Extremal functions for Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities

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Extremal functions for Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities

Jean Dolbeault & Maria J. Esteban

We consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities and weighted logarithmic Hardy inequalities which have been obtained recently as a limit case of the first ones. We discuss the ranges of the parameters for which the optimal constants are achieved by extremal functions. The comparison of these optimal constants with the optimal constants of Gagliardo-Nirenberg interpolation

inequalities and Gross’ logarithmic Sobolev inequality, both without weights, gives a general criterion for such an existence result in some particular cases.

Keywords.Sobolev spaces; Hardy-Sobolev inequality; Caffarelli-Kohn-Nirenberg inequality; logarithmic Hardy inequality; extremal functions; Kelvin transformation;

Emden-Fowler transformation; radial symmetry; symmetry breaking; existence;

compactness –AMS classification (2000):49J40; 46E35; 26D10; 58E35

1. Introduction

In this paper we discuss the existence of extremal functions in two families of interpolation inequalities introduced in [2, 4]: some of the Caffarelli-Kohn-Nirenberg inequalities and weighted logarithmic Hardy inequalities. By extremal functions, we mean functions for which the inequalities, written with their optimal constants, become equalities. Existence of extremal functions is a crucial issue for the study of several qualitative properties like expressions of the best constants or symmetry breaking properties of the extremal functions. Before stating our results, let us recall the two families of inequalities in which we are interested.

1.1. Caffarelli-Kohn-Nirenberg interpolation inequalities

Let 2^∗ := ∞ if d = 1, 2, and 2^∗ := 2d/(d−2) if d ≥ 3. Define ϑ(p, d) :=

d(p−2)/(2p) and consider the spaceD^1,2_a (R^d) obtained by completion ofD(R^d\{0}) with respect to the norm u7→ k |x|^−a∇uk²_L2(R^d). Under the restriction θ >1/2 if d= 1, notice that θ ∈ [ϑ(p, d),1) for a given p∈ [2,2^∗) if and only if θ ∈ [0,1), p∈[2, p(θ, d)] withp(θ, d) := 2d/(d−2θ). Leta_c:= (d−2)/2.

Theorem 1.1. [2]Let d≥1. For any p∈ [2,2^∗] if d≥3 or p∈[2,2^∗) if d= 1, 2, for any θ ∈ [ϑ(p, d),1] with θ > 1/2 if d = 1, there exists a positive constant CCKN(θ, p, a) such that

Z

R^d

|u|^p

|x|^{b p} dx ²_p

≤C_CKN(θ, p, a) Z

R^d

|∇u|²

|x|²^a dx ^θ Z

R^d

|u|²

|x|^{2 (a+1)} dx ^1−θ

(1.1) for any u∈ D^1,2_a (R^d). Here a, b and p are related by b = a−a_c+d/p, with the restrictionsa≤b≤a+ 1if d≥3,a < b≤a+ 1 ifd= 2 anda+ 1/2< b≤a+ 1

1

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2 J. Dolbeault & M.J. Esteban

if d = 1. Moreover, the constants C_CKN(θ, p, a) are uniformly bounded outside a neighborhood ofa=a_c.

By a transformation of Kelvin type, namely u7→ |x|^{2 (a−a}^c⁾u(x/|x|²), the case a > a_c can be reduced to the casea < a_c. See [6] for details. For simplicity of the statements, we shall therefore assume thata < a_c.

The caseθ= 1,p∈[2,2^∗] andd≥3 has been widely discussed in the literature:

see [3, 5, 6, 8]. The caseθ <1 has been much less considered.

When extremal functions are radially symmetric, they are explicitly known: see [4, 7]. On the other hand,symmetry breaking, which means that extremal functions are not radially symmetric, has been established in [4] whend≥2 and

ϑ(p, d)≤θ <Θ(a, p, d) if a≥¯a(p, d) and ϑ(p, d)≤θ≤1 if a <¯a(p, d), with Θ(a, p, d) := _{32 (d−1)}^p−2 _p

(p+ 2)²(d²+ 4a²−4a(d−2))−4p(p+ 4) (d−1) and ¯a(p, d) := ^d−2₂ −2√

d−1/p

(p−2)(p+ 2). This region extends the one found forθ= 1 in [6, 8].

Finding extremal functions of (1.1) amounts to proving the existence of minimizers for the following variational problem

1

CCKN(θ, p, a) = inf

u∈Da^1,2(R^d)\{0}

k|x|^−a∇uk²_L2^θ(R^d)k|x|^−(a+1)uk^{2 (1−θ)}_L2(R^d)

k|x|^−buk²_L_p₍

R^d)

.

1.2. Weighted logarithmic Hardy inequalities

In [4], a new class of inequalities has been considered. These inequalities can be obtained from (1.1) by takingθ=γ(p−2) and passing to the limit asp→2+. Theorem 1.2. [4]Let d≥1,a < a_c,γ ≥d/4 andγ > 1/2 if d= 2. Then there exists a positive constant C_WLH(γ, a) such that, for any u∈ D^1,2_a (R^d) normalized byR

R^d|x|^{−2 (a+1)}|u|²dx= 1, we have Z

R^d

|u|²

|x|^{2 (a+1)} log

|x|^{2 (a}^c^−a)|u|²

dx≤2γ log

CWLH(γ, a) Z

R^d

|∇u|²

|x|²^a dx

. (1.2) Moreover, the constantsCWLH(γ, a)are uniformly bounded outside a neighborhood ofa=ac.

For this problem, a symmetry breaking result similar to the one of [3, 8] has been established in [4] for anyγ <1/4 + (a−ac)²/(d−1) when d≥2 anda <−1/2.

Finding extremal functions of (1.2) amounts to proving the existence of minimizers for the following variational problem

1

CWLH(γ, a) = inf

u∈D^1,2_a (R^d) k|x|^−(a+1)|u|k_{L2 (R}d)= 1

k|x|^−a∇uk²_L2(R^d)

e

1 2γ

R

Rd

|u|2

|x|2 (a+1) log(^|x|^{2 (ac−a)}^|u|²)^dx.

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1.3. Main results

Our aim is to prove existence of the extremal functions for inequalities (1.1) and (1.2). We shall assume that CCKN(θ, p, a) and CWLH(γ, a) are optimal, i.e.

take their lowest possible value. Cases of optimality among radial functions and further considerations on symmetry breaking will be dealt with in [7]. Existence of extremal functions for (1.1) has been studied in various papers in case θ = 1:

see primarily [3] and references therein for details. In the case of radial functions, whenθ <1 andd≥1, existence of extremal functions has been established in [4]

for anyθ > ϑ(p, d). Still in the radial case, similar results hold for (1.2) if d≥1 andγ >1/4. Notice that nonexistence of extremal functions has been proved in [4]

ford = 1 and θ =ϑ(p, d). Nonexistence of extremal functions without symmetry assumption has also been established in [3] ford ≥3,θ = 1 and a= b <0. Our main result goes as follows.

Theorem 1.3. Let d≥2 and assume thata∈(−∞, ac).

(i) For any p∈(2,2^∗)and any θ ∈ (ϑ(p, d),1),(1.1) admits an extremal function in D^1,2_a (R^d). Moreover there exists a continuous function a^∗: (2,2^∗)→ (−∞, ac) such that (1.1) also admits an extremal function in D^1,2_a (R^d) if θ=ϑ(p, d)anda∈(a^∗(p), ac).

(ii) For any γ > d/4,(1.2) admits an extremal function in D_a^1,2(R^d). Moreover there exists a^∗∗ ∈(−∞, a_c)such that (1.2) also admits an extremal function inD^1,2_a (R^d)if γ=d/4,d≥3anda∈(a^∗∗, a_c).

As we shall see below, the optimal constant whenp= 2,θ∈(0,1), is (a−ac)⁻²^θ and it is never achieved: in this case there are no extremal functions inD_a^1,2(R^d).

For a given p∈ (2,2^∗), the case θ =ϑ(p, d) deserves a more detailed analysis.

Consider the following sub-family of Gagliardo-Nirenberg interpolation inequalities, which have been extensively studied in the context of nonlinear Schr¨odinger equations (see for instance [11]),

kuk²_Lp(R^d)≤C_GN(p)k∇uk²_L₂^ϑ(p,d)₍

R^d) kuk2 (1−ϑ(p,d))

L²(R^d) ∀u∈H¹(R^d).

If uis a radial minimizer for 1/CGN(p), define un(x) := u(x+ne) for some e ∈ S^d−1. It is straightforward to check thata ϑ(p, d) + (a+ 1) (1−ϑ(p, d)) =b. Since k|x|^−a∇unk_L2(R^d)∼n^−ak∇uk_L2(R^d),k|x|^−(a+1)unk_L2(R^d)∼n^−(a+1)kuk_L2(R^d) and k|x|^−bunk_Lp(R^d) ∼n^−bkuk_Lp(R^d), it follows that CGN(p) ≤ CCKN(ϑ(p, d), p, a). A more careful expansion actually shows that

1

CCKN(ϑ(p, d), p, a) ≤k|x|^−a∇unk²_L^ϑ(p,d)2(R^d) k|x|^−(a+1)unk2 (1−ϑ(p,d)) L²(R^d)

k|x|^−bunk²_L_p₍

R^d)

= 1

CGN(p) 1 +Rn⁻²+O(n⁻⁴) asn→ ∞, for some real constantR, that can be explicitly computed:

R=R1

k|x|uk²_L2(R^d)

kuk²_L2(R^d)

+R0 (1.3)

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whereR0andR1are polynomials of degree two in terms ofa, with finite coefficients depending on p, d (but not on t). For given d ≥ 2 and p ∈ (2,2^∗), a sufficient condition forR<0 is that bothR₁andR₀ are negative, which defines an explicit interval in (−∞, a_c) for which we know thatC_GN(p)<C_CKN(ϑ(p, d), p, a). This will be discussed in Section 5.

Similar results can be proved for (1.2). In that case, we shall consider Gross’

logarithmic Sobolev inequality in Weissler’s scale invariant form (see [9, 12]) e^d²^R^R^d^|u|²^log^|u|²^dx≤CLSk∇uk²_L2(R^d) ∀u∈H¹(R^d) such thatkuk_L2(R^d)= 1, where CLS = 2/(π d e). With (un)n as above and u(x) = (2π)^−d/4 exp(−|x|²/4), we find thatC⁻¹_WLH ≤C⁻¹_LS+O(n⁻²).

In the cases θ =ϑ(p, d) and γ =d/4, if either CCKN(ϑ(p, d), p, a) =CGN(p) or CWLH(d/4, a) = CLS, we have readily found a non relatively compact minimizing sequence. This indicates the possibility of non-existence of extremal functions. On the opposite, if we have strict inequalities, we can expect an existence result and this is indeed the case.

Theorem 1.4. Under the assumptions of Theorem 1.3,

(i) if θ = ϑ(p, d) and CGN(p) < CCKN(θ, p, a), then (1.1) admits an extremal function inD^1,2_a (R^d),

(ii) if γ = d/4, d ≥ 3, and CLS < CWLH(γ, a), then (1.2) admits an extremal function in D^1,2_a (R^d). Additionnally, if a∈(a?, ac)with a?:=ac−√

Λ? and Λ?:= (d−1)e(2^d+1π)^−1/(d−1)Γ(d/2)^2/(d−1), thenCLS<CWLH(d/4, a).

In case (i), ford≥3 and p= 2^∗, according to [3], it is known thatCGN(2^∗) = CCKN(1,2^∗, a) for anya≤0. Extremal functions exist for anya≥0 and are radial, up to translations. The casea= 0 corresponds to the celebrated extremal functions of Aubin and Talenti for Sobolev’s inequality.

The criteria of Theorem 1.4 are sharp, in the following sense. Consider the case (i).

If for some a0 ∈ (−∞, ac), (1.1) admits an extremal function in D^1,2_a (R^d) with a = a0 and θ = ϑ(p, d), then for any a ∈ (a0, ac), by considering an extremal function corresponding to a0 as a test function for the inequality corresponding toa, we realize thatCCKN(ϑ(p, d), p, a)>CCKN(ϑ(p, d), p, a0). Choose now

¯

a:= inf{a∈(¯a, ac) : CGN(p)<CCKN(ϑ(p, d), p, a)}.

If ¯a > −∞, then (1.1) admits an extremal function for any a > ¯a and admits no extremal function for any a < ¯a. Similar observations hold in case (ii). See Section 5 for further comments and the proof of the sufficient condition forCLS<

CWLH(d/4, a).

This paper is organized as follows. We shall first reformulate (1.1) and (1.2) in cylindrical variables using the Emden-Fowler transformation and state some preliminary results. Sections 3 and 4 are devoted to the proofs of Theorems 1.3 and 1.4. In Section 5, we shall discuss sufficient conditions forRgiven by (1.3) to be negative and compare the results of Theorems 1.3 (i) and 1.4 (i) whenθ=ϑ(p, d).

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2. Observations and preliminary results

It is very convenient to reformulate the Caffarelli-Kohn-Nirenberg inequality in cylindrical variables as in [3]. By means of the Emden-Fowler transformation

s= log|x| ∈R, ω= x

|x| ∈S^d−1, y= (s, ω), v(y) =|x|^a^c^−au(x), Inequality (1.1) foruis equivalent to a Gagliardo-Nirenberg-Sobolev inequality on the cylinderC:=R×S^d−1:

kvk²_Lp(C)≤CCKN(θ, p, a)

k∇vk²_L2(C)+ Λkvk²_L2(C)

^θ

kvk^{2 (1−θ)}_L2(C) ∀v∈H¹(C) (2.1) with Λ := (ac−a)². Similarly, withw(y) =|x|^a^c^−au(x), Inequality (1.2) is equivalent to

Z

C

|w|²log|w|²dy≤2γ logh

CWLH(γ, a)

k∇wk²_L2(C)+ Λi

for any w ∈ H¹(C) such that kwk_L2(C) = 1. When w is not normalized in L²(C), this last inequality can also be written as

kwk²_L2(C)

CWLH(γ, a) exp

1 2γ

Z

C

|w|²

kwk²_{L2 (C)} log _|w|2

kwk²_{L2 (C)}

dy

≤ k∇vk²_L2(C)+ Λkvk²_L2(C), (2.2) for anyw∈H¹(C). We shall denote byC^∗_CKN(θ, p, a) and C^∗_WLH(γ, a) the optimal constants among radial functions for (1.1) and (1.2) respectively. Radial symmetry for (1.1) and (1.2) means that there are minimizers ofE_θ andF_γ depending only ons. In such a case,C_CKN(θ, p, a) =C^∗_CKN(θ, p, a) andC_WLH(γ, a) =C^∗_WLH(γ, a).

Radial optimal functions are explicit and the values of the optimal constants, C^∗_CKN(θ, p, a) andC^∗_WLH(γ, a), have been computed in [4]:

CCKN(θ, p, a)≥C^∗_CKN(θ, p, a) =C^∗_CKN(θ, p, ac−1) Λ^p−2^2p^−θ CWLH(γ, a)≥C^∗_WLH(γ, a) =C^∗_WLH(γ, ac−1) Λ⁻¹⁺⁴¹^γ

(2.3)

where Λ = (a−ac)², C^∗_CKN(θ, p, ac−1)

=h

2π^d/2 Γ(d/2)

i−^p−2_p h _(p−2)2

2+(2θ−1)p

i^p−2₂_p h₂₊₍₂_θ−1)p

2p θ

i^θh

4 p+2

i^6−p₂_p

Γ(p−2² +¹₂)

√πΓ(p−2² ) ^p−2_p

,

C^∗_WLH(γ, ac−1) = ₄¹_γ ⁽⁴^γ−1)

4γ−1 4γ

(2π^d+1e)⁴¹^γ

[Γ(d/2)]²¹^γ ifγ > ¹₄

and C^∗_WLH(¹₄, a_c−1) = ^[Γ(d/2)]₂_π_d+1_e² ifγ=¹₄ . Symmetry breaking means that Inequalities in (2.3) are strict.

In case (i), forp= 2, it is clear from (2.1) thatC_CKN(θ,2, a)≥Λ^−θ. Letv∈H¹(C) be a function depending only on s such that kvk_L2(C) = 1 and define v_n(y) :=

n⁻¹v(s/n) for anyn≥1,y= (s, ω)∈ C. It is therefore straightforward to observe

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that lim_n→∞(k∇vnk²_L2(C)+ Λ)^θ = Λ^θ = 1/C_CKN(θ,2, a) = 1/C^∗_CKN(θ,2, a). From (2.1), we also read that equality cannot hold for a nontrivial functionv.

The following elementary estimates will also be useful in the sequel.

Lemma 2.1. For anyx,y>0 and any η∈(0,1), we have:

(i) (1 +x)^η(1 +y)^1−η≥1 +x^ηy^1−η, with strict inequality unless x=y,

(ii) ηx^1/η+ (1−η)y^1/(1−η)≥x y, with strict inequality unlessx=yandη= 1/2.

Proof. In case (i), let f(η) :=η log(1 +x) + (1−η) log (1 +y)−log 1 +x^ηy^1−η . We observe thatf(0) =f(1) = 0 and, moreover,f(1/2)>0, since (1 +x) (1 +y)>

(1 +√

x y)²and (1 +x^ηy^1−η)²f⁰⁰(η) =−x^ηy^1−η(log(_y^x))²<0, at least ifx6=y. This proves assertion (i).

In case (ii), letf(η) :=ηx^1/η+ (1−η)y^1/(1−η)−x y. We observe thatf(1/2)≥0 (with strict inequality unlessx=y) and

f⁰⁰(η) =η⁻³x^1/η|logx|²+ (1−η)⁻³y^1/(1−η)|logy|²>0,

thus proving the second assertion.

The functionalw7→R

C|w|² log |w|²/kwk²_L2(Ω)

dy can be seen as the limit case of w7→ kwk²_Lp(C). At least from the point of view of H¨older’s inequalities, this is indeed the case, and the following estimate will be useful in the sequel.

Let Ω be an arbitrary measurable set and consider H¨older’s inequality,kwk_Lq(Ω)≤ kwk^η_L2(Ω)kwk^1−η_Lp(Ω)withη= 2 (p−q)/(q(p−2)) for anyqsuch that 2≤q≤p≤2^∗. Forq= 2, this inequality becomes an equality, withη= 1, so that we can differen- tiate with respect toqatq= 2 and obtain

Z

Ω

|w|²log _|w|2

kwk²

L2 (Ω)

≤ p

p−2kwk²_L2(Ω) log_kwk²

Lp(Ω)

kwk²

L2 (Ω)

. (2.4)

3. Proof of Theorem 1.3

In case (i), letp∈(2,2^∗) andθ ∈(ϑ(p, d),1) . In case (ii), letγ > d/4. Consider sequences (v_n)_n and (w_n)_n of functions in H¹(C), which respectively minimize the functionals

Eθ[v] :=

^θ

kvk^{2 (1−θ)}_L2(C) ,

Fγ[w] :=

k∇wk²_L2(C)+ Λ exp

−₂¹_γ Z

C

|w|² log|w|²dy

,

under the constraints kvnkL^p(C) = 1 and kwnkL²(C) = 1 for any n ∈ N. We shall first prove that these sequences are relatively compact and converge up to translations and the extraction of a subsequence towards minimizers if they are bounded in H¹(C). Next we will establish thea prioriestimates in H¹(C) needed for the proof of Theorem 1.3. Under restrictions ona, these a priori estimates are also valid for θ=ϑ(p, d) orγ=d/4 and also give an existence result for minimizers.

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3.1. Convergence of bounded minimizing sequences

Consider C as a manifold embedded in R^d+1 and denote by Br(y) the ball in R^d+1 with radiusrcentered aty. From [3, Lemma 4.1], which is an adaptation of [10, Lemma I.1, p. 231], we have the following lemma.

Lemma 3.1. [10, 3]Let r >0 andq∈[2,2^∗). If (fn)n is bounded inH¹(C)and if lim sup

n→∞

Z

Br(y)∩C

|fn|^qdy= 0 for anyy∈ C, thenlim_n→∞kfnk_Lp(C)= 0for any p∈(2,2^∗).

As a consequence of this result and of the convexity estimates of Lemma 2.1, the relative convergence of bounded sequences is a rather straightforward issue.

Proposition 3.2. Let d ≥ 2, p ∈ (2,2^∗) and θ ∈ [ϑ(p, d),1). Let (vn)n be a minimizing sequence for Eθ such that kvnkL^p(C) = 1 for any n ∈ N. If (v_n)_n is bounded inH¹(C), then(v_n)_n is relatively compact and converges up to translations and the extraction of a subsequence to a functionv∈H¹(C)such thatkvk_Lp(C)= 1 andE_θ[v] = 1/C_CKN(θ, p, a).

Proof. Up to translations and the extraction of a subsequence, (vn)n weakly converges in H¹(C), strongly in L²_loc∩L^p_loc(C) and a.e. inCtowards a functionv∈H¹(C).

By Lemma 3.1,vis non-trivial andkvk_Lp(C)6= 0.

Up to the extraction of subsequences, using

lim_n→∞kv_nk²_L2(C)=kvk²_L2(C)+ lim_n→∞kv_n−vk²_L2(C), lim_n→∞k∇vnk²_L2(C)=k∇vk²_L2(C)+ lim_n→∞k∇vn− ∇vk²_L2(C), withη=θ,

x= lim

n→∞

k∇vn− ∇vk²_L2(C)+ Λkvn−vk²_L2(C)

and y= lim

n→∞

kvn−vk²_L2(C)

kvk²_L2(C)

,

by Lemma 2.1 (i), we find that 1

CCKN(θ, p, a) = lim

n→∞Eθ[vn]≥ Eθ[v] + lim

n→∞Eθ[vn−v]

≥ 1

CCKN(θ, p, a)

kvk²_Lp(C)+ lim

n→∞kvn−vk²_Lp(C)

By the Brezis-Lieb Lemma (see [1, Theorem 1]), we know that 1 =kvnk^p_Lp(C)=kvk^p_Lp(C)+ lim

n→∞kvn−vk^p_Lp(C).

The functionf(z) :=z^2/p+ (1−z)^2/p is strictly concave so that for anyz∈[0,1], f(z)≥1 with strict inequality unless z = 0 orz = 1. Applied with z =kvk^p_Lp(C), this proves that kvk_Lp(C) = 1. Since lim_n→∞E_θ[v_n] ≥ E_θ[v], we know that v is a nontrivial extremal function for (1.1). This completes the proof.

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Proposition 3.3. Letd≥2 andγ≥d/4with strict inequality if d= 2. Let(w_n)_n be a minimizing sequence forF_γ such thatkw_nk_L2(C)= 1for anyn∈N. If(w_n)_n is bounded inH¹(C), then(w_n)_n is relatively compact and converges up to translations and the extraction of a subsequence to a functionw∈H¹(C)such thatkwk_L2(C)= 1 andFγ[w] = 1/CWLH(γ, a).

Proof. Consider now the sequence (wn)n and denote bywits weak limit in H¹(C), after translations and the extraction of a subsequence if necessary.

By (2.2) and (2.4), we know that

F_γ[w_n]≥C_WLH(γ, a)⁻¹kw_nk⁻

p γ(p−2)

L^p(R^d)

for anyp∈ (2,2^∗). If w≡0, then limn→∞kwnk_Lp(R^d) = 0 by Lemma 3.1, which contradicts the fact that (w_n)_n is a minimizing sequence andC_WLH(γ, a) is finite.

Hence we havekwk²_L2(C)6= 0.

By the Brezis-Lieb lemma and by semi-continuity, we know that 1

CWLH(γ, a) = lim

n→∞Fγ[wn]

≥h

k∇wk²_L2(C)+ Λkwk²_L2(C)+ lim

n→∞ k∇wn−wk²_L2(C)+ Λkwn−wk²_L2(C)

i

exp

−₂¹_γ lim

n→∞

Z

C

|wn|²log|wn|²dy

up to the extraction of subsequences. We may apply (2.2) to w and w_n−w. Let η := kwk²_L2(C), so that lim

n→∞kw_n−wk²_L2(C) = 1−η. We know that η ∈ (0,1]. If η <1, with

x= exp

1 2γ

R

C|w|² log

|w|² kwk²

L2 (C)

dy

,

y= exp

1

2γ lim_n→∞R

C|wn−w|²log _|w

n−w|² kwn−wk²_{L2 (C)}

dy

,

we can write 1

CWLH(γ, a) ≥

ηx¹^η + (1−η)y^1−η¹ exp

−₂¹_γ lim

n→∞

Z

C

|wn|² log|wn|²dy

.

We may then apply Lemma 2.1 (ii) and find that 1

CWLH(γ, a)≥ x y

CWLH(γ, a) exp

−₂¹_γ lim

n→∞

Z

C

|wn|²log|wn|²dy

.

According to [1, Theorem 2], we have Z

C

|w|² log|w|²dy+ lim

n→∞

Z

C

|wn−w|² log|wn−w|²dy= lim

n→∞

Z

C

|wn|²log|wn|²dy

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and, as a consequence, it follows that 1

C_WLH(γ, a) ≥ 1

C_WLH(γ, a) exph

−₂¹_γ η logη+ (1−η) log(1−η)i

This proves thatη= 1. Using (2.4), we have lim

n→∞

Z

C

|w_n−w|² log|w_n−w|²dy= 0.

Hencewis an extremal function, which completes the proof.

The remainder of this section is devoted to the a priori estimates which are needed to establish the boundedness of minimizing sequences in H¹(C).

3.2. A priori estimates for Caffarelli-Kohn-Nirenberg inequalities Lemma 3.4. Assume that d≥2,p∈(2,2^∗) andθ ∈(ϑ(p, d),1). For any ε >0, there exists anA >0 such that, for any v ∈H¹(C)with kvk_Lp(C)= 1 andEθ[v]≤

1+ε

CCKN(θ,p,a), thenkvkH¹(C)≤A.

If d ≥ 3, there exists a positive function a^∗_ε : (2,2^∗) → (−∞, ac) such that, whenevera∈(a^∗_ε(p), ac), the same conclusion holds if θ=ϑ(p, d).

Proof. By H¨older’s and Sobolev’s inequalities, for anyp∈[2,2^∗], we have kvk²_Lp(C)≤

kvk^ϑ(d,p)_L₂∗

(C)kvk^1−ϑ(d,p)_L2(C)

²

≤h

CCKN(1,2^∗, ac−1)

k∇vk²_L2(C)+ Λ0kvk²_L2(C)

iϑ(d,p)kvk2 (1−ϑ(d,p)) L²(C)

where Λ₀ := a²_c. Let t := k∇vk²_L2(C)/kvk²_L2(C) and write Λ = Λ(a) for brevity.

Because of the conditionEθ[v]≤(1 +ε)/CCKN(θ, p, a), we have

(t+ Λ)^θ=E_θ[v]kvk²_Lp(C)

kvk²_L2(C)

≤ (1 +ε)kvk²_Lp(C)

C_CKN(θ, p, a)kvk²_L2(C)

≤(1 +ε)(C_CKN(1,2^∗, a_c−1))^ϑ(d,p)

CCKN(θ, p, a) (t+ Λ0)^ϑ(d,p) . This proves thatt is bounded ifθ > ϑ(p, d).

Ifd≥3 andθ=ϑ(p, d), let

κ^θ_ε:= (1 +ε)C^∗_CKN(1,2^∗, a_c−1)^θ/(C^∗_CKN(θ, p, a_c−1)).

SinceCCKN(1,2^∗, ac−1), the best constant corresponding to Sobolev’s critical em- bedding D^1,2(R^d) ,→ L²^∗(R^d), is achieved among radial functions, by (2.3) the above condition reads

t+ Λ≤κεΛ

1 d−1 0 Λ¹⁻¹^d

t+ Λ0 ,

which again shows that t is bounded if a∈ (a^∗_ε(p), a_c), for somea^∗_ε(p) such that a_c−a^∗_ε(p)>0 is not too big.

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Sincekvk²_L2(C)(t+ Λ)^θ=Eθ[v]≤(1 +ε)/C_CKN(θ, p, a),kvkL²(C)andk∇vkL²(C)= tkvkL²(C) are also bounded as soon as t is bounded, thus establishing a bound in H¹(C).

Ifd= 2, letθ > ϑ(p,2) = ^p−2_p . For a choice ofqsuch thatη= ^q_p^(p−2)_(q−2)< θ,i.e.for q > _2−p^{p θ}_(1−θ), by H¨older’s inequality we have kvk_Lp(C)≤ kvk^η_Lq(C)kvk^1−η_L2(C). Hence, from

(t+ Λ)^θ≤ (1 +ε)kvk²_Lp(C)

CCKN(θ, p, a)kvk²_L2(C)

≤(1 +ε)(CCKN(1, q, a))^η

CCKN(θ, p, a) (t+ Λ)^η , we deduce thattis bounded. As above, this proves thatvis bounded in H¹(C).

A more careful investigation actually provides an explicit expression ofa^∗_ε(p). We get an upper bound fortif we simultaneously have

κεΛ

1 d−1

0 Λ¹⁻^d¹ and Λ< κεΛ

1 d−1

0 Λ¹⁻¹^dΛ0, that is

Λ<minn Λ0κ⁻

d d−1

ε ,Λ0κ^d_εo . Hence, ford≥3,t is bounded forε >0 small enough if

a > a^∗₀(p) :=ac−ac minn κ⁻

d 2(d−1)

0 , κ

d 2

0

o

. (3.1)

We shall comment on this bound in Section 5.

Proof of Theorem 1.3 (i).Consider a minimizing sequence (vn)n forEθ such that kvnkL^p(C) = 1. For any given ε >0, the condition Eθ[vn]≤ _C ^1+ε

CKN(θ,p,a) is satisfied fornlarge enough. By Lemma 3.4, (vn)nis bounded in H¹(C). By Proposition 3.2, we know that it converges towards a minimizer v ∈H¹(C) withkvk_Lp(C) = 1, up to translations and the extraction of a subsequence. This concludes the proof with a^∗=a^∗₀ given by (3.1).

3.3. A priori estimates for the weighted logarithmic Hardy inequalities Lemma 3.5. Assume that d≥2,γ ≥d/4 and γ >1/2 if d= 2. For any ε >0, there exists anA >0 such that, for anyw∈H¹(C)withkwkL²(C)= 1andFγ[w]≤

1+ε

C_WLH(γ,a), thenkwk_H1(C)≤A.

If d ≥ 3, there exists a^∗∗_ε ∈ (−∞, ac) such that (1.2) also admits an extremal function inD_a^1,2(R^d) ifγ=d/4,d≥3 anda∈(a^∗∗_ε , a_c).

Proof. By (2.1) and (2.4), we find that for anyα < ac and anyp∈(2,2^∗), Z

C

|w|² log _|w|2

kwk²

L2 (C)

dy≤ p

p−2kwk²_L2(C) log_kwk²

Lp(C)

kwk²

L2 (C)

≤ p

p−2kwk²_L2(C) log

CCKN(1, p, α) (t+ Λ(α))

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witht:=k∇wk²_L2(C)/kwk²_L2(C)and Λ(α) := (α−a_c)². Assuming thatkwkL²(C)= 1 andFγ[w]≤(1 +ε)/C_WLH(γ, a), using (2.3) we find that

t+ Λ

CCKN(1, p, α) (t+ Λ(α))₂¹_γ_p−2^p ≤ Fγ[w]≤ 1 +ε

CWLH(γ, a) ≤ (1 +ε) Λ(a)¹⁻^4γ¹ C^∗_WLH(γ, ac−1) , which provides a bound onwin H¹(C) if one of the two following cases:

(i) Ford≥3, if eitherγ > ^d₄ orγ=^d₄ and Λ∈(0, a²_c) is small enough (we choose α= 0,p= 2^∗ so that ₂¹_γ _p−2^p =₄^d_γ ),

(ii) Ifd= 2, for allγ > ¹₂ (we chooseα=−1, andp > ₂⁴_γ−1^γ ).

Proof of Theorem 1.3 (ii).Consider a minimizing sequence (w_n)_n forF_γ such that kwnk_L2(C) = 1. For any given ε >0, the condition Eγ[wn] ≤ _C ^1+ε

WLH(γ,a) is satisfied fornlarge enough. By Lemma 3.5, (wn)n is bounded in H¹(C). By Proposition 3.3, we know that it converges towards a minimizerw∈H¹(C) with kwkL²(C)= 1, up to translations and the extraction of a subsequence. This concludes the proof with a^∗∗= lim inf_ε→0₊ a^∗∗_ε .

4. Proof of Theorem 1.4

This section is devoted to the limit casesθ=ϑ(p, d) orγ=d/4. A sharp criterion for the existence of extremal functions for Caffarelli-Kohn-Nirenberg and weighted logarithmic Hardy inequalities is given by the comparison of their optimal constants with the optimal constants of Gagliardo-Nirenberg and Gross’ logarithmic Sobolev inequalities.

As already noted in the introduction, CGN(p) ≤CCKN(ϑ(p, d), p, a) and CLS ≤ CWLH(d/4, a) for anya∈(−∞, ac). When equality holds, compactness of minimizing sequences is lost, because of translations. Here we shall establish a compactness result for special sequences of functions made of minimizers forC_CKN(θ_n, p, a) and C_WLH(γ_n, a) withθ_n > ϑ(p, d), lim

n→∞θ_n=ϑ(p, d) andγ_n> d/4, lim

n→∞γ_n=d/4.

4.1. Compactness of sequences of extremal functions for Caffarelli- Kohn-Nirenberg inequalities approaching the limit caseθ =ϑ(p, d) Lemma 4.1. Letd≥2,p∈(2,2^∗)anda < a_c. Consider a sequence(θ_n)_nsuch that θn> ϑ(p, d) andlim_n→∞θn=ϑ(p, d). If (vn)n is a sequence of extremal functions for (2.1)written for θ=θn such thatkvnk_Lp(C)= 1 for anyn∈N, then (vn)n is bounded inH¹(C)ifCGN(p)<CCKN(ϑ(p, d), p, a). In that case, (vn)nconverges, up to translations and the extraction of a subsequence, towards a minimizerv∈H¹(C) ofE_ϑ(p,d), under the constraintkvk_Lp(C)= 1.

Proof. For brevity, let us write θ = ϑ(p, d) and recall that θ_n −θ > 0 for all n∈N. Consider first a smooth, compactly supported functionv_εsuch thatE_θ[v_ε]≤

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12 J. Dolbeault & M.J. Esteban 1/C_CKN(θ, p, a) +εandkvεkL^p(C)= 1. We have

lim inf

n→∞

1

C_CKN(θ_n, p, a) ≤lim inf

n→∞ Eθ_n[vε] =Eθ[vε]≤ 1

C_CKN(θ, p, a)+ε for anyε > 0 and can pass to the limit asε→0+. On the other hand, we know from [4] thatCCKN(θn, p, a) is bounded uniformly asn→ ∞, so that

0<lim inf

n→∞

1

C_CKN(θ_n, p, a) ≤ 1

C_CKN(ϑ(p, d), p, a) . (4.1) Consider now the sequence (vn)n of Lemma 4.1. With

tn :=k∇vnk²_L2(C)/kvnk²_L2(C),

the Euler-Lagrange equation satisfied byvn for eachn∈Nreads

−θ_n∆v_n+ (1− θ_n)t_n+ Λ

v_n=C_CKN(θ_n, p, a)⁻¹(t_n+ Λ)^1−θⁿv_n^p−1 on C. As in [3], using the translation invariance of (2.1) in thes-variable, the invariance of the functionalE_θ under rotations onS^d−1, and the fact that v_n is a minimizer, we can assume that vn is nonnegative and achieves its maximum at some fixed, given pointω_∗∈S^d−1. By the maximum principle, we know that −∆vn(0, ω_∗)≥0 and henceMn:=vn(0, ω_∗) =kvnk_L^∞_(C) is such that

M_n^p−2≥C_CKN(θ_n, p, a) (1−θ_n)t_n+ Λ

(t_n+ Λ)^θⁿ⁻¹.

After the extraction of a subsequence, we may assume that (L_n)_n converges and L:= lim_n→∞L_n ∈ ((1−θ_n)C_CKN(θ_n, p, a),+∞] where L_n := M_n^p−2t^−θ_n ⁿ. Let us consider the rescaled functionf_ndefined onC_n:=R×σ_nS^d−1byv_n(x) =M_nf_n(y), wherey =σ_nxand σ²_n = (t_n+ Λ)^1−θⁿM_n^p−2/C_CKN(θ_n, p, a). For any n∈N, the functionf_n is nonnegative, satisfies

−θ_n∆f_n+ (1−θ_n)t_n+ Λ

σ⁻²_n f_n=f_n^p−1

and reaches its maximum value, 1, at the point (0, ω_n), whereω_n =σ_nω_∗. Assume by contradiction that

n→∞lim t_n=∞.

In such a case, we know thatEθ_n[vn] ∼ t^θ_nⁿkvnk²_L2(R^d) so that kvnk²_L2(R^d) ∼t^−θ_n ⁿ andk∇vnk²_L2(R^d)=tnkvnk²_L2(R^d)∼t^1−θ_n ⁿ. Moreover, we haveM_n^p−2=Lnt^θ_nⁿ → ∞ and, by (4.1),σ_n²∼t^1−θ_n ⁿM_n^p−2∼Lntn→ ∞, andfn solves

−θn∆fn+1−θn

Ln

(1 +o(1))fn=f_n^p−1. As a consequence, ∆f_n is locally uniformly bounded.

Next we define on R^d the functions g_n(s,Π_nω) := f_n(s, ω)ρ_n(s,Π_nω), where ω∈σ_nS^d−1and Π_nis the stereographic projection ofσ_nS^d−1ontoR^d−1, considered

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as the tangent plane toS^d−1 at ω_n. The cut-off function ρ_n is smooth and such that ρ_n(x) =ρ(x/log(1 +σ_n)) for anyx∈R^d, with 0≤ρ≤1,ρ≡1 onB₁ and suppρ ⊂ B₂. Locally around (0, ω_n), Π_n converges to the identity while its first and second derivatives converge to 0. Hence we know that

k∇gnk²_L2(R^d)≤ k∇fnk²_L2(Cn)(1 +o(1)) and k∇fnk²_L2(Cn)∼σ^d−2_n M_n⁻²k∇vnk²_L2(C)

asn→ ∞. Altogether, we find that k∇g_nk²_L2(R^d)=O

t^(θ−θⁿ⁾

p p−2

n L

d−2 2 −_p−2² n

which means that (∇gn)_n is bounded in L²(R^d) and (g_n)_nconverges in H¹_loc(R^d) to a constant ifL=∞. In any case, up to the extraction of a subsequence, (g_n)_ncon- verges weakly in H¹_loc(R^d). Since ∆g_n is bounded in L^∞(R^d), by elliptic estimates, (g_n)_n strongly converges in C_loc² (R^d) to a nonnegative function g : R^d → R such that, withθ=ϑ(p, d),

−θ∆g+A g=g^p−1 inR^d, g(0) =kgk_L∞(R^d)= 1,

wheregis constant andA= 0 ifL=∞, andA= (1−θ)/Lotherwise. However, if L=∞, theng≡1 cannot be a solution. This proves thatLis finite andAtakes a finite, positive value. Moreover,kgnk²_L2(R^d)∼σ_n²t⁻¹_n k∇gnk²_L2(R^d)∼ k∇gnk²_L2(R^d) is bounded so that (g_n)_n weakly converges in H¹(R^d) tog6≡0. Hence we get

lim inf

n→∞ Eθ_n[vn] = lim inf

n→∞ (tn+ Λ)^θⁿ^−θEθ[vn]

≥lim inf

n→∞ Eθ[vn] = lim inf

n→∞

M_n² σn²^d/p

k∇fnk²_L2^θ(Cn)kfnk^{2 (1−θ)}_L2(Cn) ≥ k∇gk²_L2^θ(R^d)kgk^{2 (1−θ)}_L₂₍

R^d)

lim_n→∞kf_nk²_L_p₍

R^d)

,

eventually after extraction of a subsequence, where the latter inequality holds by semi-continuity.

Let

E_θ_n_,C_n[f] :=σ_n^{2 (ϑ}ⁿ^−ϑ)

k∇f_nk²_L2(C)+ Λ

σ²_nkf_nk²_L2(C)

^θn

kf_nk^{2 (1−θ}_L2(C)ⁿ⁾

so that E_θ_n[v_n] = E_θ_n_,C_n[f_n]/kf_nk²_Lp(Cn). Because of the change of variables, we know that Inequality (2.1) becomes

E_θ_n_,C_n[f] kfk²_Lp(Cn)

≥ 1

CCKN(ϑn, p, a) ∀f ∈H¹(C_n). (4.2) By the local strong convergence of the sequence (g_n)_n, there exists a sequence (Rn)n with lim_n→∞Rn =∞such that

n→∞lim R

C_n∩B_Rnf_n^pdy kfnk^p_Lp(Cn)

=δ and lim

n→∞

R

C_n∩(R^d\B_4Rn)f_n^pdy kfnk^p_Lp(Cn)

= 1−δ .