# Critical and Subcritical Anisotropic Trudinger–Moser Inequalities on the Entire Euclidean Spaces

Authors: Xiaoping Song; Dongliang Li; Maochun Zhu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

We investigate the subcritical anisotropic Trudinger–Moser inequality in the entire spaceℝN, obtain the asymptotic behavior of the supremum for the subcritical anisotropic Trudinger–Moser inequalities on the entire Euclidean spaces, and provide a precise relationship between the supremums for the critical and subcritical anisotropic Trudinger–Moser inequalities. Furthermore, we can prove critical anisotropic Trudinger–Moser inequalities under the nonhomogenous norm restriction and obtain a similar relationship with the supremums of subcritical anisotropic Trudinger–Moser inequalities.


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## Body

## 1. Introduction

LetΩ be a domain with finite measure in ℝN and W01,pΩ represent the completion of C∞Ω in the norm(1)uW01,pΩ=∫Ω∇up+updx1/p.The classical Sobolev embedding theorem tells us thatW01,NΩ⊆LpΩ, where p>1, but W01,NΩ⊆L∞Ω. One knows from the studies of Yudovich [1], Peetre [2], Pohozaev [3], and Trudinger [4] that W01,NΩ embeds into the Orlicz space LϕΩ, with the N-function ϕt=exptN/N−1−1. This embedding was made more precise by Moser [5] and obtained the following inequality:(2)sup∇uN≤1∫ΩexpαuN/N−1−1dx<+∞ifα≤αN,=+∞ifα>αN,where αN=NωN−11/N−1 and ωN−1 is the measure of the unit sphere in ℝN.In 2000, Adachi-Tanaka [6] obtained a sharp Trudinger–Moser inequality on ℝN:(3)supu∈W1,NℝN,∫ℝN∇uNdx≤1∫ℝNΦNαuN/N−1dx≤Cα,NuNN,iff0<α<αN,where ΦNt:=et−∑i=0N−2ti/i!. Note that inequality (3) has the subcritical form, that is, α<αN. Later, in [7, 8], Li and Ruf showed that the best exponent αN becomes admissible if the Dirichlet norm ∫ℝN∇uNdx is replaced by Sobolev norm ∫ℝNuN+∇uNdx. More precisely, they proved that(4)supu∈W1,NℝN∫ℝNuN+∇uNdx≤1∫ℝNΦNαuNN−1dx<+∞,iffα≤αN.The proofs of both critical and subcritical Trudinger–Moser inequalities (3) and (4) rely on the Pólya–Szegö inequality and the symmetrization argument. Lam and Lu [9, 10] developed a symmetrization-free method to establish the critical Trudinger–Moser inequality (see also Li, Lu, and Zhu [11]) in settings such as the Heisenberg group where the Pólya–Szegö inequality fails. Such an argument also provides an alternative proof of both critical and subcritical Trudinger–Moser inequalities (3) and (4) in the Euclidean space. In fact, the equivalence and relationship between the supremums of critical and subcritical Trudinger–Moser inequalities have been established by Lam, Lu, and Zhang [12].Theorem 1.
LetN≥2 and αN=ωN−11/N−1, where 0≤β<N and 0≤α<αN. Then, ATα,β is defined as(5)ATα,β≜sup∇uN≤11uNN−β∫ℝNΦNα1−βNuN/N−1dxxβ.Then, there exist positive constants cN,β and CN,β when α is close enough to αN; one has(6)cN,β1−α/αNN−1N−β/N≤ATα,β≤CN,β1−α/αNN−1N−β/N,and the constant αN is sharp, that is, ATαN,β=∞.Theorem 2.
LetATαN,β=∞, where 0≤β<N, and 0<a,b, and define(7)MTa,bβ≜sup∇uNa+uNb≤1∫ℝNΦNαN1−βNuN/N−1dxxβ.
Then,MTa,bβ<∞ if and only if b≤N, the constant αN is sharp, and(8)MTa,bβ=supα∈0,αN1−α/αNN−1/Naα/αNN−1/NbN−β/bATα,β,in particular,(9)MTa,bβ=supα∈0,αN1−α/αNN−1α/αNN−1N−β/NATα,β.
Trudinger–Moser inequalities have many other related extensions: concentration-compactness principle and Adimurthi–Druet-type inequalities on unbounded domains (see Do et al. [13] and Lu-Zhu [14]); extensions of the Trudinger–Moser inequality to higher-order Sobolev spaces (see Adams [15], Ruf-Sani [16], Lam-Lu [10], and Chen-Lu-Zhu [17]); Trudinger–Moser inequalities on manifolds or Heisenberg groups (see Li [18, 19]); Trudinger–Moser inequalities on Heisenberg groups (see the work of Cohn, Lam, Lu, and Zhu [9, 11, 20]).
In 2012, Wang and Xia [21] investigated a sharp Trudinger–Moser inequality involving the anisotropic Dirichlet norm ∫ΩFN∇u1/Ndx on W01,NΩ for N≥2:(10)supu∈W01,NΩ,∫ΩFN∇udx≤1∫ΩeαuN/N−1dx<+∞iff0<α≤λN=NN/N−1kN1/N−1.
Here,kN is the volume of a unit Wulff ball WF:=x∈ℝN|F0x≤1, F is convex and homogeneous of degree 1, and its polar F0 represents a Finsler metric on ℝN.
Similar to B. Ruf’s work [8], when anisotropic Dirichlet norm ∫ΩFN∇udx1/N is replaced by full anisotropic Sobolev norm ∫ΩFN∇u+uNdx1/N, Zhou [22] extended the results of Wang and Xia [21] to the entire space,(11)supu∈W1,NℝN,∫ℝNFN∇u+uNdx1N≤1∫ℝNΦNλuN/N−1dx<∞,provided λ≤λN, and the integral above will tend to infinity for any λ>λN.
In this paper, we will establish the Adachi–Tanaka-type subcritical Trudinger–Moser inequality and the equivalence relationship between the supremums of critical and subcritical Trudinger–Moser inequalities involving the anisotropic norm restriction similar as in [12].
Our main results can be stated as follows.Theorem 3.
LetN≥2, λN=NN/N−1κN1/N−1, where 0≤β<N and 0≤λ<λN. AATλ,β is defined as(12)AATλ,β≜supF∇uLN≤11uNN−β∫ℝNΦNλ1−βNuN/N−1dxFoxβ.
Ifλ is close enough to λN, then there exist constants cN,β and CN,β such that(13)cN,β1−λ/λNN−1N−β/N≤AATλ,β≤CN,β1−λ/λNN−1N−β/N,where λN is sharp, that is, AATλN,β=∞.Theorem 4.
LetN≥2, 0≤β<N, and 0<a,b. Define(14)AMTa,bβ≜supF∇uLNa+uLNb≤1∫ℝNΦNλN1−βNuN/N−1dxFoxβ,AMTβ≜AMTN,Nβ.
Then,AMTa,bβ<∞ if and only if b≤N, and λN is sharp, and we also have(15)AMTa,bβ=supλ∈0,λN1−λ/λNN−1/Naλ/λNN−1/NaN−β/bAATλ,β,in particular,(16)AMTβ=supλ∈0,λN1−λ/λNN−1λ/λNN−1N−β/NAATλ,β.


## 2. Finsler Metric and Some Useful Lemmas

Before giving the proof, for the convenience of the readers, we provide some notations and basic facts about the Finsler metric. LetF:ℝN↦ℝ be a nonnegative convex function of class C2ℝN\0 which is even and positively homogeneous of degree 1, for any ξ∈ℝNandt∈ℝ so that(17)Ftξ=tFξ.A typical example isFξ=∑iξiq1/q for q∈1,∞. For any ξ≠0, we then assume Fξ>0.Because of homogeneity ofF, there exist two constants 0<a≤b<∞ such that(18)aξ≤Fξ≤bξ.If we consider the map(19)φ:SN−1⟶ℝN,φξ=Fξξ.Its imageφSN−1 is a smooth, convex hypersurface in ℝN which is called Wulff shape of F. Let Fo be the support function of(20)K≜x∈ℝN:Fx≤1,which is defined by(21)Fox≜supξ∈kx,ξ.We can verify thatFo:ℝN↦0,+∞ is also a convex, homogeneous function of class of C2ℝN\0. The fact F in the sense is dual to Fo in that(22)Fox=supξ≠0x,ξFξ,Fx=supξ≠0x,ξFoξ.We can see thatφSN−1=x∈ℝN|Fox=1. The Lebesgue measure of WF is κN≜WF, where WF≜x∈ℝN|Fox≤1. We call Wr0≜x∈ℝN|Fox≤r a Wulff ball of radius r with center at 0. Next, according to the assumption of F, we can give some properties of the function F:iFx−Fy≤Fx+y≤Fx+Fyii1/C≤∇Fx≤Cand1/C≤∇Fox≤C for any x≠0, for some C>0iiix,∇Fx=Fxandx,∇Fox=Fox for any x≠0ivF∇Fox=1andFoF∇x=1 for any x≠0vFoxFξ∇Fox=x for any x≠0viFξtξ=sgntFξξ for any ξ≠0,t≠0In the following, we give two lemmas that will be used later.Lemma 1.
(23)AATλ,β=supF∇uLN≤1,uLN=1∫ℝNΦNλ1−βNuN/N−1dxFoxβ.Proof.
For anyu∈W1,NℝN satisfying F∇uLN≤1, we can define vx as(24)vx=uλ˜x,λ˜=uLN,and then we have(25)∇vx=λ˜∇uλ˜x.
Due to the homogeneity ofFx, we obtain(26)F∇vLN=∫ℝNFN∇uλ˜xdλ˜x1/N=F∇uLN≤1,vLN=1,and so we have(27)∫ℝNΦNλ1−βNvxN/N−1dxFoxβ=∫ℝNΦNλ1−βNuλ˜xN/N−1dxFoxβ=λ˜β−N∫ℝNΦNλ1−βNuλ˜xN/N−1dλ˜xFoλ˜xβ=uLNβ−N∫ℝNΦNλ1−βNuN/N−1dxFoxβ.
Hence,(28)AATλ,β≤supF∇vLN≤1,vLN=1∫ℝNΦNλ1−βNvN/N−1dxFoxβ≤supF∇uLN≤1uLNβ−N∫ℝNΦNλ1−βNuN/N−1dxFoxβ=AATλ,β,and the proof is finished.By Lemma 2, when we consider the sharp Trudinger–Moser inequality, we can always assumeuLN=1.Lemma 2.
The sharp subcritical Trudinger–Moser inequality is the sequence of the sharp critical Trudinger–Moser inequality. More precisely, ifAMTa,bβ is bounded, then AATλ,β is also bounded and(29)AATλ,β≤λ/λNN−1/Nb1−λ/λNN−1/NaN−β/bAMTa,bβ,in particular,(30)AATλ,β≤λ/λNN−11−λ/λNN−1N−β/NAMTβ.Proof.
Letu∈W1,NℝNwithF∇uLN≤1;uLN=1. We set(31)vx=λλNN−1/Nuλ˜x,where(32)λ˜=λ/λNN−1/Nb1−λ/λNN−1/Na1/b.Then,(33)F∇vLNa=λλNN−1/Naλ˜F∇uλ˜xLNa=λλNN−1/NaF∇uLNa≤λλNN−1/Na,vLNb=λλNN−1/Nbλ˜−buLNb=1−λλNN−1/Na.
BecauseF∇vLNa+vLNb≤1, we have(34)∫ℝNΦNλ1−βNuxN/N−1dxFoxβ=∫ℝNΦNλ1−βNuλ˜xN/N−1dλ˜xFoλ˜xβ=λ˜N−β∫ℝNΦNλN1−βNvxN/N−1dxFoxβ≤λ/λNN−1/Nb1−λ/λNN−1/NaN−β/bAMTa,bβ.


## 3. Equivalence between the Critical and Subcritical Anisotropic Trudinger–Moser Inequalities under the Homogeneous Norm Restriction

In this section, we give the asymptotic behavior of the supremum for the subcritical anisotropic Trudinger–Moser inequalities, show the equivalence between the critical and subcritical anisotropic Trudinger–Moser inequalities under the homogeneous norm restriction, and finish the Proof of Theorem 3.Proof of Theorem 3.
Assumeu∈C0∞ℝN\0, F∇uLN≤1, and uLN=1. Set(35)Ωu=x:ux>1−λλNN−11/N.
Now, we estimate the volume ofΩu:(36)Ωu=∫Ωu1dx≤∫ΩuuxN1−λ/λNN−1dx≤11−λ/λNN−1.
We rewrite(37)∫ℝNΦNλ1−βNuxN/N−1dxFoxβ,as(38)∫ℝN\Ωu+∫ΩuΦNλ1−βNuxN/N−1dxFoxβ≜I1+I2.Then,(39)I1=∫ℝN\ΩuΦNλ1−βNuxN/N−1dxFoxβ≤∫u≤1ΦNλuxN/N−1dxFoxβ≤eλ∫u≤1uNFoxβdx≤eλ∫u≤1;Fox≥1uNFoxβdx+∫u≤1;Fox<1uNFoxβdx≤CN,β11−λ/λNN−1N−β/N.
By calculation,(40)I2=∫ΩuΦNλ1−βNuxN/N−1dxFoxβ≤∫Ωuexpλ1−βNuxN/N−1dxFoxβ.
In the areaΩu, we assume(41)vx=ux−1−λλNN−11/N,where v∈W01,NΩu, and we can easily have(42)F∇vLN≤1.Set ε=λN/λ−1, for any a,b,ε>0 and p>1.We have that(43)uxN/N−1≤v+1−λλNN−11/NN/N−1≤1+εvN/N−1+1−11+εN−11/1−N1−λλNN−11/NN/N−1=λNλvN/N−1+1−λλNN−11/1−N1−λλNN−11/N−1=λNλvN/N−1+1,by using the following elementary inequality:(44)a+bp≤εbp+1−1+ε−1/p−11−pap.
Using the singular Trudinger–Moser inequality under the anisotropic norm in the bounded domain [23], we have that(45)I2≤∫Ωuexpλ1−βNuxN/N−1dxFoxβ≤∫ΩuexpλN1−βNvxN/N−1+λ1−βNdxFoxβ≤CN,βΩu1−β/N≤CN,β1−λ/λNN−1N−β/N.
Therefore,(46)AATλ,β≤CN,β1−λ/λNN−1N−β/N.
Next, we show that thatAATλN,β=∞. Set ukx:(47)ukx=0,ifFox≥1,N−βκNNk1/Nlog1Fox,ife−K/N−β<Fox<1,1NκN1/NkN−β1/N/N,if0≤Fox≤e−k/N−β.
By calculation, we have(48)ukLN=∫0e−k/N−βNκN−1kN−βN−1rN−1dr+∫e−k/N−β1N−βNκNkln1rNrN−1dr=CkN−1∫0e−k/N−βrN−1dr+C1k∫0k/N−βyNe−Nydy∼kNe−Nk/N−β+1k∼1k;F∇ukLN=N−βNκNk∫e−k/N−β1r−NrN−1κNNdr=1.
Therefore,(49)F∇ukLN=1,ukLN=O1k.
Next,(50)∫ℝNΦNλN1−βNukxN/N−1dxFoxβ≥∫0≤Fox≤e−k/N−βΦNkFoxβdx=NκNΦNk∫0e−k/N−βrN−1−βdr=NκNΦNkekN−β⟶NκNN−βk⟶∞.
That is,(51)AATλN,β=supF∇uLN≤11uNN−β∫ℝNΦNλN1−βNuN/N−1dxFoxβ⟶∞.
There exists a quite large constantM1 and when k≥M1, we have(52)ukLN=∫0e−k/N−βNκN−1kN−βN−1rN−1dr+∫e−k/N−β1N−βNκNkln1rNrN−1dr=CkN−1∫0e−k/N−βrN−1dr+C1k∫0k/N−βyNe−Nydy∼kNe−Nk/N−β+1k∼1k.
Therefore,(53)ukLNN−β≈1kN−β/N.
Now, we establish the lower bounds ofAATλ,β:(54)∫ℝNΦNλ1−βNukxN/N−1dxFoxβ≥C∫0e−k/N−βΦNλ1−βN1NκN1/N−1kN−βrN−1−βdr≥C∫0e−k/N−βΦNλλNkrN−1−βdr∼ΦNλλNke−k.
Whenλ/λN≥1/2, there exists a very large constant M2 which is independent of λ; when k≥M2, we have that(55)ΦNλλNk≈eλ/λNk.Then,(56)∫ℝNΦNλ1−βNukxN/N−1dxFoxβ≈>eλ/λNke−k=e−1−λ/λNk.
Whenλ is close enough to λN, we can always find a suitable k satisfying 1≤1−λ/λNk≤2 and(57)λ≈1−1kλN≥1−1maxM1,M2λN,or(58)maxM1,M2≤k≈11−λ/λN.Then,(59)1ukLNN−β∫ℝNΦNλ1−βNukxN/N−1dxFoxβ∼kN−β/Ne−2∼11−λ/λNN−β/N∼11−λ/λNN−1N−β/N.
Sinceλ is close enough to λN, we have(60)1−λ/λNN−11−λ/λN≈1,which is(61)AATλ,β≥cN,β1−λ/λNN−1N−β/N.


## 4. Critical Anisotropic Trudinger–Moser Inequalities under the Nonhomogenous Norm Restriction and the Relationship with the Subcritical Anisotropic Trudinger–Moser Inequalities

In this section, we prove critical anisotropic Trudinger–Moser inequalities under the nonhomogenous norm restriction and give a precise relationship between the supremums for the critical and subcritical anisotropic Trudinger–Moser inequalities under the nonhomogenous norm restriction.Proof of Theorem 4.
Letb≤Nandu∈W1,NℝN\0,withF∇uLNa+uLNb≤1. Assume(62)F∇uLN=θ∈0,1;uLNb≤1−θa.
If1/2<θ<1, let(63)vx=uλ˜xθ,where(64)λ˜=1−θa1/bθ>0.
By the simple calculation,(65)F∇vLN=∫ℝNF∇uλ˜xθNdx1/N=F∇uLNθ=1,vLNN=∫ℝNvNdx=1θN∫ℝNuλ˜xNdx=1θNλ˜NuLNN≤1−θaN/bθNλ˜N=1.
By Theorem3, we can obtain(66)∫ℝNΦNλN1−βNuxN/N−1dxFoxβ=∫ℝNΦNλN1−βNuλ˜xN/N−1dλ˜xFoλ˜xβ≤λ˜N−β∫ℝNΦNθN/N−1λN1−βNvxN/N−1dxFoxβ≤λ˜N−βAATθN/N−1λN,β≤1−θaN/bθN1−β/NCN,β1−θN/N−1λN/λNN−1N−β/N≤1−θaN/b1−θN1−β/NcN,β≤C.
Whenθ⟶1, we can use L’Hospital’s rule to estimate the last but one term 1−θaN/b/1−θN1−β/NcN,β that is finite, since b≤N.
If0<θ≤1/2, we define(67)vx=2u2x,and then we can obtain(68)F∇vLN=2F∇uLN≤1,vLN=uLN≤1.
By Theorem3, we can obtain(69)∫ℝNΦNλN1−βNuxN/N−1dxFoxβ=∫ℝNΦNλN1−βNu2xN/N−1d2xFo2xβ≤2N−β∫ℝNΦNλN1−βN2−N/N−1vN/N−1dxFoxβ≤2N−βAAT2−N/N−1λN,β≤CN,β.
Next, we will prove the constantλN1−β/N is optimal. We can establish the same sequence ukx as in (47). We recall that ukx satisfies(70)F∇ukLN=1;ukLN=O1k1/N.Set(71)wkx=λ˜kukx,where λ˜k∈0,1 satisfies(72)λ˜ka+λ˜kbukLNb=1.
Then,1≥λ˜k≥1−O1/1/kb/N1/a⟶1k⟶∞.
Thus,(73)F∇wkxLNa+wkxLNb=1and(74)∫ℝNΦNλ1−βNwkxN/N−1dxFoxβ≥∫0≤Fox≤e−k/N−βΦNλ1−βNλ˜k1NκN1/NkN−βN−1/NN/N−1dxFoxβ∼ΦNλ1−βN1NκN1/N−11N−βλ˜kN/N−1k∫0≤Fox≤e−k/N−βdxFoxβ=ΦNλλNλ˜kN/N−1k∫0≤Fox≤e−k/N−βdxFoxβ∼expkλλNλ˜kN/N−1−1,because λ˜k⟶1and so there exists a large enough k, such that λ/λNλ˜kN/N−1−1>0. At last, we have that(75)expkλλNλ˜kN/N−1−1⟶∞k⟶∞.
Now, we will prove(76)AMTa,bβ=supλ∈0,λN1−λ/λNN−1/Naλ/λNN−1/NaN−β/BAATλ,β,when AMTa,bβ<∞. By Lemma 2, we have(77)supλ∈0,λN1−λ/λNN−1/Naλ/λNN−1/NbN−β/bAATλ,β≤AMTa,bβ.
Letukx be the maximizing sequence of AMTa,bβ, i.e.,(78)uk∈W1,NℝN\0:F∇ukLNa+ukLNb≤1,∫ℝNΦNλN1−βNukxN/N−1dxFoxβ⟶AMTa,bβk⟶∞.We define vkx=ukλ˜kx/F∇ukLN, where(79)λ˜k=1−F∇ukxLNaF∇ukxLNb1/B>0.
Thus, we have(80)F∇vkLN=1;vkLN≤1,∫ℝNΦNλN1−βNukxN/N−1dxFoxβ=λ˜kN−β∫ℝNΦNF∇ukxLNN/N−1λN1−βNvkxN/N−1dxFoxβ≤λ˜kN−βAATF∇ukxLNN/N−1λN,β=1−F∇ukxLNaF∇ukxLNbN−β/bAATF∇ukxLNN/N−1λN,β≤supλ∈0,λN1−λ/λNN/N−1aλ/λNN/N−1aN−β/bAATλ,β.
Therefore, we have proved(81)AMTa,bβ=supλ∈0,λN1−λ/λNN−1/Naλ/λNN−1/NbN−β/bAATλ,β,when AMTa,bβ<∞
At last, we will prove thatAMTa,bβ<∞ if and only if b≤N. Recall that AMTa,bβ<∞ if b≤N. Here, if there exist some b>N such that AMTa,bβ<∞, then we have(82)limsupλ⟶λN−1−λλNN−1/NaN−β/bAATλ,β<∞.
SinceAMTa,bβ<∞,(83)limsupλ⟶λN−1−λλNN−1N−β/NAATλ,β<∞.
From Theorem3, we can obtain(84)liminfλ⟶λN−1−λλNN−1N−β/NAATλ,β>0.
Thus, whenλ is close enough to λN−, we have that AATλ,β∼1−λ/λNN−1β−N/N. Thus,(85)liminfλ⟶λN−1−λ/λNN−1/NaN−β/b1−λ/λNN−1N−β/N<∞,which is impossible because b>N. Hence, we complete the proof.



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