Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


IMPACT FACTOR 2018: 6.636

CiteScore 2018: 5.03

SCImago Journal Rank (SJR) 2018: 3.215
Source Normalized Impact per Paper (SNIP) 2018: 3.225

Mathematical Citation Quotient (MCQ) 2018: 3.18

Open Access
Online
ISSN
2191-950X
See all formats and pricing
More options …

Critical and subcritical fractional Trudinger–Moser-type inequalities on

Futoshi Takahashi
  • Corresponding author
  • Department of Mathematics, Osaka City University & OCAMI, Sumiyoshi-ku, Osaka, 558-8585, Japan
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-11-16 | DOI: https://doi.org/10.1515/anona-2017-0116

Abstract

In this paper, we are concerned with the critical and subcritical Trudinger–Moser-type inequalities for functions in a fractional Sobolev space H1/2,2 on the whole real line. We prove the relation between two inequalities and discuss the attainability of the suprema.

Keywords: Trudinger–Moser inequality; fractional Sobolev spaces; maximizing problem

MSC 2010: 35A23; 26D10

1 Introduction

Let ΩN, with N2, be a domain with finite volume. Then the Sobolev embedding theorem assures that W01,N(Ω)Lq(Ω) for any q[1,+). However, by a simple example we see that the embedding W01,N(Ω)L(Ω) does not hold. Instead, functions in W01,N(Ω) enjoy the exponential summability:

W01,N(Ω){uLN(Ω):Ωexp(α|u|N/(N-1))𝑑x< for any α>0};

see Yudovich [19], Pohozaev [33] and Trudinger [37]. Later, Moser [26] improved the above embedding, and obtained the following inequality, now known as the Trudinger–Moser inequality:

TM(Ω,α)=supuW01,N(Ω)uLN(Ω)11|Ω|Ωexp(α|u|N/(N-1))𝑑x{<,ααN,=,α>αN,

where

αN=NωN-11/(N-1),

and ωN-1=|SN-1| denotes the area of the unit sphere in N. On the attainability of TM(Ω,α), Carleson and Chang [5], Struwe [36], Flucher [10] and Lin [24] proved that TM(Ω,α) is attained for any 0<ααN.

On domains with infinite volume, for example on the whole space N, the Trudinger–Moser inequality does not hold as it is. However, several variants are known on the whole space. In the following, let

ΦN(t)=et-j=0N-2tjj!

denote the truncated exponential function.

First, Ogawa [27], Ogawa and Ozawa [28], Cao [4], Ozawa [29] (for small α>0) and finally Adachi and Tanaka [1] proved that the following inequality holds true, which we call Adachi–Tanaka-type Trudinger–Moser inequality:

A(N,α)=supuW1,N(N){0}uLN(N)11uLN(N)NNΦN(α|u|N/(N-1))𝑑x{<,α<αN,=,ααN

(see also do Ó [8] and Cassani, Sani and Tarsi [6] for further information). This inequality enjoys the scale invariance under the scaling u(x)uλ(x)=u(λx) for λ>0. Note that the critical exponent α=αN is not allowed for the finiteness of the supremum. Recently, it was proved by Ishiwata, Nakamura and Wadade [16] and Dong and Lu [9] that A(N,α) is attained for any α(0,αN). In this sense, the Adachi–Tanaka-type Trudinger–Moser inequality can be considered as a subcritical inequality.

On the other hand, Ruf [34] and Li and Ruf [22] proved that the following inequality holds true:

B(N,α)=supuW1,N(N)uW1,N(N)1NΦN(α|u|N/(N-1))𝑑x{<,ααN,=,α>αN.

Here,

uW1,N(N)=(uLN(N)N+uLN(N)N)1/N

is the full Sobolev norm. Note that the scale invariance (uuλ) does not hold for this inequality. Also note that the critical exponent α=αN is permitted to the finiteness. Later, Adimurthi and Yang [2] proved that for all β[0,1) and all τ>0 there holds

AN,β,τ(α)=supuW1,N(N)N(|u|N+τ|u|N)𝑑x1NΦN(α(1-β)|u|N/(N-1))|x|Nβ𝑑x{<,ααN,=,α>αN,

by a different method. Clearly, the case β=0 and τ=1 reduces to that of Ruf [34] and Li and Ruf [22].

Concerning the attainability of B(N,α), the following facts have been proved:

  • If N3, then B(N,α) is attained for 0<ααN; see [22].

  • If N=2, then there exists α*>0 such that B(2,α) is attained for α*<αα2 (=4π); see [34] (for α=α2, see [15]).

  • If N=2 and α>0 is sufficiently small, then B(2,α) is not attained; see [15].

The non-attainability of B(2,α) for α sufficiently small is attributed to the non-compactness of “vanishing” maximizing sequences, as described in [15]. Concerning the attainability of AN,β,τ(α), recently Li and Yang [21] proved that AN,β,τ(α) is attained when 0<β<1, τ>0 and ααN. This complements the results by Li and Ruf [22] and Ishiwata [15].

In the following, we focus our attention on the fractional Sobolev spaces.

Let s(0,1), p[1,+) and let ΩN be a bounded Lipschitz domain. For s>0, let us consider the space

Ls(N)={uLloc1(N):N|u|1+|x|N+s𝑑x<}.

For uLs(N), we define the fractional Laplacian (-Δ)s/2u as follows: First, for ϕ𝒮(N), the rapidly decreasing function on N is defined via the normalized Fourier transform as

(-Δ)s/2ϕ(x)=-1(|ξ|sϕ(ξ))(x)

for xN. Then for uLs(N), the fractional Laplacian (-Δ)s/2u is defined as an element of 𝒮(N), the tempered distributions on N, by the relation

ϕ,(-Δ)s/2u=(-Δ)s/2ϕ,u=(-Δ)s/2ϕu𝑑x,ϕ𝒮(N).

Note that Lp(N)Ls(N) for any p1. Also note that it could happen that supp((-Δ)s/2u)Ω even if supp(u)Ω for some open set Ω in N.

By using the above notion, we define the Bessel potential space Hs,p(Ω) for a (possibly unbounded) set ΩN as

Hs,p(N)={uLp(N):(-Δ)s/2uLp(N)},H~s,p(Ω)={uHs,p(N):u0 on NΩ}.

On the other hand, the Sobolev–Slobodeckij space Ws,p(N) is defined as

Ws,p(N)={uLp(N):[u]Ws,p(N)<},=Ws,p(N)pNN|u(x)-u(y)|p|x-y|N+spdxdy,

and for a bounded domain ΩN, we define

W~s,p(Ω)=Cc(Ω)¯Ws,p(N)

where

uWs,p(N)=(uLp(N)p+[u]Ws,p(N)p)1/p.

It is known that

W~s,p(Ω)={uWs,p(N):u0 on NΩ}

if Ω is a Lipschitz domain, Hs,p(N)=Fp,2s(N) (the Triebel–Lizorkin space), and Ws,p(N)=Bp,ps(N) (the Besov space). Thus Hs,2(N)=Ws,2(N). However in general, Hs,p(N)Ws,p(N) for p2; see [31, 17] and the references therein.

Recently, Martinazzi [25] (see also [18]) proved a fractional Trudinger–Moser-type inequality on H~s,p(Ω) as follows: Let p(1,) and s=Np for N. Then for any open ΩN with |Ω|<, it holds that

supuH~s,p(Ω)(-Δ)s/2uLp(Ω)11|Ω|Ωexp(α|u|p/(p-1))𝑑x{<,ααN,p,=,α>αN,p.

Here,

αN,p=NωN-1(Γ((N-s)/2)Γ(s/2)2sπN/2)-p/(p-1).

We note that, differently from the classical case, the attainability of the supremum is not known even for N=1 and p=2.

On the Sobolev–Slobodeckij spaces W~s,p(Ω) with sp=N, a similar fractional Trudinger–Moser inequality was also proved by Parini and Ruf [31] when N2, and Iula [17] when N=1. They proved the validity of the inequality for sufficiently small values of α>0, and the problem of the sharp exponent is still open.

In the following, we are interested in the simplest one-dimensional case, that is, we put N=1, s=12 and p=2. In this case, the Bessel potential space H1/2,2() coincides with the Sobolev–Slobodeckij space W1/2,2(), and both seminorms are related as

(-Δ)1/4uL2()2=12π[u]W1/2,2()2;

see [7, Proposition 3.6.]. Then the fractional Trudinger–Moser inequality in [25, 18] can be read as in the following proposition.

Proposition 1.1 (A fractional Trudinger–Moser inequality on H~1/2,2(I)).

Let IR be an open bounded interval. Then it holds that

supuH~1/2,2(I)(-Δ)1/4uL2(I)11|I|Ieα|u|2𝑑x{<,αα1,2=π,=,α>π.

For the fractional Adachi–Tanaka-type Trudinger–Moser inequality on the whole line, put

A(α)=supuH1/2,2(){0}(-Δ)1/4uL2()11uL2()2(eαu2-1)𝑑x.(1.1)

Then by the precedent results by Ogawa and Ozawa [28] and Ozawa [29] it is known that A(α)< for small exponent α.

On the other hand, a fractional Li–Ruf-type Trudinger–Moser inequality on H1/2,2() is already known as follows.

Proposition 1.2 ([18]).

We have

B(α)=supuH1/2,2()uH1/2,2()1(eαu2-1)𝑑x{<,απ,=,α>π.(1.2)

Here,

uH1/2,2()=((-Δ)1/4uL2()2+uL2()2)1/2

is the full Sobolev norm on H1/2,2(R).

Concerning A(α) in (1.1), a natural question is to determine the range of the exponent α>0 for which A(α) is finite. As pointed out in [14], this remained an open problem for a while. In this paper, we first prove the finiteness of the supremum in the full range of values of the exponent.

Theorem 1.3 (Full range Adachi–Tanaka-type on H1/2,2(R)).

We have

A(α)=supuH1/2,2(){0}(-Δ)1/4uL2()11uL2()2(eαu2-1)𝑑x{<,α<π,=,απ.

Ozawa [30] proved that the Adachi–Tanaka-type Trudinger–Moser inequality is equivalent to the Gagliardo–Nirenberg-type inequality, and he obtained an exact relation between the best constants of both inequalities. Actually, he proved the result for general 1<p<, and if p=2, the main result in [30] can be read as follows: Put α0=sup{α>0:A(α)<} and

β0=lim supqsupuH1/2,2(),u0uLq()q1/2(-Δ)1/4uL2()1-2/quL2()2/q.

Then it is shown that 1/α0=2eβ02; see [30, Theorem 1]. Thus, by a direct consequence of Theorem 1.3, we have the next corollary.

Corollary 1.4.

We have β0=(2πe)-1/2.

Furthermore, we obtain the relation between the suprema of both critical and subcritical Trudinger–Moser-type inequalities along the line of [20].

Theorem 1.5 (Relation).

We have

B(π)=supα(0,π)1-(α/π)(α/π)A(α).

Also we obtain how the Adachi–Tanaka-type supremum A(α) behaves when α tends to π.

Theorem 1.6 (Asymptotic behavior).

There exist C1,C2>0 such that for any α<π which is close enough to π it holds that

C11-α/πA(α)C21-α/π.

Note that the estimate from above follows from Theorem 1.5 and Proposition 1.2. On the other hand, we will see that the estimate from below follows from a computation using the Moser sequence.

Concerning the existence of maximizers of the Adachi–Tanaka-type supremum A(α) in (1.1), we have the following theorem.

Theorem 1.7 (Attainability of A(α)).

A(α) is attained for any α(0,π).

On the other hand, as for B(α) in (1.2), we have the following result.

Theorem 1.8 (Non-attainability of B(α)).

B(α) is not attained for 0<α1.

It is plausible that there exists α*>0 such that B(α) is attained for α*<απ, but we do not have a proof up to now.

Finally, we improve the subcritical Adachi–Tanaka-type inequality along the line of [9].

Theorem 1.9.

For α>0, set

E(α)=supuH1/2,2(){0}(-Δ)1/4uL2()11uL2()2u2eαu2𝑑x.

Then we have

E(α){<,α<π,=,απ.

Furthermore, E(α) is attained for all α(0,π).

Since

eαt2-1αt2eαt2

for t, Theorem 1.9 extends Theorem 1.3. In the classical case, Dong and Lu [9] used a rearrangement technique to reduce the problem to one dimension and obtained a similar inequality by estimating a one-dimensional integral. In the fractional setting H1/2,2, we cannot follow this argument and we need a new idea.

The organization of the paper is as follows: In Section 2, we prove Theorems 1.3, 1.5 and 1.6. In Section 3, we prove Theorems 1.7 and 1.8. In Section 4, we prove Theorem 1.9.

Note.

After this work was completed, the author was informed by the anonymous referee that the full range Adachi–Tanaka-type inequality is proven, among other relevant results, in the recent preprints [12, 13] by different methods.

2 Proofs of Theorems 1.3, 1.5 and 1.6

For the proofs of Theorems 1.3, 1.5 and 1.6, we prepare several lemmas.

Lemma 2.1.

Set

A~(α)=supuH1/2,2(){0}(-Δ)1/4uL2()1uL2()=1(eαu2-1)𝑑x.

Then A~(α)=A(α) for any α>0.

Proof.

For any uH1/2,2(){0} and λ>0, we put uλ(x)=u(λx) for x. Then we have

{(-Δ)1/4uλL2()=(-Δ)1/4uL2(),uλL2()2=λ-1uL2()2(2.1)

since

2π(-Δ)1/4uλL2()2=[uλ]W1/2,2()2=|u(λx)-u(λy)|2|x-y|2𝑑x𝑑y=|u(λx)-u(λy)|2|λx-λy|2d(λx)d(λy)=[u]W1/2,2()2=2π(-Δ)1/4uL2()2.

Thus for any uH1/2,2(){0} with (-Δ)1/4uL2()1, if we choose λ=uL2()2, then uλH1/2,2() satisfies

(-Δ)1/4uλL2()1anduλL2()2=1.

Thus

1uL2()2(eαu2-1)𝑑x=(eαuλ2-1)𝑑xA~(α),

which implies A(α)A~(α). The opposite inequality is trivial. ∎

Lemma 2.2.

For any 0<α<π, it holds that

A(α)(α/π)1-(α/π)B(π).

Proof.

Choose any uH1/2,2() with (-Δ)1/4uL2()1 and uL2()=1. Further, put v(x)=Cu(λx), where C2=απ(0,1) and λ=C2/(1-C2). Then by the scaling rules (2.1) we see

vH1/2,2()2=(-Δ)1/4vL2()2+vL2()2=C2(-Δ)1/4uL2()2+λ-1C2uL2()2C2+λ-1C2=1.

Also we have

(eπv2-1)𝑑x=(eπC2u2(λx)-1)𝑑x=λ-1(eπC2u2(y)-1)𝑑y=1-C2C2(eαu2(y)-1)𝑑y=1-(α/π)(α/π)(eαu2(y)-1)𝑑y.

Thus, testing B(π) by v, we see

B(π)(eπv2-1)𝑑x1-(α/π)(α/π)(eαu2(y)-1)𝑑y.

By taking the supremum for uH1/2,2() with (-Δ)1/4uL2()1 and uL2()=1, we have

B(π)1-(α/π)(α/π)A~(α).

Finally, Lemma 2.1 implies the result. ∎

Proof of Theorem 1.3.

The assertion that A(α)< for α<π follows from Lemma 2.2 and the fact that B(π)< by Proposition 1.2.

For the proof of A(π)=, we use the Moser sequence

uε={(log(1/ε))1/2if |x|<ε,log(1/|x|)(log(1/ε))1/2if ε<|x|<1,0if 1|x|,(2.2)

and its estimates

(-Δ)1/4uεL2()2=π+o(1),(2.3)(-Δ)1/4uεL2()2π(1+(Clog(1/ε))-1),(2.4)uεL2()2=O((log(1/ε))-1)(2.5)

as ε0 for some C>0. Note that uεW~1/2,2((-1,1))W1/2,2()=H1/2,2(). For estimate (2.3), we refer to [17, Proposition 2.2]. For estimate (2.4), we refer to [17, (35)]. Actually, after a careful look at the proof of [17, Proposition 2.2], we confirm that

limε0(log(1/ε))((-Δ)1/4uεL2()2-π)C

for a positive C>0, which implies (2.4). For (2.5), we compute

uεL2()2=|x|ε(log(1/ε))𝑑x+ε<|x|1(log(1/|x|)(log(1/ε))1/2)2𝑑x=2εlog(1/ε)+2log(1/ε)log(1/ε)0t2(-et)𝑑x=2εlog(1/ε)+2log(1/ε)(Γ(3)+o(1))

as ε0. Thus we obtain (2.5).

By testing A(π) by vε=uε/(-Δ)1/4uεL2(), we have

A(π)1vεL2()2(eπvε2-1)𝑑x(-Δ)1/4uεL2()2uεL2()2|x|ε(eπvε2-1)𝑑x(-Δ)1/4uεL2()2uεL2()2εexp(πlog(1/ε)(-Δ)1/4uεL2()2)(-Δ)1/4uεL2()2uεL2()2εexp(log(1/ε)1+(Clog(1/ε))-1)

since et-1(1/2)et for t large and by (2.4). Also since

t1+1/Ct-t=-1/C1+1/Ct-1Cas t,

we see

t1+1/Ct=t-1C+o(1)

as t. Put t=log(1/ε). We see

exp(log(1/ε)1+(Clog(1/ε))-1)=exp(log(1/ε)-1/C+o(1))=(1/ε)e-1/C+o(1),

which leads to

εexp(log(1/ε)1+(Clog(1/ε))-1)e-1/C+o(1)δ>0

for some δ>0 independent of ε0. Therefore, by (2.3), (2.4) and (2.5) we have for δ>0,

A(π)π+o(1)(Clog(1/ε))-1δδ(log(1/ε))

as ε0. This proves A(π)=. ∎

Proof of Theorem 1.5.

By Lemma 2.2, we have

B(π)supα(0,π)1-(α/π)(α/π)A(α).

Let us prove the opposite inequality. Let

{un}H1/2,2(),un0,(-Δ)1/4unL2()2+unL2()21

be a maximizing sequence of B(π). We may assume (-Δ)1/4unL2()2<1 for any n. Put

vn(x)=un(λnx)(-Δ)1/4unL2()(x),λn=1-(-Δ)1/4unL2()2(-Δ)1/4unL2()2>0.

Thus by (2.1) we see

(-Δ)1/4vnL2()2=1,vnL2()2=λn-1(-Δ)1/4unL2()2unL2()2=unL2()21-(-Δ)1/4unL2()21

since (-Δ)1/4unL2()2+unL2()21. Thus, setting αn=π(-Δ)1/4unL2()2<π for any n, we may test A(αn) by {vn}, which results in

B(π)+o(1)=(eπun2(y)-1)𝑑y=λn(eπ(-Δ)1/4unL2()2vn2(x)-1)𝑑xλn1vnL2()2(eαnvn2(x)-1)𝑑xλnA(αn)=1-(αn/π)(αn/π)A(αn)supα(0,π)1-(α/π)(α/π)A(α).

Here we have used a change of variables y=λnx for the second equality, and vnL2()21 for the first inequality. Letting n, we have the desired result. ∎

Proof of Theorem 1.6.

We need to prove that there exists C1>0 such that for any α<π which is sufficiently close to π it holds that

A(α)C11-α/π.

Again we use the Moser sequence (2.2) and we test A(α) by vε=uε/(-Δ)1/4uεL2(). As in the similar calculations in the proof of Theorem 1.3, we have

A(α)1vεL2()2(eαvε2-1)𝑑x(1/2)vεL2()2|x|εeαvε2𝑑xCε(log(1/ε))exp(απlog(1/ε)1+(Clog(1/ε))-1)=Cε(log(1/ε))exp(δεlog(1/ε)),

where we put

δε=απ11+(Clog(1/ε))-1(0,1).

Now, for α<π which is sufficiently close to π, we fix ε>0 small such that

11-α/πlog(1/ε)21-α/π,(2.6)

which implies

exp(-21-α/π)εexp(-11-α/π).

With this choice of ε>0, we have

A(α)Cε(log(1/ε))exp(δεlog(1/ε))=Cε(log(1/ε))(1/ε)δε(2.7)=Cε1-δε(log(1/ε)).(2.8)

Now, we estimate that

ε1-δε(exp(-21-α/π))1-δε=exp(-21-α/π(1-δε))=exp(-(21-α/π){(1-απ)+(απ)(1-11+(Clog1/ε)-1)})=exp(-2-(2(α/π)1-α/π)(11+Clog1/ε))exp(-2-(2(α/π)1-α/π)(11+C1-α/π))=e-2e-2(α/π)C+1-α/π=e-2e-f(α/π),

where f(t)=2t/(C+1-t) for t[0,1] and we have used (2.6) in the last inequality. We easily see that f(0)=0 and f(t)=2(C+1)/(C+1-t)2>0 for t>0, thus f(t) is strictly increasing in t and maxt[0,1]f(t)=f(1)=2C. Thus we have

ε1-δεe-2e-2/C=:C0,

which is independent of α. Going back to (2.7) with (2.6), we observe that

A(α)Cε1-δε(log(1/ε))CC0(log(1/ε))CC01-α/π,

which proves the result. ∎

3 Proofs of Theorems 1.7 and 1.8

For uH1/2,2(), we denote by u* its symmetric decreasing rearrangement defined as follows: For a measurable set A, let A* denote an open interval A*=(-|A|/2,|A|/2). We define u* by

u*(x)=0χ{y:|u(y)|>t}*(x)𝑑t,

where χA denote the indicator function of a measurable set A. Note that u* is nonnegative, even and decreasing on the positive line +=[0,+). It is known that

F(u*)𝑑x=F(|u|)𝑑x

for any nonnegative measurable function F:++, which is the difference of two monotone increasing functions F1, F2 with F1(0)=F2(0)=0 such that either F1|u| or F2|u| is integrable. Also the inequality of Pólya–Szegő type

|(-Δu*)1/4|2𝑑x|(-Δu)1/4|2𝑑x

holds true for uH1/2,2(); see for example [3, 32, 23].

Remark 3.1.

Note that the radial compactness lemma by Strauss [35] is violated on . More precisely, let

Hrad1/2,2()={uH1/2,2():u(x)=u(-x),x0};

then Hrad1/2,2() cannot be embedded compactly in Lq() for any q>0. To see this, let ψ0 be an even function in Cc() with supp(ψ)(-1,1), and put un(x)=ψ(x-n)+ψ(x+n). Then we see that un is an even, compactly supported smooth function, and un0 weakly in H1/2,2() as n. But {un} does not have any strong convergent subsequence in Lq() because unLq()q=2ψLq()q>0 for any n sufficient large.

However, for a sequence {un}nH1/2,2() with un even, nonnegative and decreasing on +, we have the following compactness result.

Proposition 3.2.

Assume {un}H1/2,2(R) to be a sequence such that un is even, nonnegative and decreasing on R+. Let unu weakly in H1/2,2(R). Then unu strongly in Lq(R) for any q(2,+) for a subsequence.

Proof.

Since {un}H1/2,2() is a weakly convergent sequence, we have

supnunH1/2,2()C

for some C>0. We also have un(x)u(x) a.e. x for a subsequence, thus u is even, nonnegative and decreasing on +. Now, we use the estimate below, which is referred to a simple radial lemma: If uL2() is even, nonnegative and decreasing on +, then it holds that

u2(x)12|x|-|x||x|u2(y)dy12|x|uL2()2(x0).(3.1)

Thus un2(x)C2|x| for x0 by

supnunH1/2,2()C,

and u2(x)C2|x| by the pointwise convergence. Now, set vn=|un-u|q for q>2. Then we see vn(x)0 a.e. x. Moreover,

|x|R|un-u|q𝑑x=2R|un-u|q𝑑x2q(R|un|q𝑑x+R|u|q𝑑x)CRdx|x|q/2=CR1-q/2(q/2)-10

as R since q>2. Thus {vn}n is uniformly integrable. Also, by [7, Theorem 6.9] we know that

H1/2,2()Lq0()for any q02, anduLq0()CuH1/2,2().

For any q>2, take q0 such that 2<q<q0<. Since un-u is uniformly bounded in H1/2,2(), we have un-uLq0()C, and

Ivn𝑑x=I|un-u|q𝑑x(I|un-u|q0𝑑x)q/q0|I|1-q/q0

for any bounded measurable set I. Therefore, Ivn𝑑x0 if |I|0, which implies that {vn} is uniformly absolutely continuous. Thus by Vitali’s convergence theorem (see, for example, [11, p. 187]) we obtain vn=|un-u|q0 strongly in L1(), which is the desired conclusion. ∎

Proposition 3.3.

Assume {un}H1/2,2(R) to be a sequence with (-Δ)1/4unL2(R)1. Let unu weakly in H1/2,2(R) for some u and assume un is even, nonnegative and decreasing on R+. Then we have

(eαun2-1-αun2)𝑑x(eαu2-1-αu2)𝑑x

for any α(0,π).

Proof.

A similar proposition has already appeared; see [16, Lemma 3.1] and [9, Lemma 5.5]. We prove it here for the reader’s convenience.

Put

Φα(t)=eαt2-1andΨα(t)=eαt2-1-αt2.

Note that Φα(t) is nonnegative, strictly convex and Ψα(t)=2αtΦα(t). Thus by the mean value theorem we have

|Ψα(un)-Ψα(u)|Ψα(θun+(1-θ)u)|un-u|2α|θun+(1-θ)u|Φα(θun+(1-θ)u)|un-u|2α(|un|+|u|)(θΦα(un)+(1-θ)Φα(u))|un-u|2α(|un|+|u|)(Φα(un)+Φα(u))|un-u|.

Thus we have

|Ψα(un)-Ψα(u)|𝑑x2α(|un|+|u|)(Φα(un)+Φα(u))|un-u|𝑑x2α|un|+|u|La()Φα(un)+Φα(u)Lb()un-uLc()(3.2)

by Hölder’s inequality, where a,b,c>1 and 1a+1b+1c=1 are chosen later.

First, direct calculation shows that

(Φα(t))b<ebαt2-1(t)(3.3)

for all b>1. Thus if we fix 1<b<πα so that bα<π is realized, then we have

Φα(un)+Φα(u)Lb()b(Φα(un)Lb()+Φα(u)Lb())b2b-1((Φα(un))b𝑑x+(Φα(u))b𝑑x)2b-1((ebαun2-1)𝑑x+(ebαu2-1)𝑑x)2b-1A(bα)(unL2()2+uL2()2).

Here we used (3.3) for the third inequality and Theorem 1.3 for the last inequality, the use of which is valid since (-Δ)1/4unL2()1 and (-Δ)1/4uL2()1 by the weak lower semicontinuity. Note that {un} satisfies

supnunH1/2,2()C

for some C>0. Thus we have obtained Φα(un)+Φα(u)Lb()=O(1) independent of n.

Next, we estimate the term |un|+|u|La(). Since {un} is a bounded sequence in H1/2,2(), we have by [7, Theorem 6.9] that uLq()CunH1/2,2() for any q2. Thus we see |un|+|u|La()C for some C>0 independent of n for a2. Now, note that if we choose 1<b<πα and a>2 sufficiently large, then we can find c>2 such that 1a+1b+1c=1.

By these choices and Proposition 3.2, we conclude that un-uLc()0 as n. Going back to (3), we conclude that

Ψα(un)dxΨα(u)dx(n),

which is the desired conclusion. ∎

Proof of Theorem 1.7.

We will show that A(α) in (1.1) is attained for any 0<α<π. Since A(α)=A~(α) by Lemma 2.1, we choose a maximizing sequence for A~(α):

(eαun2-1)dx=A(α)+o(1)(n).

Here {un}nH1/2,2() satisfies (-Δ)1/4unL2()1 and unL2()=1. Appealing to the use of rearrangement, moreover we may assume that un is nonnegative, even and decreasing on +. Since {un}nH1/2,2() is a bounded sequence, we have uH1/2,2() such that unu in H1/2,2(). By Proposition 3.3, we see

(eαun2-1-αun2)𝑑x=(eαu2-1-αu2)𝑑x

as n. Therefore, since unL2()2=1, we have, letting n,

A(α)=α+(eαu2-1-αu2)𝑑x.(3.4)

Next, we claim that A(α)>α for any 0<α<π. Indeed, take any u0H1/2,2() such that u00, (-Δ)1/4u0L2()1 and u0L2()=1. Then we have

A(α)=A~(α)(eαu02-1)𝑑x=α+(eαu02-1-αu02)𝑑x.

Now, since eαt2-1-αt2>0 for any t>0, we have

(eαu02-1-αu02)𝑑x>0

for u00, which results in A(α)>α, the claim.

By the claim and (3.4), we conclude that the weak limit u satisfies u0. By the weak lower semi-continuity, we have that u0 satisfies (-Δ)1/4uL2()1 and uL2()1. Thus, by (3.4) again, we see

A(α)=α+(eαu2-1-αu2)𝑑xα+1uL2()2(eαu2-1-αu2)𝑑x=α+1uL2()2(eαu2-1)𝑑x-αuL2()2uL2()2=1uL2()2(eαu2-1)𝑑x.

Thus we have shown that uH1/2,2() maximizes A(α). ∎

Next, we prove Theorem 1.8. We follow Ishiwata’s argument in [15]. Let

M={uH1/2,2():uH1/2,2()=1},Jα:M,Jα(u)=(eαu2-1)𝑑x.

Actually, we will show a stronger claim that Jα has no critical point on M for sufficiently small α>0. Assume to the contrary that there exists a critical point vM of Jα for small α>0. Then we define an orbit on M through v as

vτ(x)=τv(τx),τ(0,),wτ=vτvτH1/2M.

Note that w1=v, thus it must hold that

ddτ|τ=1Jα(wτ)=0.

By the scaling rules (2.1), we see for any p2,

vτLp()p=τp/2-1vLp()pand(-Δ)1/4vτL2()=τ(-Δ)1/4vL2().

Now, we see

Jα(wτ)=(eαwτ2-1)𝑑x=j=1αjj!vτ2j(x)(vτ22+(-Δ)1/4vτ22)j=j=1αjj!vτ2j2j(vτ22+(-Δ)1/4vτ22)j=j=1αjj!τj-1v2j2j(v22+τ(-Δ)1/4v22)j=j=1αjj!fj(τ),

where

fj(τ)=τj-1c(b+τa)j

with a=(-Δ)1/4v22, b=v22 and c=v2j2j. Since

fj(τ)=τj-2c(b+τa)j+1{-τa+(j-1)b}

and (-Δ)1/4v22+v22=1, we calculate

ddτ|τ=1Jα(wτ)=j=1[αjj!τj-2v2j2j(v22+τ(-Δ)1/4v22)j+1{-τ(-Δ)1/4v22+(j-1)v22}]τ=1-α(-Δ)1/4v22v22+j=2αj(j-1)!v2j2j=α(-Δ)1/4v22v22{-1+j=2αj-1(j-1)!v2j2j(-Δ)1/4v22v22}.

Here, we need the following lemma.

Lemma 3.4 (Ogawa and Ozawa [28]).

There exists C>0 such that for any uH1/2,2(R) and p2, it holds that

uLp()pCpp/2(-Δ)1/4uL2()p-2uL2()2.

Proof.

For p=2j, Lemma 3.4 implies

v2j2j(-Δ)1/4v22v22C(2j)j(-Δ)1/4v22j-41(j2)C(2j)j.

Thus, for 0<α1 sufficiently small (it would be enough that α<12e), Stirling’s formula j!jje-j2πj implies that

j=2αj-1(j-1)!v2j2j(-Δ)1/4v22v22j=2αj-1(j-1)!(2j)jαC

for some C>0 independent of α. Therefore we have

ddτJα(wτ)|τ=1<0

for small α, which is the desired contradiction. ∎

4 Proof of Theorem 1.9

In order to prove Theorem 1.9, first we set

F(β)=supuH1/2,2()uH1/2,2()1u2eβu2𝑑x(4.1)

for β>0. Then we obtain the following result.

Proposition 4.1.

We have F(β)< for β<π.

Proof.

We follow the proof of [18, Theorem 1.5]. Take any uH1/2,2() with uH1/2,2()1 in the admissible sets for F(β) in (4.1). By appealing to the rearrangement, we may assume that u is even, nonnegative and decreasing on +. We divide the integral

u2eβu2𝑑x=Iu2eβu2𝑑x+Iu2eβu2𝑑x=(I)+(II),

where I=(-12,12).

First, we estimate (I). By the radial lemma (3.1), we see for any k, k2,

u2k(x)(uL2()22|x|)k=uL2()2k2k1|x|kfor x0.

Thus,

Iu2k(x)𝑑xuL2()2k2kIdx|x|k=uL2()2k2k-11/2dxxk=uL2()2kk-1.

Therefore, we have

(I)=Iu2eβu2𝑑x=Iu2(1+k=1βku2kk!)𝑑x=Iu2𝑑x+k=2βk-1(k-1)!Iu2k𝑑xuL2()2+k=2βk-1(k-1)!uL2()2kk-1=uL2()2(1+k=2βk-1(k-1)(k-1)!uL2()2(k-1)).

Now by the constraint uH1/2,2()1, we have uL2()1. Also if we put

ak=βk-1(k-1)(k-1)!,

then k=2ak converges since ak+1/ak=β(k-1)/k20 as k. Thus we obtain

(I)1+k=2βk-1(k-1)(k-1)!C,

where C>0 is independent of uH1/2,2() with uH1/2,2()1.

Next, we estimate (II). Set

v(x)={u(x)-u(1/2),|x|12,0,|x|>12.

Then by the argument of [18] we know that

(-Δ)1/4vL2()2(-Δ)1/4uL2()2

and

u2(x)v2(x)(1+uL2()2)+2

for xI. Put

w=v1+uL2()2.

Then we have wH~1/2,2(I) since v0 on I, and

(-Δ)1/4wL2()2=(1+uL2()2)(-Δ)1/4vL2()2(1+uL2()2)(1-uL2()2)1.

Thus we may use the fractional Trudinger–Moser inequality (Proposition 1.1) to w to obtain

Ieπw2𝑑xC

for some C>0 independent of u. By u2w2+2 on I, we conclude that

Ieπu2𝑑xIeπ(w2+2)𝑑x=e2πIeπw2𝑑xC.

Now, since β<π, there is an absolute constant C0 such that

t2eβt2C0eπt2

for any t. Finally, we obtain

(II)=Iu2eβu2𝑑xC0Ieπu2𝑑xC0C.

Proposition 4.1 follows from the estimates (I) and (II). ∎

By using Proposition 4.1 and arguing as in the proof of Theorem 1.3 (after establishing claims similar to those in Lemma 2.1 and Lemma 2.2), it is easy to obtain the following proposition.

Proposition 4.2.

For any 0<α<β<π, we have

E(α)(11-α/β)F(β).

Since F(β)< for any β<π, this proves the first part of Theorem 1.9. For the attainability of E(α) for α(0,π) it is enough to argue as in the proof of Theorem 1.7. We omit the details.

References

  • [1]

    S. Adachi and K. Tanaka, A scale-invariant form of Trudinger–Moser inequality and its best exponent, Proc. Amer. Math. Soc. (1999), no. 1102, 148–153.  Google Scholar

  • [2]

    Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trundinger–Moser inequality in N and its applications, Int. Math. Res. Not. IMRN (2010), no. 13, 2394–2426.  Google Scholar

  • [3]

    F. J. Almgren, Jr. and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc. 2 (1989), no. 4, 683–773.  CrossrefGoogle Scholar

  • [4]

    D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in 𝐑2, Comm. Partial Differential Equations 17 (1992), no. 3–4, 407–435.  Google Scholar

  • [5]

    L. Carleson and S.-Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. (2) 110 (1986), no. 2, 113–127.  Google Scholar

  • [6]

    D. Cassani, F. Sani and C. Tarsi, Equivalent Moser type inequalities in 2 and the zero mass case, J. Funct. Anal. 267 (2014), no. 11, 4236–4263.  Google Scholar

  • [7]

    E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573.  Web of ScienceCrossrefGoogle Scholar

  • [8]

    J. A. M. B. do Ó, N-Laplacian equations in 𝐑N with critical growth, Abstr. Appl. Anal. 2 (1997), no. 3–4, 301–315.  Google Scholar

  • [9]

    M. Dong and G. Lu, Best constants and existence of maximizers for weighted Trudinger–Moser inequalities, Calc. Var. Partial Differential Equations 55 (2016), no. 4, Article ID 88.  Web of ScienceGoogle Scholar

  • [10]

    M. Flucher, Extremal functions for the Trudinger–Moser inequality in 2 dimensions, Comment. Math. Helv. 67 (1992), no. 3, 471–497.  CrossrefGoogle Scholar

  • [11]

    G. B. Folland, Real analysis, 2nd ed., Pure Appl. Math. (New York), John Wiley & Sons, New York, 1999.  Google Scholar

  • [12]

    L. Fontana and C. Morpurgo, Sharp Adams and Moser–Trudinger inequalities on n and other spaces of infinite measure, preprint (2015), https://arxiv.org/abs/1504.04678.  

  • [13]

    L. Fontana and C. Morpurgo, Sharp exponential integrability for critical Riesz potentials and fractional Laplacians on n, preprint (2017), https://arxiv.org/abs/1702.02708.  

  • [14]

    A. Iannizzotto and M. Squassina, 1/2-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl. 414 (2014), no. 1, 372–385.  CrossrefWeb of ScienceGoogle Scholar

  • [15]

    M. Ishiwata, Existence and nonexistence of maximizers for variational problems associated with Trudinger–Moser type inequalities in N, Math. Ann. 351 (2011), no. 4, 781–804.  Google Scholar

  • [16]

    M. Ishiwata, M. Nakamura and H. Wadade, On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 2, 297–314.  CrossrefGoogle Scholar

  • [17]

    S. Iula, A note on the Moser–Trudinger inequality in Sobolev–Slobodeckij spaces in dimension one, preprint (2016), https://arxiv.org/abs/1610.00933v1.  

  • [18]

    S. Iula, A. Maalaoui and L. Martinazzi, A fractional Moser–Trudinger type inequality in one dimension and its critical points, Differential Integral Equations 29 (2016), no. 5–6, 455–492.  Google Scholar

  • [19]

    V. I. Judovič, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR 138 (1961), 805–808.  Google Scholar

  • [20]

    N. Lam, G. Lu and L. Zhang, Equivalence of critical and subcritical sharp Trudinger–Moser–Adams inequalities, preprint (2015), https://arxiv.org/abs/1504.04858v1.  

  • [21]

    X. Li and Y. Yang, Extremal functions for singular Trudinger–Moser inequalities in the entire Euclidean space, preprint (2016), https://arxiv.org/abs/1612.08247.  

  • [22]

    Y. Li and B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in n, Indiana Univ. Math. J. 57 (2008), no. 1, 451–480.  Google Scholar

  • [23]

    E. H. Lieb and M. Loss, Analysis, 2nd ed., Grad. Stud. Math. 14, American Mathematical Society, Providence, 2001.  Google Scholar

  • [24]

    K.-C. Lin, Extremal functions for Moser’s inequality, Trans. Amer. Math. Soc. 348 (1996), no. 7, 2663–2671.  CrossrefGoogle Scholar

  • [25]

    L. Martinazzi, Fractional Adams–Moser–Trudinger type inequalities, Nonlinear Anal. 127 (2015), 263–278.  Web of ScienceCrossrefGoogle Scholar

  • [26]

    J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077–1092.  Google Scholar

  • [27]

    T. Ogawa, A proof of Trudinger’s inequality and its application to nonlinear Schrödinger equations, Nonlinear Anal. 14 (1990), no. 9, 765–769.  CrossrefGoogle Scholar

  • [28]

    T. Ogawa and T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem, J. Math. Anal. Appl. 155 (1991), no. 2, 531–540.  CrossrefGoogle Scholar

  • [29]

    T. Ozawa, On critical cases of Sobolev’s inequalities, J. Funct. Anal. 127 (1995), no. 2, 259–269.  CrossrefGoogle Scholar

  • [30]

    T. Ozawa, Characterization of Trudinger’s inequality, J. Inequal. Appl. 1 (1997), no. 4, 369–374.  Web of ScienceGoogle Scholar

  • [31]

    E. Parini and B. Ruf, On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces, preprint (2016), https://arxiv.org/abs/1607.07681v1.  

  • [32]

    Y. J. Park, Fractional Polya–Szegö inequality, J. Chungcheong Math. Soc. 24 (2011), no. 2, 267–271.  Google Scholar

  • [33]

    S. Pohozaev, The Sobolev embedding in the case pl=n, Proceedings of the Technical Scientic Conference on Advances of Scientic Research (1964/1965), Energetics Institute, Moscow (1965), 158–170.  Google Scholar

  • [34]

    B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in 2, J. Funct. Anal. 219 (2005), no. 2, 340–367.  Google Scholar

  • [35]

    W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), no. 2, 149–162.  CrossrefGoogle Scholar

  • [36]

    M. Struwe, Critical points of embeddings of H01,n into Orlicz spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 5, 425–464.  Google Scholar

  • [37]

    N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483.  Google Scholar

About the article

Received: 2017-05-22

Revised: 2017-08-10

Accepted: 2017-09-21

Published Online: 2017-11-16


Funding Source: Japan Society for the Promotion of Science

Award identifier / Grant number: 15H03631

Award identifier / Grant number: 26610030

Part of this work was supported by JSPS Grant-in-Aid for Scientific Research (B), no. 15H03631, and JSPS Grant-in-Aid for Challenging Exploratory Research, no. 26610030.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 868–884, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0116.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
João Marcos do Ó, Jacques Giacomoni, and Pawan Kumar Mishra
Nonlinear Differential Equations and Applications NoDEA, 2019, Volume 26, Number 4

Comments (0)

Please log in or register to comment.
Log in