Sharp Singular Trudinger–Moser Inequalities in Lorentz–Sobolev Spaces

Guozhen Lu; Hanli Tang

doi:10.1515/ans-2015-5046

Article Open Access

Sharp Singular Trudinger–Moser Inequalities in Lorentz–Sobolev Spaces

Guozhen Lu and Hanli Tang

Published/Copyright: June 18, 2016

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Advanced Nonlinear Studies Volume 16 Issue 3

Abstract

In this paper, we first establish a singular (0<β<n) Trudinger–Moser inequality on any bounded domain in ℝn with Lorentz–Sobolev norms (Theorem 1.1). Next, we prove the critical singular (0<β<n) Trudinger–Moser inequality on any unbounded domain in ℝn with Lorentz–Sobolev norms (Theorem 1.2). Then, we set up a subcritical singular (0<β<n) Trudinger–Moser inequality on any unbounded domain in ℝn with Lorentz–Sobolev norms (Theorem 1.3). Finally, we establish the subcritical nonsingular (β=0) Trudinger–Moser inequality on any unbounded domain in ℝn with Lorentz–Sobolev norms (Theorem 1.5). The constants in all these inequalities are sharp. In [9], for the proof of Theorem 1.2 in the nonsingular case β=0, the following inequality was used (see [17]):

u ∗ ⁢ ( r ) - u ∗ ⁢ ( r 0 ) ≤ 1 n ⁢ w n 1 / n ⁢ ∫ r r 0 U ⁢ ( s ) ⁢ s 1 / n ⁢ d ⁢ s s ,

where U⁢(x) is the radial function built from |∇⁡u| on the level set of u, i.e.,

∫ | u | > t | ∇ u | d x = ∫ 0 | { | u | > t } | U ( s ) d s .

The construction of such U uses the deep Fleming–Rishel co-area formula and the isoperimetric inequality and is highly nontrivial. Moreover, this argument will not work in the singular case 0<β<n. One of the main novelties of this paper is that we can avoid the use of this deep construction of such a radial function U (see remarks at the end of the introduction). Moreover, our approach adapts the symmetrization-free argument developed in [19, 21, 23], where we derive the global inequalities on unbounded domains from the local inequalities on bounded domains using the level sets of the functions under consideration.

Keywords: Singular Trudinger–Moser Inequality; Lorentz–Sobolev Spaces; Best Constants

MSC 2010: 42B35; 42B37

1 Introduction

The Trudinger–Moser inequalities can be considered as the limiting case of Sobolev inequalities. They were established by Trudinger [40] (see also [41, 35]). In 1971, Moser [33], sharpening the Trudinger inequality in [35, 40, 41], proved the following theorem.

Theorem A

Let Ω be a domain with finite measure in an n-dimensional Euclidean space ℝn, n≥2. Then there exists a sharp constant

α n = n ⁢ ( n ⁢ π n / 2 Γ ⁢ ( n 2 + 1 ) ) 1 / ( n - 1 )

such that

(1.1) 1 | Ω | ⁢ ∫ Ω exp ⁡ ( α ⁢ | f | n / ( n - 1 ) ) ⁢ 𝑑 x ≤ c 0 < ∞

for any α≤αn and any f∈C0∞⁢(Ω) with ∫Ω|∇f|ndx≤1. This constant αn is sharp in the sense that if α>αn, then the above inequality can no longer hold with some c0 independent of f.

Trudinger–Moser’s result has been studied and extended in many directions. For instance, we refer the reader to [10, 32, 25] for the sharp Trudinger–Moser inequality with mean value zero, to [3] for the singular Trudinger–Moser inequality and to [12] for the sharp affine Trudinger–Moser inequalities. Also, for the sharp Trudinger–Moser inequalities on the Heisenberg group and CR spheres, and on compact Riemannian manifolds, etc., see [5, 13, 14, 22] and [16, 26, 27], respectively. We also refer to the articles of Chang and Yang [11] for applications of such inequalities to geometric analysis and of Lam and Lu [20] for descriptions of applications of such inequalities to nonlinear PDEs.

When Ω has infinite volume, the subcritical Trudinger–Moser type inequalities for unbounded domains were proposed by Cao [7] when n=2 and do Ó [6] for the general case n≥2. These results were sharpened later by Adachi and Tanaka [1] in order to determine the best constant.

Theorem B

Theorem B ([1])

For any α<αn, there exists a positive constant Cn,α such that

∫ ℝ n ϕ n ⁢ ( α ⁢ | u | n / ( n - 1 ) ) ⁢ 𝑑 x ≤ C n , α ⁢ ∥ u ∥ n n

for all u∈W1,n⁢(ℝn), ∥∇⁡u∥n≤1, where

ϕ n ⁢ ( t ) = e t - ∑ j = 0 n - 2 t j j ! .

The constant αn is sharp in the sense that the supremum is infinity when α≥αn.

We note that in the above theorem, we only impose the restriction ∥∇⁡u∥Ln⁢(ℝn)≤1. The method in [1] requires a symmetrization argument which is not available in many other non-Euclidean settings. A symmetrization-free argument was developed to prove the subcritical Trudinger–Moser inequality on the Heisenberg group by Lam, Lu and Tang [23].

Recently, Lam, Lu and Zhang proved in [24] the precise asymptotic estimates for the following supremum.

Theorem C

Let

n ≥ 2 , α n = n ⁢ ( n ⁢ π n / 2 Γ ⁢ ( n 2 + 1 ) ) 1 / ( n - 1 ) , 0 ≤ β < n , 0 ≤ α < α n .

Denote

A T ( α , β ) = sup ∥ ∇ u ∥ n ≤ 1 1 ∥ u ∥ n n - β ∫ ℝ n ϕ n ( α ( 1 - β n ) | u | n / ( n - 1 ) ) d ⁢ x | x | β .

Then there exist positive constants c=c⁢(n,β) and C=C⁢(n,β) such that when α is close enough to αn, we have

c ⁢ ( n , β ) ( 1 - ( α α n ) n - 1 ) ( n - β ) / n ≤ A ⁢ T ⁢ ( α , β ) ≤ C ⁢ ( n , β ) ( 1 - ( α α n ) n - 1 ) ( n - β ) / n .

Moreover, the constant αn is sharp in the sense that A⁢T⁢(αn,β)=∞.

The upper bound in the above estimates for the subcritical case was obtained by an argument inspired by the work of Lam and the authors [19, 21, 23], where a local Trudinger–Moser inequality on the level sets of the functions under consideration can lead to a global one on the entire spaces, without a priori knowing the validity of the critical inequality.

The above inequality fails in the critical case α=αn. So it is natural to ask in which case the above inequality is true when α=αn. This is done by Ruf [36], and Li and Ruf [28] by using the restriction of the full norm ∥∇⁡u∥Ln⁢(ℝn)+∥u∥Ln⁢(ℝn)≤1.

Theorem D

Theorem D (see [36, 28])

For all 0≤α≤αn, we have

sup ∥ u ∥ ≤ 1 ⁡ ∫ ℝ n ϕ n ⁢ ( α ⁢ | u | n / ( n - 1 ) ) ⁢ 𝑑 x < ∞ ,

where

∥ u ∥ = ( ∫ ℝ n ( | ∇ ⁡ u | n + | u | n ) ⁢ 𝑑 x ) 1 / n .

Moreover, this constant αn is sharp in the sense that if α>αn, then the supremum is infinity.

Surprisingly, Lam, Lu and Zhang [24] have shown that the subcritical Moser–Trudinger inequality in [1] and the critical Trudinger–Moser inequality in [36, 28] are actually equivalent. We will provide another proof of the sharp critical Trudinger–Moser inequality using the subcritical one, and vice versa. Furthermore, we have shown the following precise relationship between the suprema in critical and subcritical Trudinger–Moser inequalities.

Theorem E

Let n≥2, 0≤β<n, a>0 and b>0. Denote

M ⁢ T a , b ⁢ ( β ) = sup ∥ ∇ ⁡ u ∥ n a + ∥ u ∥ n b ≤ 1 ⁡ ∫ ℝ n ϕ n ⁢ ( α n ⁢ ( 1 - β n ) ⁢ | u | n / ( n - 1 ) ) ⁢ d ⁢ x | x | β , M ⁢ T ⁢ ( β ) = M ⁢ T n , n ⁢ ( β ) .

Then M⁢Ta,b⁢(β)<∞ if and only if b≤n. The constant αn is sharp. Moreover, we have the following identity:

M ⁢ T a , b ⁢ ( β ) = sup α ∈ ( 0 , α n ) ⁡ ( 1 - ( α α n ) a ⁢ ( n - 1 ) / n ( α α n ) b ⁢ ( n - 1 ) / n ) ( n - β ) / b ⁢ A ⁢ T ⁢ ( α , β ) .

In particular, M⁢T⁢(β)<∞ and

M ⁢ T ⁢ ( β ) = sup α ∈ ( 0 , α n ) ⁡ ( 1 - ( α α n ) n - 1 ( α α n ) n - 1 ) ( n - β ) / n ⁢ A ⁢ T ⁢ ( α , β ) .

Such critical Trudinger–Moser inequalities on unbounded domains on the Heisenberg group have been established by Lam and Lu [19] where symmetrization is not available.

As far as the existence of extremal functions of Moser’s inequality is concerned, the first result was due to the work of Carleson and Chang [8], in which they proved that the supremum

sup u ∈ W 0 1 , n ( Ω ) , ∫ Ω | ∇ u | n d x ≤ 1 ⁡ 1 | Ω | ⁢ ∫ Ω exp ⁡ ( α n ⁢ | u | n / ( n - 1 ) ) ⁢ 𝑑 x

can be achieved when Ω⊂ℝn is an Euclidean ball. This result came as a surprise because it has been known that the Sobolev inequality does not have extremal functions supported on any finite ball. Subsequently, existence of extremal functions has been established on arbitrary domains by Flucher in [15] and Lin in [30], and on Riemannian manifolds by Li in [26, 27].

We now return to the discussions on the sharp Trudinger–Moser inequalities in Lorentz norms.

An improvement of (1.1) by exploiting a Lorentz space has been addressed in [4, 18]. We recall that ψ∈Lp,q⁢(ℝn), 1<p<+∞,1≤q<+∞, if the quantity

∥ ψ ∥ p , q ∗ = ( ∫ 0 + ∞ [ ψ ∗ ⁢ ( t ) ⁢ t 1 / p ] q ⁢ d ⁢ t t ) 1 / q

is finite, where ψ∗⁢(t) is the decreasing rearrangement of ψ (see Section 2 for more details). As we know, Lp,p⁢(ℝn)=Lp⁢(ℝn) and if q≤r, then Lp,r⁢(ℝn)⊂Lp,q⁢(ℝn). Alvino, Ferone and Trombetti [4] established the Trudinger–Moser inequality on bounded domain in Lorentz–Sobolev space (see also [18] for the case 1<q≤n). They set up the following theorem.

Theorem F

Let Ω be a domain with finite measure in the Euclidean space ℝn, n≥2, and 1<q≤∞. Then there exist a constant c0>0 and a sharp constant αn,q=(n⁢vn1/n)q/(q-1) (where vn is the measure of the unit ball) such that

1 | Ω | ⁢ ∫ Ω exp ⁡ ( α ⁢ | f | q / ( q - 1 ) ) ⁢ 𝑑 x ≤ c 0 < ∞

for any α≤αn,q and any f∈C0∞⁢(Ω) with ∥∇⁡u∥n,q≤1. This constant αn is sharp in the sense that if α>αn, then the above inequality can no longer hold with some c0 independent of f.

What is the situation when Ω is an unbounded domain? This question has been studied by Cassani and Tarsi in [9]. Indeed the following is shown in [9].

Theorem G

Let 1<q<∞ and n≥2. Then there exist a constant Cn,q such that for any 0<α≤αn,q and any u∈C0∞⁢(ℝn), the following inequality holds:

sup ∥ ∇ ⁡ u ∥ n , q q + ∥ u ∥ n , q q ≤ 1 ⁡ ∫ ℝ n Φ ⁢ [ α ⁢ | u | q / ( q - 1 ) ] ⁢ 𝑑 x ≤ C n , q ,

where

Φ ⁢ ( t ) = e t - ∑ j = 0 k 0 t j j ! , k 0 = [ ( q - 1 ) ⁢ n q ] .

The constant αn,q is sharp in the sense that if α>αn,q, then the supremum will become infinite.

In this paper, we will consider some sharp singular Trudinger–Moser type inequalities in a Lorentz–Sobolev space with q≠n. When q=n, the Lorentz norm ∥u∥n,q becomes the Ln norm and in this case the inequalities have been already established [1, 36, 28]. First, we will prove the following singular Trudinger–Moser inequality on a bounded domain in Lorentz–Sobolev spaces.

Theorem 1.1

Let Ω be a bounded domain in ℝn (n≥2), αn,q=(n⁢vn1/n)q/(q-1) and 0≤β<n. Then there exists a constant C⁢(n,q,β) such that for any 0≤α≤αn,q,

sup u ∈ C 0 ∞ ⁢ ( Ω ) , ∥ ∇ ⁡ u ∥ n , q ≤ 1 ⁡ 1 | Ω | 1 - β / n ⁢ ∫ Ω exp ⁡ [ ( 1 - β n ) ⁢ α ⁢ | u | q / ( q - 1 ) ] | x | β ⁢ 𝑑 x ≤ C ⁢ ( n , q , β ) .

The constant αn,q is sharp in the sense that if α>αn,q, then the supremum will become infinite.

The nonsingular case β=0 was proved in [4, 18].

Next, we will consider the following sharp singular critical Trudinger–Moser inequality on any unbounded domain in ℝn in the spirit of [36, 28].

Theorem 1.2

Let 1<q<+∞ and 0<β<n. Then there exist a constant C⁢(n,q,β) such that for any 0<α≤αn,q and any u∈W01⁢Ln,q⁢(ℝn), the following inequality holds:

sup ∥ ∇ ⁡ u ∥ n , q q + ∥ u ∥ n , q q ≤ 1 ⁡ ∫ ℝ n Φ ⁢ [ ( 1 - β n ) ⁢ α ⁢ | u | q / ( q - 1 ) ] | x | β ⁢ 𝑑 x ≤ C ⁢ ( n , q , β ) ,

where

Φ ⁢ ( t ) = e t - ∑ j = 0 k 0 t j j ! , k 0 = [ ( q - 1 ) ⁢ n q ] .

The constant αn,q is sharp in the sense that if α>αn,q, then the supremum will become infinite.

The nonsingular case β=0 was proved by Cassani and Tarsi [9], and our approach is substantially different from theirs. See more remarks at the end of this section.

We note that the inequality in the nonsingular case β=0 in [9] was proved only under the restriction of the full norm ∥∇⁡u∥n,qq+∥u∥n,qq≤1. Then, it is interesting to ask whether either the singular or nonsingular subcritical Trudinger–Moser inequalities on unbounded domains in ℝn holds only under the restriction of Lorentz–Sobolev norms ∥u∥n,q≤1.

In this paper, we will also address this issue. We will first establish the sharp subcritical singular Trudinger–Moser inequality in the spirit of Adachi and Tanaka [1].

Theorem 1.3

Let 1<q<+∞ and 0<β<n. Then there exists a constant C⁢(n,q,β) such that for any 0<α<αn,q and any u∈W01⁢Ln,q⁢(ℝn) with ∥∇⁡u∥n,q≤1, the following inequality holds:

∫ ℝ n Φ ⁢ [ ( 1 - β n ) ⁢ α ⁢ | u | q / ( q - 1 ) ] | x | β ⁢ 𝑑 x ≤ C ⁢ ( n , q , β ) ⁢ ∥ u ∥ n , q n - β ,

where

Φ ⁢ ( t ) = e t - ∑ j = 0 k 0 t j j ! , k 0 = [ ( q - 1 ) ⁢ n q ] .

The restriction α<αn,q is sharp in the sense that if α≥αn,q, then there exists a sequence {uk}1+∞⊂W01⁢Ln,q⁢(ℝn) with ∥∇⁡uk∥n,q=1 such that

1 ∥ u k ∥ n , q n - β ⁢ ∫ ℝ n Φ ⁢ [ ( 1 - β n ) ⁢ α ⁢ | u k | q / ( q - 1 ) ] | x | β ⁢ 𝑑 x → + ∞ .

Remark 1.4

The proof of Theorem 1.3 does not give rise to the sharp constant in the case β=0, since in our proof the function 1/|x|β can help us to remedy the integrability issue at infinity for β>0. So we will treat the subcritical nonsingular case when β=0 separately below by using a different argument.

Theorem 1.5

Let 1<q<+∞. Then there exist a constant C⁢(n,q) such that for any 0<α<αn,q and any u∈W01⁢Ln,q⁢(ℝn) with ∥∇⁡u∥n,q≤1, the following inequality holds:

∫ ℝ n Φ ⁢ ( α ⁢ | u | q / ( q - 1 ) ) ⁢ 𝑑 x ≤ C ⁢ ( n , q ) ⁢ ∥ u ∥ n , q n ,

where

Φ ⁢ ( t ) = e t - ∑ j = 0 k 0 t j j ! , k 0 = [ ( q - 1 ) ⁢ n q ] .

The restriction α<αn,q is sharp in the sense that if α≥αn,q, then the inequality can no longer hold with some C⁢(n,q) independent of u.

The following remarks are in order. First of all, as we have pointed out earlier, nonsingular (i.e., β=0) Trudinger–Moser type inequalities with Lorentz norms have been obtained on bounded domains by Alvino, Ferone and Trombetti [4], and by Hudson and Leckband [18], and on unbounded domains by Cassani and Tarsi [9] in the critical case. Our results extend to the singular case 0<β<n. Our Theorem 1.1 considers the case of bounded domains and Theorem 1.2 takes care of the unbounded domains. Second, our results also address the subcritical case α<αn,q for both the singular (0<β<n) and nonsingular (β=0) case. Moreover, the proofs of these two cases are different. See Theorems 1.3 and 1.5. Third, the proofs of Theorems 1.2, 1.3 and 1.5 are essentially different from that in [9]. Therein, the proof of Theorem 1.2 in the nonsingular case β=0 is based crucially on the inequality

u ∗ ⁢ ( r ) - u ∗ ⁢ ( r 0 ) ≤ v ∗ ⁢ ( r ) ,

where v is such that |∇⁡v| is dominated by |∇⁡u| in an appropriate sense. In fact, the following result from [17] has been used

u ∗ ⁢ ( r ) - u ∗ ⁢ ( r 0 ) ≤ 1 n ⁢ w n 1 / n ⁢ ∫ r r 0 U ⁢ ( s ) ⁢ s 1 / n ⁢ d ⁢ s s ,

where U⁢(x) is the radial function built from |∇⁡u| on the level set of u, i.e.,

∫ | u | > t | ∇ u | d x = ∫ 0 | { | u | > t } | U ( s ) d s .

The construction of such U uses the deep Fleming–Rishel co-area formula the isoperimetric inequality and is highly nontrivial. In particular, such optimal Fleming–Rishel type co-area formula and the isoperimetric inequality are still not available in other non-Euclidean settings such as the Heisenberg group. An essential difference between our proof of the critical Trudinger–Moser inequality with Lorentz norms and the proof given by Cassani and Tarsi [9] is that we do not use this inequality involving the function U built from |∇⁡u| on the level set of u at all. Instead, we will consider the integral on ℝn by dividing it into two parts using the level set of u, inspired by an idea of Lam and the first author for Moser–Trudinger inequalities on the Heisenberg group [19] and Adams inequalities on high order Sobolev spaces [21]. The essence of this argument is that we can establish global Trudinger–Moser inequalities on unbounded domains from local Trudinger–Moser inequalities on bounded domains. This argument to derive global inequalities from the local ones using the level sets works in several other settings such as hyperbolic spaces [31], Riemannian manifolds, etc., and can be used for both critical and subcritical Moser–Trudinger inequalities [19, 23].

The organization of the paper is as follows. In Section 2, we introduce some preliminaries about the Lorentz–Sobolev spaces. In Section 3, we will give a singular Trudinger–Moser inequality on bounded domains with the Lorentz–Sobolev space norms (Theorem 1.1). In Section 4, we will prove the singular critical Trudinger–Moser inequality for unbounded domains (Theorem 1.2). In Section 5, we obtain the best constant for the singular subcritical Trudinger–Moser inequality on unbounded domains (Theorem 1.3). Finally, in Section 6, we will give subcritical Trudinger–Moser inequality in the nonsingular case on unbounded domains with the Lorentz–Sobolev space norms (Theorem 1.5)

2 Preliminaries

Let f:ℝn→ℝ be a function such that

| { x ∈ ℝ n : | f ⁢ ( x ) | > t } | = ∫ { x ∈ ℝ n : | f ⁢ ( x ) | > t } 𝑑 x < + ∞

for every t>0. Its distribution function is defined by

d f ⁢ ( t ) = | { x : | f ⁢ ( x ) | > t } | ,

and its decreasing rearrangement f∗ is defined by

f ∗ ⁢ ( s ) = sup ⁡ { t > 0 : d f ⁢ ( t ) > s } .

Now, define f♯:ℝn→ℝ by

f ♯ ⁢ ( x ) = f ∗ ⁢ ( v n ⁢ | x | n ) ,

where vn is the volume of the unit ball in ℝn. Then, from [29], for every continuous increasing function Ψ:[0,+∞)→[0,+∞), we have that

∫ ℝ n Ψ ⁢ ( f ) ⁢ 𝑑 x = ∫ ℝ n Ψ ⁢ ( f ♯ ) ⁢ 𝑑 x .

Since f∗ is nonincreasing, the maximal function f∗∗ of the rearrangement of f∗, defined by

f ∗ ∗ := 1 s ⁢ ∫ 0 s f ∗ ⁢ 𝑑 t for ⁢ s ≥ 0 ,

is also nonincreasing and f∗≤f∗∗. For more details on the rearrangement we refer the reader to [29, 37, 39].

The Lorentz spaces Lp,q, with 1<p<+∞ and 1≤q<+∞, is defined to consist of all functions ψ such that

∥ ψ ∥ p , q ∗ = ( ∫ 0 + ∞ [ ψ ∗ ⁢ ( t ) ⁢ t 1 / p ] q ⁢ d ⁢ t t ) 1 / q < ∞ .

It is well known that when q>p the quantity ∥ψ∥p,q∗ is not a norm. But the quantity

∥ ψ ∥ p , q = ( ∫ 0 + ∞ [ ψ ∗ ∗ ⁢ ( t ) ⁢ t 1 / p ] q ⁢ d ⁢ t t ) 1 / q

is a norm for any p and q. It is easy to prove, by using Hardy’s inequalities (see [37]), that these two quantities are equivalent in the sense that

∥ ψ ∥ p , q ∗ ≤ ∥ ψ ∥ p , q ≤ C ⁢ ( p , q ) ⁢ ∥ ψ ∥ p , q ∗ .

As the classic Sobolev spaces are built up from Lebesgue spaces, similarly one can introduce Lorentz–Sobolev spaces which consist of functions having weak derivatives belonging to the Lorentz space Lp,q⁢(ℝn). For more details on Lorentz spaces, we refer the reader to the book by Stein and Weiss [37, Chapter 5, Section 3].

In particular, let 1<p<+∞ and let W01⁢Ln,q⁢(ℝn) be the closure of

{ u ∈ C 0 ∞ ⁢ ( ℝ n ) : ∥ u ∥ 1 , ( n , q ) < + ∞ } ,

under the norm

∥ u ∥ 1 , ( n , q ) q = ∥ u ∥ n , q q + ∥ ∇ ⁡ u ∥ n , q q .

We recall two important lemmas which will be used in our proof. One is the well-known Strauss radial lemma [38].

Lemma 2.1

If u∈Lp,q⁢(ℝn), 1<p<+∞, 1≤q<+∞, then

u ∗ ⁢ ( t ) ≤ ( q p ) 1 / q ⁢ ∥ u ∥ p , q t 1 / p .

Lemma 2.2

Let Ω be a bounded domain and u∈C0∞⁢(Ω). Then

u ∗ ( t ) ≤ 1 n ⁢ v n 1 / n [ ∫ t | Ω | | ∇ u | ∗ ( r ) r 1 / n d r + 1 t 1 - 1 / n ∫ 0 t | ∇ u | ∗ ( r ) d r ] .

The proof of this lemma follows easily by combining the representation formula (see, e.g., [2, Lemma 2]) and O’Neil’s lemma for rearrangement of convolution operators [34] (see the explicit statement listed in the proof of [2, Theorem 2]).

3 Singular Trudinger–Moser Inequality on Bounded Domains with Lorentz–Sobolev Norms

In this section, we will prove Theorem 1.1. To do so, we will apply some ideas used in [4, 2].

First, we state a lemma which was originally proved in [2] and the current version is taken from [21]. This is a key tool for proving the theorem.

Lemma 3.1

Let 0<α≤1, 1<p<∞, and a⁢(s,t) be a non-negative measurable function on (-∞,∞)×[0,∞] such that (a.e)

a ( s , t ) ≤ 1 when 0 < s < t 𝑎𝑛𝑑 sup t > 0 ( ∫ - ∞ 0 + ∫ t ∞ a ( s , t ) p ′ d s ) 1 / p ′ = b < ∞ .

∫ - ∞ ∞ ϕ ⁢ ( s ) p ⁢ 𝑑 s ≤ 1 for ϕ ≥ 0 ,

then there exists a constant c0=c0⁢(p,b) such that

∫ 0 ∞ e - F α ⁢ ( t ) ⁢ 𝑑 t ≤ c 0 , 𝑤ℎ𝑒𝑟𝑒 F α ⁢ ( t ) = α ⁢ t - α ⁢ ( ∫ - ∞ ∞ a ⁢ ( s , t ) ⁢ ϕ ⁢ ( s ) ⁢ 𝑑 s ) p ′ .

Now we are ready to prove Theorem 1.1.

Proof.

Using Lemma 2.2, we have

u ∗ ( t ) ≤ 1 n ⁢ v n 1 / n [ ∫ t | Ω | | ∇ u | ∗ ( r ) r 1 / n d r + 1 t 1 - 1 / n ∫ 0 t | ∇ u | ∗ ( r ) d r ] .

Then

u ∗ ( | Ω | e - s ) ≤ | Ω | 1 / n n ⁢ v n 1 / n [ ∫ 0 s | ∇ u | ∗ ( | Ω | e - r ) e - r / n d r + e s ⁢ ( 1 - 1 / n ) ∫ s + ∞ | ∇ u | ∗ ( | Ω | e - r ) e - r d r ] .

Now set

Φ ⁢ ( r ) = { | Ω | 1 / n ⁢ | ∇ ⁡ u | ∗ ⁢ ( | Ω | ⁢ e - r ) ⁢ e - r / n if r ≥ 0 , 0 if r < 0 ,

and

a ⁢ ( r , s ) = { 0 if r ≤ 0 , 1 if 0 < r ≤ s , e ( n - 1 ) ⁢ ( s - r ) / n if s < r < + ∞ .

With this notation, we have

u ∗ ⁢ ( | Ω | ⁢ e - s ) ≤ 1 n ⁢ v n 1 / n ⁢ ∫ - ∞ + ∞ Φ ⁢ ( r ) ⁢ a ⁢ ( r , s ) ⁢ 𝑑 r .

Then we will use the arrangement argument. By simple calculations, we know that

( | x | - β ) ∗ ⁢ ( t ) = ( t v n ) - β / n

and

( exp ⁡ [ ( 1 - β n ) ⁢ α ⁢ | u | q / ( q - 1 ) ] ) ∗ ⁢ ( t ) = exp ⁡ [ ( 1 - β n ) ⁢ α ⁢ ( u ∗ ⁢ ( t ) ) q / ( q - 1 ) ] .

For α≤αn,q, by the Hardy–Littlewood inequality, we have

1 | Ω | 1 - β / n ⁢ ∫ Ω exp ⁡ [ ( 1 - β n ) ⁢ α ⁢ | u | q / ( q - 1 ) ] | x | β ⁢ 𝑑 x

≤ 1 | Ω | 1 - β / n ⁢ ∫ 0 | Ω | exp ⁡ [ ( 1 - β n ) ⁢ α ⁢ ( u ∗ ⁢ ( t ) q / ( q - 1 ) ) ] ⁢ ( t v n ) - β / n ⁢ 𝑑 t

= 1 | Ω | 1 - β n ⁢ ∫ 0 + ∞ exp ⁡ [ ( 1 - β n ) ⁢ α ⁢ | u ∗ ⁢ ( e - s ⁢ | Ω | ) | q / ( q - 1 ) ] ⁢ ( e - s ⁢ | Ω | v n ) - β / n ⁢ e - s ⁢ | Ω | ⁢ 𝑑 s

= v n β / n ⁢ ∫ 0 + ∞ exp ⁡ [ ( 1 - β n ) ⁢ α ⁢ | u ∗ ⁢ ( e - s ⁢ | Ω | ) | q / ( q - 1 ) ] ⁢ e - s ⁢ ( 1 - β / n ) ⁢ 𝑑 s

≤ v n β / n ⁢ ∫ 0 + ∞ exp ⁡ [ ( 1 - β n ) ⁢ ( α ( q - 1 ) / q n ⁢ v n 1 / n ⁢ ∫ - ∞ + ∞ Φ ⁢ ( r ) ⁢ a ⁢ ( r , s ) ⁢ 𝑑 r ) q / ( q - 1 ) - ( 1 - β n ) ⁢ s ] ⁢ 𝑑 s

≤ v n β / n ⁢ ∫ 0 + ∞ exp ⁡ [ ( 1 - β n ) ⁢ ( ∫ - ∞ + ∞ Φ ⁢ ( r ) ⁢ a ⁢ ( r , s ) ⁢ 𝑑 r ) q / ( q - 1 ) - ( 1 - β n ) ⁢ s ] ⁢ 𝑑 s

≤ v n β / n ⁢ ∫ 0 + ∞ exp ⁡ [ - F 1 - β / n ⁢ ( s ) ] ⁢ 𝑑 s ,

where

F 1 - β / n ⁢ ( s ) = ( 1 - β n ) ⁢ s - ( 1 - β n ) ⁢ ( ∫ - ∞ + ∞ Φ ⁢ ( r ) ⁢ a ⁢ ( r , s ) ⁢ 𝑑 r ) q / ( q - 1 ) .

Since a⁢(r,s)≤1 for a.e 0<r<s and

∫ - ∞ 0 a ⁢ ( r , s ) q / ( q - 1 ) ⁢ 𝑑 r + ∫ s + ∞ a ⁢ ( r , s ) q / ( q - 1 ) ⁢ 𝑑 r = n ⁢ ( q - 1 ) ( n - 1 ) ⁢ q < + ∞ ,

we have

∫ - ∞ + ∞ Φ q ⁢ ( r ) ⁢ 𝑑 r = ∫ 0 + ∞ ( | Ω | 1 / n ⁢ | ∇ ⁡ u | ∗ ⁢ ( | Ω | ⁢ e - r ) ⁢ e - r / n ) q ⁢ 𝑑 r

= ∫ 0 | Ω | [ | ∇ ⁡ u | ∗ ⁢ ( t ) ⁢ t 1 / n ] q ⁢ d ⁢ t t

= ∥ ∇ ⁡ u ∥ n , q q

≤ 1 .

Now using Lemma 3.1, we obtain

∫ 0 + ∞ exp ⁡ [ - F 1 - β / n ⁢ ( s ) ] ⁢ 𝑑 s ≤ c 0 .

Therefore, when α≤αn,q,

1 | Ω | 1 - β / n ⁢ ∫ Ω exp ⁡ [ ( 1 - β n ) ⁢ α ⁢ | u | q / ( q - 1 ) ] | x | β ⁢ 𝑑 x ≤ C ⁢ ( n , q , β ) .

Now, let us prove that the constant αn,q is sharp. We can assume that Ω is a ball with |Ω|=1. Set

u k ⁢ ( x ) = { k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 if 0 ≤ | x | ≤ ( 1 v n ⁢ e - k ) 1 / n , k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 ⁢ - ln ⁡ ( v n ⁢ | x | n ) k if ⁢ ( 1 v n ⁢ e - k ) 1 / n < | x | ≤ ( 1 v n ) 1 / n , 0 if ⁢ | x | > ( 1 v n ) 1 / n .

By calculation, we have

| ∇ ⁡ u k | ⁢ ( x ) = { 0 if ⁢ 0 ≤ | x | ≤ ( 1 v n ⁢ e - k ) 1 / n , k - 1 / q v n 1 / n ⁢ | x | if ⁢ ( 1 v n ⁢ e - k ) 1 / n < | x | ≤ ( 1 v n ) 1 / n , 0 if ⁢ | x | > ( 1 v n ) 1 / n

and

| ∇ ⁡ u k | ∗ ⁢ ( s ) = { 0 if 1 - e - k ≤ s , k - 1 / q ( s + e - k ) 1 / n if 0 ≤ s < 1 - e - k .

Then

∥ ∇ ⁡ u k ∥ n , q q = 1 k ⁢ ∫ 0 1 - e - k ( s s + e - k ) q / n ⁢ d ⁢ s s = 1 k ⁢ ∫ 1 e k ( 1 - 1 y ) q / n ⁢ d ⁢ y y - 1 .

If we let

a k = ∥ ∇ ⁡ u k ∥ n , q = ( 1 k ⁢ ∫ 1 e k ( 1 - 1 y ) q / n ⁢ d ⁢ y y - 1 ) 1 / q ,

then limk→+∞⁡ak=1.

Now, choose wk⁢(x)=uk⁢(x)ak. Then wk⁢(x)∈C0+∞⁢(Ω) and ∥∇⁡wk∥n,q=1. But when α>αn,q,

∫ Ω exp ⁡ [ α ⁢ ( 1 - β n ) ] ⁢ u q / ( q - 1 ) | x | β ≥ ∫ | x | ≤ ( 1 v n ⁢ e - k ) 1 / n exp ⁡ [ α ⁢ ( 1 - β n ) ⁢ ( a k - 1 ⁢ k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 ) q / ( q - 1 ) ] | x | β ⁢ 𝑑 x

= exp ⁡ [ α α n , q ⁢ ( 1 - β n ) ⁢ a k q / ( 1 - q ) ⁢ k ] ⁢ ∫ | x | ≤ ( 1 v n ⁢ e - k ) 1 / n 1 | x | β ⁢ 𝑑 x

∼ exp ⁡ [ ( 1 - β n ) ⁢ k ⁢ ( α α n , q ⁢ a k q / ( 1 - q ) - 1 ) ]

→ + ∞ as k → + ∞ .

Therefore, we have proved that the constant αn,q is sharp. ∎

4 Critical Singular Trudinger–Moser Inequality on Unbounded Domains with Lorentz–Sobolev Norms

In this section, we will prove Theorem 1.2.

Proof of Theorem 1.2.

By the standard density argument, it suffices to prove that for any 0<β<n and u∈C0+∞⁢(ℝn) with u≥0 satisfying ∥∇⁡u∥n,qq+∥u∥n,qq≤1, there exists a constant C⁢(n,q,β) such that

∫ ℝ n Φ ⁢ [ ( 1 - β n ) ⁢ α n , q ⁢ | u | q / ( q - 1 ) ] | x | β ⁢ 𝑑 x ≤ C ⁢ ( n , q , β ) .

Set

A ⁢ ( u ) = 2 - 1 / [ q ⁢ ( q - 1 ) ] ⁢ ∥ u ∥ n , q and Ω = { x ∈ ℝ n : u > A ⁢ ( u ) } .

Then it is clear that A⁢(u)<1. By the property of rearrangement, we know that for any t∈[0,|Ω|],

(4.1) u ∗ ⁢ ( t ) > 2 - 1 / [ q ⁢ ( q - 1 ) ] ⁢ ∥ u ∥ n , q .

At the same time, using the radial Lemma 2.1, we have

(4.2) u ∗ ⁢ ( t ) ≤ ( q n ) 1 / q ⁢ ∥ u ∥ n , q t 1 / n ,

Combining inequality (4.1) and (4.2), we have

t ≤ 2 n / [ q ⁢ ( q - n ) ] ⁢ ( q n ) n / q for any ⁢ t ∈ [ 0 , | Ω | ] .

Therefore,

| Ω | ≤ 2 n / [ q ⁢ ( q - n ) ] ⁢ ( q n ) n / q .

Now, we write

∫ ℝ n Φ ⁢ [ ( 1 - β n ) ⁢ α n , q ⁢ | u | q / ( q - 1 ) ] | x | β ⁢ 𝑑 x = I 1 + I 2 ,

where

I 1 = ∫ Ω Φ ⁢ [ ( 1 - β n ) ⁢ α n , q ⁢ | u | q / ( q - 1 ) ] | x | β ⁢ 𝑑 x , I 2 = ∫ ℝ n ∖ Ω Φ ⁢ [ ( 1 - β n ) ⁢ α n , q ⁢ | u | q / ( q - 1 ) ] | x | β ⁢ 𝑑 x .

First, let us estimate I2. Since ℝn∖Ω⊂{u(x)<1}, we see that

I 2 ≤ ∫ { u ≤ 1 } 1 | x | β ⁢ ∑ j = k 0 + 1 ∞ α n , q j ⁢ ( 1 - β n ) j j ! ⁢ | u | j ⁢ q / ( q - 1 ) ⁢ d ⁢ x

≤ ∑ j = k 0 + 1 ∞ α n , q j ⁢ ( 1 - β n ) j j ! ⁢ ∫ { u ≤ 1 } 1 | x | β ⁢ | u | n ⁢ 𝑑 x

= C n , q , β ⁢ ( ∫ { u ≤ 1 , | x | ≤ 1 } 1 | x | β ⁢ | u | n ⁢ 𝑑 x + ∫ { u ≤ 1 , | x | > 1 } 1 | x | β ⁢ | u | n - β / 2 ⁢ 𝑑 x )

≤ C n , q , β ⁢ ( ∫ { | x | ≤ 1 } 1 | x | β ⁢ 𝑑 x + ∫ { | x | > 1 } | u | n - β / 2 | x | β ⁢ 𝑑 x )

≤ C ⁢ ( n , q , β ) ⁢ ( 1 + ∫ { | x | > 1 } | u | n - β / 2 | x | β ⁢ 𝑑 x ) .

Since

( χ { | x | > 1 } | x | β ) ∗ ⁢ ( t ) = ( 1 + t v n ) - β / n ,

by the Hardy–Littlewood inequality and the radial lemma, we have

∫ { | x | > 1 } | u | n - β / 2 | x | β ⁢ 𝑑 x ≤ ∫ 0 + ∞ ( u ∗ ⁢ ( t ) ) n - β / 2 ⁢ ( 1 + t v n ) - β / n ⁢ 𝑑 t

≤ ( q n ) n - β / 2 q ⁢ ∫ 0 + ∞ ∥ u ∥ n , q n - β / 2 t 1 - β / ( 2 ⁢ n ) ⁢ ( 1 + t v n ) - β / n ⁢ 𝑑 t

≤ ( q n ) n - β / 2 q ⁢ ∥ u ∥ n , q n - β / 2 ⁢ ( ∫ 0 1 1 t 1 - β / ( 2 ⁢ n ) ⁢ 𝑑 t + ∫ 1 + ∞ v n β / n t 1 - β / ( 2 ⁢ n ) + β / n ⁢ 𝑑 t )

= C ⁢ ( n , q , β ) .

Thus, we have proved that I2≤C⁢(n,q,β).

Next, to estimate I1, we set v⁢(x)=u⁢(x)-A⁢(u) in Ω, thus v∈C0∞⁢(Ω). Moreover,

| u | q / ( q - 1 ) ≤ ( | v | + A ⁢ ( u ) ) q / ( q - 1 )

≤ | v | q / ( q - 1 ) + q q - 1 ⁢ 2 q / ( q - 1 ) - 1 ⁢ ( | v | q / ( q - 1 ) - 1 ⁢ A ⁢ ( u ) + | A ⁢ ( u ) | q / ( q - 1 ) )

≤ | v | q / ( q - 1 ) + q q - 1 ⁢ 2 q / ( q - 1 ) - 1 ⁢ | v | q / ( q - 1 ) ⁢ | A ⁢ ( u ) | q q + q q - 1 ⁢ 2 q / ( q - 1 ) - 1 ⁢ ( q - 1 q + | A ⁢ ( u ) | q / ( q - 1 ) )

= | v | q / ( q - 1 ) ⁢ ( 1 + 2 1 / ( q - 1 ) q - 1 ⁢ | A ⁢ ( u ) | q ) + C q ,

where we used Young’s inequality and the following elementary inequality:

( a + b ) p ≤ a p + p ⁢ 2 p - 1 ⁢ ( a p - 1 ⁢ b + b p ) for all p ≥ 1 and a , b ≥ 0 .

Let

w ⁢ ( x ) = ( 1 + 2 1 / ( q - 1 ) q - 1 ⁢ | A ⁢ ( u ) | q ) ( q - 1 ) / q ⁢ v ⁢ ( x ) in Ω .

Then w∈C0∞⁢(Ω) and |u|q/(q-1)≤|w|q/(q-1)+Cq. Moreover, we have

∇ ⁡ w = ( 1 + 2 1 / ( q - 1 ) q - 1 ⁢ | A ⁢ ( u ) | q ) ( q - 1 ) / q ⁢ ∇ ⁡ v .

When q≥2,

∥ ∇ ⁡ w ∥ n , q q / ( q - 1 ) = ( 1 + 2 1 / ( q - 1 ) q - 1 ⁢ | A ⁢ ( u ) | q ) ⁢ ∥ ∇ ⁡ u ∥ n , q q / ( q - 1 )

≤ ( 1 + 2 1 / ( q - 1 ) q - 1 ⁢ | A ⁢ ( u ) | q ) ⁢ ( 1 - ∥ u ∥ n , q q ) 1 / ( q - 1 )

≤ ( 1 + ∥ u ∥ n , q q q - 1 ) ⁢ ( 1 - ∥ u ∥ n , q q q - 1 )

(4.3) ≤ 1 .

Here we used the inequality (1-x)p≤1-p⁢x for all 0≤x≤1, 0<p≤1.

On the other hand, when 1<q<2,

∥ ∇ ⁡ w ∥ n , q q = ( 1 + 2 1 / ( q - 1 ) q - 1 ⁢ | A ⁢ ( u ) | q ) q - 1 ⁢ ∥ ∇ ⁡ u ∥ n , q q

≤ ( 1 + 2 1 / ( q - 1 ) ⁢ | A ⁢ ( u ) | q ) ⁢ ( 1 - ∥ u ∥ n , q q )

≤ ( 1 + ∥ u ∥ n , q q ) ⁢ ( 1 - ∥ u ∥ n , q q )

(4.4) ≤ 1 .

Here we used the inequality (1+x)p≤1+p⁢x for all x≥0, 0<p≤1.

By inequality (4.3) and (4.4), we know that ∥u∥n,q≤1. So we can use Theorem 1.1 to estimate I1 as follows:

I 1 ≤ ∫ Ω exp ⁡ ( α n , q ⁢ ( 1 - β n ) ⁢ | u | q / ( q - 1 ) ) | x | β ⁢ 𝑑 x

≤ ∫ Ω exp ⁡ ( α n , q ⁢ ( 1 - β n ) ⁢ | w | q / ( q - 1 ) + C q ) | x | β ⁢ 𝑑 x

≤ C β ⁢ | Ω | 1 - β / n

≤ C n , β , q ,

since we already know that |Ω|≤2n/[q⁢(q-n)]⁢(qn)n/q.

Now, let us prove that the constant αn,q is sharp. Set

u k ⁢ ( x ) = { ( 1 - c k ) 1 / q ⁢ k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 if ⁢ 0 ≤ | x | ≤ ( 1 v n ⁢ e - k ) 1 / n , ( 1 - c k ) 1 / q ⁢ k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 ⁢ - ln ⁡ ( v n ⁢ | x | n ) k if ⁢ ( 1 v n ⁢ e - k ) 1 / n < | x | ≤ ( 1 v n ) 1 / n , 0 if ⁢ | x | > ( 1 v n ) 1 / n ,

where

c k = 1 k ⁢ ∫ 1 e k ( ( 1 - 1 y ) q / n ⁢ 1 y - 1 - 1 y ) ⁢ 𝑑 y .

As k→+∞, since

1 k ⁢ ∫ 1 e k ( 1 - 1 y ) q / n ⁢ 1 y - 1 ⁢ 𝑑 y = O ⁢ ( 1 k ) + 1 k ⁢ ∫ 2 e k ( 1 - 1 y ) q / n ⁢ 1 y - 1 ⁢ 𝑑 y

and

∫ 2 e k c y ⁢ ( y - 1 ) ⁢ 𝑑 y ≤ 1 k ⁢ ∫ 2 e k [ ( 1 - 1 y ) q / n - 1 ] ⁢ 1 y - 1 ⁢ 𝑑 y ≤ ∫ 2 e k C y ⁢ ( y - 1 ) ⁢ 𝑑 y ,

we have that

1 k ⁢ ∫ 2 e k ( 1 - 1 y ) q / n ⁢ 1 y - 1 ⁢ 𝑑 y = 1 k ⁢ ∫ 2 e k 1 y - 1 ⁢ 𝑑 y + O ⁢ ( 1 k ) = 1 + O ⁢ ( 1 k ) .

Then

1 k ⁢ ∫ 1 e k ( 1 - 1 y ) q / n ⁢ 1 y - 1 ⁢ 𝑑 y = 1 + O ⁢ ( 1 k )

and

c k = O ⁢ ( 1 k ) .

By calculation, we have

| u k | ∗ ⁢ ( s ) = { ( 1 - c k ) 1 / q ⁢ k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 if ⁢ 0 ≤ s ≤ e - k , ( 1 - c k ) 1 / q ⁢ k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 ⁢ - ln ⁡ s k if ⁢ e - k ≤ s < 1 , 0 if ⁢ s ≥ 1 ,

| ∇ ⁡ u k | ⁢ ( x ) = { 0 if ⁢ 0 ≤ | x | ≤ ( 1 v n ⁢ e - k ) 1 / n , ( 1 - c k ) 1 / q ⁢ k - 1 / q v n 1 / n ⁢ | x | if ⁢ ( 1 v n ⁢ e - k ) 1 / n < | x | ≤ ( 1 v n ) 1 / n , 0 if ⁢ | x | > ( 1 v n ) 1 / n

and

| ∇ ⁡ u k | ∗ ⁢ ( s ) = { 0 if ⁢ 1 - e - k ≤ s , ( 1 - c k ) 1 / q ⁢ k - 1 / q ( s + e - k ) 1 / n if ⁢ 0 ≤ s < 1 - e - k .

We also have

∥ ∇ ⁡ u k ∥ n , q q = ( 1 - c k ) 1 k ⁢ ∫ 0 1 - e - k ( s s + e - k ) q / n ⁢ d ⁢ s s = ( 1 - c k ) k ⁢ ∫ 1 e k ( 1 - 1 y ) q / n ⁢ d ⁢ y y - 1

and

∥ u k ∥ n , q q = ( 1 - c k ) ⁢ [ ∫ 0 e - k ( k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 ⁢ s 1 / n ) q ⁢ d ⁢ s s + ∫ e - k 1 ( k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 ⁢ - ln ⁡ s k ) q ⁢ s q / n ⁢ d ⁢ s s ]

= ( 1 - c k ) ⁢ [ n q ⁢ ( n ⁢ v n 1 / n ) - q ⁢ k q - 1 ⁢ e - k ⁢ q / n + ( n ⁢ v n 1 / n ) - q ⁢ 1 k ⁢ ∫ 0 k s q ⁢ e - s ⁢ q / n ⁢ 𝑑 s ] .

Then

∥ ∇ ⁡ u k ∥ n , q q + ∥ u k ∥ n , q q

= ( 1 - c k ) ⁢ [ 1 k ⁢ ∫ 1 e k ( 1 - 1 y ) q / n ⁢ d ⁢ y y - 1 + n q ⁢ ( n ⁢ v n 1 / n ) - q ⁢ k q - 1 ⁢ e - k ⁢ q / n + ( n ⁢ v n 1 / n ) - q ⁢ 1 k ⁢ ∫ 0 k s q ⁢ e - s ⁢ q / n ⁢ 𝑑 s ]

= ( 1 - c k ) ⁢ [ 1 + 1 k ⁢ ∫ 1 e k ( ( 1 - 1 y ) q / n ⁢ 1 y - 1 - 1 y ) ⁢ 𝑑 y + n q ⁢ ( n ⁢ v n 1 / n ) - q ⁢ k q - 1 ⁢ e - k ⁢ q / n + ( n ⁢ v n 1 / n ) - q ⁢ 1 k ⁢ ∫ 0 k s q ⁢ e - s ⁢ q / n ⁢ 𝑑 s ]

= ( 1 - c k ) ⁢ ( 1 + c k + o ⁢ ( c k ) )

= 1 + O ⁢ ( 1 k ) .

Now, choose

w k ⁢ ( x ) = u k ⁢ ( x ) ( ∥ u k ⁢ ( x ) ∥ n , q q + ∥ ∇ ⁡ u k ∥ n , q q ) 1 / q .

Then wk∈W01⁢Ln,q⁢(ℝn) and ∥wk⁢(x)∥n,qq+∥∇⁡wk∥n,qq=1. But as k→+∞, for α>αn,q,

∫ ℝ n Φ ⁢ [ α ⁢ ( 1 - β n ) ⁢ u q / ( q - 1 ) ] | x | β ⁢ 𝑑 x ≥ ∫ | x | ≤ ( 1 v n ⁢ e - k ) 1 / n Φ ⁢ [ α ⁢ ( 1 - β n ) ⁢ u q / ( q - 1 ) ] | x | β ⁢ 𝑑 x

≥ ∫ | x | ≤ ( 1 v n ⁢ e - k ) 1 / n Φ ⁢ [ α ⁢ ( 1 - β n ) ⁢ ( 1 - c k 1 + O ⁢ ( 1 k ) ) 1 / ( q - 1 ) ⁢ ( k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 ) q / ( q - 1 ) ] | x | β ⁢ 𝑑 x

∼ Φ ⁢ [ α α n , q ⁢ ( 1 - β n ) ⁢ ( 1 - c k 1 + O ⁢ ( 1 k ) ) 1 / ( q - 1 ) ⁢ k ] ⁢ exp ⁡ ( ( 1 - β n ) ⁢ e - k )

∼ exp ⁡ [ ( 1 - β n ) ⁢ k ⁢ ( α α n , q ⁢ ( 1 - c k 1 + O ⁢ ( 1 k ) ) 1 / ( q - 1 ) - 1 ) ]

→ + ∞ .

Thus, we have proved that the constant αn,q is sharp. ∎

5 Subcritical Singular Trudinger–Moser Inequality on Unbounded Domains with Lorentz–Sobolev Norms

In this section, we will prove Theorem 1.3.

Proof of Theorem 1.3.

By the standard density argument, it suffices to prove that for any 0<β<n, 0<α<αn,q and u∈C0∞⁢(ℝn) with u≥0 satisfying ∥∇⁡u∥n,q≤1, there exists a constant C⁢(n,q,β) such that

∫ ℝ n Φ ⁢ [ ( 1 - β n ) ⁢ α ⁢ | u | q / ( q - 1 ) ] | x | β ⁢ 𝑑 x ≤ C ⁢ ( n , q , β ) ⁢ ∥ u ∥ n , q n - β .

Set

Ω = { x ∈ ℝ n : u > 1 } .

Then, by the property of rearrangement, we know that u∗⁢(t)>1 for any t∈[0,|Ω|]. At the same time, using the radial lemma, we have

u ∗ ⁢ ( t ) ≤ ( q n ) 1 / q ⁢ ∥ u ∥ n , q t 1 / n ,

and thus

t ≤ ( q n ) n / q ⁢ ∥ u ∥ n , q n for any ⁢ t ∈ [ 0 , | Ω | ] .

Therefore,

| Ω | ≤ ( q n ) n / q ⁢ ∥ u ∥ n , q n .

Now, we write

∫ ℝ n Φ ⁢ [ ( 1 - β n ) ⁢ α ⁢ | u | q / ( q - 1 ) ] | x | β ⁢ 𝑑 x = I 1 + I 2 ,

where

I 1 = ∫ Ω Φ ⁢ [ ( 1 - β n ) ⁢ α ⁢ | u | q / ( q - 1 ) ] | x | β ⁢ 𝑑 x , I 2 = ∫ ℝ n ∖ Ω Φ ⁢ [ ( 1 - β n ) ⁢ α ⁢ | u | q / ( q - 1 ) ] | x | β ⁢ 𝑑 x .

First, we estimate I2. Since ℝn∖Ω⊂{u(x)<1}, we see that

I 2 ≤ ∫ { u ≤ 1 } 1 | x | β ⁢ ∑ j = k 0 + 1 ∞ α j ⁢ ( 1 - β n ) j j ! ⁢ | u | j ⁢ n / ( n - 1 ) ⁢ d ⁢ x

≤ ∑ j = k 0 + 1 ∞ α n , q j ⁢ ( 1 - β n ) j j ! ⁢ ∫ { u ≤ 1 } 1 | x | β ⁢ | u | n ⁢ 𝑑 x

= C n , q , β ⁢ ( ∫ { u ≤ 1 , | x | ≤ ∥ u ∥ n , q } 1 | x | β ⁢ | u | n ⁢ 𝑑 x + ∫ { u ≤ 1 , | x | > ∥ u ∥ n , q } 1 | x | β ⁢ | u | n - β / 2 ⁢ 𝑑 x )

≤ C n , q , β ⁢ ( ∫ { | x | ≤ ∥ u ∥ n , q } 1 | x | β ⁢ 𝑑 x + ∫ { | x | > ∥ u ∥ n , q } | u | n - β / 2 | x | β ⁢ 𝑑 x )

≤ C ⁢ ( n , q , β ) ⁢ ( ∥ u ∥ n , q n - β + ∫ { | x | > ∥ u ∥ n , q } | u | n - β / 2 | x | β ⁢ 𝑑 x ) .

Since

( χ { | x | > ∥ u ∥ n , q } | x | β ) ∗ ⁢ ( t ) = ( ∥ u ∥ n , q n + t v n ) - β / n ,

by the Hardy–Littlewood inequality and the radial lemma, we have

∫ { | x | > ∥ u ∥ n , q n } | u | n - β / 2 | x | β ⁢ 𝑑 x ≤ ∫ 0 + ∞ ( u ∗ ⁢ ( t ) ) n - β / 2 ⁢ ( ∥ u ∥ n , q n + t v n ) - β / n ⁢ 𝑑 t

≤ ( q n ) n - β / 2 q ⁢ ∫ 0 + ∞ ∥ u ∥ n , q n - β / 2 t 1 - β / ( 2 ⁢ n ) ⁢ ( ∥ u ∥ n , q n + t v n ) - β / n ⁢ 𝑑 t

≤ ( q n ) n - β / 2 q ⁢ ∥ u ∥ n , q n - β / 2 ⁢ ( ∫ 0 ∥ u ∥ n , q n ∥ u ∥ n , q - β t 1 - β / ( 2 ⁢ n ) ⁢ 𝑑 t + ∫ ∥ u ∥ n , q n + ∞ v n β / n t 1 - β / ( 2 ⁢ n ) + β / n ⁢ 𝑑 t )

= C ⁢ ( n , q , β ) ⁢ ∥ u ∥ n , q n - β .

So, we have proved that I2≤C⁢(n,q,β)⁢∥u∥n,qn-β.

Set v⁢(x)=u⁢(x)-1 in Ω. Then v∈C0∞⁢(Ω) and ∥∇g⁡v∥n,q≤1. By Theorem 1.1, we have

1 | Ω | 1 - β / n ⁢ ∫ Ω exp ⁡ ( α ⁢ ( 1 - β n ) ⁢ | v ⁢ ( x ) | q / ( q - 1 ) ) | x | β ⁢ 𝑑 x ≤ C n , q , β .

Now, put ε=αn,qα⁢(1-βn)-1>0. Using the elementary inequality

( a + b ) p ≤ ε ⁢ b p + ( 1 - ( 1 + ε ) - 1 / ( p - 1 ) ) 1 - p ⁢ a p for any a , b , ε > 0 and p > 1 ,

we have, in Ω, that

| u ⁢ ( x ) | q / ( q - 1 ) = ( 1 + v ⁢ ( x ) ) q / ( q - 1 ) ≤ ( 1 + ε ) ⁢ | v | q / ( q - 1 ) + ( 1 - 1 ( 1 + ε ) q - 1 ) 1 / ( 1 - q ) .

Set

C ε = ( 1 - 1 ( 1 + ε ) n - 1 ) 1 / ( 1 - n ) .

Hence,

I 1 = ∫ Ω Φ n ⁢ ( α ⁢ ( | u | ) q / ( q - 1 ) ) | x | β ⁢ 𝑑 x

≤ ∫ Ω exp ⁡ ( α ⁢ ( | u | ) q / ( q - 1 ) ) | x | β ⁢ 𝑑 x

= ∫ Ω exp ⁡ ( α ⁢ ( | v + 1 | ) q / ( q - 1 ) ) | x | β ⁢ 𝑑 x

≤ ∫ Ω exp ⁡ ( α ⁢ ( 1 + ε ) ⁢ | v | q / ( q - 1 ) + α ⁢ C ε ) | x | β ⁢ 𝑑 x

≤ e α ⁢ C ε ⁢ ∫ Ω exp ⁡ ( α n , q ⁢ ( 1 - β n ) ⁢ | v | q / ( q - 1 ) ) | x | β ⁢ 𝑑 x

≤ C n , q , β ⁢ | Ω | 1 - β n

≤ C α , β ⁢ ∥ u ∥ n , q n ,

since we already know that |Ω|≤(qn)n/q⁢∥u∥n,qn.

Now, we prove that the restriction α<αn,q is optimal. Set

u k ⁢ ( x ) = { C k ⁢ k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 if ⁢ 0 ≤ | x | ≤ ( 1 v n ⁢ e - k ) 1 / n , C k ⁢ k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 ⁢ - ln ⁡ ( v n ⁢ | x | n ) k if ⁢ ( 1 v n ⁢ e - k ) 1 / n < | x | ≤ ( 1 v n ) 1 / n , 0 if ⁢ | x | > ( 1 v n ) 1 / n ,

where

C k = ( 1 k ⁢ ∫ 1 e k ( 1 - 1 y ) q / n ⁢ d ⁢ y y - 1 ) - 1 / q .

By calculation, we have Ck→1 and Ckq/(q-1)⁢k-k→0 as k→+∞. Therefore,

| u k | ∗ ⁢ ( s ) = { C k ⁢ k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 if 0 ≤ s ≤ e - k , C k ⁢ k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 ⁢ - ln ⁡ s k if e - k ≤ s < 1 , 0 if s ≥ 1 ,

| ∇ ⁡ u k | ⁢ ( x ) = { 0 if ⁢ 0 ≤ | x | ≤ ( 1 v n ⁢ e - k ) 1 / n , C k ⁢ k - 1 / q v n 1 / n ⁢ | x | if ⁢ ( 1 v n ⁢ e - k ) 1 / n < | x | ≤ ( 1 v n ) 1 / n , 0 if ⁢ | x | > ( 1 v n ) 1 / n

and

| ∇ ⁡ u k | ∗ ⁢ ( s ) = { 0 if ⁢ 1 - e - k ≤ s , C k ⁢ k - 1 / q ( s + e - k ) 1 / n if ⁢ 0 ≤ s < 1 - e - k .

We also have

∥ ∇ ⁡ u k ∥ n , q q = C k q k ⁢ ∫ 0 1 - e - k ( s s + e - k ) q / n ⁢ d ⁢ s s = C k q k ⁢ ∫ 1 e k ( 1 - 1 y ) q / n ⁢ d ⁢ y y - 1 = 1

and

∥ u k ∥ n , q q = C k q ⁢ [ ∫ 0 e - k ( k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 ⁢ s 1 / n ) q ⁢ d ⁢ s s + ∫ e - k 1 ( k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 ⁢ - ln ⁡ s k ) q ⁢ s q / n ⁢ d ⁢ s s ]

∼ C k q ⁢ ( k q - 1 ⁢ e - k ⁢ q / n + 1 k ⁢ ∫ 0 k s q ⁢ e - s ⁢ q / n ⁢ 𝑑 s )

∼ 1 k as ⁢ k → + ∞ .

Thus, we have uk∈W01⁢Ln,q⁢(ℝn), ∥∇⁡uk∥n,q=1 and ∥uk∥n,q=O⁢(1k1/q). Moreover,

∫ ℝ n Φ ⁢ [ α ⁢ ( 1 - β n ) ⁢ u q / ( q - 1 ) ] | x | β ⁢ 𝑑 x = ∫ | x | ≤ ( 1 v n ⁢ e - k ) 1 / n Φ ⁢ [ α ⁢ ( 1 - β n ) ⁢ u q / ( q - 1 ) ] | x | β ⁢ 𝑑 x + ∫ ( 1 v n ⁢ e - k ) 1 / n < | x | ≤ ( 1 v n ) 1 / n Φ ⁢ [ α ⁢ ( 1 - β n ) ⁢ u q / ( q - 1 ) ] | x | β ⁢ 𝑑 x

≥ ∫ | x | ≤ ( 1 v n ⁢ e - k ) 1 / n Φ ⁢ [ α ⁢ ( 1 - β n ) ⁢ ( C k ⁢ k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 ) q / ( q - 1 ) ] | x | β ⁢ 𝑑 x

- ∑ j = 1 k 0 ∫ ( 1 v n ⁢ e - k ) 1 / n < | x | ≤ ( 1 v n ) 1 / n α j ⁢ ( 1 - β n ) j ⁢ | C k ⁢ k ( q - 1 ) / q ⁢ ( n ⁢ v n 1 / n ) - 1 ⁢ ln ⁡ ( v n ⁢ | x | n ) k | j | x | β ⁢ 𝑑 x

∼ Φ ⁢ [ α α n , q ⁢ ( 1 - β n ) ⁢ C k q / ( q - 1 ) ⁢ k ] ⁢ ∫ | x | ≤ ( 1 v n ⁢ e - k ) 1 / n 1 | x | β ⁢ 𝑑 x + ∑ j = 1 k 0 C j k j / q

∼ exp ⁡ [ ( 1 - β n ) ⁢ k ⁢ ( α α n , q ⁢ C k q / ( q - 1 ) - 1 ) ]

→ 1 as k → + ∞ ,

when α=αn,q. But,

1 ∥ u k ∥ n , q n - β ⁢ ∫ ℝ n Φ ⁢ [ ( 1 - β n ) ⁢ α n , q ⁢ | u k | q / ( q - 1 ) ] | x | β ⁢ 𝑑 x → + ∞ .

Therefore, we proved that the restriction α<αn,q is optimal. ∎

6 Subcritical Trudinger–Moser Inequality on Unbounded Domains with Lorentz–Sobolev Norms

In this section, we will prove Theorem 1.5.

Proof of Theorem 1.5.

By the standard density argument, it suffices to prove that for any 0<α<αn,q and u∈C0∞⁢(ℝn) with u≥0 satisfying ∥∇⁡u∥n,q≤1, there exists a constant C⁢(n,q) such that

∫ ℝ n Φ ⁢ ( α ⁢ | u | q / ( q - 1 ) ) ⁢ 𝑑 x ≤ C ⁢ ( n , q ) ⁢ ∥ u ∥ n , q n .

By symmetrization, we have

∫ ℝ n Φ ⁢ ( α ⁢ | u | q / ( q - 1 ) ) ⁢ 𝑑 x = ∫ 0 + ∞ Φ ⁢ ( α ⁢ | u ∗ | q / ( q - 1 ) ) ⁢ 𝑑 t .

Now set R=(qn)n/q⁢∥u∥n,qn. Using the radial lemma, we have

u ∗ ⁢ ( R ) ≤ ( q n ) 1 / q ⁢ ∥ u ∥ n , q R 1 / n = 1 .

Now, we write

∫ 0 + ∞ Φ ⁢ ( α ⁢ | u ∗ | q / ( q - 1 ) ) ⁢ 𝑑 t = I 1 + I 2 ,

where

I 1 = ∫ 0 R Φ ⁢ ( α ⁢ | u ∗ | q / ( q - 1 ) ) ⁢ 𝑑 t , I 2 = ∫ R + ∞ Φ ⁢ ( α ⁢ | u ∗ | q / ( q - 1 ) ) ⁢ 𝑑 t .

First, we estimate I2. Since k0=[q-1q⁢n], we have qq-1⁢j>n when j≥k0+1. Using the radial lemma yields

I 2 = ∑ j = k 0 + 1 ∞ α j j ! ∫ R + ∞ | u ∗ ( t ) | q / ( q - 1 ) ⁢ j d t

≤ ∑ j = k 0 + 1 ∞ α j j ! ⁢ ∫ R + ∞ [ ( q n ) 1 / q ⁢ ∥ u ∥ n , q t 1 / n ] q / ( q - 1 ) ⁢ j ⁢ 𝑑 t

≤ ∑ j = k 0 + 1 ∞ ( α ⁢ ( q n ) 1 / ( q - 1 ) ) j j ! ⁢ ∥ u ∥ n , q q / ( q - 1 ) ⁢ j ⁢ ∫ R + ∞ [ t - 1 / n ] q / ( q - 1 ) ⁢ j ⁢ 𝑑 x

= ∑ j = k 0 + 1 ∞ ( α ⁢ ( q n ) 1 / ( q - 1 ) ) j j ! ⁢ ∥ u ∥ n , q q / ( q - 1 ) ⁢ j ⁢ R 1 - q / [ n ⁢ ( q - 1 ) ] ⁢ j q ⁢ j ( q - 1 ) ⁢ n - 1

≤ C ⁢ ( n , q ) ⁢ ∥ u ∥ n , q n .

Next, let us estimate I1. Set w⁢(t)=[u∗⁢(t)-u∗⁢(R)]⁢χ[0,R]⁢(t). Then w⁢(t)≥0. Now, put ε=αn,qα-1>0. Using the elementary inequality

( a + b ) p ≤ ε ⁢ b p + ( 1 - ( 1 + ε ) - 1 / ( p - 1 ) ) 1 - p ⁢ a p for any a , b , ε > 0 and p > 1 ,

we have, when 0≤t<R, that

| u ∗ ⁢ ( t ) | q / ( q - 1 ) = ( w ⁢ ( t ) + u ∗ ⁢ ( R ) ) q / ( q - 1 ) ≤ ( 1 + ε ) ⁢ | w ⁢ ( t ) | q / ( q - 1 ) + ( 1 - 1 ( 1 + ε ) q - 1 ) 1 / ( 1 - q ) ⁢ ( u ∗ ⁢ ( R ) ) q / ( q - 1 ) .

Since u∗⁢(R)≤1, we have

I 1 = ∫ 0 R Φ ⁢ ( α ⁢ | u ∗ | q / ( q - 1 ) ) ⁢ 𝑑 t

≤ ∫ 0 R exp ⁡ ( α ⁢ | u ∗ | q / ( q - 1 ) ) ⁢ 𝑑 t

≤ ∫ 0 R exp ⁡ [ α ⁢ ( 1 + ε ) ⁢ | w ⁢ ( t ) | q / ( q - 1 ) + α ⁢ C ε ⁢ ( u ∗ ⁢ ( R ) ) q / ( q - 1 ) ] ⁢ 𝑑 t

(6.1) ≤ e α ⁢ C ε ⁢ ∫ 0 R exp ⁡ [ α n , q ⁢ | w ⁢ ( t ) | q / ( q - 1 ) ] ⁢ 𝑑 t .

Using Lemma 2.2 yields

w ( t ) = u ∗ ( t ) - u ∗ ( R ) ≤ 1 n ⁢ v n 1 / n [ ∫ t R | ∇ u | ∗ ( r ) r 1 / n d r + 1 t 1 - 1 / n ∫ 0 t | ∇ u | ∗ ( r ) d r ] .

Then

w ( R e - s ) ≤ R 1 / n n ⁢ v n 1 / n [ ∫ 0 s | ∇ u | ∗ ( R e - r ) e - r / n d r + e s ⁢ ( 1 - 1 / n ) ∫ s + ∞ | ∇ u | ∗ ( R e - r ) e - r d r ] .

Now set

Φ ⁢ ( r ) = { R 1 / n ⁢ | ∇ ⁡ u | ∗ ⁢ ( R ⁢ e - r ) ⁢ e - r / n if ⁢ r ≥ 0 , 0 if ⁢ r < 0

and

a ⁢ ( r , s ) = { 0 if r ≤ 0 , 1 if 0 < r ≤ s , e ( n - 1 ) ⁢ ( s - r ) / n if ⁢ s < r < + ∞ .

With this notation we have

w ⁢ ( R ⁢ e - s ) ≤ 1 n ⁢ v n 1 / n ⁢ ∫ - ∞ + ∞ Φ ⁢ ( r ) ⁢ a ⁢ ( r , s ) ⁢ 𝑑 r .

Then

∫ 0 R exp [ α n , q | w ( t ) | q / ( q - 1 ) ) ] d t = ∫ 0 + ∞ exp [ α n , q | w ( e - s R ) | q / ( q - 1 ) ] e - s R d s

≤ R ⁢ ∫ 0 + ∞ exp ⁡ [ ( α n , q ( q - 1 ) / q n ⁢ v n 1 / n ⁢ ∫ - ∞ + ∞ Φ ⁢ ( r ) ⁢ a ⁢ ( r , s ) ⁢ 𝑑 r ) q / ( q - 1 ) - s ] ⁢ 𝑑 s

≤ R ⁢ ∫ 0 + ∞ exp ⁡ [ ( ∫ - ∞ + ∞ Φ ⁢ ( r ) ⁢ a ⁢ ( r , s ) ⁢ 𝑑 r ) q / ( q - 1 ) - s ] ⁢ 𝑑 s

(6.2) ≤ R ⁢ ∫ 0 + ∞ exp ⁡ [ - F 1 ⁢ ( s ) ] ⁢ 𝑑 s ,

where

F 1 ⁢ ( s ) = s - ( ∫ - ∞ + ∞ Φ ⁢ ( r ) ⁢ a ⁢ ( r , s ) ⁢ 𝑑 r ) q / ( q - 1 ) .

Since a⁢(r,s)≤1 for a.e 0<r<s, and

∫ - ∞ 0 a ⁢ ( r , s ) q / ( q - 1 ) ⁢ 𝑑 r + ∫ s + ∞ a ⁢ ( r , s ) q / ( q - 1 ) ⁢ 𝑑 r = n ⁢ ( q - 1 ) ( n - 1 ) ⁢ q < + ∞ ,

we have

∫ - ∞ + ∞ Φ q ⁢ ( r ) ⁢ 𝑑 r = ∫ 0 + ∞ ( R 1 / n ⁢ | ∇ ⁡ u | ∗ ⁢ ( R ⁢ e - r ) ⁢ e - r / n ) q ⁢ 𝑑 r = ∫ 0 R [ | ∇ ⁡ u | ∗ ⁢ ( t ) ⁢ t 1 / n ] q ⁢ d ⁢ t t ≤ ∥ ∇ ⁡ u ∥ n , q q ≤ 1 .

Now using Lemma 3.1, we obtain

(6.3) ∫ 0 + ∞ exp ⁡ [ - F 1 - β / n ⁢ ( s ) ] ⁢ 𝑑 s ≤ c 0 .

So by inequalities (6.1), (6.2) and (6.3), we have

I 1 ≤ C ⁢ ( n , q ) ⁢ R = C ⁢ ( n , q ) ⁢ ∥ u ∥ n , q n .

To prove the restriction α<αn,q is optimal, we can take the same uk⁢(x) as in Section 5, i.e.,

where

C k = ( 1 k ⁢ ∫ 1 e k ( 1 - 1 y ) q / n ⁢ d ⁢ y y - 1 ) - 1 / q .

By almost the same calculation, we can prove that the constant αn,q is sharp, so we omit the details. ∎

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1301595

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11371056

Funding statement: The research of the first author was partly supported by a US NSF grant DMS-1301595 and a Simons Fellowship from the Simons Foundation. The research of the second author was partly supported by a Youth Scholars Program of Beijing Normal University (no. 2014NT30) and a grant of NNSF of China (no.11371056).

References

[1] Adachi S. and Tanaka K., Trudinger type inequalities in ℝN and their best exponents, Proc. Amer. Math. Soc. 128 (1999), 2051–2057. 10.1090/S0002-9939-99-05180-1Search in Google Scholar

[2] Adams D. R., A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. (2) 128 (1988), no. 2, 385–398. 10.2307/1971445Search in Google Scholar

[3] Adimurthi and Sandeep K., A singular Moser–Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl. 13 (2007), no. 5–6, 585–603. 10.1007/s00030-006-4025-9Search in Google Scholar

[4] Alvino A., Ferone V. and Trombetti G., Moser-type inequalities in Lorentz spaces, Potential Anal. 5 (1996), 273–299. 10.1007/BF00282364Search in Google Scholar

[5] Beckner W., Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality, Ann. of Math. (2) 138 (1993), no. 1, 213–242. 10.2307/2946638Search in Google Scholar

[6] Bezerra do Ó J. M., N-Laplacian equations in ℝN with critical growth, Abstr. Appl. Anal. 2 (1997), no. 3–4, 301–315. 10.1155/S1085337597000419Search in Google Scholar

[7] Cao D., Nontrivial solution of semilinear elliptic equation with critical exponent in ℝ2, Comm. Partial Differential Equations 17 (1992), no. 3–4, 407–435. 10.1080/03605309208820848Search in Google Scholar

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

[9] Cassani D. and Tarsi C., A Moser-type inequality in Lorentz–Sobolev spaces for unbounded domains in ℝN, Asymptot. Anal. 64 (2009), no. 1–2, 29–51. 10.3233/ASY-2009-0934Search in Google Scholar

[10] Chang S. Y. A. and Yang P., Conformal deformation of metrics on S2, J. Differential Geom. 27 (1988), 259–296. 10.4310/jdg/1214441783Search in Google Scholar

[11] Chang S. Y. A. and Yang P., The inequality of Moser and Trudinger and applications to conformal geometry, Comm. Pure Appl. Math. 56 (2003), no. 8, 1135–1150. 10.1002/cpa.3029Search in Google Scholar

[12] Cianchi A., Lutwak E., Yang D. and Zhang G., Affine Moser–Trudinger and Morrey–Sobolev inequalities, Calc. Var. Partial Differential Equations 36 (2009), no. 3, 419–436. 10.1007/s00526-009-0235-4Search in Google Scholar

[13] Cohn W. S. and Lu G., Best constants for Moser–Trudinger inequalities on the Heisenberg group, Indiana Univ. Math. J. 50 (2001), no. 4, 1567–1591. 10.1512/iumj.2001.50.2138Search in Google Scholar

[14] Cohn W. S. and Lu G., Sharp constants for Moser–Trudinger inequalities on spheres in complex space ℂn, Comm. Pure Appl. Math. 57 (2004), no. 11, 1458–1493. 10.1002/cpa.20043Search in Google Scholar

[15] Flucher M., Extremal functions for the Trudinger–Moser inequality in 2 dimensions, Comment. Math. Helv. 67 (1992), no. 3, 471–497. 10.1007/BF02566514Search in Google Scholar

[16] Fontana L., Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv. 68 (1993), 415–454. 10.1007/BF02565828Search in Google Scholar

[17] Giarrusso E. and Nunziante D., Symmetrization in a class of first-order Hamilton–Jacobi equations, Nonlinear Anal. 8 (1984), no. 4, 289–299. 10.1016/0362-546X(84)90031-2Search in Google Scholar

[18] Hudson A. and Leckband M., A sharp exponential inequality for Lorentz–Sobolev spaces on bounded domains, Proc. Amer. Math. Soc. 127 (1999), 2029–2033. 10.1090/S0002-9939-99-05147-3Search in Google Scholar

[19] Lam N. and Lu G., Sharp Moser–Trudinger inequality on the Heisenberg group at the critical case and applications, Adv. Math. 231 (2012), 3259–3287. 10.1016/j.aim.2012.09.004Search in Google Scholar

[20] Lam N. and Lu G., The Moser–Trudinger and Adams inequalities and elliptic and subelliptic equations with nonlinearity of exponential growth, Recent Developments in Geometry and Analysis, Adv. Lect. Math. (ALM) 23, International Press, Somerville (2012), 179–251. Search in Google Scholar

[21] Lam N. and Lu G., A new approach to sharp Moser–Trudinger and Adams type inequalities: A rearrangement-free argument, J. Differential Equations 255 (2013), no. 3, 298–325. 10.1016/j.jde.2013.04.005Search in Google Scholar

[22] Lam N., Lu G. and Tang H., On nonuniformly subelliptic equations of Q-sub-Laplacian type with critical growth in the Heisenberg group, Adv. Nonlinear Stud. 12 (2012), no. 3, 659–681. 10.1515/ans-2012-0312Search in Google Scholar

[23] Lam N., Lu G. and Tang H., Sharp subcritical Moser–Trudinger inequalities on Heisenberg groups and subelliptic PDEs, Nonlinear Anal. 95 (2014), 77–92. 10.1016/j.na.2013.08.031Search in Google Scholar

[24] Lam N., Lu G. and Zhang L., Equivalence of critical and subcritical sharp Moser–Trudinger–Adams inequalities, preprint 2015, http://arxiv.org/abs/1504.04858. 10.4171/RMI/969Search in Google Scholar

[25] Leckband M., Moser’s inequality on the ball Bn for functions with mean value zero, Comm. Pure Appl. Math. 58 (2005), no. 6, 789–798. 10.1002/cpa.20056Search in Google Scholar

[26] Li Y. X., Moser–Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differ. Equ. 14 (2001), no. 2, 163–192. Search in Google Scholar

[27] Li Y. X., Remarks on the extremal functions for the Moser–Trudinger inequality, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 2, 545–550. 10.1007/s10114-005-0568-7Search in Google Scholar

[28] Li Y. X. and Ruf B., A sharp Trudinger–Moser type inequality for unbounded domains in ℝn, Indiana Univ. Math. J. 57 (2008), no. 1, 451–480. 10.1512/iumj.2008.57.3137Search in Google Scholar

[29] Lieb E. and Loss M., Analysis, 2nd ed., Grad. Stud. Math. 14, American Mathematical Society, Providence, 2001. Search in Google Scholar

[30] Lin K., Extremal functions for Moser’s inequality, Trans. Amer. Math. Soc. 348 (1996), no. 7, 2663–2671. 10.1090/S0002-9947-96-01541-3Search in Google Scholar

[31] Lu G. and Tang H., Best constants for Moser–Trudinger inequalities on high dimensional hyperbolic spaces, Adv. Nonlinear Stud. 13 (2013), no. 4, 1035–1052. 10.1515/ans-2013-0415Search in Google Scholar

[32] Lu G. and Yang Y., A sharpened Moser–Pohozaev–Trudinger inequality with mean value zero in R2, Nonlinear Anal. 70 (2009), no. 8, 2992–3001. 10.1016/j.na.2008.12.022Search in Google Scholar

[33] Moser J., A sharp form of an inequality by Trudinger, Indiana Univ. Math. J. 20 (1971), 1077–1092. 10.1512/iumj.1971.20.20101Search in Google Scholar

[34] O’Neil R., Convolution operators and L⁢(p,q) spaces, Duke Math. J. 30 (1963), 129–142. 10.1215/S0012-7094-63-03015-1Search in Google Scholar

[35] Pohožaev S. I., On the eigenfunctions of the equation Δ⁢u+λ⁢f⁢(u)=0 (in Russian), Dokl. Akad. Nauk SSSR 165 (1965), 36–39. Search in Google Scholar

[36] Ruf B., A sharp Trudinger–Moser type inequality for unbounded domains in ℝ2, J. Funct. Anal. 219 (2005), no. 2, 340–367. 10.1016/j.jfa.2004.06.013Search in Google Scholar

[37] Stein E. M. and Weiss G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton University Press, Princeton, 1971. Search in Google Scholar

[38] Strauss W., Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149–162. 10.1007/BF01626517Search in Google Scholar

[39] Talenti G., Elliptic equations and rearrangements, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 3 (1976), 697–718. Search in Google Scholar

[40] Trudinger N. S., On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483. 10.1512/iumj.1968.17.17028Search in Google Scholar

[41] Yudovič V. I., Some estimates connected with integral operators and with solutions of elliptic equations (in Russian), Dokl. Akad. Nauk SSSR 138 (1961), 805–808. Search in Google Scholar

Received: 2015-12-31

Revised: 2016-04-29

Accepted: 2016-05-03

Published Online: 2016-06-18

Published in Print: 2016-08-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2015-5046

Keywords for this article

Singular Trudinger–Moser Inequality; Lorentz–Sobolev Spaces; Best Constants

Creative Commons

BY 4.0