Extremal Functions of the Singular Moser-Trudinger Inequality Involving the Eigenvalue

ZHOU Changliang and ZHOU Chunqin

School of Mathematical Sciences,Shanghai Jiao Tong University,Shanghai 200240,China.

1 Introduction

Let ??R2be a smooth bounded domain.The famous Moser-Truding erine quality[3-5]says that

for anyσ≤4π.Moreover,for any fixedu∈(?),it also holds that t

for anyσ＞ 0.In particular,the constantσ=4πis optimal in(1.1),which implies that,for anyσ＞4π,the inequality(1.1)is invalid and there exists a sequence of{u?}insuch that

Moser-Trudinger inequality(1.1),as a limit case of the Sobolev embedding,plays an important role in two-dimensional analytic and geometric problems.The further interesting subject is the existence of extremal functions to(1.1).By using the blow-up method Carleson and Chang[6]showed that the supremum is actually attained if ? is a ball.Flucher[7]generalized this result to arbitrary bounded domains in R2.See also Adimurthi-Tintarev[8],Malchiodi-Martinazzi[9]and Mancini-Sandeep[10]and the references in these papers for recent developments on this subject.

This inequality was generalized in many ways.One kind of generalization of(1.1)is the so-called singular Moser-Trudinger inequality,which was originally established by Adimurthi-Sandeep[11].They proved that

fors∈(0,4π(1-t))andt∈[0,1).Further,Csat′o-Roy[12]proved that the supremum is attained for this singular Moser-Trudinger embedding.

For the case of several singular points,Iula-Mancini[1]proved that the supremum

is finite and is attained forλ∈[0,λq(?)).Here

forq＞1,and

whereKare the different points in ?;andαi∈(-1,+∞),αi/∈Z such that

Now we describe another kind of generalization of(1.1).Tint arev[13]introduced the fi rst eigenvalue

to the Moser-Trudinger inequality.Instead of the usual sobolev norm,he take an equivalent norm of eachuin

whereβ＜λ1(?).Then he proved that the supremum

is finite.

Later,Yang-Zhu[2]extended Tintarev’result to Moser-Trudinger inequalities with a singular point,i.e.they prove the supremum

is fi nite and is attained forβ∈[0,λ1(?))andα∈(0,1).

Formore generalizations of the classical Moser-Truding erine quality(1.1),one can see for instance[12,14-22]and the reference therein.

In this paper,we want to introduce the equivalent norm(1.5)and the multiple singular points to the Moser-Trudinger inequality at the same time.The new inequality is more general than that of([1,2]).We also show the existence of the extremal functions for such stronger inequalities.Our main results are stated as following：

Theorem 1.1.Let??R2be a smooth bounded domain and V(x)be as in(1.3).Let α∈(-1,0)be fi xed and λ1(?)be defined as in(1.4).Then for any β＜λ1(?),we have

where||u||1,βis defined as in(1.5).

Theorem 1.2.Under the assumption of Theorem1.1,there exists some function

for any β＜λ1(?).

For the proof of our results,we use an important tool in geometric analysis,the blowup analysis.Since the problems involve more complicated norm and multiple singular points,not only we can’t use the symmetrization to deal with a one-dimensional inequality,but also the steps of the blow-up analysis become more delicate.Because of the presence of several singularities,it is difficult to identify the number of the blow up point and to locate the position of the blow up point when the maximizing sequence blows up.Actually,in the Section 3,we illustrate the processes of identifying the number of the blow up point and locating the position of the blow up point by combining the classification results of Chen-Li[23]and that of Prajapat-Tarantello[24].We should mention that we finally prove that the only blow up point is the singular point with the least powerαand consequently get the desired bubble.

2 Maximizers for subcritical-Moser-Trudinger functional

In this section,we will show the existence of the maximizers for Moser-Trudinger functional in the subcritical case.Let us start with two useful Lemmas.The first Lemma is an embedding Lemma of Orlicz type,i.e.

Lemma 2.1.Let V(x)be as in(1.3)and u∈W1,20(?).For any p＞0,and any γ＞0such that-1＜αγ＜0,there holdsZ

Proof.For any fixedp＞0,we takeq＞1 such that-2＜2αγq＜0.By the H¨older inequality,we have

where

The other useful Lemma is the following,which is obviously obtained from(1.2).

Lemma 2.2.For any β＞0satisfying-1＜αβ＜0,there holds

Proposition2.1.Forany ?∈(0,1+α),there exist some u?1and

In particular,u?satis fi es the following Dirichlet problem

in the distributional sense,where λ?=

Proof.For any fixed?∈(0,1+α),we letu?,jbe a maximizing sequence inwith||u?,j||1,β≤1.Sinceβ＜λ1(?),we have

which implies thatu?,jis bounded inHence their exists somesuch that up to a subsequence,

Foranyby theHlderinequality and the inequality

we have

Choosep,1+δ,ssuf fi ciently close to 1 such that

Clearly we have that

and that

Combining(2.6)and(2.7),we conclude limsupInserting(2.6)and(2.7)into(2.4),by Lemma 2.2 and Lemma 2.1,we haveV(x)e4π(1+α-?)u2?,jis uniformly bounded inLq(?)for someq＞1.

Since

andu?,j→u?strongly inLp(?)for anyp≥1 asj→+∞,we conclude that

Thus we have thatu?attains the supremum.Clearlyu?/≡0.If we supposethat||u?||1,β＜1.It follows that

which is a contradiction.Hence we have||u?||1,β=1.A straightforward calculation shows thatu?satis fi es the Euler-Lagrange equation(2.3)in the distributional sense.

Moreover,using the H¨older inequality and Lemma 2.1,forp＞1,r＞1,q＞1 such that-1＜pqα＜0 andpr(1+α)≤1+αp,we can have that

Sois bounded inLp(?)for somep＞ 1.By the standard elliptic estimation,we have thatu?∈C0

We also have the following crucial observation.

Lemma 2.3.

Proof.Obviously,

On the other hand,we have by Proposition 2.1,

Which implies

Hence the result holds.

3 Blow-up analysis

In this section,we will develop the blow-up analysis when the sequenceu?blows up when?→0.Sinceu?is bounded in(?)from the before section,we can assume

without loss of the generality

Now denoteis bounded,then for anyby the Lebesgue dominated convergence theorem we have

Henceu0is the desired maximizer.

In the following,we can assumeM?→+∞ as?→0.We may also assumex?→P∈?.Here and in the sequel,we do not distinguish sequence and subsequence,the reader can recognize it from the context.

In the following,we distinguish two cases(the concentration pointPlies in the interior of ? or on the boundary of ?)to analyze the asymptotic behavior ofu?.

Firstly,by an inequalityet≤1+tet,

This leads to liminf?→0λ?＞0.

case 1.Plies in the interior of ?.

we can prove the concentration-compactness Theorem foru?near the blow-up point.

Theorem 3.1.u0≡0,and|?u?|2dx?δP,where δPdenotes the Dirac measure centered at the point P.

Proof.Supposeu0/≡0,then we have

Forp＞1,δ＞0,s＞1,t＞1 such that-1＜ptα＜0 andby using H¨older inequality again,it follow from Lemma 2.1 and Lemma 2.2 to get

i.e.is uniformly bounded inLp(?)for somep＞1.By the standard elliptic estimation,we haveu?uniformly bounded in(?),and thenu?uniformly bounded inwhich contradictsM?→+∞as?→0.Henceu0≡0.

Sinceandu?→ 0 strongly inLq(?)for anyq＞ 1.Assume|?u?|2dx?μin the sense of measure.Ifμ/=δP,we can choose a cut-off functionψ(x)∈(?),which is supported inBr0(P)?? and equal to 1 inBr0/2(P)for some smallr0＞0 such that Z

for someη＞ 0 provided that?is suf fi ciently small.Then by Lemma 2.1,Lemma 2.2 and Hlder inequality,V(x)e4π(1+α)(ψu?)2is uniformly bounded inLs(Br0(P))for somes＞1.Then the standard elliptic estimate on the Euler-Lagrange equation(2.3)impliesu?is uniformly bounded inBr0/2(P),which contradictsM?→+∞.Therefore,|?u?|2dx?δP.

Next we will locate the position of the pointP

Subcase 1.P∈?{p1,p2,···,pm}

Let

Sinceu?→0 inLq(?)for anyq≥1,then we have

Choose a ballBδ(P)such thatfor all 1≤i≤m.

Denote

define the blowing up functions

A direct computation gives

inB0,?.Note that|ψ?(x)|≤1,applying the standard elliptic estimates to above equations,In particular

Liouville-type theorem impliesψ0≡1 in R2.

On the other hand,we have in any ballBR(0)

Applying the standard elliptic estimates to the equation ofφ?,we also haveφ?→φinIn particular,

On one hand,we have for anyR＞0,

This leads to

On the other hand,in view of(3.3)and(3.4),a result of Chen and Li[23]implies thatφis radially symmetric

and we get

The contradiction between(3.4)and(3.5)indicates that Subcase 1 can not occur.This impliesP∈{p1,p2,···,pm}.

Subcase 2.P∈{p1,p2,···,pm}.SetP=pi(i=1,2,···,m).Let

Note thatu?→0 strongly inLq(?)for anyq＞1.Fixτ＜1+αand choosesclose to 1 such that-α+τs＜1,then

We obtain

We now distinguish two steps to proceed.

Step 1.

Setand defineBi,?={x∈R2：x?+t?x∈Bδi(pi)},whereBδi(pi)??,someδi＞ 0 andpj/∈Bδi(pi)for allj/=i.It is clear thatt?→ 0.We also assume thatV(x)=gi(x)|x-pi|2αisuch thatgi(pi)＞0.Denote the blowing up functions as

A straightforward calculation shows

inBi,?,wherey=x?+t?x.Since 0≤w?≤1 andwhereo?(1)→0 inBRfor anyR＞0,we have by applying elliptic estimates to(3.6)thatw?→win

Sincew≤1 andw(0)=1,the Liouville theorem leads tow=1.Also we have

Clearly

and

Applying the elliptic estimates to(3.7),we have thatv?→vinwherevsatisfi es

A result of Chen and Li[23]implies thatvis radially symmetric and

So we get contradiction,which indicates that the step 1 can not occur.

Step 2.for some constantC.define

It follows that(x)is a distributional solution to the equation inBi,?.

Sinceρ?→ 0 andB2,?→ R2,we can assumefor somex?∈R2.Applying elliptic estimates to(3.8),we have thatwhereξ(x)is a distributional harmonic function on R2.Sinceξ(x)≤limsup?→0ξ?(x)≤ 1 for allx∈R2andξ(0)=lim?→0ξ?(0)=1,the Liouville theorem implies thatξ(x)=1 on R2.Hence we conclude

Clearly,φ?is a distributional solution to

inandapplying elliptic estimates to(3.10),we have thatφ?(x)→φ(x)inwhereφ(x)is a solution to

If we lety=x?+ρ?xwith|x+x?|≤R,then for any fi xedR＞|x?|+1,there holds|y-pi|≤2Rρ?.Combining(3.9)and Fatou’s lemma,we have

Hence

By the classi fi cation result of J.Prajapat,G.Tarantello[24],we have

Noticing that

and combining(3.13)and(3.14),we have thatx?=0 and thus

It follows that

The equality holds if and only ifαi=α.So we have that the blow up pointPis the only one point in{pj}withαj=α.Without loss of generality,we can assume that the blow up pointP=0 andV(x)=g(x)|x|2α.Then we have

defineu?,L=min{u?,M?/L}.Similar to[21],we have the following：

Lemma 3.1.For any L＞1,there holds

Proof.In view of the equation(2.3)and by Theorem 3.1,we choosea test function and then obtain

TakingR→+∞,we have form(3.17)that

Noticing that

we get the conclusion.

As a consequence of Lemma 3.1,we also have the following：

Lemma 3.2.There holds

Proof.For anyL＞1,there holds

By Lemmas 3.1 and 2.2,V(x)e4π(1+α-?)u2?,Lis bounded inLq(?)for someq＞1.Noticing also thatu?,Lconvergence to 0 almost everywhere,V(x)e4π(1+α-?)u2?converges toV(x)inL1(?).Taking?→0 in(3.18),we obtain

LetL→1,we conclude the Lemma.

It follows Lemma 3.2 that

For otherwise,we haveλ?/→0 as?→0.be such that||v||1,β=1.Then we have by Lemma 3.2 that

This is impossible sincev/≡0.Thus(3.19)holds.

The following Lemma will be used in Section 5.

Lemma 3.3.

Proof.On the one hand,

On the other hand,

which gives

Combining(3.20),(3.21),Lemma 3.2 and Lemma 2.3,we get the result.

In order to investigate the convergence behavior ofu?away fromP=0,we need the following Lemma.

Lemma 3.4.We have

Proof.We divide ? into three parts

for someL＞1 and anyR＞0.Denote the integral on the above domains byI1,I2andI3respectively.Since

then we have

ForI2,it is easy to get

ForI3,since

then we have

Putting(3.23)-(3.25)together,the result holds.

Letg?=M?u?.It is clear thatg?satis fi es the following equation

Eq.(3.22)and its proof show thatconverges to the Dirac operatorδ0inD′(?).This suggests thatg?should tend to the corresponding Green’s functionG.That is con fi rmed as follows.

Lemma 3.5.g?is uniformly bounded in(?)for any1＜q＜2.Furthermore,g??G weaklywhere G satis fi es

in the distributional sense.

Proof.It follows from(3.22)thatis uniformly bounded inL1(?).We claim thatg?is uniformly bounded inL1(?).To see this,we suppose on the contraryand we have||f?||L1(?)=1.Then applying a result of Struwe([26,Theorem 2.2]),we have thatf?is uniformly bounded infor any 1＜q＜2,in particularf?→fstrongly inL1(?).Since

inL1(?),fis a distributional solution to-?f+βf=0 in ?,which leads tof≡0.This contradicts

and con fi rms our claim.Now sinceis uniformly bounded inL1(?),applying Struwe([26,Theorem 2.2])again,we have thatg?is uniformly bounded(?)for any 1＜q＜2.Hence there existfor any 1＜q＜2.Since(3.22)implies thatin sense of measure.ThenGis a distributional solution to(3.27).Sobolev embeding impliesinLt(?)for anyt＞1.

Moreover,using the H¨older inequality and Lemma 2.1,forp＞1,r＞1,q＞1 such that-1＜pqα＜0 andpr(1+α)≤1+αp,we can have that

Sois uniformly bounded inLp(?)for somep＞ 1.By using the standard elliptic estimation to the(3.26),theng?is uniformly bounded inThe Sobolev embeding Theorem impliesg?→Guniformly in

Obviously,Gtakes the form

whereCGis a constant and

Up to now,we have described the convergence behavior ofu?nearPand away fromPwhen the concentration pointPis in the interior of ?.

Case 2.Plies on??.

Lemma 3.6.Let d?=dist(x?,??),and r?be defined in(3.2).There holds r?/d?→0.

Proof.We suppose on the contrary that there existsR＞0 such thatd?≤Rr?.Take somey?∈?? such thatd?=|x?-y?|.SetBδ=Bδ(P)∩? with someδ＞0 such thatpi/∈Bδfor all 1≤i≤mand

ThenLetIn thissatis fi es By a re fl ection argument,we have-?v=0 in R2.Forwe can easily get that=1.Then we haveThis contradicts to the fact

By Lemma 3.6,we haveandBδ,?→R2as?→0.define the blowing up functions

Similar to Subcase1satisfy

By the classify of solution,we have

and we get

Thus we get a contradiction.So the case don’t occur.

4 Proof of theorem

In this section,we show extremal functions exist.To this purpose,we need to derive the upper bound fi rst.We also need the following similar result motivated by the arguments by Carleson and Chang in[6].

De fi nition 4.1.is a normalized concentrating sequenceif

Then we call u?as a normalized concentrating sequences,and x0is called as a blow up point.

Theorem 4.1.Assume thatis a normalized concentrating sequence with ablow up point,the origin.Then for α∈(-1,0],we have

Proof.Letis the schwarz symmetry rearrangement of the functionu?(x).Setψ?(x)=then we have

Moreover,by Hardy-Little wood inequality,

Nowweuseafact in[6]：Foranynormalized concentrating sequenceit holds

Hence(4.1)holds.

Set

Motivated by the arguments in[6]and[27],we first compute the upper bound ofT0ifu?blows up.Our Lemma is the following：

Lemma 4.1.Iflimsup?→0||u?||∞=∞,then

Proof.

whereoδ(1)→0 asδ→0.Hence we obtain

whereo?(1)→0 as?→0.LetThenby(4.3)and the fact that

we have

By Theorem 4.1,

Then if we setg(ξ)=withξwe have

Now we focus the estimate on bubbling domainBy Step 2,φ?→φinand whenceu?=M?+o?(R),whereo?(R)→ 0 as?→ 0 for any fi xedR＞ 0.Together with the factM?u?→Gstrongly inwe have

and

Notice that

Putting(4.5)-(4.7)together,we obtain onBRρ?(x?)

Therefore,we have

Takingδ→0,by the continuity ofg(x)near the origin,we have

Then by the Lemma 3.3,we have

The proof is completed by using Lemma 2.3.

The proof of Theorem1.1If the sequence||u?||L∞(?)is uniformly bounded,by Lemma 2.3,we can easily getT0＜+∞.While if||u?||L∞(?)→+∞,then by Lemma 4.1 we also getT0＜+∞. □

Next we construct a sequence to get the lower bound estimate,which is an opposite toT0.Thus we get a contradiction and consequently we complete the proof of Theorem 1.2.

Lemma 4.2.There holds

Proof.define a sequence of functions on ? by whereGis the Green function andψis the regular part as(3.28),satisfying thatη=1 onBR?andbandCare constants depending only on?to be determined later.Here and in the sequel,Brstands for a ball centered at 0 with radiusr.ClearlyB2R?? ? provided that?is sufficiently small.In order to assure thatwe set

which gives

Noting thatψ(x)=O(|x|)asx→0,we have|?(ηψ)|=O(1)as?→0.It follows that

Integration by parts gives

This leads to

Also we have

Hence

Note that

andO(R?log(R?))If we choose the suitable constantCsuch that

the we must have

Combine(4.10)and(4.11),we obtain

In view of(4.11)and(4.12),there holds onBR?,

which together with the estimate

leads to

On the other hand,since

we obtain

Therefore

In view ofwe haveAnd thus we obtain

provided that?＞0 is chosen suf fi ciently small.

Acknowledgments

The authors are supported partially by NSFC of China(No.11271253 and No.11771285).The authorsalso thank reviewersfor theirhelpful commentsonearlier draft ofthis paper.

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