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Available online atwww.sciencedirect.com

Ann. I. H. Poincaré – AN 31 (2014) 297–314

www.elsevier.com/locate/anihpc

On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form

Michinori Ishiwata

a

, Makoto Nakamura

b

, Hidemitsu Wadade

c,

aDepartment of Industrial System, Faculty of Symbiotic Systems Science, Fukushima University, Kanayagawa, Fukushima, 960-1296, Japan bMathematical Institute, Tohoku University, Sendai, 980-8578, Japan

cDepartment of Applied Physics, Waseda University, Tokyo, 169-8555, Japan Received 8 May 2012; received in revised form 10 March 2013; accepted 12 March 2013

Available online 22 March 2013

Abstract

In this article, we establish the weighted Trudinger–Moser inequality of the scaling invariant form including its best constant and prove the existence of a maximizer for the associated variational problem. The non-singular case was treated by Adachi and Tanaka (1999)[1]and the existence of a maximizer is a new result even for the non-singular case. We also discuss the relation between the best constants of the weighted Trudinger–Moser inequality and the Caffarelli–Kohn–Nirenberg inequality in the asymptotic sense.

©2013 Elsevier Masson SAS. All rights reserved.

MSC:primary 46E35; secondary 35J20

Keywords:Weighted Trudinger–Moser inequality; Existence of maximizer; Caffarelli–Kohn–Nirenberg inequality with asymptotics

1. Introduction and main results

In this article, we shall establish the weighted Trudinger–Moser type inequality with its sharp constant and consider the existence of a maximizer associated with the (weighted) Trudinger–Moser type inequality. As is well- known, a function inH1,N(RN) (N2)could have a local singularity and this causes the failure of the embedding H1,N(RN)L(RN)although the continuous embeddingH1,N(RN) Lq(RN)holds for allNq <∞.

As one of the characterizations of the critical embedding inH01,N(Ω), Moser[17]and Trudinger[25]established the following: for any bounded domainΩ⊂RNwithN2, there exists a positive constantC=C(N )such that, for any 0< ααN:=N ω

1 N1

N1, whereωN1denotes the surface area of the unit ball inRN, there holds

Ω

expαu(x)N

dxC|Ω| (1.1)

* Corresponding author.

E-mail address:hidemitsuwadade@gmail.com(H. Wadade).

0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2013.03.004

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for alluH01,N(Ω)with ∇u LN(Ω)1, whereN:= NN1. Concerning the existence of a maximizer associated with (1.1), Carleson and Chang [6] showed the existence of a maximizer when Ω is a ball,N 2 andααN, although the problem suffers from a lack of the compactness whenα=αN, see also Struwe[24]for related works.

After that, the existence of a maximizer was proved for any bounded domain by Flucker[8] whenN=2 andα= α2=4πand by Lin[16]whenN2 andα=αN.

There are several extensions of (1.1). As a scaling invariant form inRN, Adachi and Tanaka[1]proved the follow- ing: forN2 and 0< α < αN, there exists a positive constantC=C(N, α)such that the inequality

RN

ΦNαu(x)N

dxC u N

LN(RN) (1.2)

holds for alluH1,N(RN)with ∇u LN(RN)1, where ΦN(t):=

j=N1

tj

j! fort0. (1.3)

In[1], it was also proved that (1.2) fails ifααN. The inequality (1.2) was originally obtained by Ogawa[19]with sufficiently smallαwhenN=2. Furthermore, (1.2) was extended to general critical Sobolev spaces in Ogawa and Ozawa[20]and Ozawa[22], see also Kozono, Sato and Wadade[14], Nagayasu and Wadade[18]and Ozawa[21]

for related works. Moreover, another kind of the Trudinger–Moser type inequality is known. ForN2, there exists a positive constantC=C(N )such that for any 0< ααN, there holds

RN

ΦN

αu(x)N

dxC (1.4)

for alluH1,N(RN)with u H1,N(RN)1. Li and Ruf[15]obtained (1.4) with the best constantαN for N3, where the authors also proved the existence of a maximizer associated with (1.4) when α=αN. For N=2, (1.4) was proved by Cao[5], and Ruf[23]showed the sharpness ofα=α2(=4π ). In Ruf[23]and in Ishiwata[13], the existence of a maximizer associated with (1.4) was considered whenN=2 and it was also verified in[13]that the non-existence of a maximizer occurs whenN=2 andαis sufficiently small.

Keeping the historical remarks above in mind, we investigate (1.2) of the scaling invariant form into two directions.

Our first aim is to extend (1.2) to the weighted inequality as follows: forN2 and−∞< st < N,

RN

ΦNαu(x)Ndx

|x|t C u

N (Nt) Ns

LN(RN;|x|−sdx) (1.5)

for all uLN(RN; |x|sdx)∩ ˙H1,N(RN)with ∇u LN(RN)1 with some positive constants α andC, where LN(RN; |x|sdx)denotes the weighted Lebesgue space endowed with the norm

u LN(RN;|x|sdx):=

RN

u(x)N|x|sdx 1

N

,

see Theorem1.1. This generalization from (1.2) to (1.5) is naturally motivated by the special case of the Caffarelli–

Kohn–Nirenberg inequality obtained in[4], which states that, forN2,−∞< st < N andN q <∞, there exists a positive constantC=C(N, s, t, q)such that the inequality

u Lq(RN;|x|tdx)C u

N (Nt) q(Ns)

LN(RN;|x|−sdx)u 1

N (Nt) q(Ns)

LN(RN) (1.6)

holds for alluLN(RN; |x|sdx)∩ ˙H1,N(RN). We can regard (1.5) as a critical version of (1.6).

As another direction, we consider the existence of a maximizer associated with (1.5), which has not been discussed even for the non-singular inequality (1.2) as far as we know. In Theorem1.3, we shall establish the existence of a maximizer to the above variational problem.

Next, we investigate the constant C in the Caffarelli–Kohn–Nirenberg inequality (1.6) including its asymptotic sharp constant, see Theorem1.5. Among others, in Ozawa[21], the author gave the explicit relation between the

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constants appearing in (1.2) and the non-singular case s=t=0 of (1.6). We shall revisit the strategy developed in[21]and apply it to the weighted inequalities obtained in Theorem1.1.

Now we are in a position to give our main results. We often assume the condition for the exponents as follows:

N2, −∞< st < N and 0< α < αN,t:=(Nt )ω

1 N1

N1. (1.7)

Theorem 1.1.(i)Assume(1.7). Then there exists a positive constantC=C(N, s, t, α)such that the inequality

RN

ΦN

αu(x)Ndx

|x|t C u

N (Nt) N−s

LN(RN;|x|sdx) (1.8)

holds for all radially symmetric functionsuLN(RN; |x|sdx)∩ ˙H1,N(RN)withu LN(RN)1, whereΦN is defined by(1.3).

(ii)Assume(1.7). The constantαN,t is sharp for the weighted Trudinger–Moser type inequality(1.8). Indeed, the inequality(1.8)fails ifααN,t.

We can remove the assumption of the radial symmetry on the functions for the special cases=0 and 0t < N in Theorem1.1by virtue of the rearrangement argument, which may not work for the cases=0, see Section 2 and Appendix A for the details. As a result, we obtain the following corollary of Theorem1.1.

Corollary 1.2.(i)Assume(1.7)withs=0. Then there exists a positive constantC=C(N, t, α)such that the inequal-

ity

RN

ΦN

αu(x)Ndx

|x|t C u Nt

LN(RN) (1.9)

holds for alluH1,N(RN)withu LN(RN)1.

(ii)Assume(1.7)withs=0. IfααN,t, then the inequality(1.9)fails.

Remark. The non-singular case t =0 in Corollary 1.2 coincides with the result in Adachi and Tanaka[1, The- orems 0.1–0.2]. Moreover, the inequality (1.9) in Corollary1.2 was obtained as a particular case of the result in Nagayasu and Wadade[18, Corollary 1.3]which does not include the consideration for its sharp constant with respect toα. However, both of Corollary1.2(i) and (ii) are new results for the singular case 0< t < N.

Next, we shall discuss the existence of a maximizer associated with the Trudinger–Moser type inequality (1.8). We define the sharp constantμN,s,t,α(RN)for (1.8) by

μN,s,t,α RN

:= sup

uX1,Ns,rad, u LN (RN )=1

FN,s,t,α(u),

where

FN,s,t,α(u):= RNΦN

α|u(x)|Ndx

|x|t

u

N (N−t) Ns

LN(RN;|x|sdx)

, (1.10)

and the function spacesXs1,N andXs,rad1,N denote the weighted Sobolev spaces defined by X1,Ns :=LN

RN; |x|sdx

∩ ˙H1,N RN

, X1,Ns,rad:=

uXs1,N; uis radially symmetric ,

respectively. Though the variational problem associated with μN,s,t,α(RN) is a subcritical one from the view- point of the exponent α since α < αN,t, the problem suffers from a lack of the compactness due to the scaling invariance of the inequality. To explain the non-compactness, let us assume there exists a strongly convergent

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maximizing sequence(wn)forμN,s,t,α(RN)inX1,Ns,rad with ∇wn LN(RN)=1 (otherwise we already have the non- compactness). Thus(wn)satisfies FN,s,t,α(wn)μN,s,t,α(RN) andwnw as n→ ∞ strongly in Xs1,N. It is easy to see that w=0. Now let us define a sequence(un)byun(x):=wnnx)for x∈RN withn)satisfying λn→ ∞as n→ ∞. Then by the scaling invariance, we obtainFN,s,t,α(un)=FN,s,t,α(wn)μN,s,t,α(RN)and ∇un LN(RN)= ∇wn LN(RN)=1. Furthermore, since un LN(RN;|x|sdx)→0 asλn→ ∞, (un) is bounded in X1,Ns,radand up to a subsequenceunconverges 0 weakly inX1,Ns . However, we see asn→ ∞,

un X1,Ns := un LN(RN;|x|sdx)+ ∇un LN(RN)

un LN(RN)= ∇wn LN(RN)= ∇w LN(RN)+o(1) >0,

which implies thatuncannot converge to 0 strongly inXs1,N, and then we have the non-compactness ofFN,s,t,αat the levelμN,s,t,α(RN).

In spite of this difficulty, we can prove the following existence result by using a suitable renormalization argument.

Theorem 1.3.(i)Assume(1.7). ThenμN,s,t,α(RN)is attained.

(ii)LetN2and0< α < αN=αN,0=N ω

1 N1

N1, and define for any domainD⊂RN,

˜

μN,α(D):= sup

uH01,N(D), u LN (D)=1

DΦN

α|u(x)|N dx u N

LN(D)

,

whereH01,N(D)denotes the completion ofCc (D)over the norm · H1,N(D). Thenμ˜N,α(D)is attained if and only ifD=RN.

We can remove the assumption of the radial symmetry on functions in Theorem1.3(i) whens=0 and 0t < N, see LemmaA.1in Appendix A. As a consequence, we obtain the following corollary.

Corollary 1.4.Assume(1.7)withs=0. Then μN,0,t,α

RN

= sup

uH1,N(RN), u LN (RN )=1

RNΦN

α|u(x)|Ndx

|x|t

u Nt

LN(RN)

is attained.

Remark.(i) As far as we know, Corollary1.4is a new result even for the non-singular caset=0, which corresponds to (1.2).

(ii) Theorem 1.3(ii) shows thatμ˜N,α(D)admits a maximizer only when D=RN. We here consider the case s=t=0 andDis a radially symmetric domain with 0∈D. Let

˜

μN,t,α(D):= sup

uX1,Nt,rad(D), u LN (D)=1

DΦN

α|u(x)|Ndx

|x|t

u N

LN(D;|x|−tdx)

,

where Xt,rad1,N (D) denotes the closure of the class of radially symmetric functions in Cc (D) over the norm · LN(D;|x|−tdx)+ ∇ · LN(D). Then Theorem1.3(i) yields the attainability of μ˜N,t,α(RN) sinceμ˜N,t,α(RN)= μN,t,t,α(RN)by definition. On the other hand,μ˜N,t,α(D)is not attained ifD=RN. Indeed, assume thatμ˜N,t,α(D) is attained byu0X1,Nt,rad(D)for some radially symmetric domainD=RNwith 0∈D. By using the transformation v(x):=Nt

N

NN1

˜ u

|x|NNt

withu(x)= ˜u(|x|), see (2.3) and (2.4), we obtainμ˜N,t,α(D)= ˜μN,0, N

Ntα(D), where˜ D˜ := {x∈RN; |x|N−tN = |y|for someyD}. On the other hand, since 0∈ ˜D, by the scaling and the rearrange- ment argument, it holds˜N,t,α(D)=˜N,0, N

Ntα(D)˜ = ˜μN, N

Ntα(D), which implies that˜ μ˜N, N

Ntα(D)˜ is attained by

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v0(x):=Nt

N

N−1

N u˜0

|x|NNt

H01,N(D)˜ withu0(x)= ˜u0(|x|). However, this is a contradiction to Theorem1.3(ii) sinceD=RNyieldsD˜=RN.

(iii) On the contrary, when (s, t)=(0,0)and 0∈/D, we can guess that the corresponding assertion to Theo- rem1.3(ii) is not true. For instance, the attainability for the best constant of the Caffarelli–Kohn–Nirenberg inequal- ity (1.6) was extensively studied, see e.g. Chern and Lin[7], Ghoussoub and Kang[9], Ghoussoub and Robert[10,11]

and Hsia, Lin and Wadade[12], where the authors proved the existence of a maximizer associated with the Caffarelli–

Kohn–Nirenberg inequality forDwith 0∈∂D(e.g. the half-space) when(s, t)=(0,0).

Next, we investigate the asymptotic behavior of the constant for the Caffarelli–Kohn–Nirenberg inequality (1.6) with respect to the exponentq. Let us recall (1.6)

u Lq(RN;|x|tdx)C u

N (Nt) q(Ns)

LN(RN;|x|sdx)u 1

N (Nt) q(Ns)

LN(RN)

for alluXs1,N, whereCdepends only onN, s, tandq. Hereafter we fixN,sandt. It is known thatC=Cqbehaves likeCqβqN1 asq→ ∞for someβ >0, see e.g., Nagayasu and Wadade[18]. Now we define

βN,t :=lim sup

q→∞ sup

uXs1,N\{0}

u Lq(RN;|x|tdx)

qN1 u

N (N−t) q(Ns)

LN(RN;|x|sdx)u 1

N (N−t) q(Ns)

LN(RN)

,

which we call the asymptotic best constant of (1.6), see Section 4 for the precise definition ofβN,t. The next theorem givesβN,t explicitly and we obtain

βN,t =

N−1 eN (Nt )ω

1 N1

N1

N1

N

.

More precisely, under the assumption

N2, −∞< st < N and β > βN,t, (1.11)

we obtain:

Theorem 1.5.Assume(1.11). Then there exists a positive constantq0=q0(N, s, t, β)N such that, for anyqq0 the inequality

u Lq(RN;|x|tdx)βqN1 u

N (Nt) q(Ns)

LN(RN;|x|sdx)u 1

N (Nt) q(Ns)

LN(RN) (1.12)

holds for alluXs,rad1,N . Furthermore, the constantβN,t is sharp for(1.12). Indeed,(1.12)fails if0< β < βN,t in the above asymptotic sense.

In particular, the cases=0 and 0t < N in Theorem1.5yields the following corollary by virtue of the rear- rangement argument.

Corollary 1.6.Assume(1.11)withs=0. Then there exists a positive constantq0=q0(N, t, β)N such that, for anyqq0the inequality

u Lq(RN;|x|tdx)βqN1 u

Nt q

LN(RN)u 1

Nt q

LN(RN) (1.13)

holds for alluH1,N(RN). Furthermore, the constantβN,t is sharp for(1.13). Indeed, the inequality(1.13)fails if 0< β < βN,t in the above asymptotic sense.

Remark.In fact, we shall show the exact relation between the constantsαin (1.8) andβin (1.12) explicitly given by β= 1

eNα

N1

, which was established in Ozawa[21]for the non-singular case of the critical Sobolev space with the fractional derivatives. Then Theorem1.5will be proved by notingβN,t= 1

eNαN,t

N1 .

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Here, we describe the organization of this article. Section 2, Section 3 and Section 4 are devoted to prove Theo- rem1.1, Theorem1.3and Theorem1.5, respectively. Moreover, we shall collect several lemmas in Appendix A for the proof of the main theorems.

2. Proof of Theorem1.1

First, we consider the case s=t in Theorem 1.1(i), and the general case st can be obtained by using the Caffarelli–Kohn–Nirenberg interpolation inequality (1.6). The cases=t in Theorem1.1(i) can be rewritten as fol- lows. Throughout this paper,Cis a positive constant independent of the functionuand may vary from line to line.

Proposition 2.1.Assume(1.7)withs=t. Then there exists a positive constantC=C(N, t, α)such that the inequality

RN

ΦN

αu(x)Ndx

|x|t C u NLN(RN;|x|−tdx) (2.1)

holds for alluX1,Nt,radwithu LN(RN)1.

Once Proposition2.1has been established, Theorem1.1(i) withs < t will be its immediate consequence through the Caffarelli–Kohn–Nirenberg inequality (1.6) withq=N. Indeed, by combining the inequality

u LN(RN;|x|tdx)C u

NN−ts

LN(RN;|x|sdx)u 1

NN−ts

LN(RN) (2.2)

with the Trudinger–Moser type inequality (2.1), we obtain Theorem1.1(i) withs < t.

Proof of Proposition 2.1. Let 0< α < αN,t and let uX1,Nt,rad with ∇u LN(RN)1. We define the functionvH1,N(RN)through the formula such as forx∈RN,

v(x):=

Nt N

N−1N

˜ u

|x|NNt

, (2.3)

whereu(x)= ˜u(|x|)forx∈RN. Then direct computations show that

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

u LN(RN)= ∇v LN(RN), u LN(RN;|x|tdx)= N

Nt 1

N

v LN(RN),

RN

ΦNαu(x)Ndx

|x|t = N Nt

RN

ΦN N

Ntαv(x)N dx.

(2.4)

Thus substituting (2.4) into (2.1), we see that (2.1) is transferred to the non-singular form equivalently in terms of the functionv. Since 0<NNtα <NNtαN,t =N ω

1 N1

N1, by the Trudinger–Moser type inequality (1.2), we have

RN

ΦNαu(x)Ndx

|x|t = N Nt

RN

ΦN N

Ntαv(x)N dx

C v NLN(RN)=C u NLN(RN;|x|tdx),

where we used ∇v LN(RN)1, and we finish the proof of Proposition2.1. 2

Next, we proceed to the proof of the optimality for the constantαN,t stated in Theorem1.1(ii). In order to show this, we shall construct a sequence of functions inXs,rad1,N so that the corresponding functional diverges.

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Proof of Theorem1.1(ii). Fork∈N, we define a sequenceukof radially symmetric functions inX1,Ns,radby

uk(x):=

⎧⎪

⎪⎨

⎪⎪

0 if|x|1,

Nt

ωN−1k

N1 log1

|x|

ifeNkt <|x|<1, 1

ωN1

1

N k

Nt

N1

if 0|x|eNkt.

For the non-singular case t =0, this sequence of functions ukH1,N(RN) was used in Adachi and Tanaka [1, Theorem 0.2]. Then direct computations show that ∇uk LN(RN)=1 for allk∈Nand

⎧⎪

⎪⎨

⎪⎪

uk LN(RN;|x|sdx)=o(1) ask→ ∞,

RN

ΦN

αN,tuk(x)Ndx

|x|t ωN1

NN(k)ek= ωN1 N−1

1−ek

N2 j=0

kj j!

.

Thus we have

RNΦN

αN,t|uk(x)|Ndx

|x|t

uk

N (N−t) Ns

LN(RN;|x|sdx)

1−ekN2

j=0 kj j!

o(1) → ∞

ask→ ∞, which implies that (1.8) fails whenα=αN,t, and we finish the proof of Theorem1.1(ii). 2

At the end of this section, we shall give a proof of Corollary1.2below. Since a particular cases=0 and 0t < N in Theorem1.1(ii) directly implies Corollary1.2(ii), it remains to prove Corollary1.2(i).

Proof of Corollary1.2(i). By takings=0 and 0t < Nin Theorem1.1(i), we see that (1.9) holds for all radially symmetric functionsuH1,N(RN)with ∇u LN(RN)1. To remove the radially symmetric condition for the func- tions, we utilize the Schwarz symmetrization. Letu#H1,N(RN)be the Schwarz symmetrization ofuH1,N(RN).

Then we have, for anyqN, u Lq(RN;|x|tdx)u#

Lq(RN;|x|tdx), (2.5)

and ∇u#

LN(RN)u LN(RN). (2.6)

Among others, we refer to Almgren and Lieb[2, Theorem 2.7]for (2.6) and Bennett and Sharpley[3]for abundant information on the Schwarz symmetrization. We will prove (2.5) in Appendix A for the completeness of the paper, see LemmaA.1.

Since the integral on the left-hand side of (1.9) consists of a countable sum of the weighted Lebesgue norms u Lq(RN;|x|tdx)withqN through Taylor’s expansion, by using (2.5), (2.6) and Theorem1.1(i) withs=0, we see for anyuH1,N(RN)with ∇u LN(RN)1,

RN

ΦNαu(x)Ndx

|x|t =

j=N1

αj j! u Nj

LNj(RN;|x|tdx)

j=N1

αj

j!u#Nj

LNj(RN;|x|−tdx)=

RN

ΦNαu#(x)Ndx

|x|t Cu#Nt

LN(RN)=C u Nt

LN(RN), which completes the proof of Corollary1.2(i). 2

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3. Proof of Theorem1.3

We distinguish two casess=t ands=t. The former cases=t is more difficult to deal with compared to the case s=t due to the non-compactness of the embeddingXs,rad1,N LN(RN; |x|sdx)fors < N. Keeping this difficulty in mind, we first prepare the following lemma.

Lemma 3.1.Assume(1.7)and let(un)be a bounded sequence ofX1,Ns,radwithun LN(RN)=1. Moreover, assume un u weakly inX1,Ns

asn→ ∞. Then it holds asn→ ∞,

RN

ΦN

αun(x)N

αN1

(N−1)!un(x)N dx

|x|t

RN

ΦNαu(x)N

αN1

(N−1)!u(x)N dx

|x|t. (3.1)

Proof. LetΨN,k,α(τ ):=eατNk j=0αj

j!τNj for τ 0, whereN 2, k∈N∪ {0}and 0< α < αN,t. Then the desired convergence (3.1) can be rewritten as

RN

ΨN,N1,α

|un|dx

|x|t

RN

ΨN,N1,α

|u|dx

|x|t asn→ ∞. A direct computation shows forτ 0,

ΨN,N 1,α(τ )= αN

N−1τN11ΨN,N2,α(τ ).

Thus, by the mean value theorem and the convexity of the functionΨN,N2,α, we see for someθ∈ [0,1], ΨN,N1,α

|un|

ΨN,N1,α

|u| ΨN,N 1,α

θ|un| +(1θ )|u|

|unu|

= αN N−1

θ|un| +(1θ )|u| 1

N1ΨN,N2,α

θ|un| +(1θ )|u|

|unu|

αN

N−1

θ|un| +(1θ )|u|N−11

θ ΨN,N2,α

|un|

+(1θ )ΨN,N2,α

|u|

|unu|

αN

N−1

|un| + |u|N−11

ΨN,N2,α

|un|

+ΨN,N2,α

|u|

|unu|. (3.2)

Take the numbersa, b, c >1 satisfying 1a+1b+1c =1, and then by (3.2) and the Hölder inequality, we have

RN

ΨN,N1,α

|un|

ΨN,N1,α

|u|dx

|x|t

αN

N−1

RN

|un| + |u| 1

N1

ΨN,N2,α

|un|

+ΨN,N2,α

|u|

|unu|dx

|x|t

αN

N−1|un| + |u|N11

LN−1a (RN;|x|tdx)

×ΨN,N2,α

|un|

+ΨN,N2,α

|u|

Lb(RN;|x|−tdx) unu Lc(RN;|x|−tdx). (3.3) We now use

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