Sharp Singular Trudinger–Moser Inequalities Under Different Norms

1 Introduction

The Trudinger–Moser and Adams inequalities are the replacements for the Sobolev embeddings in the limiting case. When Ω⊂ℝN is a bounded domain and k⁢p<N, it is well known that W0(Ω)k,p⊂Lq(Ω) for all q with 1≤q≤N⁢pN-k⁢p. However, we have W0k,Nk⁢(Ω)⊈L∞⁢(Ω). For example, one could check that for any N>1, the unbounded function log⁡log⁡(1+1|x|) is in W1,N⁢(B1⁢(0)). Nevertheless, in this situation, Trudinger [58] proved that W01,N⁢(Ω)⊂LφN⁢(Ω), where LφN⁢(Ω) is the Orlicz space associated with the Young function

for some α>0 (see also Yudovich [18] and Pohozaev [54]):

These results were refined in the 1971 paper [51] by J. Moser. In fact, Moser showed that the best possible constant α in the above theorem is

where ωN-1 is the area of the surface of the unit N-ball, in the sense that if α>αN, then the above inequality can no longer hold with some c0 independent of u. This inequality is by now known as the Trudinger–Moser inequality. This type of result has been studied and extended in many directions.

It can be noted that the Trudinger–Moser–Adams inequalities are senseless when the domains have infinite volume. Thus, it is interesting to investigate versions of the Trudinger–Moser–Adams inequalities in this setting. We will state here two such results:

We have the following remarks: There are attempts to extend the Trudinger–Moser inequality to the singular case by Adimurthi and Sandeep in [2], to infinite volume domains by Cao [6], Ogawa [52, 53] in dimension two and by do Ó [3], Adachi and Tanaka [1], and Kozono, Sato and Wadade [19] in higher dimension. Interestingly enough, the Trudinger–Moser-type inequality can only be established for the subcritical case when only the seminorm ∥∇⁡u∥N is used in the restriction of the function class. Indeed, (1.1) has been proved in [3] and [1] if α<αN. Moreover, their results are actually sharp in the sense that the supremum is infinity when α≥αN. To achieve the critical case α=αN, Ruf [55] and then Li and Ruf [37] need to use the full form in W1,N, namely, (∥u∥NN+∥∇⁡u∥NN)1N. Again αN is sharp. In all the above works, the symmetrization argument is very crucial. An alternative proof of (1.2) without using symmetrization has been given by the first two authors in [24, Section 6]. Different proofs of (1.1) and (1.2) have also been given without using symmetrization in settings such as on the Heisenberg group or high and fractional order Sobolev spaces where symmetrization is not available (see the work by the first two authors [23], and by Tang and the first two authors, [27] which extend the earlier work by Cohn and Lu [9] on finite domains). The arguments developed in these works [23, 24, 27] are from local inequalities to global ones using the level sets of functions under consideration. This argument has also be used in other contexts (see [8, 12, 28, 31, 32, 33, 40, 41, 42, 43, 46, 45, 59, 60]). In the recent paper [30], it was proved that these two versions of the above sharp subcritical and critical Trudinger–Moser-type inequalities are indeed equivalent. Indeed, an identity of these suprema was also given. Moreover, it was also showed in [30] that for a,b>0,

We also refer the reader to [57] for a generalization of the equivalence result in [30] to the equivalence of critical and subcritical suprema of the Trudinger–Moser inequalities in Lorentz–Sobolev norms.

We also mention that extremal functions for Trudinger–Moser inequalities on bounded domains were first established by Carleson and Chang in their celebrated work [7], and has been extended in a number of settings by de Figueiredo, do Ó, and Ruf [10], Flucher [14], Lin [39], and on Riemannian manifolds by Y. X. Li [34, 35], and on infinite volume domains by Ruf [55], Li and Ruf [37], Li and Ndiaye [36], Lu and Yang [47], Zhu [61], Ishiwata [16] and Ishiwata, Nakamura and Wadade [17], Dong and Lu [13], Lam [22, 20, 21], Lam, Lu and Zhang [29], Dong, Lam and Lu [12], Lu and Zhu [48]. In particular, using the identity which exhibits the relationship between the suprema of the critical and subcritical Trudinger–Moser inequalities developed in [30], the authors established in [29] the existence and nonexistence of extremal functions for a large class of Trudinger–Moser inequalities on the entire spaces without using the blow-up analysis of the associated Euler–Lagrange equations. The symmetry of extremal functions have also been established for the first time in the literature. Moreover, they evaluate precisely the values of the suprema for the first time for a class of Trudinger–Moser inequalities for certain small parameters. (The main results of [29] were already described in [13] as Theorems C, D and E.) The method developed in [29] has been successfully used in [22, 20, 21].

The main purpose of this paper is to study several sharp versions of the Trudinger–Moser inequalities on the Sobolev-type spaces DN,q⁢(ℝN),q≥1, the completion of C0∞⁢(ℝN) under the norm ∥∇⁡u∥N+∥u∥q. Our results have a close connection to the study of the well-known Caffarelli–Kohn–Nirenberg inequalities first proved in [5]. See [12] and also [11, 26] for related results. We note that when q=N, then DN,q⁢(ℝN)=W1,N⁢(ℝN).

In this paper, we will always assume that

In the study of the Trudinger–Moser-type inequalities, we need to choose a function ΦN,q,β that behaves like an exponential function at infinity, and has reasonable power near 0 by the Sobolev embeddings. Recall that by the Caffarelli–Kohn–Nirenberg inequalities, we have DN,q⁢(ℝN)↪Lp⁢(ℝN;d⁢x|x|β) continuously with p>q⁢(1-βN) (p≥q if β=0) (see Section 2). Hence it is natural to consider the following function:

We are now ready to state our main results. We will adapt the notations of constants TMCa,b⁢(q,N,β) to denote the suprema of the critical Trudinger–Moser inequalities and TMSC⁢(q,N,α,β) to denote the suprema of the subcritical Trudinger–Moser inequalities with the corresponding parameters.

As our first aim in this paper, we will prove that

As a consequence of Theorem 1.1 in the special case p=q=N and the fact that there exists Cα,N>0 such that for all u,

we have the singular Trudinger–Moser-type inequality in the spirit of Adachi and Tanaka [1]:

Another consequence of our Theorem 1.1 is the following inequality:

We note here that when α↑αN-, TMSC⁢(q,N,α,β)→∞. Hence Corollary 1.2 is subcritical in this sense. Thus we may ask what the magnitude of TMSC⁢(q,N,α,β) is when α↑αN-, and what the critical version of Corollary 1.2 is. Our next goal is to provide the answers for these questions. Indeed, we will show that:

Having studied the Trudinger–Moser inequalities under the norm constraint ∥∇⁡u∥Na+∥u∥qb≤1 (for b≤N), it is natural to ask what kind of inequalities will hold in the case b>N. In particular, if we like to have the norm constraint ∥∇⁡u∥Na+∥u∥qb≤1 (for b>N), then can we still have a valid Trudinger–Moser-type inequality? Indeed, we will answer the above question by the following versions of the Trudinger–Moser inequality on DN,q⁢(ℝN) which are new even when q=N. We remark that the following results (Theorem 1.3, Theorem 1.4, Corollary 1.3, Corollary 1.4, Theorem 1.5) are Trudinger–Moser inequalities of exact growth. These type of inequalities were studied earlier by Ibrahim, Masmoudi and Nakanishi [15], Lam and Lu [25], Masmoudi and Sani [49, 50], Lu and Tang [41], Lu, Tang and Zhu [44]. Our results below are improved versions of the aforementioned ones.

An equivalent version of Theorem 1.3 is the following:

As consequences of our results, we have:

Since it can be verified easily that the constant C⁢(N,p,q,β,a,k) in Theorem 1.4 tends to a constant C⁢(N,p,q,β,a) as k→∞, by Fatou’s lemma, we get:

Our last aim of this paper is to extend further the above proposition. Namely, we will study a Trudinger–Moser inequality with exact growth:

The organization of the paper is as follows. In Section 2, we will recall and establish some necessary lemmas which are required to prove our main theorems. Section 3 provides the proof of Theorem 1.1, a subcritical Trudinger–Moser inequality. In Section 4, we establish the critical Trudinger–Moser inequalities, i.e., Theorem 1.2. Section 5 offers the proofs of Theorems 1.3, 1.4, 1.5. In Section 6, we justify the sharpness of Theorems 1.1, 1.2, 1.3, 1.4, 1.5.

2 Some Useful Preliminaries

In this section, we introduce some useful results that will be used in our proofs. We first recall the definition of rearrangement and some useful inequalities. Let Ω⊂ℝN, N≥2, be a measurable set. We denote by Ω# the open ball BR⊂ℝN centered at 0 of radius R>0 such that |BR|=|Ω|.

Let u:Ω→ℝ be a real-valued measurable function that vanishes at infinity, that is, |{x:|u⁢(x)|>t}| is finite for all t>0. The distribution function of u is the function

and the decreasing rearrangement of u is the right-continuous, nonincreasing function u∗ that is equi-measurable with u:

It is clear that supp⁡u∗⊆[0,|Ω|]. We also define

Moreover, we define the spherically symmetric decreasing rearrangement of u:

where σN is the volume of the unit ball in ℝN.

Then we have the following important result that could be found in [38]:

Now, we recall a compactness lemma of Strauss [4, 56].

Using Lemma 2.3, we will study the continuity and compactness of the embeddings from Drad1,N⁢(ℝN)∩Lq⁢(ℝN) into La⁢(ℝN) and La⁢(ℝN;d⁢x|x|β). More precisely, we have the following lemma:

Now, we will prove a variant of [42, Lemma 2.2]:

Using the above lemma, we can now prove a Radial Sobolev inequality in the spirit of Ibrahim, Masmoudi and Nakanishi [15]: