Weighted interpolation inequalities: a perturbation approach

approach

Jean Dolbeault · Matteo Muratori · Bruno Nazaret

October 4, 2016

Abstract We study optimal functions in a family of Caffarelli-Kohn-Niren- berg inequalities with a power-law weight, in a regime for which standard symmetrization techniques fail. We establish the existence of optimal func- tions, study their properties and prove that they are radial when the power in the weight is small enough. Radial symmetry up to translations is true for the limiting case where the weight vanishes, a case which corresponds to a well-known subfamily of Gagliardo-Nirenberg inequalities. Our approach is based on a concentration-compactness analysis and on a perturbation method which uses a spectral gap inequality. As a consequence, we prove that optimal functions are explicit and given by Barenblatt-type profiles in the perturbative regime.

Keywords Functional inequalities · Weights · Optimal functions · Best constants·Symmetry ·Concentration-compactness·Gamma-convergence Mathematics Subject Classification (2010) 49J40· 46E35 · 35B06 · 26D10

J. Dolbeault

Ceremade, CNRS UMR n^◦7534 and Universit´e Paris-Dauphine, PSL?, Place de Lattre de Tassigny, 75775 Paris C´edex 16, France. E-mail:dolbeaul@ceremade.dauphine.fr

M. Muratori

Dipartimento di MatematicaFrancesco Brioschi, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. E-mail:matteo.muratori@polimi.it

B. Nazaret

SAMM, Universit´e Paris 1, 90, rue de Tolbiac, 75634 Paris C´edex 13, France.

E-mail:Bruno.Nazaret@univ-paris1.fr

1 Introduction

This paper is devoted to a special class of Caffarelli-Kohn-Nirenberg interpo- lation inequalitiesthat were introduced in [10] and can be written as

kwk_2p,γ ≤C_γ k∇wk^ϑ₂ kwk^1−ϑ_p+1,γ ∀w∈ D(R^d), (1.1) whereD(R^d) denotes the space of smooth functions onR^dwith compact sup- port,Cγ is thebest constant in the inequality,

d≥3, γ∈(0,2), p∈ 1,2^∗_γ/2

with 2^∗_γ:= 2d−γ

d−2 (1.2) and

ϑ:= 2^∗_γ(p−1)

2p 2^∗_γ−p−1= (d−γ) (p−1)

p d+ 2−2γ−p(d−2). (1.3) The norms are defined by

kwk_q,γ:=

Z

R^d

|w|^q|x|^−γdx ^1/q

and kwk_q :=kwk_q,0 .

The optimal constantCγ is determined by the minimization of the quotient Qγ[w] := k∇wk^ϑ₂ kwk^1−ϑ_p+1,γ

kwk_2p,γ .

When γ = 0, Inequalities (1.1) become a particular subfamily of the well-known Gagliardo-Nirenberg inequalities introduced in [30,43]. In that case, optimal functions have been completely characterized in [13]. Gagliardo- Nirenberg inequalities have attracted lots of interest in the recent years: see for instance [28] and references therein, or [3].

We have two main reasons to consider such a problem. First of all, opti- malityamong radial functions is achieved by

w?(x) := 1 +|x|^2−γ−p−1¹ ∀x∈R^d (1.4) up to multiplications by a constant and scalings as we shall see later. It is re- markable that the functionw?clearly departs from standard optimal functions that are usually characterized using the conformal invariance properties of the sphere and the stereographic projection, like for instance in [3, Section 6.10].

If d= 1, it is elementary to prove that optimal functions for (1.1) are of the form (1.4) up to multiplication by constants and scalings. The cased= 2 is not considered in this paper. In any higher dimension d≥3, even with a radial weight of the form|x|^−γ, there is no simple symmetry result that would allow us to identify the optimal functions in terms ofw_?. In other words, it is not known if equality holds in

inf

w∈D^?(R^d)\{0}Q_γ[w] =: C^?_γ−1

=Q_γ[w_?]≥(C_γ)⁻¹:= inf

w∈D(R^d)\{0}Q_γ[w], (1.5)

whereD^?(R^d) denotes the subset ofD(R^d) which is spanned by radial smooth functions,i.e., smooth functions which depend only on|x|. Our main result is a first step in this direction.

Theorem 1.1 Let d≥3. For anyp∈(1, d/(d−2)), there exists a positive γ^∗ such that equality holds in (1.5) for allγ∈(0, γ^∗).

A slightly stronger result is given in Theorem2.1.

We remark that optimal functions forQ_γcan be assumed to be nonnegative and satisfy, up to multiplications by a constant and scalings, the semilinear equation

−∆w+|x|^−γ w^p−w^2p−1

= 0. (1.6)

This will be discussed in Section2. However, the classical result of B. Gidas, W.M. Ni and L. Nirenberg in [31] does not allow us to decide if a positive solution of (1.6) has to be radially symmetric. So far, it is not known yet if the result can be deduced from a symmetrization method either, even for a minimizer of Qγ. We shall say that symmetry breaking occurs if C^?_γ < Cγ. Whether this happens for someγ ∈(0,2) andpin the appropriate range, or not, is an open question.

The symmetry result of Theorem1.1has very interesting consequences, and here is a second motivation for this paper. Let us consider the fast diffusion equation with weight

u_t+|x|^γ∇ · u∇u^m−1

= 0, (t, x)∈R⁺×R^d, (1.7) with initial condition u(t = 0,·) = u₀ ∈ L¹(R^d,|x|^−γdx), u₀ ≥ 0 and m ∈ (m₁,1), where

m1:= 2d−γ−2 2 (d−γ) .

From the point of view of the well-posedness of the Cauchy problem and of the long-time behaviour of the solutions, such an equation, in the porous media case (namely form >1), has been studied in [45,46]. In this paper, we consider the fast diffusion regime (casem <1). In particular, it can be shown that the mass M := R

R^du|x|^−γdx is independent of t. Let us introduce the time-dependent rescaling

u(t, x) =R^γ−dv

(2−γ)⁻¹logR , x R

, (1.8)

withR=R(t) defined by dR

dt = (2−γ)R(m−1)(γ−d)−1+γ, R(0) = 1. The solution is explicit and given by

R(t) =

1 + (2−γ) (d−γ) (m−mc)t(d−γ) (m−mc)¹ where mc:= d−2 d−γ.

After changing variables we obtain that the rescaled function v solves the Fokker-Planck-type equation

v_t+|x|^γ∇ ·h

v∇ v^m−1− |x|^2−γi

= 0 (1.9)

with initial conditionv(t= 0,·) =u0. The convergence of the solution of (1.7) towards a self-similar solution of Barenblatt type as time goes to∞is replaced by the convergence ofv towards a stationary solution of (1.9) given by

B(x) := C+|x|^2−γ_m−1¹ , whereC >0 is uniquely determined by the condition

Z

R^d

B|x|^−γdx=M .

A straightforward computation shows that thefree energy, orrelative entropy F[v] := 1

m−1 Z

R^d

v^m−B^m−mB^m−1(v−B)

|x|^−γdx is nonnegative and satisfies

d

dtF[v(t,·)] =− I[v(t,·)], (1.10) where theFisher informationis defined by

I[v] := m 1−m

Z

R^d

v

∇v^m−1−(2−γ) x

|x|^γ

2

dx .

As a consequence of Theorem 1.1, an elementary computation shows that theentropy – entropy production inequality

(2−γ)²F[v]≤ I[v] (1.11) holds if γ ∈ (0, γ^∗). More precisely, (1.11) is equivalent to (1.1) if we take w=v^m−1/2,p= 1/(2m−1) and perform a scaling. Accordingly, notice that B^m−1/2 is equal to w_? up to a scaling and a multiplication by a constant.

This generalizes toγ∈(0, γ^∗) the results obtained in [13] for the caseγ= 0.

Whether (2−γ)²is the best constant in (1.11) is a very natural question. The answer is yes if γ = 0 and no if γ > 0, as long as symmetry (no symmetry breaking) holds. An answer has been provided in [8, Proposition 11], which relies on considerations on the linearization. Key ideas are provided by the weighted fast diffusion equation whose large time asymptotics are governed by the linearized problem. These asymptotics are studied in [9]. Recent progresses on the issue of symmetry breaking, which partially rely on the present paper, have been achieved in [25].

A consequence of (1.11) is the exponential convergence of the solution v of (1.9) toB.

Corollary 1.2 Letd≥3,m∈(1−1/d,1)andγ∈(0, γ^∗), whereγ^∗is defined as in Theorem1.1forp= 1/(2m−1). Ifvis a solution to(1.9)with nonneg- ative initial datum u0 such that u0 ∈L¹(R^d,|x|^−γdx) and R

R^du^m₀ |x|^−γdx+ R

R^du0|x|^2−2γdx is finite, then

F[v(t,·)]≤ F[u0]e^{−(2−γ)2t} ∀t >0.

The above exponential decay is actually equivalent to (1.11) and henceforth to (1.1) with Cγ =C^?_γ as can be checked by computing _dt^dF[v(t,·)] att = 0.

The free energyF[v] is a measure of the distance betweenv andB. Exactly as in the case γ = 0, one can undo the change of variables (1.8) and write an intermediate asymptotics result based on the Csisz´ar-Kullback inequality.

The method is somewhat classical and will not be developed further in this paper. See for instance [13,28] for more details. To prove Corollary 1.2, one has to show that the massM is conserved along the flow defined by (1.9) and that (1.10) holds: this can be done as in [6] whenγ= 0.

Before entering in the details, let us mention that the case of the porous media equation withm >1 has been more studied than the fast diffusion case m <1. We refer the reader e.g.to [35,33,34] for some recent results relating suitable functional inequalities with the asymptotic properties of the solutions.

Let us introduce some basic notations for the functional spaces. We define the spaces L^q_γ(R^d) and Hp,γ(R^d), respectively, as the space of all measurable functionswsuch thatkwk_q,γis finite and the space of all measurable functions w, with∇wmeasurable, such thatkwk_Hp,γ(Rd):=kwk_p+1,γ+k∇wk₂is finite.

Moreover, we denote as H^?_p,γ(R^d) the subspace of Hp,γ(R^d) spanned by ra- dial functions. A simple density argument shows, in particular, thatD(R^d) is dense inHp,γ(R^d) (Lemma2.2), so that inequality (1.1) holds for any function in H_p,γ(R^d) and, as a consequence, H_p,γ(R^d) is continuously embedded into L^2p_γ (R^d).

The proof of Theorem 1.1is based on a perturbation method which relies on the fact that, by the results in [13], the optimal functions in the caseγ= 0 are radial up to translations. Our strategy is adapted from [26], except that we have no Emden-Fowler type transformation that would allow us to get rid of the weights. This has the unpleasant consequence that a fully developed analysis of the convergence of any optimal function for (1.1) is needed, based on a concentration-compactness method, as γ↓0. We prove that the limit is the radial solution, namely the only one centered at the origin, to the limit problem, although the limit problem is translation invariant. Then we are able to prove that the optimal functions are themselves radially symmetric for γ > 0 sufficiently small. As a consequence,C^?_γ =Cγ for any suchγ, and the optimal functions are all given by (1.4) up to a multiplication by a constant and a scaling. Here are the main steps of our approach.

1. We work with the non-scale-invariant form of (1.1) that can be written as Eγ[w] := 1

2 k∇wk²₂+ 1

p+ 1 kwk^p+1_p+1,γ−Jγ kwk^{2p θ}_2p,γ^γ ≥0, (1.12)

where Jγ denotes the optimal constant and θγ :=d+ 2−2γ−p(d−2)

d−γ−p(d+γ−4) ∈(0,1). (1.13) A simple scaling argument given in Section 2shows that (1.1) and (1.12) are equivalent, and relatesJ_γ andC_γ: see (2.6).

2. In Section 2 we establish the compactness of the embedding of Hp,γ(R^d) into L^2p_γ (R^d), which implies the existence of at least one optimal func- tionw_γ for (1.12). This optimal function solves (1.6) up to a multiplication by a constant and a scaling. Notice that optimal functions for (1.12) are not necessarily unique, even up to multiplication by constants. For simplicity we shall pick one optimal function for eachγ >0, denote it bywγ, but each time we use this notation, one has to keep in mind that it is not a priori granted that wγ is uniquely defined. We also adopt the convention that w0 denotes the uniqueradial minimizer, having a suitably prescribed L^2p norm, corresponding to γ= 0 (see [13] for details). Up to a multiplication by a constant and a scaling, w0 is equal tow? given by (1.4), withγ= 0.

Next, in Section 3, we prove integrability and regularity estimates for so- lutions to (1.6). We point out that C^1,α regularity can be expected only if γ ∈ (0,1), as it can be easily guessed by considering the function w?, which involves|x|^2−γ (see Remark3.3).

3. Inequality (1.12) meansEγ[w]≥ Eγ[w_γ] = 0 =E0[w₀]. Hence we know that Eγ[w_γ]− E0[w_γ]≤0≤ Eγ[w₀]− E0[w₀].

The concentration-compactness analysis of Section4shows that, up to the extraction of subsequences, lim_γ→0w_γ(·+y_γ) =w₀(·+y₀), wherey_γ ∈R^d is a suitable translation. By passing to the limit asγ↓0, we obtain

lim sup

γ→0

Z

R^d

1

2pw_γ^2p|x|^−γ−_p+1¹ w_γ^p+1

log|x|dx≤lim

γ→0

Eγ[w0]− E0[w0]

γ .

If the r.h.s. was explicit, finite, this would allow us to deduce that{y_γ}is bounded. This is not the case becauseJ_γappears in the expression ofE_γ. To circumvent this difficulty, we can use a rescaled Barenblatt-type function in place of w0 and get an equivalent formulation in which the r.h.s. stays bounded. This is done in Section5.

4. Inspired by selection principles in Gamma-convergence methods as in [1], we infer thaty0minimizes the function

y7→

Z

R^d

−w^p+1₀ p+ 1+w^2p₀

2p

!

log|x+y|dx .

The minimum turns out to be attained exactly at y = 0, so that {wγ} converges tow0. The detailed analysis is carried out in the proof of Propo- sition5.3.

5. We prove Theorem 1.1 by contradiction in Section 6, using the method of [27,26]. Angular derivatives of wγ are nontrivial if wγ is not radial. By differentiating (1.6), one finds that the angular derivatives ofwγ belong to the kernel of a suitable operator. Passing to the limit as γ ↓ 0, we get a contradiction with a spectral gap property of the limit operator.

Inequality (1.1) is a special case of theCaffarelli-Kohn-Nirenberg inequal- ities. Because these inequalities involve weights, symmetry and symmetry breaking are key issues. However, only special cases have been studied so far.

We refer to [18] for a review, to [20,21] for some additional numerical inves- tigations, and to [22,24] for more recent results. Concerning the existence of optimal functions in the Hardy-Sobolev case p= (d−γ)/(d−2), the reader is invited to read [11,19]. We may observe that inequality (1.1) has three endpoints for which symmetry is known: the case of the Gagliardo-Nirenberg inequalitiescorresponding toγ= 0, the case of the Hardy-Sobolev inequalities withp= (d−γ)/(d−2) andϑ= 1 and, as a special case, theHardy inequality with (p, γ) = (1,2): see for instance [26].

Symmetry issues are difficult problems. In most of the cases, symmetry breaking is proved using a spherical harmonics expansion as in [11,29,14] and linear instability, although an energy method has also been used in [17]. For symmetry, there is a variety of methods which, however, cover only special cases. Moving plane methods as in [12] or symmetrization techniques like in [36, 4,17] can be applied to establish that the optimal functions are given by (1.4), up to multiplications by a constant and scalings, in a certain range of the parameters. Symmetry has also been proved by direct estimates,e.g., in [17, 23], and recently in [24] using rigidity estimates based on heuristics arising from entropy methods in nonlinear diffusion equations. Beyond the range covered by symmetrization and moving planes techniques, the best established method relies on perturbation techniquesthat have been used in [47,37,27,26]. In the present paper, we shall argue by perturbation, with new difficulties due to the translation invariance of the limiting problem.

2 Preliminary results

2.1 Interpolation

We denote by ˙H¹(R^d) the closure of D(R^d) w.r.t. the norm w 7→ k∇wk₂. Assume that d ≥ 3. It is well known that for all γ ∈ [0,2] there exists a positive constantCHS such that the Hardy-Sobolev inequality

kwk₂∗

γ,γ ≤CHS k∇wk₂ ∀w∈H˙¹(R^d) (2.1) holds, where the exponent 2^∗_γ has been defined in (1.2). Let 2^∗ = 2^∗₀ = _d−2^2d . Forγ= 0 andγ= 2, we recover respectively the Sobolev inequality

kwk₂∗ ≤CSk∇wk₂ ∀w∈H˙¹(R^d) (2.2)

and the Hardy inequality

kwk_2,2≤CH k∇wk₂ ∀w∈H˙¹(R^d). (2.3) A H¨older interpolation shows thatkwk₂∗

γ,γ≤ kwk

γ 2

d−2 d−γ

2,2 kwk

d 2

2−γ d−γ

2^∗ and hence CHS≤C

γ 2

d−2 d−γ

H C

d 2

2−γ d−γ

S

for any γ ∈ [0,2]. The best constant in (2.2) has been identified in [2,48]

and it is well known thatCH= 2/(d−2). According to [12,36,26], symmetry holds so that CHS is easy to compute using the optimal functionw? defined by (1.4) with p = 2^∗_γ/2 = (d−γ)/(d−2), for any γ ∈ (0,2]. The H¨older interpolation kwk_2p,γ ≤ kwk^ϑ₂∗

γ,γ kwk^1−ϑ_p+1,γ with ϑ as in (1.3) shows that the Caffarelli-Kohn-Nirenberg inequality (1.1) holds with

Cγ≤ C_HS^ϑ

. (2.4)

2.2 Scalings and Euler-Lagrange equation Consider the following functional:

Gγ[w] := 1

2 k∇wk²₂+ 1

p+ 1 kwk^p+1_p+1,γ ∀w∈ Hp,γ(R^d). Inequality (1.12) amounts to

Jγ= 1 M^θγ infn

Gγ[w] : w∈ Hp,γ(R^d), kwk^2p_2p,γ =Mo

for anyM >0 and we shall consider the problem of the existence of an optimal functionw_γ, that is, the existence ofw_γ such that

Gγ[wγ] =JγM^θγ, wγ ∈ Hp,γ(R^d), kwγk^2p_2p,γ =M , (2.5) whereθγ is defined by (1.13). Let us check thatJγ is independent ofM. The scaling defined by

w^λ(x) :=λ^d−γ2p w(λ x) ∀x∈R^d, withλ >0, is such that

w^λ

_2p,γ=kwk_2p,γ, while a change of variables gives G_γ[w^λ] = 1

2λ^{d−γp −(d−2)}k∇wk²₂+ 1

p+ 1λ^{−p−12p (d−γ)} kwk^p+1_p+1,γ . An optimization onλ >0 shows that

min

λ>0Gγ[w^λ] =κ

kwk^2p_2p,γ Q^2p_γ [w]θγ

for some positive, explicit constantκwhich continuously depends onp,dand γ∈[0, d−(d−2)p). This proves that

J_γ=κC⁻²_γ ^{p θγ} (2.6)

is indeed independent ofM.

It is clear from the above analysis that, up to a multiplication of the function w by a constant, we can fix M > 0 arbitrarily. We shall further- more assume thatwγ is nonnegative without loss of generality because|wγ| ∈ Hp,γ(R^d),Gγ[wγ] =Gγ[|wγ|] andk |wγ| k_2p,γ =kwγk_2p,γ. By standard argu- ments,wγ satisfies the semilinear Euler-Lagrange equation

−∆w_γ+|x|^−γ w^p_γ−µ w^2p−1_γ

= 0 in R^d

in distributional sense, for some positiveLagrange multiplier µ=µ(M). From the scaling properties ofGγ, we find that

µ(M) = 2p θ_γJ_γM^θγ−1. Hence we can always takeµequal to 1 by choosing

M = (2p θγJγ)^1/(1−θγ) . (2.7) From now on,w_γ denotes a solution to(2.5)satisfying the above mass condi- tion and solving the equation

−∆wγ+|x|^−γ w_γ^p−w^2p−1_γ

= 0 in R^d. (2.8)

In Section5, however, we will use a scaling in order to argue with a different choice of mass forγ= 0.

Using a change of variables and uniqueness results that can be found for instance in [44] (also see earlier references therein), we know that the radial ground state, that is, anyradial positive solution converging to 0 as|x| → ∞, is actually unique (see Lemma6.1). Let us define

η:=d−γ−p(d−2), a_γ := (2−γ)η

(p−1)² and b_γ := η² p(p−1)² for anyγ∈[0,2). The functionw?defined by (1.4) solves

−∆w_γ+ a_γ

|x|^γ

w_γ^p−a_γ bγ

w_γ^2p−1

= 0. The rescaled function

w_γ^?(x) = aγ

b_{γ p−1}¹

w_?

b⁻

1 2−γ

γ x

solves (2.8) and is explicitly given by w^?_γ(x) :=

aγ

bγ+|x|^2−γ _p−1¹

. (2.9)

With these preliminaries in hand, we can state a slightly stronger version of Theorem1.1.

Theorem 2.1 Let d≥3. Then for any p∈ (1, d/(d−2)) there exists γ^∗ = γ^∗(p, d) ∈ (0, d−(d−2)p) such that wγ exists and is equal to w_γ^? for all γ∈(0, γ^∗).

Recall thatw_γ is a solution to (2.5) such that (2.7) and (2.8) hold. All other solutions to (2.5) can be deduced using multiplication by constants. We shall prove various intermediate results in Sections3–5 and complete the proof of Theorem2.1in Section6. As mentioned above, it is not restrictive to work with wγ ≥ 0 and we shall therefore consider only nonnegative functions, without further notice. Theorem2.1 is stronger than Theorem 1.1 because it charac- terizes all optimal functions and not only the value of the optimal constant.

Notice that the existence result ofwγdoes not require restrictions onγ∈(0,2):

see Proposition2.5.

2.3 Density and compactness results

Lemma 2.2 Let p,dsatisfy (1.2). ThenD(R^d\ {0})is dense in Hp,γ(R^d).

Proof Let us consider some functionw∈ Hp,γ(R^d). The weight|x|^−γis, locally inR^d\ {0}, bounded and bounded away from zero. By standard mollification arguments, one can pick a sequence of functions{wn} ⊂ D(R^d\ {0}) such that

n→∞lim kw−wnk_p+1,γ = 0, lim

n→∞k∇w− ∇wnk₂= 0

ifwis compactly supported inR^d\ {0}. Otherwise a simple truncation shows that it is not restrictive to assume, in addition, thatw∈L^∞(R^d). Letξbe a smooth function such that

0≤ξ(x)≤1 ∀x∈R^d, ξ(x) = 1 ∀x∈B₁, ξ(x) = 0 ∀x∈B₂^c, whereBr:=Br(0) and consider

w_n :=ξ_nw , ξ_n(x) := (1−ξ(n x))ξ(x/n) ∀x∈R^d,

for anyn≥2. It is clear that limn→∞kw−wnk_p+1,γ= 0 by dominated conver- gence. As for∇wn =w∇ξn+ξn∇w, we get that limn→∞k∇w−ξn∇wk₂= 0 again by dominated convergence so that density is proven as soon as we es- tablish that limn→∞kw∇ξnk₂= 0. This follows from

kw∇ξnk²₂≤ |S^d−1| k∇ξk²_∞kwk²_∞ n² Z 2/n

1/n

r^d−1dr+k∇ξk²_∞ n²

Z

n≤|x|≤2n

w²dx . Sinced≥3, the first term in the r.h.s. vanishes asn→ ∞. As for the second term, we get

Z

n≤|x|≤2n

w²dx≤(2n)^p+12γ Z

n≤|x|≤2n

w²|x|^−p+12γ dx

≤(2n)^p+12γ kwk²_p+1,γ

|S^d−1| Z 2n

n

r^d−1dr

p−1 p+1

by H¨older’s inequality and the r.h.s. goes to zero as n → ∞because _p+1^2γ + d^p−1_p+1 <2 for anyp <(d−γ)/(d−2). ut Lemma 2.3 Let d ≥ 3 and γ ∈ [0,2). Then H˙¹(R^d) is locally compactly embedded inL^q_γ(R^d)for allq∈[1,2^∗_γ).

Locally compactly embedded means that for any bounded sequence {wn} ⊂ H˙¹(R^d), the sequence {χ wn} is relatively compact in L^q_γ(R^d) for any charac- teristic functionχ of a compact set inR^d.

Proof As a direct consequence of (2.1) and (2.3), we have the equi-integrability estimates

Z

B_r

|w|^q|x|^−γdx≤ Z

B_r

w²|x|^−γdx

2∗ γ−q 2∗

γ−2Z

B_r

|w|^2∗γ|x|^−γdx ₂^q−2∗

γ−2

≤ r^(2−γ)

2∗ γ−q 2∗

γ−2 C

2²

∗γ−q 2∗

γ−2

H C

2^∗_γ ^q−2

2∗ γ−2

HS k∇wk^q₂ ∀r >0 for anyq∈[2,2^∗_γ), and

Z

B_r

|w|^q|x|^−γdx≤ Z

B_r

w²|x|^−γdx ^q₂

|S^d−1| Z r

0

s^d−1−γds 1−^q₂

≤ r^{(2−γ)q2+(d−γ)}(^1−q₂)C_H^q k∇wk^q₂

1

d−γ|S^d−1|^1−q₂

∀r >0 for anyq∈[1,2]. The result follows from the well-known local compactness of subcritical Sobolev embeddings (see,e.g., [32, Theorem 7.22]). ut Proposition 2.4 Letp,dsatisfy(1.2). ThenHp,γ(R^d)is compactly embedded inL^2p_γ (R^d).

Proof By definition,H_p,γ(R^d) is continuously embedded in ˙H¹(R^d) andlocally compactly embedded in L^2p_γ (R^d) by Lemma2.3. Using H¨older’s inequality and then Sobolev’s inequality (2.2) we get

Z

B^c_R

|w|^2p

|x|^γ dx≤R⁻

γ q0

Z

B_R^c

|w|^p+1

|x|^γ dx

!^1/q Z

B^c_R

|w|^2∗dx

!^1/q0

≤R^−qγ0 kwk

p+1 q

p+1,γ (C_Sk∇wk₂)²

∗

q0 ∀R >0, with q = ²₂^∗∗^−p−1−2p and q⁰ = _q−1^q , for any w ∈ Hp,γ(R^d), which is an equi- integrability property at infinity. Thenglobal compactness follows. ut

2.4 Existence of optimal functions

Proposition 2.5 Let p, d satisfy (1.2). For any M > 0, the minimization problem (2.5)admits at least one solution.

Proof The functional Gγ is coercive and weakly lower semi-continuous on Hp,γ(R^d). SinceHp,γ(R^d) is a reflexive Banach space, the existence of a solu- tion of (2.5) follows by the compact embedding result of Proposition2.4. ut Remark 2.6 Let p,dsatisfy (1.2). We have Hp,0(R^d)⊂ Hp,γ(R^d)because

Z

R^d

|w|^p+1

|x|^γ dx≤ Z

B1

|w|^2∗γ

|x|^γ dx

!^p+1₂∗

γ Z

B1

1

|x|^γ dx ^1−p+1₂∗

γ + Z

B₁^c

|w|^p+1dx

for all w∈ H_p,0(R^d).

3A priori regularity estimates

The aim of this section is to provide regularity estimates of the solutions to the Euler-Lagrange equation (2.8). The first result is based on a Moser iterative method, in the spirit of [40,41]. We recall that 2^∗_γ := 2^d−γ_d−2 and use the notation 2^∗= 2^∗₀. To anyq≥2^∗, we associateζ:= ²_q^∗+ 1−²_q^{∗ 2∗}γ−2

2^∗_γ−2p. Lemma 3.1 Letd≥3,p∈(1,2^∗/2)andγ∈

0, d−(d−2)p

. With the above notations, any solutionw_γ to(2.8) satisfies

kwγk_q ≤Ck∇wγk^ζ₂ ∀q∈[2^∗,∞]

for some positive constant C which depends continuously onγ, q and pand has a finite limit C(∞)as q→ ∞.

Proof Let us set ε0:= 2^∗_γ−2p. For any A >0, after multiplying (2.8) by the test function (wγ ∧A)^1+ε0 and integrating by parts in R^d, and then letting A→ ∞, we obtain the identity:

4 (1 +ε0) (2 +ε₀)²

Z

R^d

∇w^1+ε_γ ^0/2

2

dx+ Z

R^d

w_γ^p+1+ε0|x|^−γdx= Z

R^d

w_γ^2p+ε0|x|^−γdx . By applying the Hardy-Sobolev inequality (2.1) to the functionw=wγ^1+ε0/2, we deduce that

kwγk^2+ε_2p+ε⁰

1,γ ≤ (2 +ε0)²

4 (1 +ε0)C²_HS kwγk^2p+ε_2p+ε⁰

0,γ

with 2p+ε1= 2^∗_γ(1 +ε0/2). Let us define the sequence{εn}by the recursion relationεn+1:= 2^∗_γ(1 +εn/2)−2pfor anyn∈N, that is,

ε_n= ²

∗ γ−2p 2^∗_γ−2

h 2^∗_γ ²

∗ γ

2

n

−2i

∀n∈N,

and takeqn= 2p+εn. If we repeat the above estimates withε0 replaced by εn andε1 replaced byεn+1, we get

kwγk^2+ε_2p+εⁿ

n+1,γ≤ (2 +εn)²

4 (1 +εn)C²_HS kwγk^q_qⁿ

n,γ . (3.1)

Hence, by iterating (3.1), we obtain the estimate kwγk_q

n,γ≤Cn kwγk^ζ₂ⁿ∗

γ,γ with ζn= ²

∗ γ

2

n 2^∗_γ qn

where the sequence{Cn} is defined byC0=CHS and C_n+1^2+εn= (2 +εn)²

4 (1 +ε_n)C²_HSC_n^qn ∀n∈N.

The sequence {Cn} converges to a limit C∞. Letting n → ∞ we get the uniform bound

kwγk_∞≤C∞ kwγk^ζ₂^∞∗ γ,γ

where ζ_∞ = ²

∗ γ−2

2^∗_γ−2p = lim_n→∞ζn. The proof is completed using the H¨older interpolation inequalitykwγk_q≤ kwγk²₂^∗∗^/qkwγk¹⁻²_∞ ^∗/q, (2.1) and (2.2). ut Lemma 3.2 Letd≥3,p∈(1,2^∗/2),q∈[1,∞)andγ∈

0, d−(d−2)p such that 2p q γ < d. Any solutionwγ to(2.8)is bounded in W^2,q_loc(R^d), and locally relatively compact in C^1,α(R^d) with α= 1−d/q, for any q > d. Moreover, there exists a positive constant C_γ,q,p,d such that

kwγk_C1,α(^B1(x₀))≤Cγ,q,p,d

kwγk_Lq(B2(x0))+kwγk^2p−1_L(2p−1)q(B2(x0))

+kwγk^p_Lp q(B₂(x₀))+kwγk^2p−1_L2p q(B₂(x₀))+kwγk^p_L2p q(B₂(x₀))

(3.2) for any x0∈R^d, andlim sup_γ→0+Cγ,q,p,d<∞.

Proof For any domain Ω ∈ R^d, we obtain from the Euler-Lagrange equa- tion (2.8) that

2^1−qk∆wγk^q_Lq(Ω)

≤ Z

Ω

w_γ^(2p−1)qdx+ Z

Ω

w^pq_γ dx+ Z

B₁

|x|^−2γpqdx _2p¹ Z

Ω

w^2pq_γ dx ^2p−1_2p

+ Z

B₁

|x|^−2γqdx ¹₂Z

Ω

w_γ^2pqdx ¹₂

≤2^1−qC^q

kwγk^(2p−1)q_L(2p−1)q(Ω)+kwγk^pq_Lp q(Ω)+kwγk^(2p−1)q_L2p q(Ω)+kwγk^pq_L2p q(Ω)

with 2^1−qC^q = max

1, R

B₁|x|^−2γpqdx_2p¹ ,

R

B₁|x|^{−2γ q}dx¹₂

. As a con- sequence, for any domainΩ⊂R^d and any solutionw_γ to (2.8), we have

k∆wγk_Lq(Ω)

≤C

kwγk^2p−1_L(2p−1)q(Ω)+kwγk^p_Lp q(Ω)+kwγk^2p−1_L2p q(Ω)+kwγk^p_L2p q(Ω)

. (3.3) By the Calder´on-Zygmund theory, we know that there exists a positive con- stantC⁰=C⁰(q, d) such that, for any functionw∈W^2,q_loc(B2(x0))∩L^q(B2(x0)) with∆w∈L^q(B2(x0)), the inequality

kwk_W2,q(B1(x0)) ≤C⁰

kwk_Lq(B2(x0))+k∆wk_Lq(B2(x0))

holds. See for instance [32, Theorem 9.11].A priori we do not know whether wγ ∈W^2,q_loc(B2(x0)), but we can considerwγ,ε:=wγ∗ρε, where{ρε}is a family of mollifiers depending onε >0, apply the Calder´on-Zygmund estimate towγ,ε

and then pass to the limit as ε →0 with wγ,ε →wγ and ∆wγ,ε → ∆wγ in L^q(B2(x0)), because of Lemma3.1and of the above estimate onk∆wγk_Lq(Ω). Estimate (3.2) and the local relative compactness are then consequences of the standard Sobolev embeddings: see,e.g., [32, Theorem 7.26]. ut Remark 3.3 Thanks to the uniform bound provided by Lemma3.1, the Euler- Lagrange equation(2.8)implies that|x|^γ∆wγ∈L^∞(R^d), for allγ andpcom- plying with (1.2). Hence, if γ∈(0,1)then wγ ∈C^1,α for allα <1−γ, while w_γ ∈C^0,α for allα <2−γ ifγ∈[1,2).

The optimal regularity for solutions to (2.8) can be estimated by the reg- ularity of the function w^?_γ defined in (2.9). This solution is precisely of class C^1,1−γ forγ∈(0,1) and of classC^0,2−γ forγ∈[1,2).

4 Concentration-compactness analysis and consequences

In this section we shall make use of a suitable variant of the concentration- compactness principle as stated in [38,39]. We consider the minimization prob- lemJ_γ as defined in Section2.2in the limitγ↓0. Our goal is to prove that the solutionsw_γ to problem (2.5) approximate, up to translations, the function

w0(x) :=

a0

b₀+|x|² p−1¹

∀x∈R^d (4.1)

defined in (2.9). Notice indeed that in the limit case γ = 0 the problem is invariant under translations, which is the major source of difficulties in this section. We will put the emphasis on the differences with the standard results of the concentration-compactness method and refer to [42, Section 3.3.1] for fully detailed proofs. Our goal is to establisha priori estimates on translations and get a uniform upper bound on the optimal functions asγ↓0.

Proposition 4.1 Let p∈(1,2^∗/2). Then we have

γ→0limJγ =J0. (4.2)

Let{γn} ⊂ 0, d−(d−2)p

be a decreasing sequence such thatlim_n→∞γ_n= 0.

For anyn∈N, we consider solutionsw_γnto problem(2.5)satisfying(2.8)with γ=γ_n. Then

n→∞lim Z

R^d

w_γ^2p_n|x|^−γndx= Z

R^d

w₀^2pdx=M

and up to the extraction of a subsequence, there exists {yn} ⊂R^d such that v_n :=w_γn(·+y_n)→w₀ strongly inH˙¹(R^d),

where either {y_n} is bounded or|y_n| → ∞ and`:= lim_n→∞|y_n|^γn= 1.

Proof According to (2.4) and (2.6),Jγ is bounded away from 0 asγ↓0. Using w0as a test function yields

lim sup

γ→0

J_γ≤J₀. (4.3)

Hence, up to the extraction of a subsequence, {Jγ_n} converges to a finite positive limit that we shall denote byJ. According to (2.7),

M_n:= (2p θ_γnJ_γn)^{1/(1−θγn)}=kw_γnk^2p_2p,γ

n (4.4)

converges asn→ ∞to M := (2p θ0J)^1/(1−θ0). Let us define fn(x) :=wγ_n(x)|x|^−γn2p

and consider the sequence{f_n^2p}. To prove (4.2), we have indeed to establish the strong convergence of the sequence in L¹(R^d). Our proof relies on the concentration-compactness method.

A simple estimate based on H¨older’s inequality rules out theconcentration scenario. Consider indeed a functionw∈H˙¹(R^d). We get that

Z

Br(x0)

|w|^2p|x|^−γdx≤ Z

Br(x0)

|x|^{−γ q}dx

!1/q

kwk^2p₂∗

with q = d/ d−p(d−2)

. By standard symmetrization techniques, the r.h.s. (with w = wγ_n) is maximal when x0 = 0 and therefore bounded by O(r^d/q−γ) using Sobolev’s inequality (2.2), (4.3) and (4.4), uniformly asγ↓0.

Hence we know that

r→0lim sup x₀∈R^d

n∈N Z

B_r(x₀)

f_n^2pdx= 0. (4.5)