) .

Following [4, 18], we associate to any u∈H0s⁢(Ω) the unique s-harmonic function, denoted by s-h⁢(u), in H0⁢Ls⁢(C), the unique solution of the following problem:

{ div ⁡ ( t 1 - 2 ⁢ s ⁢ ∇ ⁡ v ) = 0 in ⁢ C , v = 0 on ⁢ ∂ L ⁡ C , v = u on ⁢ Ω × { 0 } ;

(see [4, 18] for the explicit expression of s-h⁢(u)). It follows that As is expressed by the following map:

u = ∑ k = 1 ∞ b k e k ↦ A s ( u ) = ∂ ν s ( s - h ( u ) ) / Ω × { 0 } ,

where ν denotes the unit outward normal vector to C on Ω×{0}, and for any v∈H0⁢Ls⁢(C) and any x∈Ω we have

∂ ν s ⁡ ( v ) ⁡ ( x , 0 ) = - c s ⁢ lim t → 0 + ⁡ t ⁢ ∂ ⁡ v ∂ ⁡ t ⁢ ( x , t ) and c s := Γ ⁢ ( s ) 2 1 - 2 ⁢ s ⁢ Γ ⁢ ( 1 - s ) .

In this way, problem (1.1) is equivalent to the following local problem:

(2.1) { div ⁡ ( t 1 - 2 ⁢ s ⁢ ∇ ⁡ v ) = 0 in ⁢ C , v > 0 in ⁢ C , v = 0 on ⁢ ∂ L ⁡ C , ∂ ν s ⁡ ( v ) = v n + 2 ⁢ s n - 2 ⁢ s on ⁢ Ω × { 0 } .

Therefore, if v satisfies (2.1), then u⁢(x)=v⁢(x,0):=tr⁡(v)⁢(x) for all x∈Ω is a solution of (1.1).

Notice that

H 0 s ⁢ ( Ω ) = { u = tr ⁡ ( v ) : v ∈ H 0 ⁢ L s ⁢ ( C ) ⁢ with ⁢ div ⁡ ( t 1 - 2 ⁢ s ⁢ ∇ ⁡ v ) = 0 ⁢ in ⁢ C } .

In order to present the variational structure associated to (2.1), we introduce the following Hilbert space constructed by all s-harmonic functions in H0⁢Ls⁢(C): More precisely, let

ℋ = { v ∈ H 0 ⁢ L s ⁢ ( C ) : div ⁡ ( t 1 - 2 ⁢ s ⁢ ∇ ⁡ v ) = 0 ⁢ in ⁢ C } .

For all v∈ℋ, we set

∥ v ∥ 2 := | v | 2 = ∫ C t 1 - 2 ⁢ s ⁢ | ∇ ⁡ v | 2 ⁢ 𝑑 x ⁢ 𝑑 t = c s - 1 ⁢ ∫ Ω × { 0 } ∂ ν s ⁡ v ⁢ ( x , 0 ) . v ⁢ ( x , 0 ) ⁢ d ⁢ x ,

and for all v,w∈ℋ, we set

〈 v , w 〉 = 〈 v , w 〉 H 0 ⁢ L s ⁢ ( C ) = c s - 1 ⁢ ∫ Ω × { 0 } ∂ ν s ⁡ v ⁢ ( x , 0 ) ⁢ w ⁢ ( x , 0 ) ⁢ 𝑑 x .

The first Euler–Lagrange functional is

I : ℋ → ℝ , v ↦ c s 2 ⁢ ∥ v ∥ 2 - n - 2 ⁢ s 2 ⁢ n ⁢ ∫ Ω | v ⁢ ( x , 0 ) | 2 ⁢ n n - 2 ⁢ s ⁢ 𝑑 x

and its positive critical points are the unique solutions of (2.1). Since 2⁢nn-2⁢s is the critical Sobolev exponent of the Sobolev trace embedding v∈ℋ↦tr⁡(v)∈Lp⁢(Ω) (which is continuous, but not compact for p=2⁢nn-2⁢s), the functional I is of class C1 and fails the Palais–Smale condition. Moreover, it is not lower bounded. Due to these considerable constraints, we will consider another functional as follows: Let Σ be the sphere of ℋ defined by

Σ = { v ∈ ℋ : ∥ v ∥ = c s - 1 / 2 } .

Set

J 1 : Σ → ℝ , v ↦ J 1 ⁢ ( v ) = sup λ ≥ 0 ⁡ I ⁢ ( λ ⁢ v ) .

Lemma 2.1.

For all v∈Σ, there exists a unique λ⁢(v)>0 such that J1⁢(v)=I⁢(λ⁢(v)⁢v).

Proof.

Let v∈Σ such that

∂ ∂ ⁡ λ ⁢ I ⁢ ( λ ⁢ v ) = λ - λ n + 2 ⁢ s n - 2 ⁢ s ⁢ ∫ Ω | v ⁢ ( x , 0 ) | 2 ⁢ n n - 2 ⁢ s ⁢ 𝑑 x .

Therefore,

∂ ∂ ⁡ λ ⁢ I ⁢ ( λ ⁢ v ) = 0 ⇔ λ = 0 ⁢ or ⁢ λ = λ ⁢ ( v ) := 1 ( ∫ Ω | v ⁢ ( x , 0 ) | 2 ⁢ n n - 2 ⁢ s ⁢ 𝑑 x ) n - 2 ⁢ s 4 ⁢ s .

Since I(λv)|λ=0=0 and I⁢(λ⁢v)→-∞ as λ→+∞, the maximum of the map λ↦I⁢(λ⁢v) is achieved at λ⁢(v). ∎

Lemma 2.2.

Let v∈Σ. We have the following equivalence:

v ⁢ is a critical point of ⁢ J 1 ⇔ λ ⁢ ( v ) ⁢ v ⁢ is a critical point of ⁢ I .

Moreover, J1 can be expressed for all v∈Σ by

J 1 ⁢ ( v ) = ( 1 2 - n - 2 ⁢ s 2 ⁢ n ) ⁢ 1 ( ∫ Ω | v ⁢ ( x , 0 ) | 2 ⁢ n n - 2 ⁢ s ⁢ 𝑑 x ) n - 2 ⁢ s 2 ⁢ s .

Proof.

Using Lemma 2.1, for all v∈Σ we have

J 1 ′ ⁢ ( v ) = I ′ ⁢ ( λ ⁢ ( v ) ⁢ v ) ⁢ ( v ) . λ ′ ⁢ ( v ) + λ ⁢ ( v ) ⁢ I ′ ⁢ ( λ ⁢ ( v ) ⁢ v ) .

Observe that

∂ ∂ ⁡ λ ⁢ I ⁢ ( λ ⁢ v ) = I ′ ⁢ ( λ ⁢ v ) ⁢ ( v ) .

Therefore, we get I′⁢(λ⁢(v)⁢v)⁢(v)=0, and hence

J 1 ′ ⁢ ( v ) = λ ⁢ ( v ) ⁢ I ′ ⁢ ( λ ⁢ ( v ) ⁢ v ) .

The expression of J1 follows from the definition of λ⁢(v). ∎

The Sobolev trace embedding continuity implies that J1 is lower bounded. But it is more convenient for us to work with

( 1 2 - n - 2 ⁢ s 2 ⁢ n ) - 2 ⁢ s n - 2 ⁢ s ⁢ J 1 2 ⁢ s n - 2 ⁢ s .

Therefore, in what follows we will consider the Euler–Lagrange functional

J ⁢ ( v ) = 1 ∫ Ω | v ⁢ ( x , 0 ) | 2 ⁢ n n - 2 ⁢ s ⁢ 𝑑 x , v ∈ Σ .

By Lemma 2.2, if v is a positive critical point of J, then λ⁢(v)⁢v is a solution of (2.1).

In Section 3, we introduce the almost solutions family of problem (2.1) and a useful expansion of J which provides the proof of Theorem 1.1.

3 Asymptotic Expansion

For x,y∈Ω and t>0, let G~⁢((x,t),y) denote the s-harmonic extension of Green’s function of the fractional Dirichlet Laplacian As. It satisfies

{ div ⁡ ( t 1 - 2 ⁢ s ⁢ ∇ ⁡ G ~ ⁢ ( ⋅ , y ) ) = 0 in ⁢ C , G ~ ⁢ ( ⋅ , y ) = 0 on ⁢ ∂ L ⁡ C , ∂ ν s ⁡ G ~ ⁢ ( ⋅ , y ) = δ y on ⁢ Ω × { 0 } .

We have

G ~ ⁢ ( ( x , t ) , y ) = c ^ ∥ ( x - y , t ) ∥ ℝ n + 1 n - 2 ⁢ s - H ~ ⁢ ( ( x , t ) , y ) ,

where c^ is a fixed constant defined in (3.3) and H~ is the regular part of G~. The latter satisfies

{ div ⁡ ( t 1 - 2 ⁢ s ⁢ ∇ ⁡ H ~ ⁢ ( ⋅ , y ) ) = 0 in ⁢ C , H ~ ⁢ ( ( x , t ) , y ) = c ^ ∥ ( x - y , t ) ∥ ℝ n + 1 n - 2 ⁢ s on ⁢ ∂ L ⁡ C , ∂ ν s ⁡ H ~ ⁢ ( ⋅ , y ) = 0 on ⁢ Ω × { 0 } .

Following [8, 11, 12], we see that the family of functionals δ(a,λ), a∈Ω and λ>0, defined by

δ ( a , λ ) ⁢ ( x ) = λ n - 2 ⁢ s 2 ( 1 + λ 2 ⁢ | x - a | 2 ) n - 2 ⁢ s 2 , x ∈ ℝ n ,

is the only solution of

{ A s ⁢ u = c 0 ⁢ u n + 2 ⁢ s n - 2 ⁢ s in ⁢ ℝ n , u > 0 in ⁢ ℝ n , lim | x | → ∞ ⁡ u ⁢ ( x ) = 0 ,

where c0 is a fixed positive constant which depends only n and s.

Let δ^(a,λ) be the s-harmonic extension of δ(a,λ) in ℝ+n+1. In what follows, it is more convenient to work with δ~(a,λ), a∈Ω and λ>0, defined by

δ ~ ( a , λ ) = γ ^ ⁢ δ ^ ( a , λ ) ,

where

γ ^ = c s - 1 2 ⁢ ∥ δ ^ ( a , λ ) ∥ D s ⁢ ( ℝ + n + 1 ) - 1 := ( c s ⁢ ∫ ℝ n + 1 t 1 - 2 ⁢ s ⁢ | ∇ ⁡ δ ^ ( a , λ ) | 2 ⁢ 𝑑 x ⁢ 𝑑 t ) - 1 2

is a fixed constant independent of a and λ. Therefore,

∥ δ ~ ( a , λ ) ∥ D s ⁢ ( ℝ + n + 1 ) = c s - 1 2 ,

tr ⁡ ( δ ~ ( a , λ ) ) = γ ^ ⁢ δ ( a , λ ) on ⁢ ℝ n

and

{ div ⁡ ( t 1 - 2 ⁢ s ⁢ ∇ ⁡ δ ~ ( a , λ ) ) = 0 in ⁢ ℝ + n + 1 , ∂ ν s ⁡ δ ~ ( a , y ) = γ 0 ⁢ δ ~ ( a , y ) n + 2 ⁢ s n - 2 ⁢ s on ⁢ ℝ n × { 0 } ,

where

γ 0 = c 0 ⁢ γ ^ - 4 ⁢ s n - 2 ⁢ s .

The family P⁢δ~(a,λ), a∈Ω and λ>0, of almost solutions of (2.1) is defined as the family of unique solutions of the following problem:

{ div ⁡ ( t 1 - 2 ⁢ s ⁢ ∇ ⁡ P ⁢ δ ~ ( a , λ ) ) = 0 in ⁢ C , P ⁢ δ ~ ( a , λ ) = 0 on ⁢ ∂ L ⁡ C , ∂ ν s ⁡ P ⁢ δ ~ ( a , y ) = ∂ ν s ⁡ δ ~ ( a , y ) = γ 0 ⁢ δ ~ ( a , y ) n + 2 ⁢ s n - 2 ⁢ s on ⁢ Ω × { 0 } .

Next, we introduce the best constant of Sobolev. Let

ı : H 0 ⁢ L s ⁢ ( C ) → L 2 ⁢ n n - 2 ⁢ s ⁢ ( Ω ) , v ↦ tr ⁡ ( v ) ,

be the Sobolev trace embedding. The best constant of Sobolev is given by

S = ∥ tr ⁡ δ ~ ( a , λ ) ∥ L 2 ⁢ n n - 2 ⁢ s ⁢ ( ℝ n ) ∥ δ ~ ( a , λ ) ∥ D s ⁢ ( ℝ + n + 1 ) = c s 1 2 ⁢ ∥ tr ⁡ δ ~ ( a , λ ) ∥ L 2 ⁢ n n - 2 ⁢ s ⁢ ( ℝ n )

since

∥ δ ~ ( a , λ ) ∥ D s ⁢ ( ℝ + n + 1 ) = c s - 1 2 .

Notice that S is independent of a and λ; see [19].

Observe that

inf v ∈ Σ J ( v ) = c s n n - 2 ⁢ s S - 2 ⁢ n n - 2 ⁢ s = : S ~ .

Therefore,

(3.1) S ~ = ( ∥ tr ⁡ δ ~ ( a , λ ) ∥ L 2 ⁢ n n - 2 ⁢ s ⁢ ( ℝ n ) 2 ⁢ n n - 2 ⁢ s ) - 1 = 1 ∫ ℝ n ( tr ⁡ δ ~ ( a , λ ) ) 2 ⁢ n n - 2 ⁢ s ⁢ 𝑑 x .

Remark 3.1.

The equation S~=γ0 holds. Indeed,

∥ δ ~ ( a , λ ) ∥ D s ⁢ ( ℝ + n + 1 ) 2 = ∫ ℝ + n + 1 t 1 - 2 ⁢ s ⁢ | ∇ ⁡ δ ~ ( a , λ ) | 2 ⁢ 𝑑 x ⁢ 𝑑 t = c s - 1 ⁢ ∫ ℝ n ∂ ν s ⁡ δ ~ ( a , λ ) ⁢ ( x , 0 ) ⁢ δ ~ ( a , λ ) ⁢ ( x , 0 ) ⁢ 𝑑 x .

Thus,

c s - 1 = c s - 1 ⁢ γ 0 ⁢ ∫ ℝ n δ ~ ( a , λ ) ⁢ ( x , 0 ) 2 ⁢ n n - 2 ⁢ s ⁢ 𝑑 x ,

and therefore

γ 0 = 1 ∫ ℝ n ( tr ⁡ δ ~ ( a , λ ) ) 2 ⁢ n n - 2 ⁢ s ⁢ 𝑑 x .

Remark 3.2.

S ~ is achieved in the case of Ω=ℝn. But for a bounded domain Ω, S~ is never achieved by J; see [9].

In order to give the expansion of the Euler–Lagrange functional J associated to problem (2.1), we introduce the following notations: Let K be a compact set in Ω. For any λ>0 and p∈ℕ*, we define

g λ , p : Δ p - 1 × K p → Σ , ( α , a ) ↦ c s - 1 2 ⁢ ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ ) ∥ ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ ) ∥ ,

where Δp-1={α=(α1,…,αp):αi∈(0,1) for all i, and ∑i=1pαi=1} and a=(a1,…,ap). Without loss of generality, we can assume that ai≠aj for all i≠j. In what follows, we set da:=mini≠j⁡|ai-aj|. We then have the following expansion of the functional J∘gλ,p.

Proposition 3.3.

Let p∈N* and λ>0. For all (α,a)∈Δp-1×Kp, we have

J ( g λ , p ( α , a ) ) = S ~ ( ∑ i = 1 p α i 2 ) n n - 2 ⁢ s ∑ i = 1 p α i 2 ⁢ n n - 2 ⁢ s { 1 - c 2 λ n - 2 ⁢ s [ ∑ j = 1 p ( α j 2 ∑ i = 1 p α i 2 - 2 ⁢ α j 2 ⁢ n n - 2 ⁢ s ∑ i = 1 p α i 2 ⁢ n n - 2 ⁢ s ) H ~ ( ( a j , 0 ) , a j )

+ ∑ k ≠ j ( 2 ⁢ α k ⁢ α j n + 2 ⁢ s n - 2 ⁢ s ∑ i = 1 p α i 2 ⁢ n n - 2 ⁢ s - α j ⁢ α k ∑ i = 1 p α i 2 ) G ~ ( ( a i , 0 ) , a j ) ] + O ( 1 ( λ ⁢ d a ) n - s ) } .

Here,

c 2 = n n - 2 ⁢ s ⁢ S ~ ⁢ c 1 ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ′ ,

where c1 is defined in Lemma 3.4 and

c ′ = ∫ ℝ n d ⁢ z ( 1 + | z | 2 ) n + 2 ⁢ s 2 .

The proof of Proposition 3.3 requires the following lemma.

Lemma 3.4.

For all a∈K⋐Ω and λ>0, we have

P ⁢ δ ~ ( a , λ ) ⁢ ( x , t ) = δ ~ ( a , λ ) ⁢ ( x , t ) - c 1 ⁢ H ~ ⁢ ( ( x , t ) , a ) λ n - 2 ⁢ s 2 + O ⁢ ( 1 λ n + 2 - 2 ⁢ s 2 )

for all x∈Ω and t>0. Here,

c 1 := S ~ ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ^ ⁢ c ′ .

Proof.

Let φ(a,λ):=δ~(a,λ)-P⁢δ~(a,λ). It satisfies

{ div ⁡ ( t 1 - 2 ⁢ s ⁢ ∇ ⁡ φ ( a , λ ) ) = 0 in ⁢ C , φ ( a , λ ) = δ ~ ( a , λ ) on ⁢ ∂ L ⁡ C , ∂ ν s ⁡ φ ( a , λ ) = 0 on ⁢ Ω × { 0 } .

Therefore, the functional

ϕ ( a , λ ) = φ ( a , λ ) - c 1 ⁢ H ~ ⁢ ( ⋅ , a ) λ n - 2 ⁢ s 2

satisfies

{ div ⁡ ( t 1 - 2 ⁢ s ⁢ ∇ ⁡ ϕ ( a , λ ) ) = 0 in ⁢ C , ϕ ( a , λ ) = δ ~ ( a , λ ) - c 1 ⁢ c ^ λ n - 2 ⁢ s 2 ⁢ ∥ ( x - a , t ) ∥ n - 2 ⁢ s on ⁢ ∂ L ⁡ C , ∂ ν s ⁡ ϕ ( a , λ ) = 0 on ⁢ Ω × { 0 } .

Thus, by the maximum principle we get

∥ ϕ ( a , λ ) ∥ L ∞ ⁢ ( C ) ≤ ∥ δ ~ ( a , λ ) - c 1 ⁢ c ^ λ n - 2 ⁢ s 2 ⁢ ∥ ( x - a , t ) ∥ n - 2 ⁢ s ∥ L ∞ ⁢ ( ∂ L ⁡ C ) .

From another part, using the property

( ( x , t ) , y ) ↦ c ^ ∥ ( x - y , t ) ∥ n - 2 ⁢ s

of Green’s function on ℝ+n+1, which is the singular part of G~, we have

δ ~ ( a , λ ) ⁢ ( x , t ) = ∫ ℝ n c ^ ∥ ( x - y , t ) ∥ n - 2 ⁢ s ⁢ ∂ ν s ⁡ ( δ ~ ( a , λ ) ⁢ ( y , 0 ) ) ⁡ d ⁢ y

= S ~ ⁢ ∫ ℝ n c ^ ∥ ( x - y , t ) ∥ n - 2 ⁢ s ⁢ δ ~ ( a , λ ) n + 2 ⁢ s n - 2 ⁢ s ⁢ ( y , 0 ) ⁢ 𝑑 y .

Recall that

δ ~ ( a , λ ) ⁢ ( y , 0 ) = tr ⁡ ( δ ~ ( a , λ ) ) ⁢ ( y ) = γ ^ ⁢ δ ( a , λ ) ⁢ ( y ) for all ⁢ y ∈ ℝ n .

Therefore,

δ ~ ( a , λ ) ⁢ ( x , t ) = S ~ ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ^ ⁢ ∫ ℝ n 1 ∥ ( x - y , t ) ∥ n - 2 ⁢ s ⁢ λ n + 2 ⁢ s 2 ( 1 + λ 2 ⁢ | y - a | 2 ) n + 2 ⁢ s 2 ⁢ 𝑑 y .

A change of variables z=λ⁢(y-a) yields

δ ~ ( a , λ ) ⁢ ( x , t ) = S ~ ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ^ ⁢ 1 λ n - 2 ⁢ s 2 ⁢ ∫ ℝ n 1 ∥ ( x - a - z λ , t ) ∥ n - 2 ⁢ s ⁢ d ⁢ z ( 1 + | z | 2 ) n + 2 ⁢ s 2 .

For any x∈∂⁡Ω and t≥0, we expand

1 ∥ ( x - a - z λ , t ) ∥ n - 2 ⁢ s

around x-a. Using the fact that |x-a|≥da>0 for all x∈∂⁡Ω, we get

δ ~ ( a , λ ) ⁢ ( x , t ) = S ~ ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ^ ⁢ 1 λ n - 2 ⁢ s 2 ⁢ 1 ∥ ( x - a , t ) ∥ n - 2 ⁢ s ⁢ ∫ ℝ n d ⁢ z ( 1 + | z | 2 ) n + 2 ⁢ s 2 ⁢ 𝑑 y + O ⁢ ( 1 λ n + 2 - 2 ⁢ s 2 )

= c 1 λ n - 2 ⁢ s 2 ⁢ ∥ ( x - a , t ) ∥ n - 2 ⁢ s + O ⁢ ( 1 λ n + 2 - 2 ⁢ s 2 ) .

This concludes the proof of Lemma 3.4. ∎

Proof of Proposition 3.3.

We have

J ⁢ ( g λ , p ⁢ ( α , a ) ) = c s n n - 2 ⁢ s ⁢ ∥ ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ ) ∥ 2 ⁢ n n - 2 ⁢ s ∫ Ω ( ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ ) ⁢ ( x , 0 ) ) 2 ⁢ n n - 2 ⁢ s ⁢ 𝑑 x = c s n n - 2 ⁢ s ⁢ N D

and

N n - 2 ⁢ s n = c s - 1 ⁢ ∫ Ω ∂ s ∂ ⁡ ν ⁢ ( ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ ) ) . ( ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ ) ) ⁢ d ⁢ x

= c s - 1 ⁢ S ~ ⁢ [ ∑ i = 1 p α i 2 ⁢ ∫ Ω δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s ⁢ P ⁢ δ ~ ( a i , λ ) + ∑ j ≠ i α i ⁢ α j ⁢ ∫ Ω δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s ⁢ P ⁢ δ ~ ( a j , λ ) ] .

Using Lemma 3.4, we have

∫ Ω δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s ⁢ P ⁢ δ ~ ( a i , λ ) = ∫ Ω δ ~ ( a i , λ ) 2 ⁢ n n - 2 ⁢ s - c 1 λ n - 2 ⁢ s 2 ⁢ ∫ Ω H ~ ⁢ ( ( x , 0 ) , a i ) ⁢ δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s + O ⁢ ( 1 λ n + 2 - 2 ⁢ s 2 ⁢ ∫ ℝ n δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s )

= S ~ - 1 + O ⁢ ( 1 λ n ) - c 1 ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s λ n - 2 ⁢ s ⁢ [ ∫ ℝ n H ~ ⁢ ( ( a i + z λ , 0 ) , a i ) ⁢ d ⁢ z ( 1 + | z | 2 ) n + 2 ⁢ s 2 + O ⁢ ( 1 λ 2 ⁢ s ) ] + O ⁢ ( 1 λ n + 1 - 2 ⁢ s )

(3.2) = S ~ - 1 - c 1 ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ′ λ n - 2 ⁢ s ⁢ H ~ ⁢ ( ( a i , 0 ) , a i ) + O ⁢ ( 1 λ n - 2 ⁢ s ) .

Moreover, for any i≠j we have

∫ Ω δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s ⁢ P ⁢ δ ~ ( a j , λ ) = ∫ ℝ n δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s ⁢ P ⁢ δ ~ ( a j , λ ) + O ⁢ ( 1 λ n ) .

Using again Lemma 3.4, we have

∫ Ω δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s ⁢ P ⁢ δ ~ ( a j , λ ) = ∫ ℝ n δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s ⁢ δ ~ ( a j , λ ) - c 1 λ n - 2 ⁢ s 2 ⁢ ∫ ℝ n H ~ ⁢ ( ( x , 0 ) , a j ) ⁢ δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s + O ⁢ ( 1 λ n + 1 - 2 ⁢ s )

= ∫ ℝ n δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s ⁢ δ ~ ( a j , λ ) - c 1 ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ′ λ n - 2 ⁢ s ⁢ H ~ ⁢ ( ( a j , 0 ) , a j ) + O ⁢ ( 1 λ n + 1 - 2 ⁢ s ) .

Using a similar computation as in [1], we have

∫ ℝ n δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s ⁢ δ ~ ( a j , λ ) = γ ^ 0 2 ⁢ n n - 2 ⁢ s ⁢ c ′ λ n - 2 ⁢ s ⁢ | a i - a j | n - 2 ⁢ s + O ⁢ ( 1 ( λ ⁢ | a i - a j | ) n - s ) .

Therefore,

∫ Ω δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s ⁢ P ⁢ δ ~ ( a j , λ ) = c 1 ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ′ λ n - 2 ⁢ s ⁢ ( γ ^ / c 1 | a i - a j | n - 2 ⁢ s - H ~ ⁢ ( ( a j , 0 ) , a j ) ) + O ⁢ ( 1 ( λ ⁢ d a ) n - s ) .

For

(3.3) c ^ = ( S ~ ⁢ c ′ ) - 1 2 ⁢ γ ^ - 2 ⁢ n n - 2 ⁢ s ,

which is a universal constant, we have γ^/c1=c^. Thus,

(3.4) ∫ Ω δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s ⁢ P ⁢ δ ~ ( a j , λ ) = c 1 ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ′ λ n - 2 ⁢ s ⁢ G ~ ⁢ ( ( a i , 0 ) , a j ) + O ⁢ ( 1 ( λ ⁢ d a ) n - s ) .

From (3.2) and (3.4) we deduce

N n - 2 ⁢ s n = c s - 1 ⁢ [ ∑ i = 1 p α i 2 - S ~ ⁢ c 1 ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ′ λ n - 2 ⁢ s ⁢ ( ∑ i = 1 p α i 2 ⁢ H ~ ⁢ ( ( a i , 0 ) , a i ) - ∑ j ≠ i α i ⁢ α j ⁢ G ~ ⁢ ( ( a i , 0 ) ⁢ a j ) ) ] + O ⁢ ( 1 ( λ ⁢ d a ) n - s ) ,

and thus

N = c s - n n - 2 ⁢ s ⁢ ( ∑ i = 1 p α i 2 ) n n - 2 ⁢ s ⁢ [ 1 - n n - 2 ⁢ s ⁢ S ~ ⁢ c 1 ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ′ ( ∑ i = 1 p α i 2 ) ⁢ λ n - 2 ⁢ s ⁢ ( ∑ i = 1 p α i 2 ⁢ H ~ ⁢ ( ( a i , 0 ) , a i ) - ∑ j ≠ i α i ⁢ α j ⁢ G ~ ⁢ ( ( a i , 0 ) ⁢ a j ) ) ]

(3.5) + O ⁢ ( 1 ( λ ⁢ d a ) n - s ) .

Next, we expand D. For any i=1,…,p, let Bi=B⁢(ai,min⁡(da2,ℓ)) be the ball of center ai and radius min⁡(da2,ℓ), where ℓ=d⁢(K,∂⁡Ω). Then

D = ∫ Ω ( ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ ) ) 2 ⁢ n n - 2 ⁢ s = ∫ ∪ B i ( ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ ) ) 2 ⁢ n n - 2 ⁢ s + ∫ Ω ∖ ⁣ ∪ B i ( ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ ) ) 2 ⁢ n n - 2 ⁢ s .

On Bi, we have

∑ k = 1 p α k ⁢ P ⁢ δ ~ ( a k , λ ) = α i ⁢ δ ~ ( a i , λ ) + ∑ k ≠ i α k ⁢ P ⁢ δ ~ ( a k , λ ) - α i ⁢ φ ( a i , λ ) ,

where φ(ai,λ)=δ~(ai,λ)-P⁢δ~(ai,λ). Observe that

( ∑ k = 1 p α k ⁢ P ⁢ δ ~ ( a k , λ ) ) 2 ⁢ n n - 2 ⁢ s = α i 2 ⁢ n n - 2 ⁢ s ⁢ δ ~ ( a i , λ ) 2 ⁢ n n - 2 ⁢ s + 2 ⁢ n n - 2 ⁢ s ⁢ α i n + 2 ⁢ s n - 2 ⁢ s ⁢ δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s ⁢ ( ∑ k ≠ i α k ⁢ P ⁢ δ ~ ( a k , λ ) - α i ⁢ φ ( a i , λ ) )

+ O [ ( ∑ k ≠ i α k P δ ~ ( a k , λ ) - α i φ ( a i , λ ) ) 2 ⁢ n n - 2 ⁢ s

+ ( α i δ ~ ( a i , λ ) ) 4 ⁢ s n - 2 ⁢ s inf ( ( α i δ ~ ( a i , λ ) ) 2 , ( ∑ k ≠ i α k P δ ~ ( a k , λ ) - α i φ ( a i , λ ) ) 2 ) ] .

Therefore,

∫ B i ( ∑ k = 1 p α k ⁢ P ⁢ δ ~ ( a k , λ ) ) 2 ⁢ n n - 2 ⁢ s = α i 2 ⁢ n n - 2 ⁢ s ⁢ ∫ B i δ ~ ( a i , λ ) 2 ⁢ n n - 2 ⁢ s + 2 ⁢ n n - 2 ⁢ s ⁢ α i n + 2 ⁢ s n - 2 ⁢ s ⁢ ∫ B i δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s ⁢ ( ∑ k ≠ i α k ⁢ P ⁢ δ ~ ( a k , λ ) - α i ⁢ φ ( a i , λ ) ) + O ⁢ ( 1 ( λ ⁢ d a ) n - s ) .

A computation similar to (3.2)–(3.5) yields

∫ B i ( ∑ k = 1 p α k ⁢ P ⁢ δ ~ ( a k , λ ) ) 2 ⁢ n n - 2 ⁢ s = α i 2 ⁢ n n - 2 ⁢ s ⁢ S ~ - 1 - 2 ⁢ n n - 2 ⁢ s ⁢ c 1 ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ′ ⁢ α i 2 ⁢ n n - 2 ⁢ s λ n - 2 ⁢ s ⁢ H ~ ⁢ ( ( a i , 0 ) , a i )

+ 2 ⁢ n n - 2 ⁢ s ⁢ c 1 ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ′ ⁢ α i n + 2 ⁢ s n - 2 ⁢ s λ n - 2 ⁢ s ⁢ ∑ k ≠ i G ~ ⁢ ( ( a i , 0 ) , a k ) + O ⁢ ( 1 ( λ ⁢ d a ) n - s ) .

From another part,

∫ Ω ∖ ⋃ i = 1 p B i ( ∑ k = 1 p α k ⁢ P ⁢ δ ~ ( a k , λ ) ) 2 ⁢ n n - 2 ⁢ s ≤ ∫ Ω ∖ ⋃ i = 1 p B i c ⁢ ∑ k = 1 p α k 2 ⁢ n n - 2 ⁢ s ⁢ ( δ ~ ( a k , λ ) - φ ( a k , λ ) ) 2 ⁢ n n - 2 ⁢ s .

Using Lemma 3.4, we have

∫ Ω ∖ ⋃ i = 1 p B i ( ∑ k = 1 p α k ⁢ P ⁢ δ ~ ( a k , λ ) ) 2 ⁢ n n - 2 ⁢ s ≤ c ⁢ ∑ k = 1 p ∫ Ω ∖ ⋃ i = 1 p B i δ ~ ( a k , λ ) 2 ⁢ n n - 2 ⁢ s + O ⁢ ( 1 λ n ) ≤ c ⁢ 1 ( λ ⁢ d a ) n + O ⁢ ( 1 λ n ) .

Thus,

D = S ~ - 1 ⁢ ∑ i = 1 p α i 2 ⁢ n n - 2 ⁢ s - 2 ⁢ n n - 2 ⁢ s ⁢ c 1 ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ′ λ n - 2 ⁢ s ⁢ ( ∑ i = 1 p α i 2 ⁢ n n - 2 ⁢ s ⁢ H ~ ⁢ ( ( a i , 0 ) , a i ) - ∑ k ≠ i α i n + 2 ⁢ s n - 2 ⁢ s ⁢ α k ⁢ G ~ ⁢ ( ( a i , 0 ) , a k ) ) + O ⁢ ( 1 ( λ ⁢ d a ) n - s )

= S ~ - 1 ⁢ ∑ i = 1 p α i 2 ⁢ n n - 2 ⁢ s ⁢ [ 1 - 2 ⁢ n n - 2 ⁢ s ⁢ S ~ ∑ i = 1 p α i 2 ⁢ n n - 2 ⁢ s ⁢ c 1 ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ′ λ n - 2 ⁢ s ⁢ ( ∑ i = 1 p α i 2 ⁢ n n - 2 ⁢ s ⁢ H ~ ⁢ ( ( a i , 0 ) , a i ) - ∑ k ≠ i α i n + 2 ⁢ s n - 2 ⁢ s ⁢ α k ⁢ G ~ ⁢ ( ( a i , 0 ) , a k ) ) ]

(3.6) + O ⁢ ( 1 ( λ ⁢ d a ) n - s ) .

Therefore, from (3.5) and (3.6) we get

J ( g λ , p ( α , a ) ) = S ~ ( ∑ i = 1 p α i 2 ) n n - 2 ⁢ s ∑ i = 1 p α i 2 ⁢ n n - 2 ⁢ s { 1 - n n - 2 ⁢ s S ~ ∑ i = 1 p α i 2 c 1 ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ′ λ n - 2 ⁢ s

× [ ∑ j = 1 p α j 2 ⁢ H ~ ⁢ ( ( a j , 0 ) , a j ) - ∑ k ≠ j α j ⁢ α k ⁢ G ~ ⁢ ( ( a j , 0 ) , a k ) ] + 2 ⁢ n n - 2 ⁢ s ⁢ S ~ ∑ i = 1 p α i 2 ⁢ n n - 2 ⁢ s ⁢ c 1 ⁢ γ ^ n + 2 ⁢ s n - 2 ⁢ s ⁢ c ′ λ n - 2 ⁢ s

× [ ∑ j = 1 p α j 2 ⁢ n n - 2 ⁢ s H ~ ( ( a j , 0 ) , a j ) - ∑ k ≠ j α j 2 ⁢ n n - 2 ⁢ s α k G ~ ( ( a j , 0 ) , a k ) ] } + O ( 1 ( λ ⁢ d a ) n - s ) .

This concludes the proof of Proposition 3.3. ∎

The above expansion is of course useful when λ⁢da is very large. The next proposition provides an estimate of J without involving λ⁢da.

Proposition 3.5.

For any p∈N* and λ>0, for all (α,a)∈Δp-1×Kp we have

J ⁢ ( g λ , p ⁢ ( α , a ) ) ≤ S ~ n n - 2 ⁢ s ⁢ { ∫ Ω ( ∑ i = 1 p α i ⁢ δ ~ ( a i , λ ) ) 2 ⁢ n n - 2 ⁢ s ∫ Ω ( ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ ) ) 2 ⁢ n n - 2 ⁢ s } 1 2 ⁢ ( ∑ i = 1 p ∫ Ω β i ⁢ δ ~ ( a i , λ ) 2 ⁢ n n - 2 ⁢ s ) 2 ⁢ s n - 2 ⁢ s ,

where

β i = α i ⁢ δ ~ ( a i , λ ) ∑ k = 1 p α k ⁢ δ ~ ( a k , λ ) .

Proof.

Following the proof of Proposition 3.3, we have

N n - 2 ⁢ s n = c s - 1 ⁢ S ~ ⁢ ∫ Ω ( ∑ i = 1 p α i ⁢ δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s ) ⁢ ( ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ ) )

≤ c s - 1 ⁢ S ~ ⁢ [ ∫ Ω ( ∑ i = 1 p α i ⁢ δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s ) 2 ⁢ n n + 2 ⁢ s ] n + 2 ⁢ s 2 ⁢ n ⁢ [ ∫ Ω ( ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ ) ) 2 ⁢ n n - 2 ⁢ s ] n - 2 ⁢ s 2 ⁢ n .

We have

∫ Ω ( ∑ i = 1 p α i ⁢ δ ~ ( a i , λ ) n + 2 ⁢ s n - 2 ⁢ s ) 2 ⁢ n n + 2 ⁢ s ≤ [ ∫ Ω ( ∑ i = 1 p α i ⁢ δ ~ ( a i , λ ) ) 2 ⁢ n n - 2 ⁢ s ] n - 2 ⁢ s n + 2 ⁢ s ⁢ [ ∫ Ω ( ∑ i = 1 p β i ⁢ δ ~ ( a i , λ ) 8 ⁢ n ⁢ s n 2 - 4 ⁢ s 2 ) n + 2 ⁢ s 4 ⁢ s ] 4 ⁢ s n + 2 ⁢ s

≤ [ ∫ Ω ( ∑ i = 1 p α i ⁢ δ ~ ( a i , λ ) ) 2 ⁢ n n - 2 ⁢ s ] n - 2 ⁢ s n + 2 ⁢ s ⁢ [ ∑ i = 1 p ∫ Ω β i ⁢ δ ~ ( a i , λ ) 2 ⁢ n n - 2 ⁢ s ] 4 ⁢ s n + 2 ⁢ s .

Therefore,

N ≤ c s - n n - 2 ⁢ s ⁢ S ~ n n - 2 ⁢ s ⁢ [ ∫ Ω ( ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ ) ) 2 ⁢ n n - 2 ⁢ s ] 1 2 ⁢ [ ∫ Ω ( ∑ i = 1 p α i ⁢ δ ~ ( a i , λ ) ) 2 ⁢ n n - 2 ⁢ s ] 1 2 ⁢ [ ∑ i = 1 p ∫ Ω β i ⁢ δ ~ ( a i , λ ) 2 ⁢ n n - 2 ⁢ s ] 2 ⁢ s n - 2 ⁢ s .

Hence Proposition 3.5 follows. ∎

Corollary 3.6.

For any p∈N* and ε>0, there exists λ1=λ⁢(p,ε) such that for all λ≥λ1 and (α,a)∈Δp-1×Kp we have

J ⁢ ( g λ , p ⁢ ( α , a ) ) ≤ ( p + ε ) 4 ⁢ s n - 2 ⁢ s ⁢ S ~ .

Proof.

Using (3.1), we have

∫ Ω ∑ i = 1 p β i ⁢ δ ~ ( a i , λ ) 2 ⁢ n n - 2 ⁢ s ≤ p ⁢ S ~ - 1 + O ⁢ ( 1 λ n ) .

Therefore, by Proposition 3.5 we obtain

J ⁢ ( g λ , p ⁢ ( α , a ) ) ≤ S ~ n n - 2 ⁢ s ⁢ { ∫ Ω ( ∑ i = 1 p α i ⁢ δ ~ ( a i , λ ) ) 2 ⁢ n n - 2 ⁢ s ∫ Ω ( ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ ) ) 2 ⁢ n n - 2 ⁢ s } 1 2 ⁢ ( p 2 ⁢ s n - 2 ⁢ s ⁢ S ~ - 2 ⁢ s n - 2 ⁢ s + O ⁢ ( 1 λ 2 ⁢ n ⁢ s n - 2 ⁢ s ) ) .

Thus, Corollary 3.6 follows from the above estimate and Lemma 3.4. ∎

Corollary 3.7.

There exist p0∈N* and λ0>0 such that for all (α,a)∈Δp0-1×Kp0 and all λ≥λ0 we have

J ⁢ ( g λ , p 0 ⁢ ( α , a ) ) ≤ p 0 4 ⁢ s n - 2 ⁢ s ⁢ S ~ .

Proof.

We divide the proof into two steps.

Step 1: We assume that there exists an index i, 1≤i≤p for p≥2, such that αi is small enough.

Without loss of generality, we may assume that α1≪1. According to the proof of Proposition 3.3, we have

N n - 2 ⁢ s n = c s - 1 ⁢ [ O ⁢ ( α 1 ) + ( 1 - α 1 ) 2 ⁢ ∫ Ω ( ∑ i = 2 p α i ( 1 - α 1 ) ⁢ ∂ λ s ⁡ ( P ⁢ δ ~ ( a i , λ ) ) ) ⁢ ( ∑ i = 2 p α i ( 1 - α 1 ) ⁢ ( P ⁢ δ ~ ( a i , λ ) ) ) ] .

Moreover,

D = O ⁢ ( α 1 ) + ( 1 - α 1 ) 2 ⁢ n n - 2 ⁢ s ⁢ ∫ Ω ( ∑ i = 2 p α i ( 1 - α 1 ) ⁢ ( P ⁢ δ ~ ( a i , λ ) ) ) 2 ⁢ n n - 2 ⁢ s .

Thus,

J ⁢ ( g λ , p ⁢ ( α , a ) ) ≤ J ⁢ [ g λ , p - 1 ⁢ ( ( α 2 1 - α 1 , … , α p 1 - α 1 ) , ( a 2 , … , a p ) ) ] + O ⁢ ( α 1 ) .

Therefore, Corollary 3.7 follows for α1 small enough by using Corollary 3.6.

Step 2: We assume that αi, for all i, 1≤i≤p, is lower bounded by a fixed constant ε0>0.

Assume that da=|ai0-aj0|. If λ⁢da is large enough, by using the expansion of Proposition 3.3, there exists d1>0 such that if da≤d1, we have

J ⁢ ( g λ , p ⁢ ( α , a ) ) ≤ p 2 ⁢ s n - 2 ⁢ s ⁢ S ~ .

This is a consequence of the fact that G~⁢((ai0,0),aj0)→+∞ as |ai0-aj0|→0.

From another part, if λ⁢da is upper bounded, by using Lemma 3.4, Proposition 3.5 and the fact that

∫ Ω δ ~ ( a i 0 , λ ) δ ~ ( a i 0 , λ ) + ε 0 ⁢ δ ~ ( a j 0 , λ ) ⁢ δ ~ ( a j 0 , λ ) 2 ⁢ n n - 2 ⁢ s ≤ S ~ - 1 ⁢ ( 1 - ε 0 ) ,

Corollary 3.7 follows in this case.

Lastly, if da≥d1, using the fact that there exist c~>0 and γ~>0 such that

H ~ ⁢ ( ( a , 0 ) , a ) ≤ c ~ for all ⁢ a ∈ K ,

G ~ ⁢ ( ( a , 0 ) , b ) ≤ γ ~ for all ⁢ a , b ∈ K ,

from Proposition 3.3 we derive that there exist c¯>0 and γ¯>0 such that

J ⁢ ( g λ , p ⁢ ( α , a ) ) ≤ p 2 ⁢ s n - 2 ⁢ s ⁢ ( S ~ + 2 λ n - 2 ⁢ s ⁢ ( c ¯ - p ⁢ γ ) ) + O ⁢ ( 1 ( λ ⁢ d a ) n - s ) .

Therefore, by taking p0 such that c¯-p⁢γ<0 and λ large enough, Corollary 3.7 follows. ∎

4 Proof of Theorem 1.1

We provide the proof of Theorem 1.1 by prescribing the loss of compactness of our problem. Since the exponent 2⁢nn-2⁢s is the critical exponent for the Sobolev trace embedding, the functional J associated to problem (2.1) fails to satisfy the Palais–Smale condition. Arguing as [16], by the following proposition we describe the Palais–Smale sequences.

Proposition 4.1.

Assume that (2.1) has no solution. Let (vk)k be a sequence in Σ+:={v∈Σ:v≥0} such that J⁢(vk)→c and ∂⁡J⁢(vk)→0. There exists p∈N* and a subsequence of (vk)k denoted again (vk)k such that vk∈V⁢(p,εk), where εk→0 as k→+∞, and

V ( p , ε ) = { u ∈ Σ + : there exist ( a 1 , … , a p ) ∈ Ω p , ( λ 1 , … , λ p ) ∈ [ 1 ε , ∞ ) p and ( α 1 , … , α p ) ∈ ℝ + p

such that ∥ u - 1 c s ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ i ) ∥ ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ i ) ∥ < ε ∥ < ε with λ i d ( a i , ∂ Ω ) > 1 ε and ε i ⁢ j < ε for all i ≠ j } .

Here,

ε i ⁢ j = 1 ( λ i λ j + λ j λ i + λ i ⁢ λ j ⁢ | a i - a j | 2 ) n - 2 ⁢ s 2 .

Remark 4.2.

If (vk)k⊂V⁢(p,εk), where εk→0, then

J ⁢ ( v k ) → p 2 ⁢ s n - 2 ⁢ s ⁢ S ~ .

The following proposition gives suitable parameters for V⁢(p,ε). The proof is similar to the one of [3, Proposition 7].

Proposition 4.3.

Let p∈N*. There exists ε>0 such that for all u∈V⁢(p,ε) the minimization problem

inf α i , a i , λ i ⁡ ∥ u - ∑ i = 1 p α i ⁢ P ⁢ δ ~ ( a i , λ i ) ∥

admits a unique solution (α¯,a¯,λ¯) modulo a parametrization on the indices set. Therefore, we write

u = ∑ i = 1 p α ¯ i ⁢ P ⁢ δ ~ ( a ¯ i , λ ¯ i ) + v

for any u∈V⁢(p,ε). Here, v∈H and ∥v∥<ε.

Next, we will use the following notations: For any p∈ℕ*, we set

b p = p 2 ⁢ n n - 2 ⁢ s ⁢ S ~ , B p ⁢ ( K ) = { ∑ i = 1 p α i ⁢ δ a i : ( α 1 , … , α p ) ∈ Δ p - 1 , ( a 1 , … , a p ) ∈ K p } , W p = J b p + 1 ,

where

J c = { u ∈ Σ : u ≥ 0 ⁢ and ⁢ J ⁢ ( u ) < c } for all ⁢ c > 0 .

Observe that

W 0 = J S ~ = ∅ and B 0 ⁢ ( K ) = ∅ .

Following the Bahri–Brezis deformation lemma [2, Lemma 17], we have that Jc2 retracts by deformation on Jc1 for any bp<c1<c2≤bp+1, provided J has no critical point.

Proof of Theorem 1.1.

Under the assumption of Theorem 1.1, there exists a cycle K of dimension k in Ω such that [K], the class of K in Hk⁢(Ω,ℤ2), is not trivial. We introduce the following lemma.

Lemma 4.4.

Assume that (2.1) has no solution. There exists a group of homomorphisms

g ~ 1 ⁣ * : H * ( B 1 ( K ) , B 0 ( K ) ) → H * ( W 1 , W 0 ) , * ∈ ℕ ,

such that g~1⁢k⁢([B1⁢(K),B0⁢(K)])≠0. Here H*⁢(M,N) denotes the homology group associated to the topological pair (M,N) if M⊂N.

Proof.

For λ>0 large enough and ε>0 small enough, we consider the following continuous mappings:

π 1 : B 1 ⁢ ( K ) → K , ⁢ δ a ↦ a ,

g 1 , λ : K → J S ~ + ε ∩ V ⁢ ( 1 , ε ) , ⁢ a ↦ 1 c s ⁢ P ⁢ δ ~ ( a , λ ) ∥ P ⁢ δ ~ ( a , λ ) ∥ ,

where g1,λ is well defined by using Corollary 3.6. Consider the natural injection

ı 1 : J S ~ + ε → W 1 .

Recall that a continuous mapping f:M→N between two manifolds M and N induces a group of homomorphisms (f*)*⁣∈ℕ such that

f * : H * ⁢ ( M ) → H * ⁢ ( N ) , [ c ] ↦ [ f ⁢ ( c ) ] .

Let

g ~ 1 = ı 1 ∘ g 1 , λ ∘ π 1 .

Then (g~1⁣*)*⁣∈ℕ is a non-zero group. Indeed,

g ~ 1 ⁣ * = ı 1 ⁣ * ∘ g 1 , λ ⁣ * ∘ π 1 ⁣ * .

Observe that π1⁣* is an isomorphism since π1 is an homeomorphism. Furthermore, ı1⁣* is an isomorphism since ı1 is an homotopy equivalence (under the assumption that J has no critical point). Lastly, we claim that g1,λ⁣*⁢([K])≠0 for *=k. Indeed, if g1,λ⁣*⁢([K])=0, then (P*∘g1,λ⁣*)⁢([K])=0, where

P : J S ~ + ε ∩ V ⁢ ( 1 , ε ) → Ω , u = α ¯ ⁢ P ⁢ δ ~ ( a ¯ , λ ¯ ) + v ∥ α ¯ ⁢ P ⁢ δ ~ ( a ¯ , λ ¯ ) + v ∥ ↦ a ¯ .

Observe that P∘g1,λ⁢(K)=K. We therefore have [K]=0 in H*⁢(Ω), which is absurd. Hence our claim is valid. It is now easy to check that

g ~ 1 ⁣ * ( π 1 ⁣ * - 1 ( [ K ] ) ) = g ~ 1 ⁣ * ( [ B 1 ( K ) ] ) ≠ 0 for * = k .

The proof of Lemma 4.4 follows. ∎

Next, we extend the result of Lemma 4.4 for any p≥1.

Lemma 4.5.

Assume that (2.1) has no solution. For any p∈N*, there exists a group of homomorphisms

g ~ p ⁣ * : H * ( B p ( K ) , B p - 1 ( K ) ) → H * ( W p , W p - 1 ) , * ∈ ℕ ,

such that g~p⁢d⁢([Bp⁢(K),Bp-1⁢(K)])≠0, where d=k⁢p+p-1.

Proof.

Let ρp be the group of permutations of order p. The group ρp acts on Kp×Δp-1 by

σ ⁢ ( ( a 1 , … , a p ) , ( α 1 , … , α p ) ) = ( ( a σ ⁢ ( 1 ) , … , a σ ⁢ ( p ) ) , ( α σ ⁢ ( 1 ) , … , α σ ⁢ ( p ) ) )

for any σ∈ρp. We denote by Kp×ρpΔp-1 the associated quotient space. Set

π p : B p ⁢ ( K ) → K p × ρ p Δ p - 1 , ∑ i = 1 p α i ⁢ δ a i ↦ ( ( a σ ⁢ ( 1 ) , … , a σ ⁢ ( p ) ) , ( α σ ⁢ ( 1 ) , … , α σ ⁢ ( p ) ) ) .

For

S p = { ( a 1 , … , a p ) ∈ K p : there exists ⁢ i ≠ j ⁢ such that ⁢ a i = a j } ,

we define an ρp equivariant neighborhood Tp of Sp in Kp. Observe that the projection πp induces a map of pairs

π p : ( B p ⁢ ( K ) , B p - 1 ⁢ ( K ) ) → ( K p × ρ p Δ p - 1 , S p × Δ p - 1 ∪ ρ p K p × ∂ ⁡ Δ p - 1 ) ,

which is again an homeomorphism. Define the natural injection by

j p : ( K p × ρ p Δ p - 1 , S p × Δ p - 1 ∪ ρ p K p × ∂ ⁡ Δ p - 1 ) → ( K p × ρ p Δ p - 1 , T ¯ p × Δ p - 1 ∪ ρ p K p × ∂ ⁡ Δ p - 1 ) .

This is an homotopy equivalence since T¯p retracts by deformation on Sp. Next, we denote

K 0 p := K p ∖ T p ,

and we define the natural injection

ψ p : ( K 0 p × ρ p Δ p - 1 , ∂ ⁡ ( K 0 p × ρ p Δ p - 1 ) ) → ( K p × ρ p Δ p - 1 , T ¯ p × Δ p - 1 ∪ ρ p K p × ∂ ⁡ Δ p - 1 ) ,

which is an homotopy equivalence. For any *∈ℕ, let

φ p ⁣ * : H * ⁢ ( B p ⁢ ( K ) , B p - 1 ⁢ ( K ) ) → H * ⁢ ( K 0 p × ρ p Δ p - 1 , ∂ ⁡ ( K 0 p × ρ p Δ p - 1 ) )

be the homomorphism defined by

φ p ⁣ * = ψ p ⁣ * - 1 ∘ j p ⁣ * ∘ π p ⁣ * .

Let

g p , λ ⁣ * : H * ⁢ ( K 0 p × ρ p Δ p - 1 , ∂ ⁡ ( K 0 p × ρ p Δ p - 1 ) ) → H * ⁢ ( J p 2 ⁢ s n - 2 ⁢ s ⁢ S ~ + ε ∩ V ⁢ ( p , ε ) , J p 2 ⁢ s n - 2 ⁢ s ⁢ S ~ ∩ V ⁢ ( p , ε ) )

be the homomorphism induced by gp,λ, which is well defined by using Corollary 3.6 and the proof of Corollary 3.7. Let

ı p ⁣ * : H * ⁢ ( J p 2 ⁢ s n - 2 ⁢ s ⁢ S ~ + ε ∩ V ⁢ ( p , ε ) , J p 2 ⁢ s n - 2 ⁢ s ⁢ S ~ ∩ V ⁢ ( p , ε ) ) → H * ⁢ ( W p , W p - 1 )

be the homomorphism induced by the trivial injection. The required group of homomorphisms (g~p⁣*)*⁣∈ℕ is given by

g ~ p ⁣ * = ı p ⁣ * ∘ g p , λ ⁣ * ∘ φ p ⁣ * .

We now consider the following commutative diagram for all *≥2:

Here ∂ denotes the connecting homomorphism. Following [3, (23)–(26)], we derive that for any p≥2,

g ~ ( p - 1 ) ⁣ * ⁢ ( [ B p - 1 ⁢ ( K ) , B p - 2 ⁢ ( K ) ] ) ≠ 0 .

Then

g ~ p ⁣ * ⁢ ( [ B p ⁢ ( K ) , B p - 1 ⁢ ( K ) ] ) ≠ 0 .

In the case p-1=1, observe that g~1⁣* is exactly the one constructed by Lemma 4.4. Therefore, the fact that g~1⁣*⁢([B1⁢(K),B0⁢(K)])≠0 implies the result of Lemma 4.5. ∎

Now, if we suppose that (2.1) has no solution, by Lemma 4.5 we get

g ~ p ⁣ * ⁢ ( [ B p ⁢ ( K ) , B p - 1 ⁢ ( K ) ] ) ≠ 0 ⁢ in ⁢ H * ⁢ ( W p , W p - 1 ) for all ⁢ p ≥ 1 .

But this is impossible since from Corollary 3.7 we have

( ı p 0 ∘ g λ , p 0 ) ⁢ ( K 0 p 0 × ρ p 0 Δ p 0 - 1 , ∂ ⁡ ( K 0 p 0 × ρ p 0 Δ p 0 - 1 ) ) ⊂ ( W p 0 - 1 , W p 0 - 1 )

for some p0∈ℕ*, and thus

g ~ p 0 ⁣ * ⁢ ( [ B p 0 ⁢ ( K ) , B p 0 - 1 ⁢ ( K ) ] ) = 0 in ⁢ H * ⁢ ( W p 0 , W p 0 - 1 ) .

This finishes the proof of Theorem 1.1. ∎

Communicated by Luis Caffarelli

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Received: 2017-06-04

Revised: 2017-10-01

Accepted: 2017-10-04

Published Online: 2017-11-04

Published in Print: 2018-04-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2017-6035

Keywords for this article

Fractional PDE; Variational Method; Critical Exponent; Loss of Compactness

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