Trudinger–Moser type inequalities with logarithmic weights in fractional dimensions

Jianwei Xue; Caifeng Zhang; Maochun Zhu

doi:10.1515/ans-2023-0161

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Trudinger–Moser type inequalities with logarithmic weights in fractional dimensions

Jianwei Xue , Caifeng Zhang and Maochun Zhu

Published/Copyright: January 17, 2025

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From the journal Advanced Nonlinear Studies Volume 25 Issue 1

Abstract

The purpose of this paper is two-fold. First, we derive sharp Trudinger–Moser inequalities with logarithmic weights in fractional dimensions:

sup ∫ 0 1 w ( r ) u ′ ( r ) β + 2 d λ α 1 / ( β + 2 ) ≤ 1 ∫ 0 1 e μ α , θ , γ u β + 2 β + 1 1 − γ d λ θ < + ∞ ,

where 0 ≤ γ < 1, α = β + 1, μ α , θ , γ ≔ θ + 1 ω α 1 / α 1 − γ 1 1 − γ , w ( r ) = w 1 ( r ) = log 1 r γ β + 1 or w ( r ) = w 2 ( r ) = log e r γ β + 1 and λ _θ(E) = ω _θ ∫ _E r ^θdr for all E ⊂ R . The case γ > 1 and γ = 1 are also be considered in this part to improve our paper. Indeed, we have a continuous embedding X(w ₂) ↪ L ^∞(0, 1) for γ > 1 and a critical growth of double exponential type for γ = 1. Second, we apply the Lions type Concentration-Compactness principle for Trudinger–Moser inequalities and the precise estimate of normalized concentration limit for normalized concentrating sequence at origin to establish the existence of extremals for Trudinger–Moser inequalities when w ( r ) = w 1 ( r ) = log 1 r γ β + 1 and γ > 0 is sufficiently small.

Keywords: Trudinger–Moser inequalities; fractional dimensions; extremals; logarithmic weight

2010 MSC: 46E35; 26D10; 42B37

1 Introduction and main results

Let X = X(α, β, θ) be a weighted Sobolev space, the collection of all locally absolutely continuous functions u : ( 0,1 ] → R satisfying u(1) = 0, u ∈ L θ β + 2 and u ′ ∈ L α β + 2 , where α, θ ≥ 0 and β > −1 are real numbers, and L θ q = L θ q ( 0,1 ) is the weighted Lebesgue space defined as the set of all measurable functions u on (0,1) satisfying

u L θ q ≔ ∫ 0 1 u ( r ) q d λ θ 1 / q < ∞ , if 1 ≤ q < ∞ , e s s sup 0 < r < 1 u ( r ) < ∞ , if q = ∞ .

In the last definition, the weighted Lebesgue measure λ _θ is given by

λ θ ( E ) = ω θ ∫ E r θ d r ,

for all E ⊂ R where

ω θ = 2 π θ + 1 2 / Γ ( ( θ + 1 ) / 2 )

and Γ ( x ) = ∫ 0 ∞ t x − 1 e − t d t is the Euler gamma function. Moreover, weighted Sobolev space X is a Banach space under the norm

u X = u L θ β + 2 β + 2 + u ′ L α β + 2 β + 2 1 / β + 2 .

Since weighted Sobolev space plays an important role in studying partial differential operators, there are many papers devoted to it. In 1990s, Opic and Kufner [1] proved the following Hardy-type inequality:

∫ 0 1 r θ u ( r ) q d r 1 / q ≤ C ∫ 0 1 r α u ′ ( r ) β + 2 d r 1 / ( β + 2 ) , u ∈ X ,

if one of the following two conditions is true:

If 1 ≤ β + 2 ≤ q < + ∞, then either (a) α > β + 1 and θ ≥ α q β + 2 − q β + 1 β + 2 − 1 , or (b) α ≤ β + 1 and θ ≥ 0.
If 1 ≤ q < β + 2 < + ∞, then either (c) α > β + 1 and θ > α q β + 2 − q β + 1 β + 2 − 1 , or (d) α ≤ β + 1 and θ ≥ 0.

In light of (1), we know that the norm  u X is equivalent to the norm  u ′ L α β + 2 for all α > 0 and β, θ ≥ 0 satisfying

θ ≥ α − β − 2 .

In addition, one finds in the Sobolev case: α − β − 1 > 0, there holds the continuous embedding:

(1.1) X ( α , β , θ ) ↪ L θ q , for all q ∈ ( 1 , p * ] and θ ≥ α − β − 2 ,

where

p * = ( θ + 1 ) ( β + 2 ) α − β − 1 .

Moreover, if we assume q < p*, embedding (1.1) becomes compact.

In this paper, we focus on the Trudinger–Moser case:

(1.2) α − β − 1 = 0 ,

which is known as the limiting case for the S. case. Once (1.2) holds, p* goes to infinity and X(α, β, θ) can be compactly embedded into L θ q where θ ≥ 0, q ∈ (1, + ∞). However, X ⊈ L θ ∞ when q = +∞. Therefore it is natural to find a maximal growth function φ(u) such that u ∈ X implies ∫ 0 1 φ ( u ) d λ θ < ∞ . To deal with this question, de Oliveira and do Ó [2] proved that the maximal growth is of exponential type by showing a fractional dimensions version of Trudinger–Moser inequality. Indeed, they stated

Theorem A.

([2]) Let α ≥ 1 and β, θ ≥ 0 be real numbers and assume that (1.2) holds. Then

sup u ∈ S ∫ 0 1 e μ u ( β + 2 ) / ( β + 1 ) d λ θ ≤ C α B 1 θ iff μ ≤ μ α , θ ,

where

S = u ∈ X ; u ′ L α β + 2 ≤ 1

and

μ α , θ = ( θ + 1 ) ω α 1 / α , B 1 θ = ∫ 0 1 d λ θ = ω θ θ + 1 .

Theorem A represents an extension to the following classical Trudinger–Moser inequality ([3], [4]): let Ω be a smooth bounded domain in R N ,

(1.3) sup u ∈ W 0 1 , N ( Ω ) , ‖ ∇ u ‖ N ≤ 1 ∫ Ω e μ | u | N N − 1 d x < ∞ iff μ ≤ μ N ≔ N ω N − 1 1 N − 1 ,

where

W 0 1 , N ( Ω ) = u ∈ C 0 ∞ ( Ω ) | ‖ ∇ u ‖ N < ∞ ̄ with ‖ ∇ u ‖ N ≔ ∫ Ω | ∇ u | N d x 1 N .

Trudinger–Moser inequality (1.3) has led to many related results: – extensions of Trudinger–Moser inequality to unbounded domains, see Li-Ruf [5], [6], – extensions of Trudinger–Moser inequality to higher order Sobolev spaces: see Adams [7], Lam-Lu [8], Ruf-Sani [9], – Trudinger–Moser inequalities on manifolds, see Li [10], [11], Fontana [12], – Trudinger–Moser inequalities on Heisenberg groups, see the work of Cohn, Lam, Li, Lu and Zhu [13], [14], [15], and Hardy–Trudinger–Moser type inequalities, see Wang and Ye [16], Lu and Yang [17], [18], Liang et al. [19] and Li et al. [20], Trudinger–Moser and Adams inequalities with degenerate potentials by Chen et al. [21], [22], [23]. For other related Trudinger–Moser and Adams inequalities one can see [24], [25], [26], [27], [28], and the references therein.

An interesting problem related to the Trudinger–Moser inequalities lies in investigating the existence of extremal functions. Carleson and Chang [29] first proved the existence of extremals for classical Trudinger–Moser inequalities (1.3) when Ω is a unit ball in R N . This result was later extended to any bounded domains by Flucher [30] in R 2 and Lin [31] in R N . One can also see [5], [10], [11] for existence of extremals for the Trudinger–Moser inequalities on compact Riemannian manifold, unbounded domains, and see [18], [32], [33], [34] for the existence of extremals for higher order Trudinger–Moser inequalities in bounded and unbounded domains.

Recently, Calanchi and Ruf ([35], [36]) studied the influence of weights in the Sobolev norm. In fact, they changed the function space in which they worked by adding some weights. More precisely, let B = B 1 0 be the unit ball in R N and

w 0 ( x ) = log 1 | x | γ N − 1 or w 0 ( x ) = log e | x | γ N − 1 , γ ≥ 0 .

They considered the weighted radial Sobolev space

W 0 , r a d 1 , N B , ω 0 = u ∈ C 0 , r a d ∞ B ; u ω 0 N = ∫ B ∇ u N ω 0 x d x < ∞ ̄

and obtained the following Trudinger–Moser type inequality with logarithmic weights on B:

(1.4) sup u ∈ W 0 , r a d 1 , N ( B , ω 0 ) , u ω 0 N ≤ 1 ∫ B e μ | u | N ( N − 1 ) ( 1 − γ ) d x < + ∞ ,

for any γ ∈ [0, 1) and μ ≤ μ N , γ ≔ N ( 1 − γ ) ω N − 1 1 / ( N − 1 ) 1 / ( 1 − γ ) .

After that, Nguyen [37] proved the existence of extremals for (1.4) when γ > 0 is small enough (also see [38] for dimension 2). In the works of the third author ([39], [40]), inequality (1.4) and existence of extremals were extended to the second order case when n ≥ 4 and n is even.

In this paper, we are interested in the Trudinger–Moser type inequality with logarithmic weight in the weighted Sobolev space including fractional dimensions. We denote

X ( w ) = u ∈ X ; u ′ L α , w β + 2 < ∞ ̄ ,

where

u ′ L α , w β + 2 = ∫ 0 1 w ( r ) u ′ ( r ) β + 2 d λ α 1 / ( β + 2 ) .

We set

(1.5) w ( r ) = w 1 ( r ) = log 1 r γ β + 1 or w ( r ) = w 2 ( r ) = log e r γ β + 1 , γ ≥ 0 .

The main results of this paper can be stated as follows.

Theorem 1.1.

Let γ ∈ [0, 1), w(r) be given by (1.5), and α, β ≥ 0 satisfy (1.2). Then

( a ) ∫ 0 1 e u δ d λ θ < + ∞ , for all u ∈ X ( w ) ,

if and only if

δ ≤ δ β , γ ≔ β + 2 β + 1 1 − γ .

Furthermore, we have

(1.6) ( b ) sup u ′ L α , w β + 2 ≤ 1 ∫ 0 1 e μ u δ β , γ d λ θ < + ∞ ,

if and only if

(1.7) μ ≤ μ α , θ , γ ≔ θ + 1 ω α 1 / α 1 − γ 1 1 − γ .

Once γ ≥ 1, the question becomes different.

Theorem 1.2.

Let γ > 1, w(r) be given by (1.5), and α, β ≥ 0 satisfy (1.2). Then we have the following embedding

X ( w 2 ) ↪ L ∞ ( 0,1 ) .

For the case γ = 1, we have the critical growth of double exponential type:

Theorem 1.3.

Let γ = 1, w 2 ( r ) = log e r β + 1 , α, β ≥ 0 satisfy (1.2) and

β o ≔ β + 2 β + 1 .

Then

( a ) ∫ 0 1 e e u β o d λ θ < + ∞ , ∀ u ∈ X ( w 2 ) ,

while

( b ) sup u ′ L α , w 2 β + 2 ≤ 1 ∫ 0 1 e a e ω α 1 / α u β o d λ θ < + ∞ ,

if and only if

a ≤ θ + 1 .

In the second part of this paper, we study the existence of extremal functions for (1.6), we only consider the case w = w 1 ( r ) = log 1 r γ β + 1 for γ ∈ [0, 1). Let us denote

T M ( μ ) ≔ sup u ′ L α , w 1 β + 2 ≤ 1 ∫ 0 1 e μ u δ β , γ d λ θ .

Then, we have

Theorem 1.4.

There exists γ 0 ∈ 0,1 such that TM(μ _α,θ,γ) is attained for any γ ∈ 0 , γ 0 .

The organization of this paper is as follows. In Section 2, we derive sharp Trudinger–Moser inequalities with logarithmic weights in fractional dimensions. In order to overcome the difficulty caused by the appearance of weight w(r), we introduce a useful auxiliary lemma(see Lemma 2.1) to establish the sharp inequality in subcritical and supercritical cases and make use of Leckband’s inequality and the method of change of variables to derive similar result in the critical case. Section 3 is devoted to the existence of extremals. We first get the Lions type Concentration-Compactness principle for functions in X(w ₁). With the help of Concentration-Compactness principle and the precise estimate of normalized concentration limit J α , θ , γ δ ( 0 ) for normalized concentrating sequence at origin, we obtain some monotonicity and continuity results for TM(μ _α,θ,γ) and J α , θ , γ δ ( 0 ) which play a key role in excluding the case u _n ⇀ u ₀ ≡ 0 where { u n } n is a maximizing sequence for TM(μ _α,θ,γ). Then TM(μ _α,θ,γ) can be attained for γ > 0 small enough follows from the Concentration-Compactness principle.

2 Trudinger–Moser type inequality with logarithmic weights

This section is devoted to establishing Theorems 1.1–1.3. To this end, we start with a auxiliary lemma.

Lemma 2.1.

Let u ∈ X(w), then

( i ) u ( r ) ≤ log ⁡ r 1 − γ / β o ω α 1 / ( β + 2 ) 1 − γ 1 / β o u ′ L α , w β + 2 , ∀γ < 1, for w = w 1 ( r ) = log ⁡ r γ β + 1 ; while for w = w 2 ( r ) = log e r γ β + 1 , we have

( i i ) u ( r ) ≤ log ( e / r ) 1 − γ − 1 1 / β o ω α 1 / ( β + 2 ) 1 − γ 1 / β o u ′ L α , w β + 2 , γ ≠ 1,

( i i i ) u ( r ) ≤ log 1 / β o ( log ( e / r ) ) ω α 1 / ( β + 2 ) u ′ L α , w β + 2 , γ = 1.

Proof.

(i) Since u(1) = 0, from Hölder’s inequality, one has

u ( r ) ≤ ∫ r 1 t α / ( β + 2 ) log ⁡ t γ α / ( β + 2 ) u ′ ( t ) t − α / ( β + 2 ) log ⁡ t − γ α / ( β + 2 ) d t ≤ ∫ r 1 t α log ⁡ t γ α u ′ ( t ) β + 2 d t 1 / ( β + 2 ) ∫ r 1 t − 1 log ⁡ t − γ d t 1 / β o = 1 ω α 1 / ( β + 2 ) ω α ∫ r 1 t α log ⁡ t γ α u ′ ( t ) β + 2 d t 1 / ( β + 2 ) − log ⁡ t 1 − γ 1 − γ r 1 1 / β o ≤ log ⁡ r 1 − γ / β o ω α 1 / ( β + 2 ) 1 − γ 1 / β o u ′ L α , w β + 2 .

By a similar argument, we obtain (ii). We next prove (iii).

u ( r ) ≤ ∫ r 1 t α / ( β + 2 ) log e / t α / ( β + 2 ) u ′ ( t ) t − α / ( β + 2 ) log e / t − α / ( β + 2 ) d t ≤ ∫ r 1 t α log e / t α u ′ ( t ) β + 2 d t 1 / ( β + 2 ) ∫ r 1 t − 1 log e / t − 1 d t 1 / β o = 1 ω α 1 / ( β + 2 ) ω α ∫ r 1 t α log e / t α u ′ ( t ) β + 2 d t 1 / ( β + 2 ) ⁡ log 1 / β o ( log ( e / r ) ) ≤ log 1 / β o ( log ( e / r ) ) ω α 1 / ( β + 2 ) u ′ L α , w β + 2 .

□

In the following, we prove the Trudinger–Moser inequalities with logarithmic weights in the subcritical and supercritical cases:

Proof of Theorem 1.1 (Subcritical and supercritical). We observe that

X ( w 2 ) ↪ X ( w 1 ) , for γ ∈ [ 0,1 ) .

Hence we only need to consider the case w 1 ( r ) = log 1 r γ β + 1 . We may assume that u ≥ 0. Set the variable t by

r = e − t / ( θ + 1 )

and define the function ψ(t) on [0, + ∞) by

(2.1) ψ ( t ) = ω α 1 / ( β + 2 ) θ + 1 β + 1 β + 2 1 − γ 1 − γ β + 1 β + 2 ⁡ u ( r ) .

Then the norm becomes

(2.2) ∫ 0 1 u ′ ( r ) β + 2 log ⁡ r γ ( β + 1 ) d λ α = ∫ 0 + ∞ ψ ′ β + 2 t γ β + 1 1 − γ β + 1 d t

and the exponential integral becomes

(2.3) θ + 1 ω θ ∫ 0 1 e μ u δ β , γ d λ θ = ∫ 0 + ∞ e μ ̄ ψ δ β , γ − t d t ,

where

μ ̄ = μ μ α , θ , γ ,

δ _β,γ and μ _α,θ,γ are given in (1.6) and (1.7).

We first prove (a), it is enough to prove that

∫ 0 + ∞ e μ ̄ ψ δ β , γ − t d t < + ∞ , for all ψ such that ∫ 0 + ∞ ψ ′ β + 2 t γ β + 1 1 − γ β + 1 d t < + ∞ .

Indeed, for all ɛ > 0, there exists T = T(ɛ) such that

∫ T + ∞ ψ ′ β + 2 t γ β + 1 1 − γ β + 1 d t < ε β + 2 .

Using Hölder’s inequality, we have

ψ ( t ) = ψ ( T ) + ∫ T t ψ ′ ( s ) d s ≤ ψ ( T ) + ∫ T t ψ ′ ( s ) s γ / β o s − γ / β o d s ≤ ψ ( T ) + ∫ T t ψ ′ ( s ) β + 2 s γ β + 1 d s 1 / β + 2 ∫ T t s − γ d s 1 / β o = ψ ( T ) + ∫ T t ψ ′ ( s ) β + 2 s γ β + 1 ( 1 − γ ) β + 1 d s 1 / β + 2 t 1 − γ − T 1 − γ 1 / β o < ψ ( T ) + ε t 1 − γ − T 1 − γ 1 / β o , for all t ≥ T .

Therefore there exists T ̄ such that

μ ̄ ψ β o 1 − γ ( t ) ≤ 1 2 t , for all t ≥ T ̄ .

This is sufficient to guarantee that the integral in (2.3) is finite.

We next prove that the condition δ ≤ δ β , γ = β o 1 − γ is necessary by contradiction.

Let δ = δ _β,γ + ɛ. For any fixed ɛ > 0, it is sufficient to test

∫ 0 + ∞ e μ ̄ ψ δ − t d t .

Denote ψ(t) by

ψ ( t ) = t 1 δ β , γ − η = t 1 − γ β o − η , if t ≥ 1 , t , if 0 < t < 1 ,

where η > 0 is chosen such that

δ β , γ + ε 1 δ β , γ − η ≔ 1 + η ̄ > 1 .

By this choice of ψ, one can easily see that

∫ 0 + ∞ ψ ′ β + 2 t γ β + 1 1 − γ β + 1 d t < C and ∫ 0 + ∞ e μ ̄ ψ δ − t d t = + ∞ .

Thus the proof of (a) is finished.

In order to prove (b), we make use of (i) of Lemma 2.1 which states that

if ∫ 0 + ∞ ψ ′ β + 2 t γ β + 1 1 − γ β + 1 d t ≤ 1 , then ψ ( t ) ≤ t 1 − γ / β o , ∀ t > 0 .

Hence, for μ ̄ < 1 , it is easy to see that

(2.4) ∫ 0 + ∞ e μ ̄ ψ δ β , γ − t d t ≤ ∫ 0 + ∞ e μ ̄ t − t d t < + ∞ .

Next, we prove that μ _α,θ,γ is sharp in the sense that the supremum in (1.6) becomes infinity if μ > μ _α,θ,γ, i.e. μ ̄ > 1 .

Case 1: w = w 1 ( r ) = log ⁡ r γ β + 1 . It is sufficient to test the first integral in (2.4) by the following functions

η k ( t ) = t 1 − γ k 1 − γ / β + 2 , if t ≤ k , k 1 − γ / β o , if t ≥ k .

Direct calculations show that

∫ 0 + ∞ η k ′ β + 2 t γ β + 1 1 − γ β + 1 d t = 1 .

If μ ̄ > 1 , we get

∫ 0 + ∞ e μ ̄ η k δ β , γ − t d t ≥ ∫ k + ∞ e μ ̄ k − t d t → + ∞ , as k → + ∞ .

Case 2: w = w 2 ( r ) = log e r γ β + 1 . We make the change of variables t = θ + 1 1 − log ⁡ r and set ψ(t) be given in (2.1).

Then the norm and exponential integral become

∫ 0 1 u ′ ( r ) β + 2 log e r γ ( β + 1 ) d λ α = ∫ 0 + ∞ ψ ′ β + 2 t γ β + 1 1 − γ β + 1 d t ,

θ + 1 ω θ e θ + 1 ∫ 0 1 e μ u δ β , γ d λ θ = ∫ 0 + ∞ e μ ̄ ψ δ β , γ − t d t .

Denote function sequence {ξ _k} by

ξ k ( t ) = 0 , 0 ≤ t ≤ θ + 1 , t 1 − γ − θ + 1 1 − γ k + θ + 1 1 − γ − θ + 1 1 − γ 1 / β + 2 , θ + 1 ≤ t ≤ k + θ + 1 , k + θ + 1 1 − γ − θ + 1 1 − γ 1 / β o , t ≥ k + θ + 1 .

One can easily verify that its norm is 1. However for μ ̄ > 1 , we have

∫ 0 + ∞ e μ ̄ ξ k δ β , γ − t d t ≥ ∫ k + θ + 1 + ∞ e μ ̄ k + θ + 1 1 − γ − θ + 1 1 − γ 1 / 1 − γ − t d t = e μ ̄ k + θ + 1 1 − γ − θ + 1 1 − γ 1 / 1 − γ − k + θ + 1 → + ∞ ,

as k → +∞.

For the continuity of our work, let us postpone the proof of the critical case ( μ ̄ = 1 ) of Theorem 1.1. Next, we give

Proof of Theorem 1.2.

In this result, we show that in the case w = w 2 ( r ) = log e r γ β + 1 and γ > 1, the functions in X(w ₂) are bounded. By (ii) in Lemma 2.1, we have

u ( r ) ≤ log ( e / r ) 1 − γ − 1 1 / β o ω α 1 / ( β + 2 ) 1 − γ 1 / β o u ′ L α , w β + 2 ≤ 2 u ′ L α , w β + 2 ω α 1 / ( β + 2 ) 1 − γ 1 / β o .

This means X(w ₂) ↪ L ^∞(0, 1). Hence, we finish the proof of Theorem 1.2.

In the following, we consider double exponential type case (γ = 1 and w 2 ( r ) = log e r β + 1 ) for the subcritical and supercritical growth:

Proof of Theorem 1.3 (Subcritical and Supercritical). Managing similar progress as the proof in the first part of Theorem 1.1, we can get (a).

For the proof of (b), set ψ ( t ) = ω α 1 / ( β + 2 ) u ( r ) with r = e ^−t. Direct computations show that

∫ 0 1 u ′ ( r ) β + 2 log e r ( β + 1 ) d λ α = ∫ 0 + ∞ ψ ′ β + 2 1 + t β + 1 d t

and

(2.5) ∫ 0 1 e a e ω α 1 / α u β o d λ θ = ω θ ∫ 0 + ∞ e a e ψ β o − θ + 1 t d t .

By (iii) of Lemma 2.1, we have

if ∫ 0 + ∞ ψ ′ β + 2 1 + t β + 1 d t ≤ 1 , then ψ ( t ) ≤ log 1 / β o ( 1 + t ) , ∀ t > 0 .

Thus, one has for a < θ + 1,

∫ 0 + ∞ e a e ψ β o − θ + 1 t d t ≤ ∫ 0 + ∞ e a ( 1 + t ) − θ + 1 t d t < + ∞ .

Next we consider the sharpness of a = θ + 1. If a > θ + 1, it is sufficient to test the right-hand side of (2.5) on the following sequence

ψ k ( t ) = log ( 1 + t ) log 1 / ( β + 2 ) ( 1 + k ) , if 0 ≤ t ≤ k , log 1 / β o ( 1 + k ) , if t ≥ k .

By direct calculations, we obtain

∫ 0 + ∞ ψ k ′ β + 2 1 + t β + 1 d t = 1

and

∫ 0 + ∞ e a e ψ k β o − θ + 1 t d t ≥ ∫ k + ∞ e a e ψ k β o − θ + 1 t d t = ∫ k + ∞ e a ( 1 + k ) − θ + 1 t d t → + ∞ ,

as k → +∞.

For the critical case a = θ + 1, we will apply the arguments of Leckband [41] in the last part of this section.

At the end of this section, we consider the critical case for Theorems 1.1 and 1.3. We first introduce the definitions of two special classes of functions, see [41], [42].

Definition 2.2.

A continuous function ρ: [0, + ∞) → [0, + ∞) is called a C*-function provided there exists a constant C _ρ < + ∞ such that for 0 < d < + ∞, we have a constant C(d) < + ∞ with

ρ ( l + d ) s ≤ C ρ ρ ( l − 1 ) s , for all l > C ( d ) and 0 < s < + ∞ .

Definition 2.3.

A C ¹-function M: [0, + ∞) → [0, + ∞) is called a C*-convex function if M is convex, M(0) = 0 and the function ρ defined by the differential equation ρ M ( t ) = M ′ ( t ) is a C*-function.

Then we give the Leckband’s Inequality which plays an important role in completing the proof of Theorems 1.1 and 1.3.

Lemma 2.4.

(Leckband’s Inequality, [41]). Let 1 ≤ q < + ∞, f ∈ L ^q([0, + ∞)) such that f L q ≤ 1 , φ : R + → R + with φ ≥ 0 is locally integrable. Set

G ( x ) = ∫ 0 x φ q / ( q − 1 ) ( y ) d y ( q − 1 ) / q and F ( x ) = ∫ 0 x f ( y ) φ ( y ) d y .

Let Φ ≥ 0 be a nonincreasing function on [0, + ∞) and M(t) be a C*-convex function. Then there exists a constant C > 0, such that

∫ 0 + ∞ Φ M ( G ( t ) ) − M ( F ( t ) ) d M ( G ( t ) ) ≤ C Φ L 1 .

Now, we devote ourselves to the proof of Theorem 1.1.

Proof of Theorem 1.1. Let q = β + 2, one can apply the Leckband’s inequality to

f ( t ) = ψ ′ ( t ) t γ 1 − γ 1 / β o and φ ( t ) = t γ 1 − γ − 1 / β o ,

where ψ ( t ) ∈ C 1 [ 0 , + ∞ ) is given by (2.1). Then by (2.2), we get

∫ 0 + ∞ f ( t ) β + 2 d t = ∫ 0 + ∞ ψ ′ β + 2 t γ β + 1 1 − γ β + 1 d t = 1 .

According to the definition of φ, one can derive that

G ( x ) = ∫ 0 x φ β o ( y ) d y 1 / β o = ∫ 0 x 1 − γ y γ d y 1 / β o = x 1 − γ / β o

and

F ( x ) = ∫ 0 x f ( y ) φ ( y ) d y = ∫ 0 x ψ ′ ( y ) d y = ψ ( x ) .

Finally, we set

M ( s ) = s β o / 1 − γ , Φ ( s ) = e − s .

Then direct calculations show that

M ( G ( t ) ) = t and M ( F ( t ) ) = ψ β o / 1 − γ ( t ) = ψ δ β , γ ( t ) .

Therefore, under these specific choices, the Leckband’s inequality becomes

∫ 0 + ∞ e ψ δ β , γ ( t ) − t d t ≤ C ∫ 0 + ∞ e − s d s = C , for all ψ with ∫ 0 + ∞ ψ ′ β + 2 t γ β + 1 1 − γ β + 1 d t ≤ 1 .

This implies (2.3) holds for μ ̄ = 1 . The proof of Theorem 1.1 is completed.

Finally, we will complete the proof of Theorem 1.3.

Proof of Theorem 1.3. Let q = β + 2 and set

f ( t ) = ψ ′ ( t ) 1 + t 1 / β o and φ ( t ) = 1 + t − 1 / β o .

Then

∫ 0 + ∞ f ( t ) β + 2 d t = ∫ 0 + ∞ ψ ′ β + 2 1 + t β + 1 d t .

According to the definition of φ, we have

G ( x ) = ∫ 0 x φ β o ( y ) d y 1 / β o = ∫ 0 x 1 1 + y d y 1 / β o = log ( 1 + x ) 1 / β o

and

F ( x ) = ∫ 0 x f ( y ) φ ( y ) d y = ∫ 0 x ψ ′ ( y ) d y = ψ ( x ) .

Denote

M ( t ) = e t β o − 1 , Φ ( s ) = e − θ + 1 s .

With direct calculations, one can obtain that

M ( G ( t ) ) = t and M ( F ( t ) ) = e ψ β o ( t ) − 1 .

Hence, the Leckband’s inequality becomes

∫ 0 + ∞ e θ + 1 e ψ β o − 1 − θ + 1 t d t ≤ C ∫ 0 + ∞ e − θ + 1 s d s = C θ ,

for all ψ with ∫ 0 + ∞ ψ ′ β + 2 1 + t β + 1 d t ≤ 1 . Obviously, This yields inequality (2.5) for a = θ + 1, the proof is finished.

3 Existence of extremals for Trudinger–Moser inequalities

This section is devoted to studying the existence of extremals for Trudinger–Moser inequality (1.6) with logarithmic weights in fractional dimensions. From now on, we only consider the case w 1 ( r ) = log 1 r γ β + 1 . For the convenience, we will give a new notation

w γ ≔ log 1 r γ β + 1 .

We first prove the existence of the maximizers in the subcritical case μ < μ _α,θ,γ.

Lemma 3.1.

Assume ɛ > 0, γ ∈ [0, 1) and α, β ≥ 0 satisfing (1.2). Then there exists a u _ɛ ∈ X(ω_γ)such that ‖ u ε ′ ‖ L α , w γ β + 2 = 1 and

T M ( μ α , θ , γ − ε ) = ∫ 0 1 e ( μ α , θ , γ − ε ) | u ε | δ β , γ d λ θ .

Proof.

For any fixed ɛ, let { u ε , j } j ⊂ X ( w γ ) be a maximizing sequence for T M μ α , θ , γ − ε . By extracting a subsequence, we can assume that u _ɛ,j ⇀ u _ɛ weakly in X(w _γ) and u _ɛ,j → u _ɛ a.e. in (0,1) as j → +∞.

Using Trudinger–Moser inequality (1.6), we know that { e ( μ α , θ , γ − ε ) | u ε , j | δ β , γ } j is bounded in L θ p for any 1 < p < μ _α,θ,γ/(μ _α,θ,γ − ɛ). Consequently, it follows from the Vitali convergence theorem that

T M μ α , θ , γ − ε = lim j → + ∞ ∫ 0 1 e ( μ α , θ , γ − ε ) | u ε , j | δ β , γ d λ θ = ∫ 0 1 e ( μ α , θ , γ − ε ) | u ε | δ β , γ d λ θ .

Finally, we prove ‖ u ε ′ ‖ L α , w γ β + 2 = 1 . From the weak lower semi-continuous property of the norm, one can derive ‖ u ε ′ ‖ L α , w γ β + 2 ≤ 1 . Once ‖ u ε ′ ‖ L α , w γ β + 2 < 1 ,

T M ( μ α , θ , γ − ε ) < ∫ 0 1 e ( μ α , θ , γ − ε ) | u ε | ‖ u ε ′ ‖ L α , w γ β + 2 δ β , γ d λ θ ≤ T M ( μ α , θ , γ − ε ) ,

which is impossible. Thus ‖ u ε ′ ‖ L α , w γ β + 2 = 1 and u _ɛ is the maximizer.□

In the following, we prove the existence of maximizers for T M μ α , θ , γ . For any function u ∈ X(w _γ) and for any 0 ≤ γ ̃ < γ , denote

(3.1) v ( r ) = μ α , θ , γ μ α , θ , γ ̃ 1 − γ ̃ / β o ⁡ u ( r ) u ( r ) γ − γ ̃ 1 − γ .

Lemma 3.2.

Let u be a function in X(w _γ) with ‖ u ′ ‖ L α , w γ β + 2 ≤ 1 , and let v be defined by (3.1). Then we have ‖ v ′ ‖ L α , w γ ̃ β + 2 ≤ 1 .

Proof.

Similar as the argument of Lemma 2.1-(i), we obtain

| v ′ ( r ) | β + 2 = μ α , θ , γ μ α , θ , γ ̃ 1 − γ ̃ β + 1 1 − γ ̃ β + 2 1 − γ β + 2 | u ′ r | β + 2 | u r | γ − γ ̃ β + 2 1 − γ ≤ 1 − γ ̃ 1 − γ ω α γ − γ ̃ 1 − γ | u ′ r | β + 2 w γ ( r ) w γ ̃ ( r ) ∫ r 1 u ′ ( t ) β + 2 w γ ( t ) t α d t γ − γ ̃ 1 − γ .

Hence

‖ v ′ ‖ L α , w γ ̃ β + 2 β + 2 = ∫ 0 1 w γ ̃ ( r ) v ′ ( r ) β + 2 d λ α ≤ 1 − γ ̃ 1 − γ ω α 1 − γ ̃ 1 − γ ∫ 0 1 u ′ ( r ) β + 2 w γ ( r ) r α ∫ r 1 u ′ ( t ) β + 2 w γ ( t ) t α d t γ − γ ̃ 1 − γ d r = − ω α 1 − γ ̃ 1 − γ ∫ 0 1 d d r ∫ r 1 u ′ ( t ) β + 2 w γ ( t ) t α d t 1 − γ ̃ 1 − γ d r = ω α 1 − γ ̃ 1 − γ ∫ 0 1 u ′ ( t ) β + 2 w γ ( t ) t α d t 1 − γ ̃ 1 − γ ≤ 1 .

The proof is finished.□

Now, we establish the Lions type Concentration-Compactness principle for Trudinger–Moser inequalities in X(w _γ) as follows.

Lemma 3.3.

Let { u n } n be a sequence in X(w _γ) such that ‖ u n ′ ‖ L α , w γ β + 2 = 1 and u _n ⇀ u ₀ weakly in X(w _γ). Then

(3.2) lim sup n → ∞ ∫ 0 1 e p μ α , θ , γ | u n | δ β , γ d λ θ < + ∞ ,

for any p < p ( u 0 ) ≔ 1 − ‖ u 0 ′ ‖ L α , w γ β + 2 β + 2 − 1 / [ ( β + 1 ) ( 1 − γ ) ] .

Proof.

If u ₀ ≡ 0, we can use (1.6) to see that (3.2) holds. Hence it remains to consider the case u ₀ ≢0.

Without loss of generality, we can assume that u _n ≥ 0. For any L > 0, we define the functions

T L ( u n ) = min { u n , L } and T L ( u n ) = u n − T L ( u n ) .

Since lim L → ∞ ‖ T L ( u 0 ) ′ ‖ L α , w γ β + 2 β + 2 = ‖ u 0 ′ ‖ L α , w γ β + 2 β + 2 , then for any p < p(u ₀), we can choose L sufficiently large such that

p 0 ≔ p 1 − ‖ T L ( u 0 ) ′ ‖ L α , w γ β + 2 β + 2 1 / [ ( β + 1 ) ( 1 − γ ) ] < 1 .

By the weakly lower semi-continuous property of the norm, one can obtain that

lim inf n → ∞ ‖ T L ( u n ) ′ ‖ L α , w γ β + 2 β + 2 ≥ ‖ T L ( u 0 ) ′ ‖ L α , w γ β + 2 β + 2 .

Hence

lim sup n → ∞ ‖ T L ( u n ) ′ ‖ L α , w γ β + 2 β + 2 = 1 − lim inf n → ∞ ‖ T L ( u n ) ′ ‖ L α , w γ β + 2 β + 2 ≤ 1 − ‖ T L ( u 0 ) ′ ‖ L α , w γ β + 2 β + 2 .

Then, there exists n ₀ such that for any n ≥ n ₀,

(3.3) p ‖ T L ( u n ) ′ ‖ L α , w γ β + 2 ( β + 2 ) / [ ( β + 1 ) ( 1 − γ ) ] ≤ p 0 + 1 2 < 1 .

Since u _n = T ^L(u _n) + T _L(u _n) and |T ^L(u _n)| ≤ L, one can employ Young’s inequality to get that

(3.4) | u n | β + 2 ( β + 1 ) ( 1 − γ ) ≤ ( 1 + ε ) | T L ( u n ) | β + 2 ( β + 1 ) ( 1 − γ ) + C ( β , γ , ε ) L β + 2 ( β + 1 ) ( 1 − γ ) .

Choosing ɛ > 0 such that (1 + ɛ)(1 + p ₀)/2 < 1, by (3.3) and (3.4), we have for any n ≥ n ₀,

(3.5) ∫ 0 1 e p μ α , θ , γ | u n | δ β , γ d λ θ ≤ ∫ 0 1 e p μ α , θ , γ ( 1 + ε ) | T L ( u n ) | δ β , γ + p μ α , θ , γ C ( β , γ , ε ) L δ β , γ d λ θ ≤ C ( β , γ , ε , p , L ) ∫ 0 1 e p μ α , θ , γ ( 1 + ε ) ‖ T L ( u n ) ′ ‖ L α , w γ β + 2 δ β , γ T L ( u n ) ‖ T L ( u n ) ′ ‖ L α , w γ β + 2 δ β , γ d λ θ ≤ C ( β , γ , ε , p , L ) ∫ 0 1 e μ α , θ , γ ( 1 + ε ) ( 1 + p 0 ) 2 T L ( u n ) ‖ T L ( u n ) ′ ‖ L α , w γ β + 2 δ β , γ d λ θ < + ∞ ,

where the last inequality comes from (1.6). The proof is finished.□

In order to obtain the existence of maximizers for T M μ α , θ , γ when γ > 0 is small enough, we need the following monotonicity results.

Lemma 3.4.

The function γ ↦ TM(μ _α,θ,γ) is decreasing on [0,1).

Proof.

Let u be any function in X(w _γ) with ‖ u ′ ‖ L α , w γ β + 2 ≤ 1 and v be the function defined by (3.1). Lemma 3.2 means that ‖ v ′ ‖ L α , w γ ̃ β + 2 ≤ 1 for any γ ̃ ≤ γ . By the definition of function v and TM(μ _α,θ,γ), one has

∫ 0 1 e μ α , θ , γ | u | δ β , γ d λ θ = ∫ 0 1 e μ α , θ , γ ̃ | v | δ β , γ ̃ d λ θ ≤ T M μ α , θ , γ ̃ .

According to the preceding estimate, we get

T M μ α , θ , γ ≤ T M μ α , θ , γ ̃ ,

for any 0 ≤ γ ̃ ≤ γ < 1 . The proof of Lemma 3.4 is finished.□

For simplicity, we set

J α , θ , γ u = ∫ 0 1 e μ α , θ , γ u δ β , γ d λ θ , u ∈ X ( w γ ) .

Let { u n } n ⊂ X ( w γ ) be a sequence such that u _n ⇀ 0 weakly in X(w _γ). We say that { u n } n is called a normalized concentrating sequence at origin (NCS) if u n ′ L α , w γ β + 2 = 1 and

lim n → ∞ ∫ ρ 1 w γ ( r ) u n ′ ( r ) β + 2 d λ α = 0 ,

for any ρ ∈ (0, 1). Next we define a normalized concentration limit by

J α , θ , γ δ 0 = sup { lim sup n → ∞ J α , θ , γ u n : { u n } n is a NCS } .

It is well known that

(3.6) J α , θ , 0 δ 0 = ω θ θ + 1 1 + e γ ̄ + Ψ ( β + 2 ) < T M ( μ α , θ , 0 ) ,

where Ψ(x) = Γ′(x)/Γ(x) is the psi-function and γ ̄ = lim m → ∞ ∑ j = 1 m 1 / j − ln ⁡ m is the Euler constant (see [2] or [43], Lemma A]). In order to provide an upper bound for J α , θ , γ δ 0 , we prove the following decreasing monotone result.

Lemma 3.5.

The function γ ↦ J α , θ , γ δ 0 is decreasing on [0,1).

Proof.

Let γ ∈ [0, 1) and { u n } n be a NCS in X(w _γ). We have

(3.7) J α , θ , γ δ 0 ≥ lim sup n → ∞ ∫ 0 1 e μ α , θ , γ | u n | δ β , γ d λ θ ≥ ∫ 0 1 d λ θ = ω θ θ + 1 .

To complete Lemma 3.5, we need to prove that

(3.8) J α , θ , γ δ 0 ≤ J α , θ , γ ̃ δ 0 , ∀ γ ̃ ≤ γ .

Let { u n } n be a NCS in X(w _γ). By extracting a subsequence, we can assume that lim n → ∞ J α , θ , γ u n exists and is equal to the upper limit of the original sequence. Let v _n be the function defined by (3.1) from u _n. According to Lemma 3.2, one can derive that v n ∈ X ( w γ ̃ ) and ‖ v n ′ ‖ L α , w γ ̃ β + 2 ≤ 1 . Hence, {v _n} is bounded in X ( w γ ̃ ) , up to a subsequence, we can assume that v _n ⇀ v in X ( w γ ̃ ) . Managing similar argument as in Lemma 2.1-(i), one can derive that

u n ( r ) ≤ log ⁡ r 1 − γ / β o ω α 1 / ( β + 2 ) 1 − γ 1 / β o ∫ r 1 w γ ( r ) u n ′ ( r ) β + 2 d λ α 1 / ( β + 2 ) ,

for any 0 < r < 1. This implies u _n → 0 uniformly on r , 1 for any r ∈ 0,1 . Hence v _n weakly converges to 0 in X ( ω γ ̃ ) as n → ∞ and uniformly converges to 0 on r , 1 . Next, we divide the proof into two cases:

Case 1: lim sup n → ∞ ‖ v n ′ ‖ L α , w γ ̃ β + 2 < 1 . There exist a ∈ (0, 1) and n ₀ such that ‖ v n ′ ‖ L α , w γ ̃ β + 2 ≤ a for any n ≥ n ₀. By (1.6), e μ α , θ , γ ̃ | v n | δ β , γ is bounded in L θ p for some p > 1. Hence, using the definition of function v _n, the Vitali convergence theorem and (3.7), we obtain

lim n → ∞ ∫ 0 1 e μ α , θ , γ | u n | δ β , γ d λ θ = lim n → ∞ ∫ 0 1 e μ α , θ , γ ̃ | v n | δ β , γ ̃ d λ θ = ω θ θ + 1 ≤ J α , θ , γ ̃ δ 0 .

Case 2: lim sup n → ∞ ‖ v n ′ ‖ L α , w γ ̃ β + 2 = 1 . Up to a subsequence, we can assume lim n → ∞ ‖ v n ′ ‖ L α , w γ ̃ β + 2 = 1 . Define v ̃ n = v n / ‖ v n ′ ‖ L α , w γ ̃ β + 2 , then v ̃ n ⇀ 0 in X ( w γ ̃ ) and | v n | ≤ | v ̃ n | . On one hand, if v ̃ n is a NCS in X ( w γ ̃ ) , then we get

lim n → ∞ J α , θ , γ u n = lim n → ∞ J α , θ , γ ̃ v n ≤ lim sup n → ∞ J α , θ , γ ̃ v ̃ n ≤ J α , θ , γ ̃ δ 0 .

On the other hand, there exist some a, ρ ∈ (0, 1) and n ≥ n ₀ such that

∫ 0 ρ | v ̃ n ′ | β + 2 w γ ̃ r d λ α ≤ a

for any n ≥ n ₀. Define the function

v ̄ n ( r ) = v ̃ n ( r ) − v ̃ n ( ρ ) , if r ≤ ρ , 0 , if ρ < r < 1 .

It is easy to check that v ̄ n ∈ X ( w γ ̃ ) and ‖ v ̄ n ′ ‖ L α , w γ ̃ β + 2 β + 2 ≤ a < 1 for any n ≥ n ₀. Choosing ɛ > 0 small enough such that ( 1 + ε ) a 1 ( β + 1 ) ( 1 − γ ̃ ) < 1 , we have v ̃ n ( r ) = v ̄ n ( r ) + v ̃ n ( ρ ) for r ∈ [0, ρ]. Hence

| v ̃ n ( r ) | β + 2 ( β + 1 ) ( 1 − γ ̃ ) ≤ ( 1 + ε ) | v ̄ n ( r ) | β + 2 ( β + 1 ) ( 1 − γ ̃ ) + C ( β , γ ̃ , ε ) | v ̃ n ( ρ ) | β + 2 ( β + 1 ) ( 1 − γ ̃ ) .

By (1.6) and the choice of a and ɛ, one can derive that e μ α , θ , γ ̃ ( 1 + ε ) | v ̄ n | δ β , γ is bounded in L θ p for some p > 1. Notice that v ̃ n ( ρ ) < C , C is a constant, we get e μ α , θ , γ ̃ | v ̃ n | δ β , γ is bounded in L θ p 0 , ρ for some p > 1. Consequently, it follows from the Vitali convergence theorem that

lim n → ∞ ∫ 0 ρ e μ α , θ , γ ̃ | v ̃ n | δ β , γ ̃ d λ θ = ω θ θ + 1 ρ θ + 1 .

On [ρ, 1], in view of Lemma 2.1-(i), one can apply the Lebesgue’s dominated convergence theorem to deduce that

lim n → ∞ ∫ ρ 1 e μ α , θ , γ ̃ | v ̃ n | δ β , γ ̃ d λ θ = ω θ θ + 1 1 − ρ θ + 1 .

Combining the last two estimates, we have

(3.9) lim n → ∞ ∫ 0 1 e μ α , θ , γ ̃ | v ̃ n | δ β , γ ̃ d λ θ = ω θ θ + 1 .

Recall that | v n | ≤ | v ̃ n | , one can infer from (3.7) that

lim n → ∞ J α , θ , γ u n = lim n → ∞ J α , θ , γ ̃ v n ≤ ω θ θ + 1 ≤ J α , θ , γ ̃ δ 0 .

This implies that (3.8) holds. The proof of Lemma 3.5 is finished.□

We need the following lemma which implies that TM(μ _α,θ,γ) as a function of γ is continuous at γ = 0.

Lemma 3.6.

It holds

lim γ → 0 T M ( μ α , θ , γ ) = T M ( μ α , θ , 0 ) .

Proof.

According to Lemma 3.4, we have

lim sup γ → 0 T M ( μ α , θ , γ ) ≤ T M ( μ α , θ , 0 ) .

Hence, it is enough to show that

lim inf γ → 0 T M ( μ α , θ , γ ) ≥ T M ( μ α , θ , 0 ) .

It is well known from [43] that TM(μ _α,θ,0) is attained by a function u 0 ∈ C 1 0,1 ⋂ X ( w 0 ) with ‖ u 0 ′ ‖ L α β + 2 = 1 . Then, it follows from Lebesgue’s dominated convergence theorem that

lim γ → 0 ∫ 0 1 | u 0 ′ r | β + 2 ω γ r d λ α = ∫ 0 1 | u 0 ′ r | β + 2 d λ α = 1 .

For any ɛ > 0, there exists γ _ɛ > 0 such that ‖ u 0 ′ ‖ L α , w γ β + 2 ≤ 1 + ε for any γ ≤ γ _ɛ. Thus, we get

T M ( μ α , θ , γ ) ≥ ∫ 0 1 e μ α , θ , γ | u 0 | 1 + ε δ β , γ d λ θ , ∀ γ ≤ γ ε .

Letting γ → 0 and using Fatou’s Lemma, we obtain

lim inf γ → 0 T M ( μ α , θ , γ ) ≥ ∫ 0 1 e μ α , θ , 0 | u 0 | 1 + ε β o d λ θ .

Letting ɛ → 0 and using again Fatou’s Lemma, we get

lim inf γ → 0 T M ( μ α , θ , γ ) ≥ ∫ 0 1 e μ α , θ , 0 | u 0 | β o d λ θ = T M ( μ α , θ , 0 ) ,

as desired.□

Based on Lemmas 3.4–3.6, we are now ready to give the proof of Theorem 1.4.

Proof of Theorem 1.4.

Using (3.6), there holds

T M ( μ α , θ , 0 ) > J α , θ , 0 δ 0 = ω θ θ + 1 1 + e γ ̄ + Ψ ( β + 2 ) .

By Lemmas 3.5 and 3.6, we see that there exists γ 0 ∈ 0,1 such that

(3.10) T M ( μ α , θ , γ ) > J α , θ , 0 δ 0 ≥ J α , θ , γ δ 0 ,

for any γ ∈ 0 , γ 0 . Let u n n be a maximizing sequence for TM(μ _α,θ,γ) where γ ∈ 0 , γ 0 . We can assume that u _n ⇀ u ₀ in X(w _γ) and u _n → u ₀ uniformly on r , 1 for any r ∈ 0,1 . Assume that u ₀ ≡ 0, if u n is a NCS, then

T M ( μ α , θ , γ ) = lim n → ∞ J α , θ , γ u n ≤ J α , θ , γ δ 0 ,

which contradicts with (3.10). Hence u n is not a NCS which implies there exist some a , ρ ∈ 0,1 and n ₀ such that

∫ 0 ρ | u n ′ | β + 2 w γ r d λ α ≤ a ,

for any n > n ₀. Using the same argument as in the proof of (3.9), we get

T M ( μ α , θ , γ ) = lim n → ∞ J α , θ , γ u n = ω θ θ + 1 ≤ J α , θ , γ δ 0 ,

which again contradicts with (3.10). Hence, we must have u ₀ ≢0. Now by Lemma 3.3, e μ α , θ , γ | u n | δ β , γ n is bounded in L θ p for some p > 1. Therefore, using the Vitali convergence theorem, it holds

T M ( μ α , θ , γ ) = lim n → ∞ ∫ 0 1 e μ α , θ , γ | u n | δ β , γ d λ θ = ∫ 0 1 e μ α , θ , γ | u 0 | δ β , γ d λ θ .

Finally, it remains to prove ‖ u 0 ′ ‖ L α , w γ β + 2 = 1 . Since u _n ⇀ u ₀ in X(w _γ), we have ‖ u 0 ′ ‖ L α , w γ β + 2 ≤ 1 by the weak lower semi-continuous property. If ‖ u 0 ′ ‖ L α , w γ β + 2 < 1 , we obtain

T M ( μ α , θ , γ ) = lim n → ∞ ∫ 0 1 e μ α , θ , γ | u n | δ β , γ d λ θ < ∫ 0 1 e μ α , θ , γ | u 0 | ‖ u 0 ′ ‖ L α , w γ β + 2 δ β , γ d λ θ ≤ T M ( μ α , θ , γ ) ,

which is impossible. Hence ‖ u 0 ′ ‖ L α , w γ β + 2 = 1 and u ₀ is the desired maximizer for TM(μ _α,θ,γ). This finishes the proof of Theorem 1.4.

Corresponding author: Maochun Zhu, School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, 210094, P.R. China, E-mail: zhumaochun2006@126.com

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: The authors state no conflict of interest.
Research funding: The second author was partly supported by a grant from the NNSF of China (No. 12001038). The third author was supported by National Natural Science Foundation of China (Grant Nos. 12071185 and 12061010).
Data availability: Not applicable.

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Received: 2024-08-13

Accepted: 2024-12-09

Published Online: 2025-01-17

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Keywords for this article

Trudinger–Moser inequalities; fractional dimensions; extremals; logarithmic weight

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