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Concentration-compactness principle associated with Adams' inequality in Lorentz-Sobolev space

  • Dongliang Li and Maochun Zhu EMAIL logo
Published/Copyright: December 31, 2022
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Advanced Nonlinear Studies
From the journal Advanced Nonlinear Studies Volume 22 Issue 1

Abstract

The concentration-compactness principle of Lions type in Euclidean space relies on the Pólya-Szegö inequality, which is not available in non-Euclidean settings. The first proof of the concentration-compactness principle in non-Euclidean setting, such as the Heisenberg group, was given by Li et al. by using a symmetrization-free nonsmooth truncation argument. In this article, we study the concentration-compactness principle of second-order Adams’ inequality in Lorentz-Sobolev space W 0 2 L 2 , q ( Ω ) for all 1 < q < ∞ . Due to the absence of the Pólya-Szegö inequality with respect to the second-order derivatives, we will use a symmetrization-free argument to study the concentration-compactness principle of second-order Adams’ inequality in Lorentz-Sobolev space. Furthermore, we show the sharpness of result by constructing a test function sequence. Our result is even new in the first-order case when q > 2 .

Keywords: concentration-compactness principle; Adams’ inequality; Lorentz-Sobolev
MSC 2010: 46E30; 46E35

1 Introduction

Let Ω ⊂ R N be a bounded domain and W 0 1 , p ( Ω ) be the usual Sobolev space denoted by the completion of C 0 ∞ ( Ω ) with the norm

‖ u ‖ W 0 1 , p ( Ω ) = ∫ Ω ( ∣ ∇ u ∣ p + ∣ u ∣ p ) d x 1 p ,

then the well-known Sobolev embedding reads as follows:

W 0 1 , p ( Ω ) ⊂ L N p N − p ( Ω ) , if 1 ≤ p < N ,

W 0 1 , p ( Ω ) ⊂ L ∞ ( Ω ) , if p > N ,

while, in the borderline case p = N , we have W 0 1 , N ( Ω ) ⊂ L q ( Ω ) , for every 1 ≤ q < ∞ . However, some examples show that W 0 1 , N ( Ω ) ⊄ L ∞ ( Ω ) .

This borderline case of the optimal Sobolev embedding is known as the Trudinger inequality [19], which was established independently by Yudovič [20], Pohožaev [18], and Trudinger [19]. In 1971, Trudinger’s inequality was sharpened by Moser in [16] by proving

(1.1) sup u ∈ W 0 1 , N ( Ω ) \ { 0 } ∥ ∇ u ∥ L N ≤ 1 ∫ Ω exp α ∣ u ∣ N N − 1 d x < + ∞ , if α ≤ α N , = + ∞ , if α > α N ,

where α N = N ω N − 1 1 N − 1 , ω N − 1 is the measure of the unit sphere in R N , which is now called the Trudinger-Moser inequality.

In 1985, Lions [12] established the concentration-compactness principle associated with the Trudinger-Moser inequality, which tells us that if a sequence { u k } ⊂ W 0 1 , p ( Ω ) satisfies ‖ ∇ u k ‖ L N = 1 and u k ⇀ u 0 , then α N can be replaced by α N p , where

p < p u = ( 1 − ∥ ∇ u ∥ L N N ) − 1 ∕ ( N − 1 ) ,

that is, if p < p u , one has

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

Indeed, the constant p u is sharp in the sense that the supremum above will become infinite when p ≥ p u (see [5]). Concentration-compactness principle plays an important role in proving the existence and the multiplicity of solutions to PDEs with critical exponential growth (see [6,13]).

In 1988, Adams [1] extended the Trudinger-Moser inequality (1.1) to the higher-order Sobolev space W 0 m , N m ( Ω ) ( 1 < m ∈ N ) and obtained

sup u ∈ W 0 m , N m ( Ω ) ⧹ { 0 } Δ m 2 u L N m ≤ 1 1 ∣ Ω ∣ ∫ Ω e ( β N , m ∣ u ∣ ) N N − m d x ≤ C ( N , m ) ,

where

Δ m 2 u = Δ l u , if m = 2 l ∇ Δ l u , if m = 2 l + 1 ( l ∈ N )

and

(1.3) β N , m = π N 2 2 m Γ m + 1 2 ω N N − m N Γ N − m + 1 2 , if m = 2 h + 1 π N 2 2 m Γ m 2 ω N N − m N Γ N − m 2 , if m = 2 h ( h ∈ N ) .

Moreover, the constant β N , m is sharp in the sense that if we replace it by any β > β N , m , the supremum above will become infinite.

Note that the proofs for the concentration-compactness principle in [5,12] depend on the Pólya-Szegö inequality in Euclidean space, which is no longer available in the higher-order case or other settings, such as Riemannian manifolds or Heisenberg groups. In a recent work [13], the authors developed a proof for the concentration-compactness principle on the Heisenberg groups without using any rearrangement inequality. The method used in [13] is a non-smooth truncation argument using the level sets of the functions under consideration. This method is inspired by the earlier works of Lam et al. [9–11]. As pointed out in [13], this same method also applies to prove the concentration-compactness of the Trudinger-Moser inequalities on Riemannian manifolds (see [14] for details). We note that Hang in [8] established the concentration-compactness principle on compact Riemannian manifolds using a smooth version of the truncation argument with respect to the higher-order derivatives.

In this work, we are concerned with the related results in the Lorentz-Sobolev space (see (2.3) for definition). The Trudinger-Moser-type inequality for Lorentz-Sobolev space was established by Alvino et al. in [3] and by Lu and Tang in [15]. Later, Alberico [2] extended the result of Angelo et al. to the high-order Lorentz-Sobolev space W 0 m L N m , q ( Ω ) . Their results can be stated as follows: let m be a positive integer satisfying 1 ≤ m < N and 1 < q < + ∞ , then

(1.4) sup u ∈ W 0 m L N m , q ( Ω ) Δ m 2 u L N m , q ≤ 1 ∫ Ω exp ( β ∣ u ∣ ) q q − 1 d x < C

if and only if β ≤ β N , m , where β N , m is the constant appearing in (1.3), and Δ m 2 u L N m , q is the Lorentz norm of Δ m 2 u (see (2.1)). Recently, the concentration-compactness principle associated with (1.4) in the first-order case was obtained by Černý [4], namely: let q ∈ ( 1 , ∞ ) and { u k } be a function sequence such that ∥ ∇ u k ∥ L N , q ≤ 1 , u k ⇀ u ∈ W 0 1 L N , q ( Ω ) . Set

(1.5) P ˜ u = Δ ( 1 − ∥ ∇ u ∥ L N , q − 1 ∕ q ) − 1 ∕ q , for ∥ ∇ u ∥ L N , q < 1 , + ∞ , for ∥ ∇ u ∥ L N , q = 1 ;

if q ∈ ( 1 , N ] , then for every p < P ˜ u , there is C > 0 such that

sup k ∫ Ω exp N N − 1 N ω N 1 N p ∣ u k ∣ 1 q − 1 d x ≤ C .

Moreover, the constant P ˜ u is sharp in the sense that the supreme above will become infinite when p ≥ P ˜ u . We remark that they cannot prove the sharpness of P ˜ u in the case q ∈ ( N , ∞ ] , because the quantity ∥ ⋅ ∥ L N , q is no longer a norm when q > N .

For other concentration-compactness principles associated with Trudinger-Moser or Adams’ inequalities on unbounded domains, compact manifolds and Heisenberg groups, one can refer to [6,13,14,17,21–23] and references therein.

In this article, modifying the smooth truncation argument in [8,14], we are able to establish the concentration-compactness principle associated with Adams’ inequality (1.4) in the second-order Lorentz-Sobolev space W 0 2 L 2 , q ( Ω ) .

Our main result reads as follows.

Theorem 1.1

Let Ω be a bounded domain in R 4 . If { u k } is a sequence in W 0 2 L 2 , q ( Ω ) satisfying ∥ Δ u k ∥ L 2 , q = 1 and u k ⇀ u ∈ W 0 2 L 2 , q ( Ω ) , then

(1.6) sup k ∫ Ω exp β 4 , 2 q q − 1 p ∣ u k ∣ q q − 1 d x < ∞

for any

0 < p < P u = 1 ( 1 − ∥ Δ u ∥ L 2 , q q ) 1 q − 1 ,

where β 4 , 2 = 4 2 π is the constant appearing in (1.3). Moreover, the constant P u is sharp for 1 < q ≤ 2 in the sense that if p ≥ P u , then the supremum in (1.6) becomes infinite; while in the case q > 2 , the constant P u is sharp in the sense that for any p > P u , the supremum in (1.6) becomes infinite.

Remark 1.2

As we pointed out earlier, the nonsmooth truncation argument using the level set of the functions under consideration was developed earlier by Li et al. on the Heisenberg group and Riemannian manifolds for Moser-Trudinger-type inequalities in [13] and [14]. In the previous result, we have proved the sharpness of P u both for cases q ∈ ( 1 , 2 ] and q ∈ ( 2 , ∞ ) . Our proof is motivated by the smooth truncation method for Adam’s inequality in [8]. The argument in this article can also be applied to show that the constant P ˜ u in (1.5) is also sharp when q > N . In view of this, our result is even new in the first-order case. However, when q ∈ ( 2 , ∞ ) and p = P u , whether the supremum in (1.6) is finite or infinite is still open.

2 Background of Lorentz-Sobolev space

In this section, we will give the definition of Lorentz-Sobolev space. Let φ : Ω → R + be a measurable function, and

μ φ ( t ) = ∣ { x ∈ Ω : φ ( x ) > t } ∣ , ∀ t ≥ 0 ,

be the distribution of φ . The decreasing rearrangement φ ∗ of φ is defined as the distribution function of μ φ ( t ) , that is,

φ ∗ ( s ) = sup { t > 0 ; μ φ ( t ) > s , 0 ≤ s ≤ ∣ Ω ∣ } ,

whereas the spherically symmetric rearrangement φ # of φ is defined as

φ # = φ ∗ ( ω N ∣ x ∣ N ) , x ∈ Ω # ,

where Ω # ⊂ R N is the open ball with center in the origin and satisfies ∣ Ω # ∣ = ∣ Ω ∣ .

We say φ belongs to Lorentz space L p , q ( Ω ) ( 1 < p < ∞ ) , which means that φ satisfies

(2.1) ∥ φ ∥ L p , q = ∫ 0 ∞ [ φ ∗ ( t ) t 1 ∕ p ] q d t t 1 ∕ q < ∞ .

Remark 2.1

The quantity ∥ ⋅ ∥ L p , q defined in (2.1) can be rewritten as follows (see [7]):

(2.2) ∥ φ ∥ L p , q = p 1 q ∫ 0 + ∞ ( μ { ∣ φ ∣ ≥ λ } ) q p λ q − 1 d λ 1 q ,

and it is a norm when q ≤ p , and when q = p and L p , q = L p . However, when q > p , ∥ ⋅ ∥ L p , q will no longer be a norm, since it does not satisfy the triangle inequality. Actually, the quantity

∥ u ∥ L ( p , q ) = q p ∫ 0 ∞ [ u ¯ ( t ) t 1 ∕ p ] q d t t 1 ∕ q

is a norm for any p and q in Lorentz space L p , q ( 1 < p < ∞ ) , where u ¯ ( t ) = 1 t ∫ 0 t u ∗ d s . It is easy to prove ∥ u ∥ L ( p , q ) is actually equivalent to ∥ u ∥ L p , q , that is,

∥ u ∥ L p , q ≤ ∥ u ∥ L ( p , q ) ≤ N N − 1 ∥ u ∥ L p , q .

Now, we define the Lorentz-Sobolev space W 0 m L N m , q ( Ω ) as

(2.3) W 0 m L N m , q ( Ω ) = c l u ∈ C 0 ∞ ( Ω ) , ∥ u ∥ W 0 m L N m , q ( Ω ) < ∞ ,

where c l stands for the closure under

∥ u ∥ W 0 m L N m , q ( Ω ) = Δ m 2 u L N m , q ( Ω ) q + ∥ u ∥ L N m , q ( Ω ) q 1 ∕ q .

3 Concentration-compactness principle of the second-order Adams’ inequality in W 0 2 L 2 , q ( Ω )

Before giving the proof of Theorem 1.1, we first give an important lemma.

Lemma 3.1

For any u ∈ W 0 2 L 2 , q ( Ω ) , β > 0 , we have

∫ Ω exp ( β ∣ u ∣ ) q q − 1 d x < ∞ .

Proof

For any ε > 0 , we can always find a v ∈ C 0 ∞ ( Ω ) such that

(3.1) ∥ u − v ∥ W 0 2 L 2 , q ( Ω ) < ε .

We donate u = v + w , then

(3.2) ∣ u ∣ q q − 1 = ∣ v + w ∣ q q − 1 ≤ 2 q q − 1 ∥ v ∥ L ∞ q q − 1 + ∣ w ∣ q q − 1 .

According to (1.4), (3.1), and (3.2), for any β > 0 , we have

∫ Ω exp ( β ∣ u ∣ ) q q − 1 d x = ∫ Ω exp ( β ∣ v + w ∣ ) q q − 1 d x ≤ exp ( 2 β ∥ v ∥ L ∞ ) q q − 1 ∫ Ω exp ( 2 β ) q q − 1 ∣ w ∣ q q − 1 d x ≤ exp ( 2 β ∥ v ∥ L ∞ ) q q − 1 ∫ Ω exp ( 2 β ∥ Δ w ∥ L 2 , q ( Ω ) ) q q − 1 ∣ w ∣ ∥ Δ w ∥ L 2 , q ( Ω ) q q − 1 d x ≤ exp ( 2 β ∥ v ∥ L ∞ ) q q − 1 ∫ Ω exp β 4 , 2 ∣ w ∣ ∥ Δ w ∥ L 2 , q ( Ω ) q q − 1 d x < ∞ .

The proof is finished.□

Now, we give the following proof of theorem.

Proof of Theorem 1.1

We will give the proof in cases 1 < q ≤ 2 and q > 2 , respectively.

Case 1: 1 < q ≤ 2

We prove the conclusion by contradiction. Assume that there exists some p ∈ ( 0 , P u ) , such that

lim k → ∞ ∫ Ω exp β 4 , 2 q q − 1 p ∣ u k ∣ q q − 1 d x = ∞ .

Thus, there exists a point x 0 ∈ Ω such that

(3.3) lim k → ∞ ∫ B r ( x 0 ) exp β 4 , 2 q q − 1 p ∣ u k ∣ q q − 1 d x = ∞

for any r > 0 , where B r ( x 0 ) denotes the ball of radius r and center x 0 . Set v k = u k − u , then by the classical Sobolev-Lorentz embedding theorem, we have v k ⇀ 0 in W 0 2 L 2 , q ( Ω ) and v k → 0 in L 2 , q ( Ω ) .

For any φ ∈ C 0 ∞ ( Ω ) , we have

∣ Δ ( φ v k ) ∣ ≤ ∣ φ Δ v k ∣ + c ( ∣ ∇ φ ∣ ∣ ∇ v k ∣ + ∣ Δ φ ∣ ∣ v k ∣ ) ≤ ∣ φ ∣ ∣ Δ u k ∣ + ∣ φ ∣ ∣ Δ u ∣ + c ( ∣ ∇ φ ∣ ∣ ∇ v k ∣ + ∣ Δ φ ∣ ∣ v k ∣ ) .

Since when 1 < q ≤ 2 , the quantity ∥ ⋅ ∥ L 2 , q is a norm, we have

∫ 0 ∞ ( ( Δ ( φ v k ) ) ∗ ( t ) ) q t q 2 − 1 d t 1 ∕ q ≤ ∫ 0 ∞ ( ( φ Δ u k ) ∗ ( t ) ) q t q 2 − 1 d t 1 ∕ q + ∫ 0 ∞ ( ( φ Δ u ) ∗ ( t ) ) q t q 2 − 1 d t 1 ∕ q + c ∫ 0 ∞ ( ( v k Δ φ ) ∗ ( t ) ) q t q 2 − 1 d t + ∫ 0 ∞ ( ( ∇ φ ∇ v k ) ∗ ( t ) ) q t q 2 − 1 d t 1 ∕ q ,

together with the elementary inequality: given q > 0 and a , b ∈ R for any ε > 0 , there exists some constant c ( ε ) such that

(3.4) ( a + b ) q ≤ ( 1 + ε ) a q + c ( ε ) b q ,

we have

(3.5) ∥ Δ ( φ v k ) ∥ L 2 , q ( Ω ) q ≤ ( 1 + ε ) ∥ φ Δ u k ∥ L 2 , q ( Ω ) q + c ( ε ) ∥ φ Δ u ∥ L 2 , q ( Ω ) q + c ( ε ) ( ∥ v k Δ φ ∥ L 2 , q ( Ω ) q + ∥ ∇ φ ∇ v k ∥ L 2 , q ( Ω ) q ) .

Since p < P u , there exists ε 0 > 0 such that

( 1 + ε 0 ) ( 1 − ∥ Δ u ∥ L 2 , q ( Ω ) q ) < p 1 − q ,

and this together with the assumption that ∥ Δ u k ∥ L 2 , q = 1 and u k ⇀ u ∈ W 0 2 L 2 , q ( Ω ) implies that there exists some k 0 and r 0 such that

∥ Δ u k ∥ L 2 , q ( B 2 r ( x 0 ) ) q ≤ ( 1 + ε 0 ) ( 1 − ∥ Δ u ∥ L 2 , q ( Ω ) q ) < p 1 − q ,

when k > k 0 and r < r 0 .

Choosing some cutoff function φ ∈ C 0 ∞ ( Ω ) satisfying φ ∣ B 2 r c ( x 0 ) = 0 and φ = 1 in B r ( x 0 ) , and letting r and ε 0 appearing in (3.5) small enough and k large enough, we can obtain

(3.6) ∥ Δ ( φ v k ) ∥ L 2 , q ( Ω ) q < p 1 − q .

Moreover, since for any ε > 0 , one has

∣ u k ∣ q q − 1 = ∣ v k + u ∣ q q − 1 ≤ ( 1 + ε ) ∣ v k ∣ q q − 1 + c ( ε ) ∣ u ∣ q q − 1 .

It follows from Hölder’s inequality that

∫ B r ( x 0 ) exp β 4 , 2 q q − 1 p ∣ u k ∣ q q − 1 d x ≤ ∫ B r ( x 0 ) exp β 4 , 2 q q − 1 p ( 1 + ε ) ∣ v k ∣ q q − 1 exp c ( ε ) ∣ u ∣ q q − 1 d x ≤ ∫ Ω exp β 4 , 2 q q − 1 p s ( 1 + ε ) ∣ φ v k ∣ q q − 1 d x 1 s ∫ Ω exp β c ( ε ) s ′ ∣ u ∣ q q − 1 d x 1 s ′ = Δ I 1 ⋅ I 2 ,

where s and s ′ satisfy 1 s + 1 s ′ = 1 .

Choosing ε small enough and s close to 1, then by (3.6) and (1.4), we have I 1 < ∞ . On the other hand, from Lemma 3.1, we obtain I 2 < ∞ .

Therefore, we have

(3.7) ∫ B r ( x 0 ) exp β 4 , 2 q q − 1 p ∣ u k ∣ q q − 1 d x < ∞ ,

which is a contradiction with the assumption (3.3).

Case 2: q > 2

As Case 1, we also prove the conclusion by contradiction. Assume that for some p ∈ ( 0 , P u ) , there exists a point x 0 ∈ Ω such that

(3.8) ∫ B r ( x 0 ) exp β 4 , 2 q q − 1 p ∣ u k ∣ q q − 1 d x = ∞

for any r and denote v k = u k − u as before.

Using the equivalence definition (2.2) of the Lorentz norm, we have

(3.9) ∥ Δ v k ∥ L 2 , q ( B r ( x 0 ) ) q = 2 ∫ 0 + ∞ ( μ { B r ( x 0 ) ∩ { ∣ Δ v k ∣ ≥ λ } } ) q 2 λ q − 1 d λ = 2 ∫ 0 M ( μ { B r ( x 0 ) ∩ { ∣ Δ v k ∣ ≥ λ } } ) q 2 λ q − 1 d λ + 2 ∫ M + ∞ ( μ { B r ( x 0 ) ∩ { ∣ Δ v k ∣ ≥ λ } } ) q 2 λ q − 1 d λ = Δ J 1 + J 2 ,

where M is some large constant. First, we estimate J 1 as follows:

(3.10) J 1 ≤ 2 ∫ 0 M ∣ B r ( x 0 ) ∣ q 2 λ q − 1 d λ ≤ C ( M ) r 2 q .

For J 2 , we have

J 2 = 2 ∫ M + ∞ ( μ { B r ( x 0 ) ∩ { ∣ Δ v k ∣ ≥ λ } } ) q 2 λ q − 1 d λ = 2 ∫ M + ∞ ( μ { B r ( x 0 ) ∩ { ∣ Δ ( u k − u ) ∣ ≥ λ } } ) q 2 λ q − 1 d λ .

For any ε 1 > 0 and some fixed ω ∈ C ∞ ( Ω ) , which will be determined later, we can find some large M such that

(3.11) ∥ Δ ω ∥ L ∞ ( Ω ) ≤ ε 1 M ,

then we have

( μ { B r ( x 0 ) ∩ { ∣ Δ ( u k − u ) ∣ ≥ λ } } ) q 2 = ( μ { B r ( x 0 ) ∩ { ∣ Δ ( u k − u + ω − ω ) ∣ ≥ λ } } ) q 2 ≤ ( μ { B r ( x 0 ) ∩ { ∣ Δ ( u k − u + ω ) ∣ ≥ ( 1 − ε 1 ) λ } } ) q 2 ,

provided λ ≥ M . So, we have

J 2 ≤ 2 ∫ M + ∞ ( μ { B r ( x 0 ) ∩ { ∣ Δ ( u k − u + ω ) ∣ ≥ ( 1 − ε 1 ) λ } } ) q 2 λ q − 1 d λ ≤ 2 ( 1 − ε 1 ) q ∫ ( 1 − ε 1 ) M + ∞ ( μ { B r ( x 0 ) ∩ { ∣ Δ ( u k − u + ω ) ∣ ≥ λ } } ) q 2 λ q − 1 d λ .

For any ε 2 > 0 , we set

A ε 2 ( λ ) = { B r ( x 0 ) ∩ { ∣ Δ ( u − ω ) ∣ ≥ ε 2 λ } } ,

and we have

{ B r ( x 0 ) ∩ { ∣ Δ ( u k − u + ω ) ∣ ≥ λ } } \ A ε 2 ( λ ) = { B r ( x 0 ) ∩ { ∣ Δ ( u k − u + ω ) ∣ ≥ λ } ; { ∣ Δ ( u − ω ) ∣ < ε 2 λ } } ⊆ { B r ( x 0 ) ∩ { ∣ Δ u k ∣ ≥ ( 1 − ε 2 ) λ } } .

Again using the elementary inequality (3.4), for any ε 3 > 0 , there exists some constant C ε 3 such that

( μ { B r ( x 0 ) ∩ { ∣ Δ ( u k − u + ω ) ∣ ≥ λ } } ) q 2 ≤ ( 1 + ε 3 ) ( μ { B r ( x 0 ) ∩ { ∣ Δ u k ∣ ≥ ( 1 − ε 2 ) λ } } ) q 2 + C ε 3 ( μ { A ε 2 ( λ ) } ) q 2 .

Hence, we have

(3.12) J 2 ≤ 2 ( 1 − ε 1 ) q ∫ ( 1 − ε 1 ) M + ∞ ( μ { B r ( x 0 ) ∩ { ∣ Δ ( u k − u + ω ) ∣ ≥ λ } } ) q 2 λ q − 1 d λ ≤ 2 ( 1 + ε 3 ) 1 ( 1 − ε 1 ) q ∫ ( 1 − ε 1 ) M + ∞ ( μ { B r ( x 0 ) ∩ { ∣ Δ u k ∣ ≥ ( 1 − ε 2 ) λ } } ) q 2 λ q − 1 d λ + 2 C ε 3 1 ( 1 − ε 1 ) q ∫ ( 1 − ε 1 ) M + ∞ ( μ { B r ( x 0 ) ∩ { ∣ Δ ( u − ω ) ∣ ≥ ε 2 λ } } ) q 2 λ q − 1 d λ = Δ Q 1 + Q 2 .

For Q 2 , there exists a constant C ε 1 , ε 2 , ε 3 such that

(3.13) Q 2 ≤ C ε 1 , ε 2 , ε 3 ∥ Δ ( u − ω ) ∥ L 2 , q ( Ω ) q .

Now, for any ε > 0 , we choose ω ∈ C ∞ ( Ω ) such that

(3.14) C ε 1 , ε 2 , ε 3 ∥ Δ ( u − ω ) ∥ L 2 , q ( Ω ) q < ε

and

sup Ω ∣ Δ ω ∣ ≤ ε 1 M ,

where M is the constant appearing in (3.11), which is independent of r .

For Q 1 , we have

(3.15) Q 1 ≤ 2 ( 1 + ε 3 ) ( 1 − ε 2 ) q ( 1 − ε 1 ) q ∫ ( 1 − ε 2 ) ( 1 − ε 1 ) M + ∞ ( μ { B r ( x 0 ) ∩ { ∣ Δ u k ∣ ≥ λ } } ) q 2 λ q − 1 d λ ≤ ( 1 + ε 4 ) ∥ Δ u k ∥ L 2 , q ( B r ( x 0 ) ) q ,

where ε 4 is a small constant depending on ε 1 , ε 2 , and ε 3 .

Combining (3.12)–(3.15), we have

(3.16) J 2 ≤ Q 1 + Q 2 ≤ ε + ( 1 + ε 4 ) ∥ Δ u k ∥ L 2 , q ( B r ( x 0 ) ) q .

Then, it follows from (3.9), (3.10), and (3.16) that

∥ Δ v k ∥ L 2 , q ( B r ( x 0 ) ) q = J 1 + J 2 ≤ C ( M ) r 2 q + ε + ( 1 + ε 4 ) ∥ Δ u k ∥ L 2 , q ( B r ( x 0 ) ) q ≤ C ( M ) r 2 q + ( 1 + ε 0 ) ( 1 − ∥ Δ u ∥ L 2 , q ( Ω ) q ) ≤ p 1 − q ,

provided r and ε 0 are small enough. Then, we can similarly prove (3.7) and obtain a contradiction with the assumption (3.8) as the same as Case 1.

Next, we prove the sharpness of the constant P u , and without loss of generality, we can assume that B 2 ( 0 ) ⊂ Ω and construct a function sequence as follows:

v k ( x ) = C k log k 32 π 2 − k 8 π 2 log k ∣ x ∣ 2 + 1 8 π 2 log k , 0 ≤ ∣ x ∣ ≤ 1 k 1 4 C k log 1 ∣ x ∣ 2 π 2 log k , 1 k 1 4 < ∣ x ∣ ≤ 1 η k , 1 < ∣ x ∣ < 2 ,

where η k ∈ C ∞ ( R 4 ) satisfies

η k ∣ ∂ B 2 ( 0 ) = 0 , ∂ η k ∂ v ∣ ∂ B 1 ( 0 ) = C k 2 π 2 log k ,

C k = ∫ 0 ω 4 k 4 k 2 π 2 log k s 1 2 q d s s + ∫ ω 4 k ω 4 2 ω 4 2 π 2 log k q d s s − 1 q .

By calculation, we can obtain

C k = ∫ 0 ω 4 k 4 k 2 π 2 log k s 1 2 q d s s + ∫ ω 4 k ω 4 2 ω 4 2 π 2 log k q d s s − 1 q = 4 k 2 π 2 log k q ∫ 0 ω 4 k s q 2 d s s + ∫ ω 4 k ω 4 2 ω 4 2 π 2 log k q d s s − 1 q = 4 k 2 π 2 log k q 2 q ω 4 k q 2 + 2 ω 4 2 π 2 log k q ∫ ω 4 k ω 4 1 d s s − 1 q = 2 ω 4 2 π 2 log k q 2 q + 1 q + log k 2 ω 4 2 π 2 log k q − 1 q = 1 2 ω 4 2 π 2 log k 2 q + 1 q + log k 1 q ∼ ( log k ) 1 2 − 1 q , as k → ∞ ,

and

(3.17) η k , ∣ ∇ η k ∣ , Δ η k = O C k log k .

Moreover, we have that

v k ∗ ( s ) = C k log k 4 π 2 + 1 2 π 2 log k − k s 2 π 2 ω 4 log k , 0 ≤ s ≤ ω 4 k ; C k ln ω 4 s 4 π 2 log k , ω 4 k < s ≤ ω 4 ; η k ∗ , ω 4 < s < 2 ω 4 ,

(3.18) Δ v k ( x ) = − 4 C k k 2 π 2 log k , 0 ≤ ∣ x ∣ ≤ 1 k 1 4 ; − C k 2 π 2 log k 2 ∣ x ∣ 2 , 1 k 1 4 < ∣ x ∣ ≤ 1 ; Δ η k , 1 < ∣ x ∣ < 2 ,

and

( Δ v k ) ∗ ( s ) = 4 C k k 2 π 2 log k , 0 ≤ s ≤ ω 4 k ; C k ω 4 2 π 2 log k 2 s , ω 4 k < s ≤ ω 4 ; ( Δ η k ) ∗ , ω 4 < s < 2 ω 4 .

Therefore, we have

∥ Δ v k ∥ L 2 , q q = ∫ 0 2 ω 4 ( Δ v k ) ∗ s 1 2 q d s s = ∫ 0 ω 4 k 4 C k k 2 π 2 log k s 1 2 q d s s + ∫ ω 4 k ω 4 C k ω 4 2 π 2 log k 2 s s 1 2 q d s s + ∫ ω 4 2 ω 4 ∣ Δ η k ∣ ∗ s 1 2 q d s s = C k q ∫ 0 ω 4 k 4 k 2 π 2 log k s 1 2 q d s s + ∫ ω 4 k ω 4 ω 4 2 π 2 log k 2 s s 1 2 q d s s + O 1 log k = 1 + O 1 log k ,

where we have used (3.17).

Normalizing the sequence v k as v ˜ k = v k ∥ Δ v k ∥ L 2 , q ( Ω ) . Set u k = u + ( 1 − δ q ) 1 q v ˜ k , where u ∈ C 0 ∞ ( B 3 ( 0 ) ) is a radial decreasing function such that

u = A > 0 on B 2 ( 0 ) and ∥ Δ u ∥ L 2 , q ( Ω ) = δ .

Note that ∥ u k ∥ L 2 , q ( Ω ) is bounded and u k → u a.e, so we have u k ⇀ u in L 2 , q since the Lorentz space L 2 , q is a reflexive space.

Now, we prove the sharpness of P u in cases 1 < q ≤ 2 and 2 < q < ∞ , respectively.

Case 1: 1 < q ≤ 2 . In this case, we have

∥ Δ u k ∥ L 2 , q ( Ω ) q = 2 ∫ 0 + ∞ ( μ { ∣ Δ u k ∣ ≥ λ } ) q 2 λ q − 1 d λ ≤ 2 ∫ 0 + ∞ μ { ∣ Δ u ∣ ≥ λ } + μ Δ ( 1 − δ q ) 1 q v ˜ k ≥ λ q 2 λ q − 1 d λ ≤ 2 ∫ 0 + ∞ ( μ { ∣ Δ u ∣ ≥ λ } ) q 2 λ q − 1 d λ + 2 ∫ 0 + ∞ μ Δ ( 1 − δ q ) 1 q v ˜ k ≥ λ q 2 λ q − 1 d λ = ∥ Δ u ∥ L 2 , q ( Ω ) q + Δ ( 1 − δ q ) 1 q v ˜ k L 2 , q ( Ω ) q = δ q + 1 − δ q = 1 .

We now show that the u k defined earlier can make the supremum in (1.6) arbitrary large when p ≥ P u :

∫ Ω exp β 4 , 2 q q − 1 P u ∣ u k ∣ q q − 1 d x ≥ ∫ B k − 1 4 ( 0 ) exp β 4 , 2 q q − 1 ∣ A + ( 1 − δ q ) 1 ∕ q v ˜ k ∣ q q − 1 ( 1 − δ q ) 1 q − 1 d x ≥ ∫ B k − 1 4 ( 0 ) exp β 4 , 2 q q − 1 ∣ C + v ˜ k ∣ q q − 1 d x = ∫ B k − 1 4 ( 0 ) exp β 4 , 2 q q − 1 C + C k log k 32 π 2 ( 1 + O ( ( log k ) − 1 ) ) 1 ∕ q q q − 1 d x = ∫ B k − 1 4 ( 0 ) exp ( 4 2 π ) q q − 1 C + ( log k ) 1 2 − 1 q log k 32 π 2 ( 1 + O ( ( log k ) − 1 ) ) − 1 ∕ q q q − 1 d x = ∫ B k − 1 4 ( 0 ) exp C ′ + ( log k ) 1 − 1 q ( 1 + O ( ( log k ) − 1 ) ) − 1 ∕ q q q − 1 d x

for some positive constants C and C ′ . Using the following elementary estimate:

( 1 + o ( 1 ) ) a = 1 − o ( 1 ) , for any a < 0 ,

we obtain

∫ B k − 1 4 ( 0 ) exp C ′ + ( log k ) 1 − 1 q ( 1 + O ( ( log k ) − 1 ) ) − 1 ∕ q q q − 1 d x , = ∫ B k − 1 4 ( 0 ) exp C ′ + ( log k ) 1 − 1 q ( 1 − O ( ( log k ) − 1 ) ) q q − 1 d x ≥ ∫ B k − 1 4 ( 0 ) exp C ″ + ( log k ) 1 − 1 q q q − 1 d x = ∫ B k − 1 4 ( 0 ) exp ( log k ) 1 − 1 q 1 + C ″ ( log k ) 1 − 1 q q q − 1 d x = ∫ B k − 1 4 ( 0 ) exp log k + C ″ q q − 1 ( log k ) 1 q d x = C ‴ k − 1 exp log k + C ″ q q − 1 ( log k ) 1 q = C ‴ exp C ″ q q − 1 ( log k ) 1 q → ∞

for some positive constants C ″ , C ‴ .

Case 2: q > 2

Using the equivalence definition (2.2) of the Lorentz norm, we have

(3.19) ∥ Δ u k ∥ L 2 , q ( Ω ) q = 2 ∫ 0 + ∞ ( μ { ∣ Δ u k ∣ ≥ λ } ) q 2 λ q − 1 d λ = 2 ∫ sup ∣ Δ u ∣ + ∞ ( μ { ∣ Δ u k ∣ ≥ λ } ) q 2 λ q − 1 d λ + 2 ∫ ( log k ) − 1 2 q sup ∣ Δ u ∣ ( μ { ∣ Δ u k ∣ ≥ λ } ) q 2 λ q − 1 d λ + 2 ∫ 0 ( log k ) − 1 2 q ( μ { ∣ Δ u k ∣ ≥ λ } ) q 2 λ q − 1 d λ = Δ Z 1 + Z 2 + Z 3 .

For Z 1 , we have

(3.20) Z 1 = 2 ∫ sup ∣ Δ u ∣ + ∞ μ Δ u + ( 1 − δ q ) 1 q v ˜ k ≥ λ q 2 λ q − 1 d λ = 2 ∫ sup ∣ Δ u ∣ + ∞ μ Δ ( 1 − δ q ) 1 q v ˜ k ≥ λ q 2 λ q − 1 d λ ≤ Δ ( 1 − δ q ) 1 q v ˜ k L 2 , q ( Ω ) q .

For Z 2 , we have

(3.21) Z 2 = 2 ∫ ( log k ) − 1 2 q sup ∣ Δ u ∣ μ { ∣ Δ u ∣ ≥ λ } + μ Δ ( 1 − δ q ) 1 q v ˜ k ≥ λ q 2 λ q − 1 d λ ≤ 2 ∫ ( log k ) − 1 2 q sup ∣ Δ u ∣ μ { ∣ Δ u ∣ ≥ λ } + μ Δ ( 1 − δ q ) 1 q v ˜ k ≥ ( log k ) − 1 2 q q 2 λ q − 1 d λ ≤ 2 ∫ ( log k ) − 1 2 q sup ∣ Δ u ∣ μ { ∣ Δ u ∣ ≥ λ } + μ Δ ( 1 − δ q ) 1 q v k ≥ ∥ Δ v k ∥ L 2 , q ( Ω ) q ( log k ) − 1 2 q q 2 λ q − 1 d λ , ( using (3.18) ) = 2 ∫ ( log k ) − 1 2 q sup ∣ Δ u ∣ μ { ∣ Δ u ∣ ≥ λ } + μ 1 ∣ x ∣ 2 O ( log k ) − 1 q ≥ 1 + O 1 log k ( log k ) − 1 2 q q 2 λ q − 1 d λ = 2 ∫ ( log k ) − 1 2 q sup ∣ Δ u ∣ μ { ∣ Δ u ∣ ≥ λ } + μ ∣ x ∣ 2 ≤ O ( log k ) − 1 2 q q 2 λ q − 1 d λ = 2 ∫ ( log k ) − 1 2 q sup ∣ Δ u ∣ μ { ∣ Δ u ∣ ≥ λ } + O ( log k ) − 1 q q 2 λ q − 1 d λ = 2 ∫ ( log k ) − 1 2 q sup ∣ Δ u ∣ ( μ { ∣ Δ u ∣ ≥ λ } ) q 2 + O ( log k ) − 1 q λ q − 1 d λ ≤ ∥ Δ u ∥ L 2 , q ( Ω ) q + O ( log k ) − 1 q ( sup ∣ Δ u ∣ ) q − ( log k ) − 1 2 = ∥ Δ u ∥ L 2 , q ( Ω ) q + O ( log k ) − 1 q .

For Z 3 , we have

(3.22) Z 3 ≤ 2 ∫ 0 ( log k ) − 1 2 q ∣ B 3 ( 0 ) ∣ q 2 λ q − 1 d λ = O ( log k ) − 1 2 .

Combining (3.19)–(3.22), we obtain

∥ Δ u k ∥ L 2 , q ( Ω ) q = Δ ( 1 − δ q ) 1 q v ˜ k L 2 , q ( Ω ) q + ∥ Δ u ∥ L 2 , q ( Ω ) q + O ( log k ) − 1 q + O ( log k ) − 1 2 = 1 + O ( log k ) − 1 q .

Now, we claim that the supremum in (1.6) can be arbitrary large if p > P u . Indeed, similar as Case 1, we have

∫ Ω exp β 4 , 2 q q − 1 p u k q q − 1 d x > ∫ Ω exp β 4 , 2 q q − 1 1 + O ( log k ) − 1 q 1 q − 1 P u u k 1 + O ( log k ) − 1 q 1 q q q − 1 d x ≥ ∫ Ω exp β 4 , 2 q q − 1 P u ∣ u k ∣ q q − 1 d x ≥ C 2 exp C 1 q q − 1 ( log k ) 1 q → ∞

as k → ∞ for some constants C 1 and C 2 , and the proof of Theorem 1.1 is finished.□

  1. Funding information: Maochun Zhu was supported by the Natural Science Foundation of China (12071185 and 12061010).

  2. Conflict of interest: Authors state no conflict of interest.

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Received: 2022-08-07
Revised: 2022-12-08
Accepted: 2022-12-09
Published Online: 2022-12-31

© 2022 the author(s), published by De Gruyter

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https://doi.org/10.1515/ans-2022-0043
Keywords for this article
concentration-compactness principle; Adams’ inequality; Lorentz-Sobolev
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