Extremal functions for the anisotropic Sobolev inequalities

Extremal functions for the anisotropic Sobolev inequalities Fonctions minimales pour des inégalités de Sobolev anisotropiques

A. El Hamidi

, J.M. Rakotoson

aLaboratoire de Mathématiques, Université de La Rochelle, Av. Michel Crépeau, 17042 La Rochelle cedex 09, France bLaboratoire de Mathématiques, U.M.R. 6086, Université de Poitiers, SP2MI, Boulevard Marie et Pierre Curie, Téléport 2,

BP 30179, 86962 Futuroscope Chasseneuil cedex, France

Received 2 September 2005; received in revised form 31 May 2006; accepted 21 June 2006 Available online 19 December 2006

Abstract

The existence of multiple nonnegative solutions to the anisotropic critical problem

− N

i=1

∂

∂x_i ∂u

∂x_i

^pi−2∂u

∂x_i

= |u|^p∗−2u inR^N

is proved in suitable anisotropic Sobolev spaces. The solutions correspond to extremal functions of a certain best Sobolev constant.

The main tool in our study is an adaptation of the well-known concentration-compactness lemma of P.-L. Lions to anisotropic operators. Furthermore, we show that the set of nontrival solutionsSis included inL^∞(R^N)and is located outside of a ball of radiusτ >0 inL^p∗(R^N).

©2006 Elsevier Masson SAS. All rights reserved.

Résumé

Nous montrons l’existence d’une infinité de solutions positives pour le problème anisotropique avec exposant critique. La mé- thode consiste à regarder la meilleure constante d’une inégalité du type Poincaré–Sobolev et à adapter le fameux principe de concentration-compacité de P. L. Lions. De plus, on montre que l’ensemble des solutionsS est contenu dansL^∞(R^N)et est localisé en dehors d’une boule de rayonτ >0 dansL^p∗(R^N).

©2006 Elsevier Masson SAS. All rights reserved.

Keywords:Quasilinear problems; Concentration-compactness; Anisotropic Sobolev inequalities

* Corresponding author.

E-mail address:rako@mathlabo.univ-poitiers.fr (J.M. Rakotoson).

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

doi:10.1016/j.anihpc.2006.06.003

1. Introduction

In this paper, the existence of nontrivial nonnegative solutions to the anisotropic critical problem

− N

i=1

∂

∂x_i ∂u

∂x_i

^pi−2∂u

∂x_i

= |u|^p∗−2u inR^N (1)

is studied, where the exponentsp_i andp^∗satisfy the following conditions pi>1,

N

i=1

1 p_i >1,

and the critical exponentp^∗is defined by p^∗:= N

_N

i=1 1 pi−1.

In the best of our knowledge, anisotropic equations with different orders of derivation in different directions, involving critical exponents were never studied before. In the subcritical case, we can refer the reader to the recent paper by I. Fragala et al. [4].

In the special casepi=2,i∈ {1,2, . . . , N}, Problem (1) is reduced to the limiting equation arising in the famous Yamabe problem [13]:

−u=u^2∗−1, u >0 inR^N. (2)

Indeed, let(M, g)be aN-dimensional Riemannian manifold andS_gbe the scalar curvature of the metricg. Consider a conformal metricg˜onMdefined byg˜:=u^N4−2gwhose scalar curvature (which is assumed to be constant) is denoted byS_g˜, whereuis a positive function inC^∞(M,R). The unknown functionusatisfies then

−_gu+ N−2

4(N−1)S_gu= N−2

4(N−1)S_g˜u^2∗−1, u >0 inM, (3)

where_g denotes the Laplace–Beltrami operator. It is clear that, up to a scaling, the limiting problem of (3) (Eq. (3) without the subcritical term _4(N^N−₋²₁₎S_gu) is exactly (2). The question of existence of minimizing solutions to (2) was completely solved by Aubin [1] and G. Talenti [9]. Their proofs are based on symmetrization theory. Notice that this theory is not relevant in our context since the radial symmetry of solutions cannot hold true because of the anisotropy of the operator.

In [5], P.-L. Lions introduced the famous concentration-compactness lemma which constitutes a powerful tool for the study of critical nonlinear elliptic equations. The concentration-compactness lemma allows an elegant and simple proof of the existence of solutions to (2) by minimization arguments. In the present work, we will adapt the concentration-compactness lemma to the anisotropic case and show that the infimum

|u|_Lp∗Inf(RN )=1

_N

i=1

1 p_i

∂u

∂x_i ^pi

pi

is achieved, of course, the functional space has to be specified.

The motivation of the present work is to give a new result which can provide extremal functions associated to the critical level corresponding to anisotropic problems involving critical exponents. Notice that the genuine extremal functions are obtained by minimization on the Nehari manifold associated to the problem and the critical level is nothing than the energy of these extremal functions.

The natural functional framework of Problem (1) is the anisotropic Sobolev spaces theory developed by [3,6,11,7, 8,10]. Then, letD^1,p(R^N)be the completion of the spaceD(R^N)with respect to the norm

u1,p:=

N

i=1

∂u

∂x_i

pi

.

It is well known that(D^1,p(R^N), · 1,p)is a reflexive Banach space which is continuously embedded inL^p∗(R^N).

In what follows, we will assume that p₊=max{p₁, p₂, . . . , p_N}< p^∗,

thenp^∗is the critical exponent associated to the operator:

N

i=1

∂

∂x_i ∂

∂x_i ^pi−2 ∂

∂x_i

.

The spaceD^1,p(R^N)can also be seen as D^1,p(R^N)=

u∈L^p∗ R^N

: ∂u

∂x_i

∈L^pi R^N

.

In the sequel, we will setp₋=min{p1, p2, . . . , p_N},p₊=max{p1, p2, . . . , p_N}andp=(p1, p2, . . . , p_n). Also, the integral symbol

will denote

R^N and · pi will denote the usual Lebesgue norm inL^pi(R^N). We denote by M(R^N)(resp.M⁺(R^N)) the space of finite measures (resp. positive finite measures) onR^N, and by · its usual norm.

2. Existence of extremal functions for a Sobolev type inequality

In this paragraph, we shall prove that a certain best Sobolev constant is achieved.

Theorem 1.Under the above assumptions onp_i,i=1, . . . , N,N2, there exists at least one functionu∈D^1,p(R^N), u0,u=0:

− N

i=1

∂

∂x_i ∂u

∂x_i

^pi−2∂u

∂x_i

=u^p∗−1 inD R^N

.

The proof will need two fundamental lemmas, the first one is a result due to M. Troisi [10]:

Lemma 1.(Troisi [10])There is a constantT₀>0depending only onpandNsuch that:

T₀up^∗ N

i=1

∂u

∂x_i

1 N

pi

and up^∗ 1 N T₀

N

i=1

∂u

∂x_i

pi

,

for allu∈D^1,p(R^N).

The second lemma is a rescaling type result ensuring the conservation of suitable norms:

Lemma 2.Letα_i=^p_p^∗_i −1,i=1, . . . , N. For everyy∈R^N,u∈D^1,p(R^N), andλ >0, if we writex=(x₁, . . . , x_N), y=(y₁, . . . , y_N),v(x)˙=u^λ,y(x)=λu(λ^α1x₁+y₁, . . . , λ^αNx_N+y_N), we get

up∗= vp∗, ∂u

∂x_i

p_i

= ∂v

∂x_i

p_i

, fori=1, . . . , N, thus,u1,p= u^λ,y1,p.

Proof. Noticing that_N

i=1α_i =p^∗, a straightforward computation with adequate changes of variables gives the result. 2

Lemma 3.Let

S= Inf

u∈D^1,p(R^N),up∗=1

_N

i=1

1 pi

∂u

∂xi

^pi

p_i

.

ThenS >0.

Proof. From Lemma 1, we obtain that ifup^∗=1, then N

i=1

∂u

∂x_i

pi

N T₀>0. (4)

Using standard argument, the infimum Inf

_N

i=1

1

p_ia^p_iⁱ, (a₁, . . . , a_n)∈R^N, N

i=1

a_iN T₀, a_i0 .

=S₁

is achieved and thus this minimum is positive. By relation (4), one concludes thatSS₁>0. 2

Corollary 1 of Lemma 3 (Sobolev type inequality). Letp₋=min(p1, . . . , p_N),p₊=max(p1, . . . , p_N)andF be the real valued function defined by

F (σ )=

σ^p+ ifσ1, σ^p− ifσ1.

Then for everyu∈D^1,p(R^N), one has SF

up^∗

N

i=1

1 pi

∂u

∂xi

^pi

p_i

˙=P (∇u).

Proof. Letube inD^1,p(R^N). Ifu=0 the inequality is true. Ifu=0, setw=u/up^∗, then from the definition ofS one has:

N

i=1

1 p_i

∂w

∂x_i ^pi

p_i

S. (5)

Sincet^pit^p+ ift >1 andt^pit^p−otherwise, the result follows from relation (5) and the definition ofF. 2 Remark 1.Along this paragraph, we only need the inequality:

Su^p_p⁺∗ P (∇u) wheneverup^∗1.

We shall call(P)the minimization problem (P) Inf

up∗=1

_N

i=1

1 p_i

∂u

∂x_i ^pi

pi

= Inf

up∗=1

P (∇u) .

Let(un)⊂D^1,p(R^N)be a minimizing sequence for the problem(P). As in [5] and Willem [12], we define the Levy concentration function:

Q_n(λ)= sup

y∈R^N

E(y,λ^α1,...,λαN)

|u_n|^p∗dx, λ >0.

HereE(y, λ^α1, . . . , λ^αN)is the ellipse defined by

z=(z1, . . . , zN)∈R^N, N

i=1

(z_i−y_i)² λ^2αi 1

withy=(y1, . . . , yN)andαi>0 as in Lemma 2. Since for everyn, limλ→0Qn(λ)=0 andQn(λ) −→

λ→+∞1. There existsλ_n>0 such thatQ_n(λ_n)=¹₂. Moreover there existsy_n∈R^Nsuch that

E(yn,λ^αn¹,...,λ^αNn )

|un|^p∗dx=1 2.

Thus by a change of variables one has forvn .

=u^λn^n,yn:

B(0,1)

|vn|^p∗dx=1 2= sup

y∈R^N

B(y,1)

|vn|^p∗dx.

Since

v_np^∗= u_np^∗, ∂v_n

∂x_i

pi

= ∂u_n

∂x_i

pi

, P (∇u_n)=P (∇v_n)

we deduce that(v_n)is bounded inD^1,p(R^N)and is also a minimizing sequence for(P). We may then assume that:

• v_n vinD^1,p(R^N),

• |_∂x^∂_i(v_n−v)|^pi μ_i inM⁺(R^N),

• |v_n−v|^p∗ νinM⁺(R^N),

• v_n→va.e. inR^N. We define:

μ= N

i=1

1 pi

μ_i,

μ_∞= lim

R→+∞lim

n

N

i=1

1 p_i

|x|>R

∂v_n

∂x_i

^pidx, (6)

ν_∞= lim

R→+∞lim

n

|x|>R

|v_n|^p∗dx. (7)

We start with some general lemmas. First by the Brezis–Lieb’s Lemma [2], direct computations give the following Lemma 4.

|v_n|^p∗|v|^p∗+ν inM⁺ R^N

.

The lemma which follows gives some reverse Hölder type inequalities connecting the measures ν, μ and μi, 1iN.

Lemma 5.Under the above statement, one has for allϕ∈C_c^∞(R^N)

|ϕ|^p∗dν _p∗¹

1

T₀ N

i=1

|ϕ|^pidμi

¹

Npi,

|ϕ|^p∗dν _p∗¹

p

1 N+_p∗¹

+ μ^{N1+p∗1 −p1+} · 1 T0

|ϕ|^p+dμ ¹

p+

.

Proof. Letϕ∈C_c^∞(R^N)and setwn=vn−v. Since

|ϕx_i|^pi|wn|^pidx −→

n→+∞0, we then have:

limn ∂

∂x_i(ϕwn)

^pidx=lim

n

|ϕ|^pi ∂w_n

∂x_i

^pidx=

|ϕ|^pidμi. (8)

Thus from Lemma 1, it follows that

|ϕ|^p∗dν _p∗¹

=lim

n

|ϕw_n|^p∗dx _p∗¹

1

T0

N

i=1

|ϕ|^pidμi

¹

Npi. (9)

On the other hand, since

|ϕ|^pidμip₊

|ϕ|^pidμp₊μ^1−ppi+

|ϕ|^p+dμ _p^pi

+ (10)

applying the estimates (9) and (10) and knowing thatN i=1

1

pi =1+_p^N∗, we deduce

|ϕ|^p∗dν _p∗¹

p

1 N+p∗¹

+ μ^{N1+p∗1 −p1+} · 1 T0

|ϕ|^p+dμ ¹

p+

.

This ends the proof. 2

We then havevp^∗1. So ifvp^∗=1 thenvis an extremal function sinceP (∇v)lim infnP (∇vn)=Sand SP (∇v). Thus, we want to show that fact, by proving that if it is not true then we have a concentration ofνat a single point and thereforev=0.

Main Lemma.

vp^∗=1.

The remainder of this section is devoted to the proof of the main Lemma Lemma 6.Ifv=0then

limn v_n−v^p_p^∗∗=1− v^p_p^∗∗<1.

Proof. From Brezis–Lieb’s Lemma we have:

limn

v_n^p_p^∗∗− v_n−v^p_p^∗∗

= v^p_p^∗∗,

Sincev_np^∗=1, we derive the result. 2 Lemma 7.

Sν^pp∗+ μ.

Proof. For largen, according to Lemma 6, we have:

|vn−v|^p∗dx1.

Thus for allϕ∈C_c^∞(R^N),|ϕ|∞1, it holds:

S

|ϕ|^p∗|vn−v|^{p∗ p}_p∗⁺

N

i=1

1 p_i

|ϕ|^pi

∂(v_n−v)

∂x_i

^pidx+on(1).

Lettingn→ +∞, one gets:

S

|ϕ|^p∗dν ^p_p∗⁺

N

i=1

1 p_i

|ϕ|^pidμiμ. (11)

Using the density ofC_c^∞(R^N)inC_c(R^N), we get then S

sup

ϕ∈C_c(R^N),|ϕ|∞=1

|ϕ|^p∗dν ^p_p∗⁺

μ,

that is the desired result. 2

Lemma 8.Letψ_R be inC¹(R),0ψ_R1,ψ_R=1 if|x|> R+1,ψ_R(x)=0 if|x|< R. Then for any γ_i >0, i=0, . . . , N, the two equalities

ν_∞= lim

R→+∞lim

n

|vn|^p∗ψ_R^γ0dx,

μ_∞= lim

R→+∞lim

n

N

i=1

1

p_i ∂v_n

∂x_i

^piψ_R^γidx

hold true, whereν_∞andμ_∞are defined by(6), (7).

Proof. As in Willem [12], one has:

|x|>R+1

|vn|^p∗dx

|vn|^p∗ψ_R^γ0dx

|x|>R

|vn|^p∗dx,

|x|>R+1

∂v_n

∂xi

^pidx ∂v_n

∂xi

ψ_R^γi

|x|>R

∂v_n

∂xi

^pidx.

We conclude with the definition ofν_∞andμ_∞. 2

Lemma 9.Letwn=vn−v. Then, for anyγi>0,i=0, . . . , N, we get ν_∞= lim

R→∞lim

n

|wn|^p∗ψ_R^γ0dx, and

μ_∞= lim

R→∞lim

n ∂wn

∂x_i

^piψ_R^γidx.

Proof. Since

R→+∞lim

|v|^p∗ψ_R^γ0= lim

R→+∞ ∂v

∂x_i

^piψ_R^γidx=0.

Thus

Rlim→∞lim

n

|w_n|^p∗ψ_R^γ0dx= lim

R→∞lim

n

|v_n|^p∗ψ_R^γ0dx=ν_∞ and

Rlim→∞lim

n

N

i=1

1

p_i ∂w_n

∂x_i

^piψ_R^γidx= lim

R→∞lim

n

N

i=1

1 p_i ∂v_n

∂x_i

^piψ_R^γidx. 2

Lemma 10.

Sν

p+

∞p∗ μ_∞.

Proof. From Lemma 6, we know that fornlarge enough, we have

ψ_R^p∗|w_n|^p∗

|w_n|^p∗dx1.

Thus by Sobolev inequality (Corollary 4 of Lemma 3), it follows S |ψ_Rw_n|^p∗dx

^p_p∗⁺

N

i=1

1 pi ∂

∂xi

(ψ_Rw_n) ^pi, S

R→+∞lim lim

n

|ψRwn|^p∗dx ^p_p∗⁺

lim

R→+∞lim

n

N

i=1

1 p_i ∂

∂x_i(ψRwn)

^pi. (12) Since

limn

N

i=1

1

p_i ∂ψ_R

∂x_i

^pi|w_n|^pi=0,

then

R→+∞lim lim

n

N

i=1

1 p_i ∂

∂x_i(ψ_Rw_n)

^pi= lim

R→+∞lim

n

N

i=1

1

p_i ∂w_n

∂x_i

^piψ_R^pi=μ_∞.

Relation (12) and Lemma 9 give:

Sν

p+

∞p∗ μ_∞. 2

Following again the arguments used in [12] we claim that:

Lemma 11.

1=lim

n v_n^p_p^∗∗= v^p_p^∗∗+ ν +ν_∞. Proof. From Lemma 4, we have:

|v_n|^p∗|v|^p∗+ν.

Thus

R→+∞lim lim

n 1−ψ_R^p∗

|v_n|^p∗dx=

|v|^p∗dx+

dν.

Rewritingv_n^p_p^∗∗as

vn^p_p^∗∗= 1−ψ_R^p∗

|vn|^p∗+

ψ_R^p∗|vn|^p∗, we obtain

limn v_n^p_p^∗∗= lim

R→+∞lim

n 1−ψ_R^p∗

|v_n|^p∗+ lim

R→+∞lim

n

ψ_R^p∗|v_n|^p∗= v^p_p^∗∗+ ν +ν_∞. 2

Next, we shall prove the following corollary:

Corollary 1 (Of Lemma 5). There exists an at most countable index set J of distinct points {x_j}j∈J ⊂R^N and nonnegative weightsa_j andb_j,j∈J, such that:

(1) ν=

j∈Jajδx_j.

(2) μ

j∈Jb_jδ_xj. (3) Sa

p+ p∗

j b_j,∀j∈J.

Proof. The proof follows essentially the concentration compactness principle of P.L. Lions [5] because we have the reverse Hölder type inequalities of Lemma 5.

Indeed, the second statement of this lemma implies that for all Borelian setsE⊂R^N, one has:

ν(E)c_μμ(E)

p∗

p+. (13)

Since the setD= {x∈R^N: μ({x}) >0}is at most countable becauseμ∈M(R^N), thereforeD= {x_j, j∈J}and b_j=. μ({x_j})satisfiesμ

j∈Jb_jδ_xj.

Relation (13) implies thatνis absolutely continuous with respect toμ, i.e.,νμand ν(B(x, r))

μ(B(x, r))c_μμ

B(x, r)^p∗

p+−1

,

provided thatμ(B(x, r))=0 (remember thatp^∗> p₊). Thus, we have:

ν(E)=

E rlim→0

ν(B(x, r)) μ(B(x, r))dμ(x), and

Dμν(x)=lim

r→0

ν(B(x, r))

μ(B(x, r))=0, μa.e. onR^N\D.

Setting a_j=D_μν(x_j)b_j, relation (13) implies thatν has only atoms that are given by{x_j}, that we have already get. 2

Letϕ∈C_c^∞(R^N),ϕ(x_j)=1,ϕ_∞=1. Then, using statement (1) of this corollary and relation (11), we have Sa

pp∗+

j S

|ϕ|^p∗dν ^p_p∗⁺

N

i=1

1 pi

|ϕ|^pidμi. (14)

We shall consider φ ∈ C_c^∞(R^N), 0 φ1, support(φ)⊂B(0,1), φ(0)=1. We fix j ∈J and set x_j = (x_j,1, . . . , x_j,N), q_i =p_ip^∗/(p^∗−p_i), i =1, . . . , N. Then α_i .

= _q¹_i satisfy _N

k=1α_k−α_iq_i =0. For ε >0, we define, for everyz∈R^N,z=(z1, . . . , zN):

φ_ε(z)=φ

z₁−x_j,1

ε^α1 , . . . ,z_N−x_j,N ε^αN

. (15)

Thus we have:

∂φ_ε

∂x_i

^qi= ∂φ

∂x_i

^qi(z)dz (16) and then

∂φε

∂xi

^pi|v|^pi ∂φ

∂xi

^qidz

1−_p∗^pi

B(xj,maxiε

1q i)

|v|^p∗dz ^pi_p∗

−→ε→00. (17)

Lemma 12.Letxj∈Dandφεbe the function defined above associated toxj. Then:

Sa

p+ p∗

j lim

ε→0lim

n

N

i=1

1 p_i

φ_ε^pi

∂v_n

∂x_i ^pidx.

Proof. Since 0φε1 then

φ^pε^∗|vn|^p∗dx1. From Corollary 4 of Lemma 3, it follows S

φ_ε^p∗|v_n|^p∗dx ^p_p∗⁺

N

i=1

1 p_i ∂

∂x_i(φ_εv_n)

^pi. (18) From relation (17), we have

εlim→0 ∂φ_ε

∂x_i

^pi|v|^pidx=0. (19)

Since

n→+∞lim ∂φ_ε

∂x_i

^pi|vn−v|^pidx=0, (20)

then one has:

εlim→0lim

n

N

i=1

1 p_i ∂

∂x_i(φεvn)

^pidx=lim

ε→0lim

n

N

i=1

1 p_i ∂v_n

∂x_i

^piφ_ε^pidx. (21)

From relations (18) and (21), knowing that|v_n|^p∗|v|^p∗+ν(see Lemma 4), we obtain Sa

pp∗+

j lim

ε→0lim

n

N

i=1

1 p_i

φ_ε^pi

∂vn

∂x_i

^pidx. 2

Lemma 13.Assume that N

i=1

1 p_i

∂v_n

∂x_i

^piμ˜ inM⁺ R^N

.

Then

(1) For allj ∈J,Sa

pp∗+

j limε→0μ(support˜ φ_ε)(one has supportφ_ε⊂B(x_j,maxiε

1 qi)).

(2) ˜μSν^pp∗+ +P (∇v).

(3) S=limn→+∞P (∇vn)˙= ˜μ +μ_∞P (∇v)+Sν^pp∗+ +μ_∞. Proof. From Lemma 12, sinceφε^piφεand

limn

N

i=1

1 pi

φ_ε^pi

∂vn

∂xi

^pidx

φ_εdμ,˜ one obtains

Sa

pp∗+

j lim

ε→0

φ_εdμ˜lim

ε→0μ˜

B

x_j; max

1iN

ε

1 qi

. (22)

This shows that{x_j}j∈J are all atomic points ofμ˜ and since_N

i=1 1

pi|_∂x^∂v_i|^piis orthogonal to the atomic part ofμ, one˜ deduces from relation (22) that

˜

μS

j∈J

a

pp∗+

j δ_xj+ N

i=1

1 p_i

∂v

∂x_i

^pi. (23)

This implies in particular that:

˜μS

j∈J

a

p+ p∗

j +P (∇v). (24)

Since ^p_p⁺_∗ <1 one has

j∈J

a_j ^p_p∗⁺

j∈J

a

pp∗+

j . (25)

Asν=

j∈Jajδx_j, it holds ν =

j∈J

aj, (26)

which means, combining relations (24) to (26), that:

˜μSν^pp∗+ +P (∇v).

For the last statement, we argue as before:

S=lim

n P (∇v_n)

= lim

R→+∞lim

n

R^N

(1−ψ_R) N

i=1

1 p_i

∂v_n

∂x_i

^pidx+ lim

R→+∞lim

n

ψ_R

N

i=1

1 p_i

∂v_n

∂x_i ^pidx,

whereψ_R=1 on|x|> R+1, 0ψ_R1,ψ_R=0 if|x|< R,ψ_R∈C(R).

By the definition ofμ, one has:˜

R→+∞lim lim

n

(1−ψR) N

i=1

1 p_i

∂v_n

∂x_i

^pidx=lim

R

(1−ψR)dμ˜ = ˜μ, and (see Lemma 8):

R→+∞lim lim

n

ψ_R

N

i=1

1 pi

∂vn

∂xi

^pidx=μ_∞,

thus, by the preceding statements:

S= ˜μ +μ_∞P (∇v)+Sν^pp∗+ +μ_∞. 2

Lemma 14.Ifvp^∗<1thenν =1,ν_∞=0andv=0.

Proof. From Lemma 10, we know that Sν

pp∗+

∞ μ_∞.

And by Corollary 4 of Lemma 3, we have Sv^p_p⁺∗ P (∇v).

From the last statement of Lemma 13 and the above inequalities we deduce that:

SS

v^p_p^∗∗^p_p∗⁺

+ ν^pp∗+ +ν

pp∗+

∞ .

Thus we obtain, due to Lemma 11, that v^p_p^∗∗^p_p∗⁺

+ ν^pp∗+ +ν

p+

∞p∗

1=

v^p_p^∗∗+ ν +ν_∞^p_p∗⁺ .

Using the inequality

v^p_p^∗∗+ ν +ν_∞^p_p∗⁺

v

pp∗+

p^∗ + ν^pp∗+ +ν

pp∗+

∞ ,