Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem

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Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem

Solutions changeant de signe du type « tour de bulles » du problème de Dirichlet pour l’équation − u = | u |

^{− ε} u

Angela Pistoia

, Tobias Weth

aDipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, via A. Scarpa 16, 00161 Roma, Italy bMathematisches Institut, Universität Giessen, Arndtstr. 2, 35392 Giessen, Germany

Received 14 October 2005; accepted 2 March 2006 Available online 20 September 2006

Abstract

We consider the problem−u= |u|^p−1−εuinΩ,u=0 on∂Ω, whereΩis a bounded smooth domain inR^Nsymmetric with respect tox₁, . . . , x_N, which contains the origin,N3,p=^N_N⁺₋²₂andεis a positive parameter. Asεgoes to zero, we construct sign changing solutions with multiple blow up at the origin. These solutions have, asεgoes to zero, more and more annular-shaped nodal domains.

©2006 Elsevier Masson SAS. All rights reserved.

Résumé

Nous considérons le problème−u= |u|^p−1−εudansΩ,u=0 sur∂Ω, oùΩest un domaine borné deR^N symétrique par rapport àx₁, . . . , x_N qui contient l’origine,N3,p=^N_N⁺₋²₂ etεest un paramètre positif. Quandε→0, nous construisons des solutions changeant de signe qui ressemblent à une superposition de transitoires centrées dans l’origine. Ces solutions admettent, quandε→0, de plus en plus de domaines nodeaux ressemblant à un anneau.

©2006 Elsevier Masson SAS. All rights reserved.

MSC:35J60; 35J25

Keywords:Critical Sobolev exponent; Sign changing solution; Multiple blow up

* Corresponding author.

E-mail addresses:[email protected] (A. Pistoia), [email protected] (T. Weth).

1 The author is supported by the M.I.U.R. National Project “Metodi variazionali e topologici nello studio di fenomeni non lineari”.

0294-1449/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2006.03.002

1. Introduction

We consider the problem −u= |u|^p−1−εu inΩ,

u=0 on∂Ω, (1.1)

whereΩ is a bounded smooth domain inR^N,N3,p=^N_N⁺₋²₂ andεis a positive parameter.

First, let us consider the critical case, i.e.ε=0. Pohozaev proved in [39] that problem (1.1) has no solutions ifΩ is starshaped. On the other hand, Kazdan and Warner observed in [32] that (1.1) has a positive radial solution when Ω is an annulus. In [5] Bahri and Coron proved that (1.1) has a positive solution, provided thatΩ has nontrivial topology. To our knowledge there are only few results about existence of sign changing solutions of problem (1.1) in the critical case. In [30] the authors provide existence and multiplicity of sign changing solutions in specific cases, like for instance in the case of tori. In [19] the authors obtain existence and multiplicity results for sign changing solutions in domains with small holes and in some contractible domains with an involution symmetry. We also recall the classical result of Ding [25] who showed that (1.1) has infinitely many sign changing solutions in the whole space Ω=R^N. Concerning sign changing solutions for different problems with critical growth, we refer to the papers [1,2, 15,17,18,27,31,34].

In this paper, we deal with the slightly subcritical case, i.e. ε >0. In order to state old and new results, it is useful to recall some well known definitions. We denote byGthe Green’s function of the Laplacian with Dirichlet boundary condition on ∂Ω and by H its regular part, i.e. G(x, y)=C_N|x−y|^2−N−H (x, y),x, y∈Ω, where C_N=1/(N−2)ωN andω_N denotes the surface area of the unit sphere inR^N. Observe that _xH=0 inΩ×Ω andG=0 on∂(Ω×Ω). The leading termH (x, x)of the regular part of the Green’s function is called the Robin’s function ofΩatx.

It is well known that problem (1.1) has always a positive least energy solutionu_εwhich is obtained by solving the variational problem

inf

u²=

Ω

|∇u|²: u∈H¹₀(Ω),

Ω

|u|^p+1−ε=1

.

In [13,26,29,42,41] it was proved that, asε goes to zero, u_ε blows up and concentrates at a single pointξ inΩ, which is a critical point (a minimum point) of the Robin’s function. Conversely, it was shown in [41,37] that, ifξ is a “stable” critical point of the Robin’s function, then problem (1.1) has forεsmall a solution which blows up atξ asεgoes to zero. In general, problem (1.1) can have positive solutions which concentrate simultaneously at different pointsξ1, . . . , ξkofΩ,k2, asεgoes to zero. This was analysed in [40,6,37]: the condition which ensures existence and multiplicity of solutions which blows up at more than one point involve both the Green’s function and Robin’s function. As far as the existence of positive solutions to (1.1) is concerned, we want to point out the fact that ifuε

solves (1.1) and blows up at some pointsξ1, . . . , ξ_kofΩ, then necessarily eachξ_iis a simple blow up point (see [33]).

More precisely, the profile of the solutionu_εnear each blow up pointξ_i can be approximated as u_ε(x)∼ λ_i√

ε

(λ²_iε+ |x−ξ_i|²)^(N−2)/2,

withα_N= [N (N−2)]^(N−2)/4, for some constantλ_i which depends only onNandk. Roughly speaking, we can also say thatu_εhasksimple positive bubbles.

The existence of one sign changing solution to (1.1) forε∈(0, p−1)was first proved in [16] and [8]. Later, multiple sign changing solutions and their nodal properties were studied in [7,9]. In all these papers, the authors consider a larger class of nonlinearities with superlinear and subcritical growth. In particular, in [7,9] it is shown that, for fixedε∈(0, p−1), problem (1.1) has a sequence of sign solutions±uⁿ_ε, withuⁿ_ε → ∞asngoes to∞, and such thatuⁿ_ε has at mostn+1 nodal domains. Recently in [10], the authors proved the existence ofN pairs±u^{(j )}_ε , j=1, . . . , N, of sign changing solutions such that eachu^{(j )}_ε blows up positively at a pointξ₁^{(j )}∈Ω and negatively at a pointξ₂^{(j )}∈Ω, withξ₁^{(j )}=ξ₂^{(j )}, asεgoes to 0. We point out that each of these solutions considered in [10] has

one simple positive bubble and one simple negative bubble.More precisely the profile ofu^{(j )}ε near each blow up point ξ_i^{(j )},i=1,2, can be approximated whenεgoes to 0 as

u^{(j )}_ε (x)∼α_N(−1)ⁱ λi√ ε

(λ²_iε+ |x−ξ_i^{(j )}|²)^(N−2)/2 ,

withα_N= [N (N−2)]^(N−2)/4, for some constantλ_i which depends only onN.

In this paper, we focus on a new phenomenon: we observe the presence ofsign changing bubble towers constituted by superposition of positive bubbles and negative bubbles of different blow up orders. This stands in strong contrast to the fact that positive solutions to (1.1) can only have simple bubbles in the subcritical case (see [33]). We assume thatΩis a bounded smooth domain inR^Nsymmetric with respect tox1, . . . , xN, which contains the origin. Our main result reads:

Theorem 1.1.For any integerk1, there existsε_k>0such that for anyε∈(0, εk)there exists a pair of solutions u_εand−u_εto problem(1.1)such that

u_ε(y)=α_N k i=1

(−1)ⁱ

Miε^2i−1N−2 M_i²ε^22iN−−12 + |y|²

^N−2

2

1+o(1),

whereα_N:= [N (N−2)]^N−24 ,M₁, . . . , M_kare positive constants depending only onNandkando(1)→0uniformly on compact subsets ofΩ, asε→0. Moreover,u_εis even with respect to the variablesx1, . . . , x_N.

It seems that this is the first result dealing with sign changing bubble tower solutions for superlinear boundary value problems close to the critical exponent. The asymptotic expansion and some energy estimates derived in the course of the proof allow to draw interesting consequences concerning the number and shape of the nodal domains of the solutionu_ε. More precisely, we have

Theorem 1.2.Forε >0sufficiently small, the solutionu_εconstructed in Theorem1.1has preciselyknodal domains Ω_ε¹, . . . , Ω_ε^k such that Ωε^j contains the sphere Sε^j = {y ∈R^N: |y| =ε^2j−1N−2}, and (−1)^juε>0 on Ωε^j for all j. Consequently,0∈Ω_ε^k, andΩ_ε¹is the only nodal domain ofuwhich touches the boundary∂Ω.

In particular, we obtain solutions with arbitrarily many annular shaped nodal domains asεgoes to 0.

The proof of Theorem 1.1 relies on a form of Lyapunov–Schmidt procedure (see [4]), which reduces the construc- tion of the searched solutions to a finite dimensional variational problem, in a general scheme already followed in the study of bubble towers in [20,21,24].

Let us point out that the situation turns out to be very different in the slightly supercritical case, i.e.ε <0. We recall that, if the domainΩ has a small hole, problem (1.1) haspositivesolutions blowing up at two or three points, see [22] and [38]. Moreover, ifΩ has some symmetries, problem (1.1) has solutions blowing up at an arbitrary number of points (see [23,35,38]). An interesting nonexistence result obtained in [11] states that, for any domainΩ, there are no positive solutions to (1.1) blowing up at a single point asεgoes to zero. A natural question then arises: is it possible to construct a sign changing bubble tower solution to (1.1) (as in Theorem 1.1) whenεis negative and small enough? We conjecture that the answer is negative. Indeed this is suggested by the estimates obtained in the present paper. More precisely, in order to detect sign changing bubble tower solutions in the slightly supercritical case, we could reduce the problem in the same way to a finite dimensional one, but for small negativeεthe reduced functional (the function I˜_εdefined in Lemma 4.1 below) does not have any critical points (as follows from Lemma 4.2 and estimate (4.24)).

It is also worth to compare Theorem 1.1 with recent results on positive bubble tower solutions for thesupercritical Dirichlet problem

⎧⎨

⎩

−u=u^p+ε+λu inΩ,

u >0 inΩ,

u=0 on∂Ω,

(1.2) whereΩis a bounded smooth domain inR^N,N4,εandλare positive parameters. In [20] the authors considered the case whenΩis a ball and they proved the existence of radial solutions to (1.2), which have a multiple bubble at the

origin, providedεandλare small enough. Recently, the result was extended in [28], where the authors constructed solutions to problem (1.2), which have multiple blow up at finitely many points which are the critical points of a function whose definition involves the Green’s function. Successively, existence of bubble tower solutions was established for the Neumann supercritical problem

⎧⎨

⎩

−u+u=u^p+ε inΩ,

u >0 inΩ,

u=0 on∂Ω,

(1.3) whenΩ is even with respect toN−1 variables,N3 and 0 is a point in∂Ωwith positive mean curvature. In [24]

the authors proved the existence of solutions to (1.3) which resemble the form of a superposition of bubbles centered at 0.

The proof of Theorem 1.1 relies on a form of Lyapunov–Schmidt procedure (see [4]), which reduces the construc- tion of the searched solutions to a finite-dimensional variational problem, in a general scheme already followed in the study of bubble towers in [20,21,24]. The paper is organized as follows. In Section 2 we collect basic tools, and we introduce a change of coordinates. In Section 3 we apply a finite-dimensional reduction method to the transformed problem. We like to warn the reader that in this section we did not repeat the proofs of the estimates required for the reduction procedure, referring the reader to [20,21,24]. In Section 4 we derive an asymptotic expansion for the reduced energy functional. Here we decided to include the details, since the expansion shows crucial differences in comparison with [20,21,24]. Finally, in Section 5 we complete the proof of Theorem 1.1 and we prove Theorem 1.2.

2. Preliminaries

It is well known (see [3,14,43]) that the functions w_μ(y)=α_N μ^N−22

(μ²+ |y|²)^N−22

, μ >0,

withα_N:= [N (N−2)]^N−24 , are the only radial solutions of the equation

−u=u^p inR^N.

We defineπμto be the unique solution to the problem π_μ=0 inΩ,

π_μ= −w_μ on∂Ω.

We remark that the functionP w_μ:=w_μ+π_μis the projection onto H¹₀(Ω)of the functionw_μ, i.e.

−P w_μ=w^p_μ inΩ, P w_μ=0 on∂Ω.

It is well known that the following expansion holds π_μ(y)= −α_Nμ^N−22H (0, y)+o

μ^N−22 uniformly inΩ. (2.1)

Let us consider parametersμ1> μ2>· · ·> μk. We look for a solution to (1.1) of the form u(y)=

k i=1

(−1)ⁱ

w_μi(y)+π_μi(y) +ψ (y), (2.2)

where the rest termψis a small function which is even with respect to the variablesy₁, . . . , y_N.

As in [20,21,24], we rewrite this problem in different variables. We consider spherical coordinatesy=y(ρ, Θ) centered at the origin given byρ= |y|andΘ=_|^y_y|. We define the transformation

v(x, Θ)=T(u)(x, Θ):=

p−1 2

_p−1² e^−xu

e^−p−21xΘ .

We denote by D the subset ofS=R×S^N−1 where the variables (x, Θ) vary. After these changes of variables, problem (1.1) becomes

L₀(v)=c_εe^−εx|v|^p−1−εv inD,

v=0 on∂D, (2.3)

where cε=

p−1 2

_p−^2ε₁

and

L₀(v)= − p−1

2 2

_SN−1v−v+v. (2.4)

L0is the transformed operator associated to−. Here and in what follows,=_∂x^∂ and_SN−1 denotes the Laplace–

Beltrami operator onS^N−1. We observe then that

T(w_μ)(x, Θ)=W_ξ(x):=W (x−ξ ), where

W (x):=

4N N−2

^N−₄² e^−x

1+e^{−N4−2x −}

N−2

2 , withμ=e^−p−21ξ. (2.5)

W is the unique solution of the problem

⎧⎨

⎩

W−W+W^p=0 inR, W(0)=0, W >0,

W (x)→0 asx→ ±∞.

(2.6) We see also that setting

Π_ξ=T(π_μ), withμ=e^−p−21ξ, (2.7)

thenΠ_ξ solves the boundary problem L₀(Π_ξ)=0 inD,

Π_ξ= −W_ξ on∂D. (2.8)

We note the useful fact that this transformation leaves the associated energies invariant (up to a constant). Indeed, the energy functional associated to problem (2.3) is

Iε(v):=1 2

p−1 2

2

D

|∇Θv|²dxdΘ+1 2

D

|v|²+ |v|² dxdΘ− c_ε p+1−ε

D

e^−εx|v|^p+1−εdxdΘ (2.9) and the energy functional associated to problem (1.1) is

J_ε(u):=1 2

Ω

|∇u|²dy− 1 p+1−ε

Ω

|u|^p+1−εdy. (2.10)

Then we have the identity I_ε(v)=

2 N−2

N−1

J_ε(u), v=T(u). (2.11)

Let us consider points 0< ξ₁< ξ₂<· · ·< ξ_k. We look for a solution to (2.3) of the form v(x, Θ)=

k i=1

(−1)ⁱ

W (x−ξi)+Πξ_i(x, Θ) +φ(x, Θ), (2.12)

where the rest termφis a small function which is symmetric with respect to the variablesΘ1, . . . , ΘN. A crucial remark is thatv(x)∼_k

i=1(−1)ⁱW (x−ξ_i)solves (2.3) if and only if (going back in the change of variables)

u(y)∼α_N k

i=1

(−1)ⁱ e⁻

2ξi N−2

e⁻

4ξi N−2 + |y|²

^N−2

2

solves (1.1). Therefore, the ansatz given forv provides (for large values of theξi’s) a sign changing bubble-tower solution for (1.1).

Let us write

W_i(x):=W (x−ξ_i), Π_i:=Π_ξi, V_i=W_i+Π_i, V :=

k i=1

(−1)ⁱV_i. (2.13)

We consider the ansatzv=V +φ. In terms ofφ, problem (2.3) becomes L(φ)=N (φ)+R inD,

φ=0 on∂D, (2.14)

where

L(φ):=L₀(φ)−c_εe^−εxf_ε(V )φ, (2.15)

N (φ):=c_εe^−εx

f_ε(V+φ)−f_ε(V )−f_ε(V )φ

, (2.16)

R:=cεe^−εxfε(V )− k

i=1

(−1)ⁱW_i^p. (2.17)

Here, we setf_ε(s):= |s|^p−1−εs.

3. The reduction method

Rather than solving (2.14) directly, we consider first the following intermediate problem: given points ξ :=

(ξ1, . . . , ξk)∈R^k find a functionφ symmetric with respect to the variables Θ1, . . . , ΘN such that for certain con- stantsci

⎧⎨

⎩

L(φ)=N (φ)+R+_k

i=1c_iZ_i inD,

φ=0 on∂D,

DZ_iφdxdΘ=0 ifi=1, . . . , k

(3.1) where theZ_i’s are defined as follows. Let

z_i(y)=μ_i ∂

∂μ_iw_μi(y) fori=1, . . . , k, withμ_i=e^−N2−2ξi. Eachzi solves the linearized problem (see [12])

−z=pw^p_μ⁻¹

i z inR^N.

LetP z_i be the projections onto H¹₀(Ω)of the functionz_i, i.e.P z_i=z_i inΩ,P z_i=0 on∂Ω. LetZ_i(x, Θ):=

T(P z_i)(x, Θ). ThenZ_i solves

L0(Zi)=pW_i^p−1W_i inD,

Zi=0 on∂D.

In order to solve problem (3.1), it is necessary to understand first its linear part. Given a functionh, we consider the problem of findingφsuch that for certain real numbersci the following is satisfied

⎧⎨

⎩

L(φ)=h+_k

i=1c_iZ_i inD,

φ=0 on∂D,

DZ_iφdxdΘ=0 ifi=1, . . . , k

(3.2)

where the linear operatorLis defined in (2.15). We need uniformly bounded solvability in proper functional spaces for problem (3.2), for a proper range of theξ_i’s. To this end, it is convenient to introduce the following norm. Given a small but fixed number 0< σ <1, we define:

g_∗:= sup

(x,Θ)∈D

_k

i=1

e^{−(1−σ )|x−ξi|} ₋1

g(x, Θ).

Although this norm depends onσ and the numbers 0< ξ₁<· · ·< ξ_k, we do not indicate this dependence in our notation. In fact, different choices ofσ andξ =(ξ₁, . . . , ξ_k)lead to equivalent norms. LetC_∗ be the Banach space of all continuous functionsg:D→Rwhich are symmetric with respect to the variablesΘ₁, . . . , Θ_N and for which g_∗<+∞.

Arguing exactly as in Propositions 1 and 2 in [20] and in Propositions 5.1 and 5.2 in [21], we obtain the following result.

Proposition 3.1.There existε₀>0,R₀, R₁>0andC >0such that ifε∈(0, ε0)and ifξ=(ξ₁, . . . , ξ_k)satisfies R₀< ξ₁, R₀< min

1i<k(ξ_i+1−ξ_i), ξ_k<R1

ε , (3.3)

then for anyh∈C∗problem(3.2)admits a unique solutionTε(ξ, h)∈C∗, with T_ε(ξ, h)

∗Ch_∗ and |c_i|Ch_∗.

Moreover, the mapξ→Tε(ξ, h), with values inL(C_∗), is of classC¹and D_ξT_ε(ξ, h)

L(C∗)C

uniformly inξ satisfying conditions(3.3).

Now, we are ready to solve problem (3.1). We shall do this after restricting conveniently the range of the parameters ξ_i. Let us consider for a numberMlarge but fixed, the following conditions:

ξ1>1 2log

1 Mε

, min

1i<k(ξi+1−ξi) >log 1

Mε

, ξk< klog 1

Mε

. (3.4)

Arguing exactly as in Proposition 3 in [20] and in Lemma 6.1 in [21], we prove the following result.

Proposition 3.2. There existsε0>0 and C >0 such that for any ε∈(0, ε0)and for any ξ =(ξ1, . . . , ξk)which satisfies(3.4)there exists a unique solutionφ=φ(ξ ), c=(c1(ξ ), . . . , ck(ξ ))to problem(3.1)which satisfiesφ∗ Cε. Moreover, the mapξ→φ(ξ )is of classC¹for the · ∗norm andD_ξφ∗Cε.

4. Estimates for the reduced functional

In this section, we fix a large numberMand assume that conditions (3.4) hold true forξ=(ξ1, . . . , ξk). According to the results of the previous section, our problem has been reduced to that of finding pointsξi so that the constants c_i which appear in (3.1), for the solutionφgiven by Proposition 3.2, are all equal to zero. Thus, we need to solve the system of equations

ci(ξ )=0 for anyi=1, . . . , k. (4.1)

If (4.1) holds, thenv=V+φwill be a solution to (2.14) or equivalently to (2.3). This system turns out to be equivalent to a variational problem, related to the functional (2.9) associated to problem (2.3). Indeed, by the same (standard) arguments as given on p. 301 in [20], the following result is proved.

Lemma 4.1.The functionV +φis a solution to(2.3)ifξ is a critical point of the function ξ→ ˜Iε(ξ ):=Iε(V+φ),

whereV =V (ξ )is given by(2.13),φ=φ(ξ )is given by Proposition3.2, andI_εis defined in(2.9).

The following estimate is crucial for finding critical points ofI˜ε. It can be proved exactly as Lemma 4 in [20] and Lemma 6.2 in [24].

Lemma 4.2.The following expansion holds:

I˜_ε(ξ )=I_ε(V )+o(ε),

where the termo(ε)is uniform over all points satisfying constraints(3.4), for some givenM >0.

We make the following choices for the pointsξ_i: ξ₁= −1

2logε+logΛ₁, (4.2)

ξ_i+1−ξ_i= −logε−logΛ_i+1, i=1, . . . , k−1, (4.3)

where theΛ_i’s are positive parameters. For notational convenience, we also setΛ:=(Λ₁, . . . , Λ_k).

The advantage of the above choice is the validity of the expansion of the functional (2.9) given in the following lemma.

Proposition 4.3.For anyδ >0, there existsε0>0such that for anyε∈(0, ε0)the following expansion holds I_ε(V )=ka₄+εΨ_k(Λ)−k²−2k+2

2 a₂εlogε+ka₁ε+εR_ε(Λ) where

Ψ_k(Λ):=a5H (0,0)

Λ²₁ +ka₂logΛ₁+ k

i=2

a₃Λ_i−(k−i+1)a2logΛ_i

and asε→0the term R_ε converges to0 uniformly on the set of Λ_i’s withδ < Λ_i < δ⁻¹,i=1, . . . , k. Here a_i, i=0, . . . ,5, are positive constants depending only onN.

The proof of this expansion relies on arguments inspired by [20,21,24]. For the convenience of the reader, we present the details here. As a first step, we collect some asymptotic estimates in the following lemma.

Lemma 4.4.For fixedδ >0andδ < Λ_i< δ⁻¹,i=1, . . . , k, the following estimates hold asεgoes to zero:

D

|V|^p+1=kω_N−1

R

W^p+1(x)dx+o(1), (4.4)

D

|V|^p+1− |V|^p+1−ε =kω_N−1

R

W^p+1(x)logW (x)dx+o(1), (4.5)

D

x|V|^p+1= _k

j=1

ξ_j

ω_N−1

R

W^p+1(x)dx+o(1). (4.6)

Here ω_N−1 is the surface area of S^N−1. Moreover, considering the numbers χ₁=0, χ_l= ^ξl−1₂^+ξl,l =2, . . . , k, χ_k+1= +∞and settingD_l= {(x, Θ)∈D: χ_lx < χ_l+1}, we have fori, j, l=1, . . . , k

Dl

W_i^pW_j=o(ε) ifi=l, (4.7)

Dl

W_l^pW_j=a₃e^{−|ξj−ξl|}+o(ε) ifj=l, (4.8)

Dl

V_l^p+1− |V|^p+1+(p+1)V_l^p

j=l

(−1)^l+jV_j

=o(ε), (4.9)

Dl

W_i^p−V_i^p V_j=o(ε) ifi=j . (4.10)

Here we have seta3:=(_N^4N₋₂)^N−42ωN−1

Re^xW^p(x)dx.

Proof. Throughout this proof,C stands for a generic constant depending only onN andkwhose value may change in every step of the calculation. From (2.5) we directly deduce the estimate

W (x)Ce^−|x| inD (4.11) which will be frequently used in the following. Combining the assumptionδ < Λ_i< δ⁻¹with (4.2) and (4.3), we get

ξ1>−log √

ε δ

and ξi+1−ξi>−log ε

δ

fori=1, . . . , k−1. (4.12)

We start by verifying (4.7), first fori=landj=l. In this case (4.11) implies

D_l

W_i^pWjC

χ_l+1

χl

e^{−p|x−ξi|}e^−|x−ξj| C(χ_l+1−χ_l)max

e^{−p|χl−ξi|},e^{−p|χl+1−ξi|} max

e^{−|χl−ξj|},e^{−|χl+1−ξj|} Cξl+1−ξl−1

2 e^{−p|ξl2−ξi|}e^{−|ξl−ξj|2} C(−logε)ε^(p+1)/2=o(ε).

Next we considerj=l. By (4.11) we get fori < l

Dl

W_i^pW_lC

χl+1

χ_l

e^{−p(x−ξi)}e^x−ξldx=Ce^{−p(ξl−ξi)}

ξl+1−ξl

2

ξl−1−ξl 2

e^−(p−1)xdx

Ce^{−p(ξl−ξi)}e^{(p−1)ξl−ξl2−1} Ce^{−p+21(ξl−ξl−1)}Cε^p+21 =o(ε).

Fori > l, we find similarly

Dl

W_i^pW_lC

χl+1

χ_l

e^p(x−ξi)e^−(x−ξl)dx=Ce^p(ξl−ξi)

ξl+1−ξl

2

ξl−1−ξl 2

e^(p−1)xdx

Ce^p(ξl−ξi)e^{(p−1)ξl+1−ξl2} Ce^{−p+21(ξl+1−ξl)}=o(ε),

and thus (4.7) is proved in all cases. Next we derive (4.8) forj < l, using the definition ofW given in (2.5):

Dl

W_l^pW_j=ω_N−1

ξl+1−ξl

2

ξl−1−ξl 2

W^p(x)W (x+ξ_l−ξ_j)dx+o(ε)

=ωN−1

4N N−2

^N−2

4

e^{−(ξl−ξj)}

ξl+1−ξl

2

ξl−1−ξl 2

W^p(x)e^−x

1+e^{−N4−2(x+ξl−ξj) −}

N−2

2 dx+o(ε)

=ωN−1

4N N−2

^N−2₄

e^{−(ξl−ξj)}

1+o(1)

R

W^p(x)e^−xdx+o(ε)

=a3e^{−|ξj−ξl|}+o(ε).

The proof forj > lis similar, sinceW (−x)=W (x)for allx∈R. In particular, (4.7) and (4.8) yield

Dl

W_i^pW_j=O(ε) ifi=lorj=l. (4.13)

Next we show (4.4), and we setW=k

i=1(−1)ⁱWi andΠ=k

i=1(−1)ⁱΠi, so thatW=V −Π. From (2.1), (2.7) and (4.12) we infer

0W_i−V_i= −Π_iCe^−ξiC√

ε inDfor alli. (4.14)

Hence the mean value theorem implies

D

W^p+1− |V|^p+1

(p+1)

D

_k

i=1

W_i p

Π=o(1), (4.15)

and, by (4.13),

D

_k

l=1

W_l^p+1−W^p+1

= k

l=1

Dl

W_l^p+1−

W_l+

j=l

(−1)^l+jW_j ^p+1

+o(1)

(p+1) k l=1

D_l

_k

i=1

Wi

p

j=l

Wj

+o(1)

C

k l,i=1

Dl

W_i^p

j=l

W_j

+o(1)=o(1). (4.16)

Combining (4.15) and (4.16) we get

D

|V|^p+1= k

l=1

D

W_l^p+1+o(1)=kωN−1

R

W^p+1(x)dx+o(1),

which shows (4.4). Since the proof of (4.5) is similar, we omit it. To show (4.6), we use again (4.13) and (4.14) to estimate

D

x|V|^p+1=

D

xW^p+1+o(1)= k

l=1

Dl

xW_l^p+1+o(1)

=ω_N−1 k l=1

ξl+1−ξl

2

ξl−1−ξl 2

(x+ξ_l)W^p+1(x)dx+o(1)

=ω_{N−1 k}

l=1

ξ_l

R

W^p+1(x)dx+k

R

xW^p+1(x)dx+o(1)

=ω_{N−1 k}

l=1

ξ_l

R

W^p+1(x)dx+o(1).

Here we used thatW (−x)=W (x)for allx∈R. It remains to show (4.9) and (4.10). Via a Taylor expansion, we get

D_l

V_l^p+1− |V|^p+1+(p+1)V_l^p

j=l

(−1)^l+jV_j

=

Dl

V_l^p+1−

V_l+

j=l

(−1)^l+jV_j

^p+1+(p+1)V_l^p

j=l

(−1)^l+jV_j

p(p+1)

2

Dl

_k

j=1

V_j

p−1 _k

j=l

V_j 2

Cmax

i,j j=l

Dl

V_i^p−1V_j²

Cmax

i,j j=l Dl

V_i^pV_j ^p−1

p

Dl

V_j^{p+1 1}

p

Cmax

i,j j=l D_l

W_i^pW_j ^p−_p¹

D_l

W_j^{p+1 1}_p

=O ε^p−1p o

ε^p1 =o(ε),

where (4.7) and (4.13) are used in the last line. Hence (4.9) is proved. Finally, by (4.13), (4.14) and the mean value theorem, we get fori=j

0

D_l

W_i^p−V_i^p Vjp

D_l

W_i^p−1(Wi−Vi)Vj

p

D_l

W_i^pV_j ^p−1

p

D_l

(W_i−V_i)^pV_j 1/p

=O ε

p−1

p O√

ε =o(ε), so that (4.10) holds. 2

Proof of Proposition 4.3 (completed). Using (4.4) and (4.5), we find I_ε(V )=I₀(V )+ 1

p+1

D

|V|^p+1− cε

p+1−ε

D

e^−εx|V|^p+1−ε

=I₀(V )− 1 p+1

D

e^−εx−1 |V|^p+1+ 1

p+1− 1 p+1−ε

D

e^−εx|V|^p+1−ε + 1−cε

p+1−ε

D

e^−εx|V|^p+1−ε+ 1 p+1

D

e^−εx

|V|^p+1− |V|^p+1−ε

=I₀(V )− 1 p+1

D

e^−εx−1 |V|^p+1+kεa₁+o(ε), (4.17)

where

a1:=ω_N−1 p+1

1

p+1+ 2

p−1logp−1

2 R

W^p+1(x)dx+

R

W^p+1(x)logW (x)dx

,

and from (4.6) we deduce

− 1 p+1

D

e^−εx−1 |V|^p+1=ε 1 p+1

D

x|V|^p+1+o(ε)=εa₂ k j=1

ξ_j+o(ε), (4.18)