Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals

Lu Chen; Bohan Wang; Maochun Zhu

doi:10.1515/ans-2022-0067

Article Open Access

Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals

Lu Chen , Bohan Wang and Maochun Zhu

Published/Copyright: May 29, 2023

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From the journal Advanced Nonlinear Studies Volume 23 Issue 1

Abstract

Let I be a bounded interval of R and λ 1 ( I ) denote the first eigenvalue of the nonlocal operator ( − Δ ) 1 4 with the Dirichlet boundary. We prove that for any 0 ⩽ α < λ 1 ( I ) , there holds

sup u ∈ W 0 1 2 , 2 ( I ) , ‖ ( − Δ ) 1 4 u ‖ 2 2 − α ∥ u ∥ 2 2 ≤ 1 ∫ I e π u 2 d x < + ∞ ,

and the supremum can be attained. The method is based on concentration-compactness principle for fractional Trudinger-Moser inequality, blow-up analysis for fractional elliptic equation with the critical exponential growth and harmonic extensions.

Keywords: fractional; extremals; Trudinger-Moser inequalities

MSC 2010: 46E30; 46E35

1 Introduction

Let Ω be a bounded domain in R n ; it is well known that the analogue of optimal Sobolev embedding for W 0 1 , n ( Ω ) (the Sobolev space consisting of functions vanishing on the boundary ∂ Ω ) can be given by the famous Trudinger-Moser inequality [30,33], which can be stated in the following form:

(1.1) sup u ∈ W 0 1 , n ( Ω ) , ‖ ∇ u ‖ n n ≤ 1 ∫ Ω exp ( α ∣ u ∣ n n − 1 ) d x < + ∞ , iff α ≤ α n = n w n − 1 1 n − 1 ,

where ω n − 1 is the area of unit sphere in R n . So far, Trudinger-Moser inequalities have also been generalized in many other directions such as the Trudinger-Moser inequalities on the unbounded domain, C-R spheres, compact Riemannian manifolds, Heisenberg group, and Trudinger-Moser inequalities in higher-order Sobolev spaces. We refer the interested readers to [1,6,7,10,14,17,23,31] and references therein.

In 1985, Lions [19] established the following concentration-compactness principle associated with inequality (1.1).

Theorem A

For any u k ∈ W 0 1 , n ( Ω ) with ∥ ∇ u k ∥ n ≤ 1 and u k ⇀ u 0 ≠ 0 in W 0 1 , n ( Ω ) , then there exists some p > 1 such that exp α n ∣ u k ∣ n n − 1 is bounded in L p ( Ω ) .

This conclusion gives more precise information and is stronger than Trudinger-Moser inequality (1.1) when the function sequence u k ⇀ u 0 ≠ 0 . Later, the authors of [4] developed a new approach to obtain and sharpen Theorem A. The result in [4] reads as follows:

Theorem B

For any u k ∈ W 0 1 , n ( Ω ) with ∥ ∇ u k ∥ n ≤ 1 , and u k ⇀ u 0 ≠ 0 in W 0 1 , n ( Ω ) , then

(1.2) sup k ∫ Ω exp α n p ∣ u k ∣ n n − 1 d x < ∞ ,

if p < p n = 1 ( 1 − ∥ ∇ u 0 ∥ n n ) 1 n − 1 . Moreover, p n is sharp in the sense that for p ≥ p n , there exists a sequence { u k } ⊂ W 0 1 , n ( Ω ) with ∣ ∣ ∇ u k ∣ ∣ n = 1 such that the supreme of (1.2) is infinite.

Note that the proofs for the concentration-compactness principle in [4,19] depend on the Polya-Szego inequality in the Euclidean space, which is no longer available in the higher-order case or other settings, such as Riemannian manifolds or the Heisenberg groups. In a recent work [17], the authors obtained the concentration-compactness principle on the Heisenberg groups through a rearrangement-free argument by considering the level sets of the functions under consideration. We remark that this argument is inspired by the works [22,23] and can avoid using any rearrangement inequality (see also [27]). For other related work on the concentration-compactness principle associated with Trudinger-Moser-type inequalities, one can refer to [15,18,37] and references therein.

Inspired by the concentration-compactness principle for Trudinger-Moser inequality, Adimurthi and Druet [2] obtained an improved Trudinger-Moser inequality involving the L 2 norm on bounded domains of R 2 by the method of blow-up analysis.

Theorem C

For any 0 < α < λ 1 ( Ω ) , there holds

(1.3) sup u ∈ W 0 1 , 2 ( Ω ) , ∥ ∇ u ∥ 2 ⩽ 1 ∫ Ω e 4 π u 2 ( 1 + α ∥ u ∥ 2 2 ) d x < + ∞ ,

where

(1.4) λ 1 ( Ω ) = inf u ∈ W 0 1 , 2 ( Ω ) ∫ Ω ∣ ∇ u ∣ 2 d x ∫ Ω u 2 d x

denotes the first eigenvalue of the Laplacian operator with the Dirichlet boundary. Furthermore, if α ⩾ λ 1 , the supremum is infinite.

This result was further extended to higher-order case or unbounded domains as well in [5,8,20,25,35,38,38]. We remark that in the work of [5], the authors can extend to (1.3) to the entire Euclidean space and the Heisenberg group by a simple scaling approach, which can avoid applying the complicated blow-up analysis often used in the literature to deal with such sharpened inequalities.

Recently, Wang and Ye [34] proved the following Trudinger-Moser inequality involved in the Hardy term:

(1.5) sup ∫ B ∣ ∇ u ∣ 2 d x − ∫ B u 2 ( 1 − ∣ x ∣ 2 ) 2 d x ≤ 1 ∫ B e 4 π u 2 d x < ∞ ,

where B denotes the unit ball in R 2 . In a recent article [21], Lu and Yang gave a rearrangement-free argument of (1.5) and confirmed that the conjecture for the Hardy-Trudinger-Moser inequality given in [34] indeed holds for any bounded and convex domain in R 2 via the Riemann mapping theorem. We state the result as follows:

Theorem D

Let Ω be a proper and convex bounded domain in R 2 and u ∈ C 0 ∞ ( Ω ) be such that

∫ Ω ∣ ∇ u ∣ 2 d x − 1 4 ∫ Ω u 2 d ( x , ∂ Ω ) 2 d x ≤ 1 ,

where d ( x , ∂ Ω ) = min { ∣ x − x ′ ∣ : x ′ ∈ ∂ Ω } . Then, there exists a constant C, which is independent of u such that

∫ Ω e 4 π u 2 d x ≤ C .

A higher-dimensional version of Theorem D has been recently established by Liang et al. in [24]. Hardy-Adams-type inequality on hyperbolic balls has also been established in [16] (see also references therein).

In the recent work [32], Tintarev modified the Hardy-type Trudinger-Moser inequality (1.5) and proved

(1.6) sup ∫ Ω ∣ ∇ u ∣ 2 d x − ∫ Ω V ( x ) u 2 d x ≤ 1 ∫ Ω e 4 π u 2 d x < ∞

for some class of V ( x ) > 0 , including the case of (1.5). In particular, when V ( x ) = α with 0 ⩽ α < λ 1 , then one has

(1.7) sup ∫ Ω ∣ ∇ u ∣ 2 d x − α ∫ Ω ∣ u ∣ 2 d x ⩽ 1 ∫ Ω e 4 π u 2 d x < + ∞ .

We remark that (1.7) is stronger than (1.3).

In this note, we are concerned with the Trudinger-Moser inequality on the line R . In order to state the related results, we first introduce the concept of fractional Laplacian and the related fractional Sobolev space. For any s ∈ ( 0 , 1 ) and φ ∈ S ( R n ) (the Schwarz space), the fractional Laplacian ( − Δ ) s φ is given by:

( − Δ ) s φ ( x ) = F − 1 ( ∣ ξ ∣ 2 s F φ ( ξ ) ) ( x ) ,

where F and F − 1 denote the Fourier transform and inverse Fourier transform, respectively. Let us consider the space:

L s ( R n ) = u ∈ L loc 1 ( R n ) : ∫ R n ∣ u ∣ 1 + ∣ x ∣ n + 2 s d x < + ∞ ,

fractional operator ( − Δ ) s can be defined on u ∈ L s ( R n ) ∩ S ′ ( R n ) through

⟨ φ , ( − Δ ) s u ⟩ = ⟨ ( − Δ ) s φ , u ⟩ .

We also define the fractional Sobolev space W s , p ( R n ) and W 0 s , p ( Ω ) ( s ∈ ( 0 , + ∞ ) and p ∈ [ 1 , + ∞ ) ) by

W s , p ( R n ) = u ∈ L p ( R n ) : ( − Δ ) s 2 u ∈ L p ( R n ) , W 0 s , p ( Ω ) = { u ∈ W s , p ( R n ) : u = 0 on R n \ Ω } .

Iula et al. [26] established the following fractional Trudinger-Moser inequality on some interval of line.

Theorem E

For any I ⊂ ⊂ R , when α ⩽ π , it holds that

sup u ∈ W 0 1 2 , 2 ( I ) , ∥ u ∥ L 2 ( I ) 2 ⩽ 1 ∫ I ( e α u 2 − 1 ) d x < + ∞ .

This result was further extended to the general fractional Sobolev space W 0 s , p ( Ω ) with s p = n by Martinazzi in [28].

Based on the aforementioned results, a natural problem arises. Whether the fractional analogue for Trudinger-Moser inequality of Tintarev type (1.7) on R still holds. In this article, we deal with this problem. Our main result states the following.

Theorem 1.1

Let I be a bounded interval of R and λ 1 ( I ) be the first eigenvalue of ( − Δ ) 1 4 with Dirichlet boundary. For any 0 ≤ α < λ 1 ( I ) , there holds

(1.8) sup u ∈ W 0 1 2 , 2 ( I ) , ‖ u ‖ 1 2 , α 2 ⩽ 1 ∫ I e π u 2 d x < + ∞ ,

where ∥ u ∥ 1 2 , α 2 = ( − Δ ) 1 4 u 2 2 − α ∥ u ∥ 2 2 .

As the application of Theorem 1.1, one can easily obtain fractional Trudinger-Moser inequality involving the L 2 norm. Indeed, denote v = ( 1 + α ‖ u ‖ L 2 2 ) 1 2 u ; direct computation gives

‖ ( − Δ ) 1 4 v ‖ L 2 2 − α ‖ v ‖ L 2 2 ≤ ( 1 + α ∥ u ∥ L 2 2 ) ( 1 − α ∥ u ∥ L 2 2 ) ≤ 1 .

This together with the fractional Trudinger-Moser inequality (1.8) implies

Corollary 1.2

For any 0 ≤ α < λ 1 ( I ) , there holds

sup u ∈ W 0 1 2 , 2 ( I ) , ( − Δ ) 1 4 u 2 2 ≤ 1 ∫ I e π u 2 ( 1 + α ‖ u ‖ 2 2 ) d x < + ∞ .

Once we establish the fractional Trudinger-Moser inequality (1.8), it remains to ask whether or not there exist extremals for this inequality. The earlier study of extremals for Trudinger-Moser inequality can date back to Carleson and Chang’s work in [3]. They used the rearrangement and ODE technique to obtain the existence of extremals for classical Trudinger-Moser inequality in a unit ball of R n . Later, their results were also extended by Flucher [9] to bounded domain in R 2 and by Lin [11] to bounded domain in R n , respectively. Existence of extremals for Trudinger-Moser inequality on compact manifold has also been established by Li in [12,13]. There are also some existence results of extremals for Trudinger-Moser inequality of Tintarev type (see [36]).

Recently, Mancini and Martinazzi [29] established the existence result for the fractional Trudinger-Moser inequality on an interval in R by the harmonic extensions and commutator estimates. In this article, we will also show that fractional Trudinger-Moser inequality (1.8) of Tintarev type also has extremals. This result reads as:

Theorem 1.3

Under the assumption of Theorem 1.1, (1.8) can be attained by some function u 0 ∈ W 0 1 2 , 2 ( I ) ∩ C 1 ( I ¯ ) , with ∥ u 0 ∥ 1 2 , α 2 = 1 .

Remark 1.4

The proof of Theorem 1.1 will be included in the proof of Theorem 1.3.

It is easily observed that the existence of extremals for fractional Trudinger-Moser inequality of Tintarev type on a bounded interval is equivalent to that in a symmetrical interval. For simplicity, in this context, we may assume I ≔ ( − 1 , 1 ) .

This article is organized as follows. In Section 2, we study the existence of extremals for subcritical fractional Trudinger-Moser inequality of Tintarev type and give the maximizing sequences for (1.8). In Section 3, we analyze the asymptotic behavior of the maximizing sequences near and away from blow-up point when the blow-up phenomenon arises. In Section 4, we derive the Carleson-Chang-type upper bound for (1.8) through capacity estimates. Finally, in Section 5, we prove the existence of extremals for fractional Trudinger-Moser inequality of Tintarev type by constructing an appropriate test function sequence such that the upper bound can be surpassed.

2 Subcritical fractional Trudinger-Moser inequality of Tintarev type and the maximizing sequences

In this section, we establish the existence of extremals for subcritical fractional Trudinger-Moser inequality of Tintarev type and give the maximizing sequences for critical fractional Trudinger-Moser inequality of Tintarev type (1.8).

Denote

(2.1) C ε ≔ sup u ∈ W 0 1 2 , 2 ( I ) , ∥ u ∥ 1 2 , α 2 ⩽ 1 ∫ I e ( π − ε ) u 2 d x .

We will show that there exists u ε ∈ W 0 1 2 , 2 ( I ) such that

(2.2) C ε = ∫ I e ( π − ε ) u ε 2 d x .

For this purpose, we need the following Lions’ type concentration-compactness principle.

Lemma 2.1

Let u ε ∈ W 0 1 2 , 2 ( I ) , with ∥ u ε ∥ 1 2 , α = 1 and u ε ⇀ u 0 ≢ 0 in W 0 1 2 , 2 ( I ) . Then, for any p < 1 1 − ∥ u 0 ∥ 1 2 , α 2 , it holds

(2.3) limsup ε → 0 ∫ I e π p u ε 2 d x < + ∞ .

Proof

By the compactness of the Sobolev embedding theorem, we obtain ∥ u ε ∥ 2 2 → ∥ u 0 ∥ 2 2 when ε → 0 . Since u ≢ 0 and ∥ u ε ∥ 1 2 , α 2 = 1 , we can easily calculate that

lim ε → 0 ( − Δ ) 1 4 ( u ε − u 0 ) 2 2 = 1 − ‖ u 0 ‖ 1 2 , α 2 < 1 .

On the other hand, a direct computation gives that for any p , there holds

∫ I e π p u ε 2 d x ⩽ ∫ I e π p ( 1 + δ ) ( u ε − u 0 ) 2 + π p 1 + 1 δ u 0 2 d x ⩽ ∫ I e π p ( 1 + δ ) ( u ε − u 0 ) 2 ⋅ r d x 1 ∕ r ⋅ ∫ I e π p 1 + 1 δ u 0 2 ⋅ s d x 1 ∕ s ,

where δ > 0 , r > 1 , s > 1 with 1 r + 1 s = 1 . For any p < 1 1 − ∥ u 0 ∥ 1 2 , α 2 , we can choose δ close to 0 and r close to 1 such that the last term in the aforementioned inequality can be controlled by:

∫ I exp π ( u ε − u 0 ) 2 ( − Δ ) 1 4 ( u ε − u 0 ) 2 2 d x 1 ∕ r ⋅ ∫ I e π p 1 + 1 δ u 0 2 ⋅ s d x 1 ∕ s ,

which is finite as a direct consequence of fractional Trudinger-Moser inequality.

Obviously, inequality (2.3) is obtained.□

Lemma 2.2

C ε could be achieved by some function u 0 ∈ W 0 1 2 , 2 ( I ) .

Proof

Let u j ∈ W 0 1 2 , 2 ( I ) be a maximizing sequence for C ε , i.e., ∥ u j ∥ 1 2 , α = 1 and

lim j → ∞ ∫ I e ( π − ε ) u j 2 d x = C ε .

Since 0 ⩽ α < λ 1 ( I ) and ∫ I ( − Δ ) 1 4 u j 2 d x − α ∫ I ∣ u j ∣ 2 d x = 1 , we obtain that { u j } is bounded in W 0 1 2 , 2 ( I ) . Therefore, we obtain

u j ⇀ u ε in W 0 1 2 , 2 ( I ) , u j → u ε in L 2 ( I ) , u j → u ε a.e. in I .

If u ε ≡ 0 , then lim j → ∞ ‖ ( − Δ ) 1 4 u j ‖ 2 2 = 1 . By fractional Trudinger-Moser inequality, for some p > 1 , e ( π − ε ) u j 2 is bounded in L p ( I ) . If u ε ≢ 0 , in view of Lemma 2.1, we can also deduce that e ( π − ε ) u j 2 is bounded in L p ( I ) for some p > 1 . It follows from Vitali convergence theorem that e ( π − ε ) u j 2 → e ( π − ε ) u ε 2 in L 1 ( I ) . This strong convergence combines with the monotonicity of e ( π − ε ) t about t , which implies that

sup u ∈ W 0 1 2 , 2 ( I ) , ∥ u ∥ 1 2 , α 2 ⩽ 1 ∫ I e ( π − ε ) u 2 d x = ∫ I e ( π − ε ) u ˜ ε 2 d x ≥ ∫ I e ( π − ε ) u ε 2 d x ,

where u ˜ ε ≔ u ε ‖ u ε ‖ 1 2 , α 2 = 1 and ∥ u ˜ ε ∥ 1 2 , α = 1 ; then, the proof is accomplished.□

Using Lemma 2.1, we can see that subcritical Trudinger-Moser inequality (2.1) could be achieved by u ε . Obviously, u ε satisfies the following Euler-Lagrange equation:

(2.4) λ ε = ∫ I u ε 2 e ( π − ε ) u ε 2 d x , ( − Δ ) 1 4 u ε 2 2 − α ∥ u ε ∥ 2 2 = 1 , 1 λ ε u ε e ( π − ε ) u ε 2 = ( − Δ ) 1 2 u ε − α u ε in I .

Furthermore, we also have

Lemma 2.3

It holds

(2.5) lim ε → 0 C ε = sup u ∈ W 0 1 2 , 2 ( I ) , ∥ u ∥ 1 2 , α 2 = 1 ∫ I e π u 2 d x .

Proof

It is obvious that

lim ε → 0 C ε ⩽ sup u ∈ W 0 1 2 , 2 ( I ) , ∥ u ∥ 1 2 , α 2 = 1 ∫ I e π u 2 d x .

For any u ∈ W 0 1 2 , 2 ( I ) , with ∥ u ∥ 1 2 , α = 1 , according to Fatou’s lemma, we obtain

∫ I e π u 2 d x ⩽ liminf ε → 0 ∫ I e ( π − ε ) u 2 d x ⩽ liminf ε → 0 C ε .

This implies that

sup u ∈ W 0 1 2 , 2 ( I ) , ∥ u ∥ 1 2 , α 2 = 1 ∫ I e π u 2 d x ⩽ liminf ε → 0 C ε .

Consequently, we obtain (2.5).□

Now, we are in position to pick up u ε as the maximizing sequences for Trudinger-Moser inequality (1.8). Assuming u ε ∈ C ∞ ( I ) ∩ C 0 , 1 2 ( I ¯ ) , which is monotonically decreasing about the origin. Since u ε is bounded in W 0 1 2 , 2 ( I ) , Banach-Alaoglu theorem and fractional compact Sobolev embedding theorem directly give that

u ε ⇀ u in W 0 1 2 , 2 ( I ) u ε → u in L 2 ( I ) u ε → u a.e. in I .

Let

c ε = u ε ( 0 ) = max I u ε .

If c ε is bounded, then e ( π − ε ) u ε 2 is also bounded, which implies that e ( π − ε ) u ε 2 converges to e π ∣ u ∣ 2 in L 1 ( I ) . Hence, for any v ∈ W 0 1 2 , 2 ( I ) ∩ C 1 ( I ¯ ) with ∥ v ∥ 1 2 , α ⩽ 1 , we have

lim ε → 0 ∫ I e ( π − ε ) u ε 2 d x = ∫ I e π ∣ u ∣ 2 d x .

This combines with Lemma 2.1; we can deduce that u is the desired extremal function.

If c ε is unbounded, without loss of generality, we claim that the weak limit of u ε is equal to zero. In fact, if u 0 ≢ 0 , by concentration-compactness principle Lemma 2.1, we know that e π u ε 2 is bounded in L p ( I ) for any p < 1 1 − ∥ u 0 ∥ 1 2 , α 2 . This together with elliptic estimates gives that u ε is bounded in I , which contradicts c ε → ∞ .

We also further claim ( − Δ ) 1 4 u ε 2 d x ⇀ δ 0 in the sense of measure. Otherwise, we can find some r 0 > 0 , B r 0 ( 0 ) ⊂ I , and a cut-off function φ ∈ C 0 ∞ ( B r 0 ( 0 ) ) , with 0 ⩽ φ ⩽ 1 , and φ = 1 on B r 0 2 ( 0 ) . We have

limsup ε → 0 ∫ B r 0 ( 0 ) ( − Δ ) 1 4 ( φ u ε ) 2 d x < 1 .

By fractional Trudinger-Moser inequality, e ( π − ε ) ( φ u ε ) 2 is bounded in L p ( I ) for some p > 1 . Applying elliptic estimates, we obtain that u ε is uniformly bounded on B r 0 2 ( 0 ) , which is a contradiction.

3 Asymptotic behavior of the maximizing sequences u ε

In this section, we will study asymptotic behavior of the maximizing sequences u ε near and far away from the blow-up point.

Set

r ε = λ ε ( π − ε ) c ε 2 e ( π − ε ) c ε 2 .

Define I ε = { x ∈ R : r ε x ∈ I } and η ε ( x ) ≔ 2 ( π − ε ) c ε ( u ε ( r ε x ) − c ε ) . If r ε → 0 , similar to the proof of Theorem 1.3 in [29], we obtain η ε ( x ) → η 0 ( x ) = ln 1 1 + ∣ x ∣ 2 and

(3.1) ∫ R e η 0 ( x ) d x = ∫ R 1 1 + ∣ x ∣ 2 d x = π .

This describes the asymptotic behavior of u ε around origin. Now, we start to analyze the asymptotic behavior of u ε away from the blow-up point 0. For this purpose, we need the following lemma.

Lemma 3.1

For any A > 1 , there holds

(3.2) limsup ε → 0 ‖ ( − Δ ) 1 4 u ε A ‖ 2 2 ≤ 1 A ,

where u ε A ≔ min u ε , c ε A .

Proof

As mentioned earlier, we have

1 + α ∥ u ε ∥ 2 2 = ( − Δ ) 1 4 u ε 2 2 = ∥ ∇ u ˜ ε ∥ 2 2

and

‖ ( − Δ ) 1 4 u ε ‖ 2 2 = ∫ I u ε ( − Δ ) 1 2 u ε d x = ∫ I u ε u ε λ ε e ( π − ε ) u ε 2 + α u ε d x .

Set u ¯ ε A ≔ min u ˜ ε , c ε A , where u ˜ ε is the harmonic extension of u ε to R + 2 = R × ( 0 , + ∞ ) given by the Poisson integral:

u ˜ ( x , y ) ≔ 1 π ∫ R y u ( ξ ) y 2 + ( x − ξ ) 2 d ξ , y > 0 .

Obviously, u ¯ ε A is a general extension of u ε A on R + 2 ; by Dirichlet energy principle, we have

( − Δ ) 1 4 u ε A 2 2 ⩽ ∫ R + 2 ∣ ∇ u ¯ ε A ∣ 2 d x d y .

Using integration by parts and the harmonicity of u ˜ ε , we obtain

∫ R + 2 ∣ ∇ u ¯ ε A ∣ 2 d x d y = ∫ R + 2 ∇ u ¯ ε A ∇ u ˜ ε d x d y = − ∫ R u ε A ( x ) ∂ u ˜ ε ( x , 0 ) ∂ y d x = ∫ R u ε A ( − Δ ) 1 2 u ε d x .

Denote

v ε A ≔ u ε − c ε A + = u ε − u ε A ,

as ε → 0 , we have

lim ε → 0 ∫ R v ε A ( − Δ ) 1 2 u ε d x = lim ε → 0 ∫ R v ε A u ε λ ε e ( π − ε ) u ε 2 + α u ε d x ⩾ lim R → ∞ lim ε → 0 1 − 1 A ∫ − R r ε R r ε u ε c ε λ ε e ( π − ε ) u ε 2 ( x ) d x + lim ε → 0 ∫ R α u ε ⋅ v ε A d x = lim R → ∞ lim ε → 0 1 − 1 A ∫ − R R u ε c ε λ ε r ε e ( π − ε ) u ε 2 ( r ε ⋅ ) d x = lim R → ∞ lim ε → 0 1 − 1 A ∫ − R R u ε e ( π − ε ) u ε 2 ( r ε ⋅ ) ( π − ε ) c ε e π c ε 2 d x = 1 − 1 A ,

where we use the fact: u ε → 0 in L 2 ( I ) in the last equality.

Since ∥ u ε ∥ 1 2 , α = 1 , we obtain that

lim ε → 0 ∫ R ( v ε A ( − Δ ) 1 2 u ε + u ε A ( − Δ ) 1 2 u ε ) d x = lim ε → 0 ∫ R u ε ( − Δ ) 1 2 u ε d x = 1 + lim ε → 0 α ∥ u ε ∥ 2 2 = 1 .

Combining the aforementioned estimates, we deduce that

∫ R u ε A ( − Δ ) 1 2 u ε d x ⩽ 1 A .

Then, we conclude (3.2).□

Lemma 3.2

We have

(3.3) lim ε → 0 ∫ I e ( π − ε ) u ε 2 d x = limsup ε → 0 λ ε c ε 2 + ∣ I ∣ .

Moreover,

(3.4) lim ε → 0 c ε λ ε = 0 .

Proof

We split

∫ I e ( π − ε ) u ε 2 d x = ∫ I ∩ u ε ⩽ c ε A e ( π − ε ) u ε 2 d x + ∫ I ∩ u ε > c ε A e ( π − ε ) u ε 2 d x ≕ I 1 + I 2 .

As the consequence of Lemma 3.1, we have ( π − ε ) ∥ ∇ u ε A ∥ 2 2 < π as ε → 0 . This together with fractional Trudinger-Moser inequality, for some p > 1 , we can deduce that

∫ I exp ( π − ε ) ∥ ∇ u ε A ∥ 2 2 ⋅ p ( u ε A ) 2 ∥ ∇ u ε A ∥ 2 2 d x ≤ C .

With the help of Vitali convergence theorem, we obtain

(3.5) lim ε → 0 I 1 = ∫ I e π u 0 2 d x = ∣ I ∣ .

A similar calculation together with Lemma 3.1 leads to

(3.6) lim ε → 0 I 2 ⩽ lim ε → 0 λ ε A 2 c ε 2 ∫ I ∩ u ε > c ε A u ε 2 λ ε e ( π − ε ) u ε 2 d x = lim ε → 0 λ ε A 2 c ε 2 .

Let A approach to 1 from the aforementioned equation, we have that

lim ε → 0 I 2 ⩽ lim ε → 0 λ ε c ε 2 ,

together with the estimate of I 1 , yields that lim ε → 0 ∫ I e ( π − ε ) u ε 2 d x ⩽ ∣ I ∣ + lim ε → 0 λ ε c ε 2 . To finish the proof of Lemma 3.2, it remains to prove that:

lim ε → 0 ∫ I e ( π − ε ) u ε 2 d x ≥ limsup ε → 0 λ ε c ε 2 + ∣ I ∣ .

Indeed, this is a consequence of the following calculation as ε → 0 and L → ∞ ,

lim L → ∞ lim ε → 0 ∫ I e ( π − ε ) u ε 2 d x = lim L → ∞ lim ε → 0 ∫ B L r ε e ( π − ε ) u ε 2 d x + ∫ I \ B L r ε e ( π − ε ) u ε 2 d x ≥ lim L → ∞ lim ε → 0 ∫ B L r ε e ( π − ε ) u ε 2 d x + ∣ I \ B L r ε ∣ = lim L → ∞ lim ε → 0 λ ε c ε 2 + ∣ I ∣ .

Therefore, we conclude that

lim ε → 0 ∫ I e ( π − ε ) u ε 2 d x = limsup ε → 0 λ ε c ε 2 + ∣ I ∣ ,

together with lim ε → 0 C ε > ∣ I ∣ , obtain lim ε → 0 c ε λ ε = 0 . Then, we finish the proof of Lemma 3.2.□

Lemma 3.3

Set f ε ≔ c ε λ ε u ε e ( π − ε ) u ε 2 , then for any ϕ ∈ C ( I ¯ ) ,

lim ε → 0 ∫ I f ε ϕ d x = ϕ ( 0 ) .

Proof

Divide

I = u ε > c ε A \ ( − R r ε , R r ε ) ∪ ( − R r ε , R r ε ) ∪ u ε ⩽ c ε A .

Using the change of variables, we have

lim R → ∞ lim ε → 0 ∫ − R r ε R r ε u ε 2 λ ε e ( π − ε ) u ε 2 d x = lim R → ∞ lim ε → 0 ∫ − R R u ε 2 ( r ε ⋅ ) λ ε r ε e ( π − ε ) u ε 2 ( r ε ⋅ ) d x = lim R → ∞ lim ε → 0 1 π − ε ∫ − R R u ε 2 ( r ε ⋅ ) c ε 2 e ( π − ε ) [ u ε 2 ( r ε ⋅ ) − c ε 2 ] d x = 1 π lim R → ∞ ∫ − R R e η 0 d x = 1 ,

then

lim R → ∞ lim ε → 0 ∫ u ε > c ε A \ ( − R r ε , R r ε ) f ε ϕ d x ⩽ lim R → ∞ lim ε → 0 A ∥ ϕ ∥ L ∞ ( I ) ∫ u ε > c ε A \ ( − R r ε , R r ε ) u ε 2 λ ε e ( π − ε ) u ε 2 d x = lim R → ∞ lim ε → 0 A ∥ ϕ ∥ L ∞ ( I ) ∫ I u ε 2 λ ε e ( π − ε ) u ε 2 d x − ∫ − R r ε R r ε u ε 2 λ ε e ( π − ε ) u ε 2 d x = A ∥ ϕ ∥ L ∞ ( I ) 1 − lim R → ∞ lim ε → 0 ∫ − R r ε R r ε u ε 2 λ ε e ( π − ε ) u ε 2 d x = 0 .

Similarly,

lim R → ∞ lim ε → 0 ∫ − R r ε R r ε f ε ϕ d x = lim R → ∞ lim ε → 0 ∫ − R R u ε c ε λ ε ϕ ( r ε x ) r ε e ( π − ε ) u ε 2 ( r ε x ) d x = ϕ ( 0 ) π lim R → ∞ ∫ − R R e η 0 d x = ϕ ( 0 ) .

By Vitali convergence theorem and Lemma 3.2,

lim ε → 0 ∫ u ε ⩽ c ε A f ε ϕ d x ≤ lim ε → 0 c ε λ ε ∥ ϕ ∥ L ∞ ( I ) ∫ I u ε A e ( π − ε ) ( u ε A ) 2 d x ⩽ lim ε → 0 c ε λ ε ∥ ϕ ∥ L ∞ ( I ) ∫ I e A ( π − ε ) ( u ε A ) 2 d x 1 A ∫ I ( u ε A ) A ′ d x 1 A ′ = 0 ,

where 1 A + 1 A ′ = 1 .

Finally, we deduce ∫ I f ε ϕ d x → ϕ ( 0 ) as ε → 0 .□

Next, we are in position to show that the asymptotic behavior of u ε away from origin; we claim that the following lemmas hold.

Lemma 3.4

As ε → 0 , we have v ε ≔ c ε u ε − G 0 → 0 in L loc ∞ ( I ¯ \ { 0 } ) ∩ L 1 ( I ) , where G 0 ( y ) = G x ( y ) ∣ x = 0 and G x ( y ) = − 1 π ln ∣ x − y ∣ + A + g x ( y ) is the Green’s function of ( − Δ ) 1 2 on I with singularity at x, which satisfies the following equations:

(3.7) ( − Δ ) 1 2 G x ( y ) − α G x ( y ) = δ x , x ∈ I G x ( y ) = 0 , x ∈ R \ I .

Proof

By equation (2.5), c ε u ε satisfies ( − Δ ) 1 2 − α ( c ε u ε ) = f ε . Arguing as what we did in Lemma 3.3, we obtain ∥ f ε ∥ L 1 ( R ) → 1 as ε → 0 . On the other hand, for any K ⊂ ⊂ I \ { 0 } , by Lemma 3.1 and fractional Trudinger-Moser inequality, we derive that u ε λ ε e ( π − ε ) u ε 2 ∈ L p ( K ) for some p > 1 .

According to elliptic regularity theorem for fractional equation, we have u ε → u 0 = 0 in L ∞ ( K ) . Then, it follows from the definition of f ε and (3.4) of Lemma 3.2 that f ε → 0 in L ∞ ( K ) as ε → 0 . Using Green’s representation formula, we deduce that

∣ v ε ( x ) ∣ = ∫ I G x ( y ) f ε ( y ) d y − G 0 ( y ) ⩽ ∫ I ∣ G x ( y ) − G 0 ( y ) ∣ f ε ( y ) d y + ∣ ∥ f ε ∥ L 1 ( I ) − 1 ∣ ∣ G 0 ( y ) ∣

where the expression of G _x(y) can refer to [29].

Given x ,

∣ G x ( y ) − G 0 ( y ) ∣ = 1 π ln ∣ y ∣ ∣ x − y ∣ + g x ( y ) − g 0 ( y ) ⩽ 1 π ln ∣ y ∣ ∣ x − y ∣ + ∣ g x ( y ) − g 0 ( y ) ∣ ⩽ 1 π ln x ∣ y ∣ − y ∣ y ∣ + C ( δ ) ∣ y ∣ β ⩽ 1 π ∣ y ∣ ∣ x ∣ + C ( δ ) ∣ y ∣ β ,

where g x ( y ) ∈ C β for some β ∈ ( 0 , 1 ) . Then, for any ε ∈ 0 , σ 2 and ∣ x ∣ ⩾ σ , we can write

lim ε → 0 ∣ v ε ( x ) ∣ ≤ lim ε → 0 ∫ B δ ∣ x ∣ ∣ G x ( y ) − G 0 ( y ) ∣ f ε ( y ) d y + ∫ I \ B δ ∣ x ∣ ∣ G x ( y ) − G 0 ( y ) ∣ f ε ( y ) d y + o ε ( 1 ) ≤ lim ε → 0 δ π + δ β ∣ x ∣ β ∥ f ε ( y ) ∥ L 1 ( B δ ∣ x ∣ ) + 1 π ∣ x ∣ + C ( δ ) ∥ f ε ( y ) ∥ L 1 ( I \ B δ ∣ x ∣ ) ≤ lim ε → 0 δ π + δ β ∣ x ∣ β + 1 π σ + C ( δ ) ∥ f ε ( y ) ∥ L 1 ( I \ B δ ∣ x ∣ ) .

Since δ can be arbitrarily small, we derive that lim ε → 0 sup x ∈ K ∣ v ε ( x ) ∣ = 0 . Now, we will show that lim ε → 0 ‖ v ε ( y ) ‖ L 1 ( I ) = 0 . In fact, we just need to prove that lim δ → 0 lim ε → 0 ‖ v ε ( y ) ‖ L 1 ( B δ ) = 0 since we have deduced that lim ε → 0 ∣ v ε ( x ) ∣ = 0 in L loc ∞ ( I \ { 0 } ) . By Green’s representation formula, we have

lim δ → 0 lim ε → 0 ∫ B δ ∣ v ε ( x ) ∣ d x ≤ lim δ → 0 lim ε → 0 ∫ B δ ∫ I ∣ G x ( y ) − G 0 ( y ) ∣ f ε ( y ) d y d x = lim δ → 0 lim ε → 0 ∫ B δ f ε ( y ) ∫ B δ ∣ G x ( y ) − G ( x ) ∣ d x d y ≤ lim δ → 0 C ( δ ) = 0 .□

Since u ˜ ε is the harmonic extension for u ε , with the help of Lemma 3.4, we can obtain the asymptotic estimate u ˜ ε in R + 2 ¯ \ { ( 0 , 0 ) } .

Lemma 3.5

c ε u ˜ ε → G ˜ in C loc 0 ( R ¯ + 2 \ { ( 0 , 0 ) } ) ∩ C loc 1 ( R ¯ + 2 ) , where

(3.8) G ˜ ( x , y ) = − 1 π ln ∣ x 2 + y 2 ∣ + A + h ( x , y ) , h ( 0 , 0 ) = 0 .

Proof

Denote v ˜ ε ≔ c ε u ˜ ε − G ˜ ; obviously, v ˜ ε is the harmonic extension for v ε . Hence, for any K ⊂ ⊂ R ¯ + 2 \ { ( 0 , 0 ) } and δ < d i s t ( K , ( 0 , 0 ) ) 2 ≔ d , we have

sup ( x , y ) ∈ K v ˜ ε ( x , y ) ≤ 1 π ∫ − δ δ y v ε ( ξ ) ( x − ξ ) 2 + y 2 d ξ + 1 π ∫ I \ ( − δ , δ ) y v ε ( ξ ) ( x − ξ ) 2 + y 2 d ξ ≔ I 1 + I 2 .

For I 1 , applying v ε → 0 in L 1 ( I ) from Lemma 3.4, we obtain

lim ε → 0 ∣ I 1 ∣ ≤ lim ε → 0 1 π ∫ B δ y ∣ v ε ( ξ ) ∣ ∣ ( x − ξ ) 2 + y 2 ∣ d ξ ≤ lim ε → 0 4 diam ( K ) π d 2 ∫ B δ ∣ v ε ( ξ ) ∣ d ξ = 0 .

For I 2 , since v ε → 0 in L loc ∞ ( I ¯ \ { 0 } ) , we derive that

lim ε → 0 ∣ I 2 ∣ ⩽ 1 π lim ε → 0 ∥ v ε ∥ L ∞ ( I \ ( − δ , δ ) ) ∫ R y ( x − ξ ) 2 + y 2 d ξ = lim ε → 0 ∥ v ε ∥ L ∞ ( I \ ( − δ , δ ) ) = 0 .

Combining the estimates of I 1 and I 2 , we accomplish the proof of Lemma 3.5.□

4 Upper-bound estimate for C π when the concentration-compactness phenomenon arises

In this part, we try to eliminate the concentration-compactness phenomenon of u ε . We will carry out the capacity estimates to derive an upper bound for inequality (1.8).

Lemma 4.1

If c ε → ∞ , there holds

sup u ∈ W 0 1 2 , 2 ( I ) , ‖ u ‖ 1 2 , α 2 ⩽ 1 ∫ I e π u 2 d x ⩽ ∣ I ∣ + 2 π e π A .

Proof

Given a large enough L > 0 and a small enough δ > 0 , set

a ε ≔ inf B L r ε ∩ R + 2 u ˜ ε , b ε ≔ sup ∂ B δ ∩ R + 2 u ˜ ε , υ ε ≔ ( u ˜ ε ∧ a ε ) ∨ b ε .

Due to ∥ ∇ u ˜ ε ∥ 2 2 − α ∥ u ε ∥ 2 2 = 1 , we have

(4.1) ∫ ( B δ \ B L r ε ) ∩ R + 2 ∣ ∇ u ˜ ε ∣ 2 d x d y = 1 + α ∥ u ε ∥ 2 2 − ∫ R + 2 \ B δ + ∫ R + 2 ∩ B L r ε ∣ ∇ u ˜ ε ∣ 2 d x d y .

The left-hand side is not less than

inf u ˜ ∣ R + 2 ∩ B L r ε = a ε u ˜ ∣ R + 2 ∩ ∂ B δ = b ε ∫ ( B δ \ B L r ε ) ∩ R + 2 ∣ ∇ u ˜ ∣ 2 d x d y = ∫ ( B δ \ B L r ε ) ∩ R + 2 ∣ ∇ Φ ˜ ε ∣ 2 d x d y = π ( a ε − b ε ) 2 ln δ − ln ( L r ε ) ,

where

Φ ˜ ε = b ε − a ε ln δ − ln ( L r ε ) ln ∣ x 2 + y 2 ∣ 2 + a ε ln δ − b ε ln ( L r ε ) ln δ − ln ( L r ε )

is the solution to set of equations

Δ Φ ˜ ε = 0 in R + 2 ∩ B δ \ B L r ε Φ ˜ ε = a ε in R + 2 ∩ B L r ε Φ ˜ ε = b ε in R + 2 ∩ ∂ B δ ∂ Φ ˜ ε ∂ y = 0 in ∂ R + 2 ∩ ( B δ \ B L r ε ) .

From Proposition 2.2 in [29],

a ε = c ε + − 1 π ln L + O ( L − 1 ) + o ( 1 ) c ε .

By Lemma 3.5 and the denotation of G ˜ ( x , y ) , we obtain

b ε = − 1 π ln δ + A + O ( δ ) + o ( 1 ) c ε .

Next, we start to calculate the right hand of equality (4.1); applying η ε ( x ) → η 0 ( x ) = ln 1 1 + ∣ x ∣ 2 , we have

lim ε → 0 c ε 2 ∫ R + 2 ∩ B L r ε ∣ ∇ u ˜ ε ∣ 2 d x d y = 1 π ln L 2 + O ln L L .

For the integral ∫ R + 2 \ B δ ∣ ∇ u ˜ ε ∣ 2 d x d y , we can write

liminf ε → 0 ∫ R + 2 \ B δ ∣ ∇ u ˜ ε ∣ 2 d x d y ⩾ 1 c ε 2 ∫ R + 2 \ B δ ∣ ∇ G ˜ ∣ 2 d x d y = 1 c ε 2 ∫ R + 2 \ ∂ B δ − ∂ G ˜ ∂ n G ˜ d σ + ∫ ( R × 0 ) \ B δ − ∂ G ˜ ( x , y ) ∂ y y = 0 G ( x ) d x d y = 1 c ε 2 ∫ R + 2 \ ∂ B δ 1 π δ + O ( 1 ) − ln δ π + A + O ( δ ) d σ + α ‖ G ( x ) ‖ 2 2 = − ln δ π + A + O ( δ ln δ ) + α c ε 2 ‖ G ( x ) ‖ 2 2 .

Gathering the aforementioned estimates, we conclude that

π ( a ε − b ε ) 2 ln δ − ln ( L r ε ) ⩽ 1 + α ‖ u ε ‖ 2 2 − − ln δ π + A + O ( δ ln δ ) + α ∥ G ( x ) ∥ 2 2 + 1 π ln L 2 + O ln L L c ε 2 .

Taking the estimates of a ε and b ε into the last equality, we obtain

π ( a ε − b ε ) 2 = π c ε 2 − 2 ln L + O ( L − 1 ) + 2 ln δ − 2 π A + O ( δ ) + o ( 1 ) + O ( ln 2 δ + ln 2 L ) c ε 2 ⩽ ln δ L + ln c ε 2 λ ε + ln ( π − ε ) + ( π − ε ) c ε 2 × 1 − − ln δ π + A + O ( δ ln δ ) + 1 π ln L 2 + O ln L L c ε 2 = ln δ − ln L + ln c ε 2 λ ε + ln ( π − ε ) + ( π − ε ) c ε 2 + ( π − ε ) ln δ π − A − 1 π ln L 2 + O ( δ ln δ ) + O ln L L + O ( ln 2 δ ) + O ( ln 2 L ) + O ( 1 ) c ε 2 ,

which implies that

ln λ ε c ε 2 ⩽ π A + ln 2 π + O ( δ ln δ ) + O ln L L + o ε ( 1 ) .

Then, letting ε → 0 and L → ∞ , we obtain

limsup ε → 0 ln λ ε c ε 2 ⩽ π A + ln 2 π ,

together with Lemma 3.2 ends the proof of Lemma 4.1.□

5 Existence of the extremals

In this section, we construct a test function whose energy can exceed the upper bound ∣ I ∣ + 2 π e π A to show the existence of extremals for the fractional Trudinger-Moser inequality of Tintarev type (1.8).

Lemma 5.1

There exists a sequence of function ϕ ε ∈ W 0 1 2 , 2 ( I ) with ∥ ϕ ε ∥ 1 2 , α 2 ≤ 1 such that

(5.1) ∫ I e π ϕ ε 2 d x > ∣ I ∣ + 2 π e π A ,

when ε is small enough.

Proof

Define

Γ L ε ≔ { ( x , y ) ∈ R + 2 : G ˜ ( x , y ) = γ L ε ≔ min R + 2 ∩ ∂ B L ε G ˜ } ,

and I L ε ≔ { ( x , y ) ∈ R + 2 : G ˜ ( x , y ) > γ L ε } . The accurate formula of G ˜ gives

(5.2) γ L ε = − ln L ε π + A + O ( L ε ) .

Let

Ψ ε ( x , y ) = c − ln 1 + y ε 2 + x 2 ε 2 + B 2 π c , ( x , y ) ∈ R + 2 ∩ B L ε ( 0 , − ε ) γ L ε c , ( x , y ) ∈ I L ε \ B L ε ( 0 , − ε ) G ˜ ( x , y ) c , ( x , y ) ∈ R + 2 \ I L ε ,

where c , B , and L depend only on ε and will be determined later. The choice of B is to make sure the continuity on R + 2 ∩ ∂ B L ε ( 0 , − ε ) , so

c − ln L 2 + B 2 π c = γ L ε c .

In view of (5.2), one can deduce that

(5.3) B = 2 π c 2 + 2 ln ε − 2 π A + O ( L ε ) .

Choosing suitable constant c such that ∇ Ψ ε 2 2 − α ∥ ϕ ε ∥ 2 2 = 1 , where ϕ ε ( x ) ≔ Ψ ε ( x , 0 ) . Direct calculation leads to

∫ R + 2 ∩ B L ε ( 0 , − ε ) ∣ ∇ Ψ ε ∣ 2 d x d y = 1 ( 2 π c ) 2 ∫ R + 2 ∩ B L ε ( 0 , − ε ) ∇ ln 1 + y ε 2 + x 2 ε 2 2 d x d y = 1 ( 2 π c ) 2 ∫ R + 2 ∩ B L ( 0 , − 1 ) ∣ ∇ ln ( ( 1 + y ) 2 + x 2 ) ∣ 2 d x d y = 1 c 2 1 π ln L 2 + O ln L L ,

∫ R + 2 ∩ I L ε ∣ ∇ Ψ ε ∣ 2 d x d y = 1 c 2 ∫ R + 2 ∩ I L ε ∣ ∇ G ˜ ∣ 2 d x d y = 1 c 2 ∫ R + 2 ∩ ∂ I L ε ∂ G ˜ ∂ n G ˜ d σ + 1 c 2 ∫ ( R × { 0 } ) \ I ¯ L ε − ∂ G ˜ ( x , y ) ∂ y y = 0 G ( x ) d x = 1 c 2 − ln L ε π + A + O ( L ε ln L ε ) + α ∥ G ( x ) ∥ 2 2

and

∫ I ∣ ϕ ε ∣ 2 d x = ∫ B ε L 2 − 1 c − ln 1 + x 2 ε 2 + B 2 π c 2 d x + ∫ I \ B ε L 2 − 1 G 2 c 2 d x = ∫ B ε L 2 − 1 c − ln 1 + x 2 ε 2 + 2 π c 2 + 2 ln ε − 2 π A + O ( L ε ) 2 π c 2 d x + ∫ I \ B ε L 2 − 1 G 2 c 2 d x = ∫ B ε L 2 − 1 2 π G + O ( L ε ln L ε ) 2 π c 2 d x + ∫ I \ B ε L 2 − 1 G 2 c 2 d x = 1 c 2 ∫ I G 2 d x + O ( L ε ln L ε ) .

Then, it holds

(5.4) − ln 2 ε + π A + O ( L ε ln L ε ) + O ln L L = π c 2

and c → + ∞ as ε → 0 . Combining this with (5.3), we derive that

(5.5) B = − 2 ln 2 + O ( L ε ln L ε ) + O ln L L .

Using the definition of ϕ ε , together with (5.4) and (5.5), yields that

∫ B ε L 2 − 1 e π ϕ ε 2 d x = ε ∫ B L 2 − 1 exp π ( c − ln ( 1 + x 2 ) + B 2 π c ) 2 d x > ε ∫ B L 2 − 1 e π c 2 − B ⋅ 1 1 + x 2 d x = 2 e π A + O ( L ε ln L ε ) + O ln L L π ( 1 + O ( L − 1 ) ) = 2 π e π A + O ( L ε ln L ε ) + O ln L L

and

∫ I \ B ε L 2 − 1 e π ϕ ε 2 d x ⩾ ∫ I \ B ε L 2 − 1 ( 1 + π ϕ ε 2 ) d x = ∣ I \ B ε L 2 − 1 ∣ + 1 c 2 ∫ I \ B ε L 2 − 1 π G 2 d x > ∣ I ∣ − 2 ε L + 1 c 2 ∫ I \ B ε L 2 − 1 π G 2 d x ≕ ∣ I ∣ − 2 ε L + ν L ε c 2 ,

with ν L ε > ν 1 2 > 0 for L ε < 1 2 . We can see that π c 2 = − ln ε + O ( 1 ) by equality (5.4), and when we choose L = ln 2 ε ,

O ( L ε ln L ε ) + O ln L L − 2 ε L = O ln ln ε ln 2 ε = o 1 c 2 ,

which can conclude that

∫ I e π ϕ ε 2 d x > ∣ I ∣ + 2 π e π A + ν 1 2 c 2 + o 1 c 2 > ∣ I ∣ + 2 π e π A .

Dirichlet energy principle gives ‖ ∇ ϕ ˜ ε ‖ 2 2 ≤ ‖ ∇ Ψ ε ‖ 2 2 , that is to say,

‖ ϕ ε ‖ 1 2 , α 2 ≤ ‖ ∇ Ψ ε ‖ 2 2 − α ‖ ϕ ε ‖ 2 2 = 1 .

Hence, the proof of Lemma 5.1 is accomplished when ε is small enough.□

Funding information: Lu Chen was partly supported by the National Natural Science Foundation of China (No. 12271027) and a grant from Beijing Institute of Technology (No. 2022CX01002). Maochun Zhu was supported by Natural Science Foundation of China (No. 12071185).
Conflict of interest: The authors state that there is no conflict of interest.

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Received: 2022-08-31

Revised: 2023-04-08

Accepted: 2023-04-16

Published Online: 2023-05-29

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fractional; extremals; Trudinger-Moser inequalities

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