Weighted critical exponents of Sobolev-type embeddings for radial functions

Jiabao Su; Cong Wang

doi:10.1515/ans-2022-0006

Artikel Open Access

Weighted critical exponents of Sobolev-type embeddings for radial functions

Jiabao Su und Cong Wang

Veröffentlicht/Copyright: 20. April 2022

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Aus der Zeitschrift Advanced Nonlinear Studies Band 22 Heft 1

Abstract

In this article, we prove the upper weighted critical exponents for some embeddings from weighted Sobolev spaces of radial functions into weighted Lebesgue spaces. We also consider the lower critical exponent for certain embedding.

Keywords: embeddings; critical exponents; compactness

MSC 2010: Primary: 35A23; 35B33; 35J20; 35J62; 46E35

1 Introduction

We assume that the real functions A , V , and Q satisfy the following assumptions:

A ∈ C ( 0 , ∞ ) , A ( r ) > 0 , and there exist ℓ , ℓ 0 ∈ R such that
lim r → ∞ A ( r ) r ℓ > 0 , lim r → 0 A ( r ) r ℓ 0 > 0 .
V ∈ C ( 0 , ∞ ) , V ( r ) > 0 , and there exist a , a 0 ∈ R such that
liminf r → ∞ V ( r ) r a > 0 , liminf r → 0 V ( r ) r a 0 > 0 .
Q ∈ C ( 0 , ∞ ) , Q ( r ) > 0 , and there exist b , b 0 ∈ R such that
limsup r → ∞ Q ( r ) r b < ∞ , limsup r → 0 Q ( r ) r b 0 < ∞ .

We assume N ∈ N and 1 < p < N + min { ℓ , ℓ 0 } and then define two numbers q ∗ and q ∗ in the following way.

q ∗ = p + p 2 ( b − a ) p ( N − 1 + a ) + ( ℓ − a ) , b ⩾ a > ℓ − p , p ( N + b ) N + ℓ − p , b ⩾ ℓ − p , a ⩽ ℓ − p , p , b ⩽ max { a , ℓ − p } , q ∗ = p ( N + b 0 ) N + ℓ 0 − p , b 0 ⩾ ℓ 0 − p , a 0 ⩾ ℓ 0 − p , p + p 2 ( b 0 − a 0 ) p ( N − 1 + a 0 ) + ( ℓ 0 − a 0 ) , ℓ 0 − p ⩾ a 0 > − p ( N − 1 ) + ℓ 0 p − 1 , b 0 ⩾ a 0 , ∞ , a 0 ⩽ − p ( N − 1 ) + ℓ 0 p − 1 , b 0 ⩾ a 0 .

Let C 0 ∞ ( R N ) denote the collection of smooth functions with compact support and

C 0 , r ∞ ( R N ) = { u ∈ C 0 ∞ ( R N ) ∣ u is radial } .

Let D r 1 , p ( R N ; A ) (resp., D 1 , p ( R N ; A ) ) be the completion of C 0 , r ∞ ( R N ) (resp., C 0 ∞ ( R N ) ) under

‖ u ‖ A = ∫ R N A ( x ) ∣ ∇ u ∣ p d x 1 p .

Define for p > 1

L p ( R N ; V ) = u : R N → R ∣ u is Lebesgue measurable, ∫ R N V ( x ) ∣ u ∣ p d x < ∞ .

Then define

W r 1 , p ( R N ; A , V ) = D r 1 , p ( R N ; A ) ∩ L p ( R N ; V ) , W 1 , p ( R N ; A , V ) = D 1 , p ( R N ; A ) ∩ L p ( R N ; V ) .

They are, under conditions ( A ) and ( V ) , reflexive Banach spaces with the norm

‖ u ‖ A , V = ∫ R N A ( x ) ∣ ∇ u ∣ p + V ( x ) ∣ u ∣ p d x 1 p .

Define for q ⩾ 1

L q ( R N ; Q ) = u : R N → R ∣ u is Lebesgue measurable, ∫ R N Q ( x ) ∣ u ∣ q d x < ∞ .

In [17], Su and Wang proved the following embedding theorem.

Theorem I

Assume ( A ) , ( V ) , and ( Q ) with 1 < p < N + min { ℓ 0 , ℓ } and q ∗ , q ∗ being well defined. The embedding

(1.1) W r 1 , p ( R N ; A , V ) ↪ L q ( R N ; Q )

is continuous for q ∗ ⩽ q ⩽ q ∗ when q ∗ < ∞ and for q ∗ ⩽ q < ∞ when q ∗ = ∞ . The embedding (1.1) is compact for q ∗ < q < q ∗ .

If b < max { a , ℓ − p } and b 0 > min { a 0 , ℓ 0 − p } , then the embedding

(1.2) W r 1 , p ( R N ; A , V ) ↪ L p ( R N ; Q )

is compact.

We first obtain a new observation that (1.2) will be compact on W 1 , p ( R N ; A , V ) .

Theorem 1.1

Assume ( A ) , ( V ) , and ( Q ) with 1 min { a 0 , ℓ 0 − p } . The embedding

(1.3) W 1 , p ( R N ; A , V ) ↪ L p ( R N ; Q )

is compact.

One of the main results in this article is the following theorem on the upper weighted critical exponent of the embedding (1.1).

Theorem 1.2

Assume ( A ) , ( V ) , and ( Q ) with 1 q ∗ . The embedding

(1.4) W r 1 , p ( R N ; A , V ) ↪ L q ∗ ( R N ; Q )

is not compact.

Now we investigate the lower critical exponent of the above embedding. When ( A ) , ( V ) , and ( Q ) are assumed, we have that q ∗ = p is exactly the lower critical exponent of the embedding (1.1) in some case.

Theorem 1.3

Assume ( A ) , ( V ) , and ( Q ) with 1 − N , b 0 > − N , and b = a ⩾ σ − p . There is no embedding from W r 1 , p ( R N ; A , V ) into L q ( R N ; Q ) for any q < p . The embedding

(1.5) W r 1 , p ( R N ; A , V ) ↪ L p ( R N ; Q )

is not compact.

Let A ( x ) = ∣ x ∣ σ , V ( x ) = ∣ x ∣ β , and Q ( x ) = ∣ x ∣ α , set

(1.6) q ∗ ( σ , α , β ) = p , if α ⩽ β , p + p 2 ( α − β ) p ( N − 1 + β ) + σ − β , if α > β ,

(1.7) q ∗ ( σ , α , β ) = p ( N + α ) N + σ − p .

We consider the following embeddings.

(1.8) W r 1 , p ( R N ; ∣ x ∣ σ , ∣ x ∣ β ) ↪ L q ( R N ; ∣ x ∣ α ) ,

(1.9) W 1 , p ( R N ; ∣ x ∣ σ , ∣ x ∣ β ) ↪ L q ( R N ; ∣ x ∣ α ) .

From Theorem I, Theorems 1.1, 1.2, and 1.3, we can obtain the following conclusions (Table 1).

Table 1

1 < p < N + σ

q	β , α	Embedding
q ∗ ( σ , α , β ) ⩽ q ⩽ q ∗ ( σ , α , β )	β ⩾ σ − p α ⩾ σ − p	(1.8) holds
q ∗ ( σ , α , β ) < q < q ∗ ( σ , α , β )	β ⩾ σ − p α ⩾ σ − p	(1.8) is compact
q = q ∗ ( σ , α , β )	β ⩾ σ − p α ⩾ σ − p	q ∗ ( σ , α , β ) is upper critical exponent of (1.8), (1.8) is noncompact
q = p	β = α ≥ σ − p	p is lower critical exponent of (1.8), (1.8) is noncompact
q = p	σ − p < α < β	(1.9) is compact

Furthermore, taking the special numbers, we can obtain the classical critical exponents (Tables 2 and 3).

Table 2

The upper critical exponents

p , σ	β	α	q ∗ ( σ , α , β )
1 < p < N , σ = 0	β = 0	α = − p	Upper Hardy critical exponent, see [7]
1 < p < N , σ = 0	β = 0	− p < α < 0	Upper Hardy-Sobolev critical exponent, see [7]
1 < p < N , σ = 0	β = 0	α = 0	Upper Sobolev critical exponent, see [6,14] for p = 2
1 < p < N , σ = 0	β = 0	α > 0	Upper Hénon-Sobolev critical exponent, see [19]
p − N < σ < ∞	β ≥ σ − p	σ − p ⩽ α ⩽ σ N N − p for 1 < p < N ;	Upper CKN critical exponent, see [3]
p > 1		α ⩾ σ − p for p ⩾ N , p > 1

Table 3

The lower critical exponents

p , σ	β	α	q ∗ ( σ , α , β )
1 < p < N , σ = 0	β = 0	α = 0	p is lower Sobolev critical exponent, see [6,14] for p = 2

In the case that A ≡ 1 , V and Q satisfy assumptions ( V ) and ( Q ) , the conclusions of Theorem I have been established in [15,16]. For α > 0 , the upper Hénon-Sobolev critical exponent is raised in [19] due to a reason that Hénon [9] first established a model of semilinear elliptic equation

(1.10) − Δ u = ∣ x ∣ α u q − 1 , u > 0 in B , u = 0 on ∂ B

on the unit ball B ≔ { x ∈ R N : ∣ x ∣ < 1 } with α > 0 in studying the rotating stellar structures in 1973. Smets et al. in [13] first revealed the symmetric breaking phenomenon of the ground state of (1.10) for 2 < q < 2 ∗ = 2 N N − 2 . Sinzeff and Willem in [12, Theorem 2.3] investigated the ground state of the equation

(1.11) − Δ u + ∣ x ∣ β u = ∣ x ∣ α u q − 1 , u > 0 in R N , u ∈ W r 1 , 2 ( R N ; ∣ x ∣ β )

for β ⩾ 0 , α ⩾ 0 and max 2 , 2 + 4 ( α − β ) 2 ( N − 1 ) + β < q < 2 ( N + α ) N − 2 . Indeed, the power q in (1.11) lies strictly between q ∗ ( σ , α , β ) and q ∗ ( σ , α , β ) given by (1.6) and (1.7) with σ = 0 and p = 2 .

It follows from Theorem I and Theorem 1.2 that q ∗ is exactly the upper critical exponent of the embedding (1.1) in the case a 0 , b 0 ⩾ ℓ 0 − p . In particular, the number q ∗ ( σ , α , β ) defined by (1.7) is exactly the upper critical exponent of the embedding (1.8).

The well-known Caffarelli-Kohn-Nirenberg (CKN) inequality [3] (see also [11]) reads as

(1.12) ∫ R N ∣ x ∣ − t q ∣ u ∣ q d x p q ⩽ C s , t ∫ R N ∣ x ∣ − p s ∣ ∇ u ∣ p d x , ∀ u ∈ C 0 ∞ ( R N ) ,

where

(1.13) − ∞ < s < N − p p , 0 ⩽ t − s ⩽ 1 , q = p N N − p + p ( t − s ) , for 1 < p < N ,

(1.14) − ∞ < s < N − p p , N − p p < t − s ⩽ 1 , q = p N N − p + p ( t − s ) , for p ⩾ N , p > 1 .

By the density, (1.12) holds for all u ∈ D 1 , p ( R N ; ∣ x ∣ − p s ) .

Let us reset α = − t q and σ = − p s . Then (1.12) is reformulated as

(1.15) ∫ R N ∣ x ∣ α ∣ u ∣ q ∗ ( σ , α ) d x p q ∗ ( σ , α ) ⩽ C s , t ∫ R N ∣ x ∣ σ ∣ ∇ u ∣ p d x , ∀ u ∈ D 1 , p ( R N ; ∣ x ∣ σ ) ,

where

(1.16) q ∗ ( σ , α ) ≔ p ( N + α ) N + σ − p ,

and (1.13), (1.14) can be reformulated, respectively, as

(1.17) p − N < σ < ∞ , σ − p ⩽ α ⩽ σ N N − p , for 1 1 .

For p − N < σ < ∞ , from the proof of Theorem 1.2 we see that q ∗ ( σ , α ) coincides the upper critical exponent, given by (1.7), of the embedding (1.8). We name the number q ∗ ( σ , α ) as the CKN critical exponent.

We remark here that in the case that p − N < σ < ∞ and 1 < p < N , the region of α for the CKN inequality is

(1.18) σ − p ⩽ α ⩽ σ N N − p .

Su and Wang in [17] proved inequality (1.15) for radial functions which extended the region of α for the CKN inequality to the case σ − p ⩽ α < ∞ . The case p = 2 was obtained in Catrina and Wang [5, Theorem 1.3].

Theorem II

[17] Assume 1 < p < N + σ and σ − p ⩽ α < ∞ . Then

(1.19) C ˜ ∫ R N ∣ x ∣ α ∣ u ∣ q ∗ ( σ , α ) d x p q ∗ ( σ , α ) ⩽ ∫ R N ∣ x ∣ σ ∣ ∇ u ∣ p d x , ∀ u ∈ D r 1 , p ( R N ; ∣ x ∣ σ ) ,

where

C ˜ = ω N p + α − σ N + α N + σ − p p − 1 ( p − 1 ) ( p + α − σ ) N + α N + σ − p p p ( N + σ − p ) N + α ,

and ω N is the surface area of the unit sphere in R N .

For the special case σ = 0 and α = − p , C ˜ = N − p p p is exactly the best Hardy constant. In general case, the constant C ˜ in (1.19) may not be optimal. We investigate the best constant for (1.19) in Theorem II by an alternative way. We define

(1.20) S σ , α = inf u ∈ D r 1 , p ( R N ; ∣ x ∣ σ ) ⧹ { 0 } ∫ R N ∣ x ∣ σ ∣ ∇ u ∣ p d x ∫ R N ∣ x ∣ α ∣ u ∣ q ∗ ( σ , α ) d x p q ∗ ( σ , α ) .

Theorem 1.4

Assume 1 < p < N + σ and σ − p < α < ∞ . Then

(1.21) S σ , α = ( N + α ) 1 p + α − σ p + α − σ N + α N + σ − p p − 1 p − 1 ( p − 1 ) ω N B N + α p + α − σ , ( p − 1 ) ( N + α ) p + α − σ p + α − σ N + α ,

where B ( ⋅ , ⋅ ) is the Beta function. Furthermore, S σ , α can be uniquely (up to dilations) achieved by the extremal function

(1.22) U ε , σ , α ( x ) = ε N + σ − p p ( p − 1 ) ( N + σ − p ) p − 1 ( N + α ) ( p − 1 ) p − 1 N + σ − p p ( p + α − σ ) ε p + α − σ p − 1 + ∣ x ∣ p + α − σ p − 1 N + σ − p p + α − σ w i t h ε > 0 .

We remark here that the best constant (1.21) and the extremal function (1.22) with the CKN critical exponent (1.16) have been widely considered. In [4], the authors established a transformation in investigating the best constant, the existence or nonexistence, and the symmetry of the extremal function for p = 2 in the setting (1.12)–(1.14). In [2], the symmetry-breaking phenomenon of the extremal functions for CKN inequality with general p was first revealed. The result of Theorem 1.4 extends the CKN critical exponent (1.16) in radial case to a large region of the power α of the weight ∣ x ∣ α . This means that for all α > σ − p , the best constant (1.20) is achieved exactly by the function (1.22) and can also be written explicitly as (1.21). The function U ε , σ , α given by (1.22) is a solution of the weighted critical equation

(1.23) − div ( ∣ x ∣ σ ∣ ∇ u ∣ p − 2 ∇ u ) = ∣ x ∣ α u q ∗ ( σ , α ) − 1 , u > 0 , in R N , u ∈ D r 1 , p ( R N ; ∣ x ∣ σ ) .

From (1.19) and (1.20) we have that S σ , α ⩾ C ˜ for α > σ − p .

For the problems restricted to the unit ball B (or a bounded spherically symmetric domain in R N ), there is only the upper critical exponent for the embeddings being concerned and we only need the conditions on the weights A and Q near the origin.

We make the following assumptions.

A ∈ C ( 0 , 1 ) , A ( r ) > 0 , there exists ℓ 0 > p − N such that lim r → 0 A ( r ) r ℓ 0 > 0 .
Q ∈ C ( 0 , 1 ) , Q ( r ) > 0 , and there exists b 0 ∈ R such that limsup r → 0 Q ( r ) r b 0 < ∞ .

Let W 0 , r 1 , p ( B ; A ) be the completion of C 0 , r ∞ ( B ) with respect to ‖ u ‖ = ∫ B A ( x ) ∣ ∇ u ∣ p d x 1 / p . Define q ⋆ = p ( N + b 0 ) N + ℓ 0 − p . Following the arguments in the proof of [17, Theorem 3.2], we have

Theorem 1.5

Assume ( A ′ ) and ( Q ′ ) with 1 < p < N + ℓ 0 and b 0 ⩾ ℓ 0 − p . The embedding

(1.24) W 0 , r 1 , p ( B ; A ) ↪ L q ( B ; Q )

is continuous for 1 ⩽ q ⩽ q ⋆ . The embedding (1.24) is compact for 1 ⩽ q < q ⋆ .

For the weighted critical exponent on the unit ball, we have a similar result.

Theorem 1.6

Assume ( A ′ ) and ( Q ′ ) with 1 q ⋆ . The embedding

W 0 , r 1 , p ( B ; A ) ↪ L q ⋆ ( B ; Q )

is not compact.

Now, let A ( x ) = ∣ x ∣ σ and Q ( x ) = ∣ x ∣ α , we consider the embedding,

(1.25) W 0 , r 1 , p ( B ; ∣ x ∣ σ ) ↪ L q ( B ; ∣ x ∣ α ) .

Based on Theorems 1.5 and 1.6, we obtain the special results (Table 4).

Table 4

1 < p < N + σ

q	α	Embedding
1 ⩽ q ⩽ q ∗ ( α , β )	α ⩾ σ − p	(1.25) holds
1 ⩽ q < q ∗ ( α , β )	α ⩾ σ − p	(1.25) is compact
q = q ∗ ( α , β )	α ⩾ σ − p	q ∗ ( α , β ) is upper critical exponent of (1.25), (1.25) is noncompact

We remark here that the embedding theorems obtained above are very important and are basic tools for applying the variational methods in the quasilinear elliptic equations of different types. For example, the compact embedding (1.9) with σ = β = 0 has been applied to study a class of weighted critical quasilinear elliptic problems in [20].

There are a few of interesting open problems on the weighted critical exponents for the embeddings. Is the upper embedding exponent q ∗ for the second case critical and is the lower exponent q ∗ critical in Theorem I? In particular, is the lower exponent q ∗ ( σ , α , β ) defined by (1.6) for the case α > β critical? We will keep working on these problems.

The article is organized as follows. In Section 2, we give the proof of Theorem 1.1. In Section 3, we give the proofs of Theorems 1.2, 1.3, 1.4, and 1.6.

2 Proof of Theorem 1.1

In this section, we give the proof of Theorem 1.1. We follow the arguments in [17].

Proof of Theorem 1.1

To prove the compactness of the embedding (1.3), we only need to prove that for any sequence { u k } ⊂ W 1 , p ( R N ; A , V ) ,

(2.1) u k ⇀ 0 weakly in W 1 , p ( R N ; A , V )

implies { u k } ⊂ L p ( R N ; Q ) and

(2.2) lim k → ∞ ∫ R N Q ( x ) ∣ u k ∣ p d x = 0 .

By (2.1), there exists M > 0 such that

(2.3) ‖ u k ‖ A , V p ⩽ M , ∀ k ∈ N .

By ( A ), ( V ), and ( Q ), there exist R 0 > r 0 > 0 , for some C 0 > 0 ,

(2.4) A ( x ) ⩾ C 0 ∣ x ∣ ℓ , V ( x ) ⩾ C 0 ∣ x ∣ a , Q ( x ) ⩽ C 0 ∣ x ∣ b , for ∣ x ∣ ⩾ R 0 , A ( x ) ⩾ C 0 ∣ x ∣ ℓ 0 , V ( x ) ⩾ C 0 ∣ x ∣ a 0 , Q ( x ) ⩽ C 0 ∣ x ∣ b 0 , for 0 < ∣ x ∣ ⩽ r 0 .

We will prove that for any ε > 0 there exist R > R 0 and 0 < r < r 0 such that the integrals

(2.5) ∫ R N ⧹ B R Q ( x ) ∣ u k ∣ p d x < ε 3 ,

(2.6) ∫ B r Q ( x ) ∣ u k ∣ p d x < ε 3 .

In the following we will use C to denote various constants.

We first consider the case out of a big ball. Since b < max { a , ℓ − p } , it holds that either b < a or b < ℓ − p .

Let b < a . For R > R 0 , we have by (2.3) and (2.4) that
(2.7) ∫ R N ⧹ B R Q ( x ) ∣ u k ∣ p d x ⩽ C 0 ∫ R N ⧹ B R ∣ x ∣ b ∣ u k ∣ p d x ⩽ R b − a ∫ R N V ( x ) ∣ u k ∣ p d x ⩽ M R b − a .
Let b < ℓ − p . We choose a smooth cutoff function ϕ ∈ C 0 ∞ ( R N ) satisfying
(2.8) ϕ ( x ) = 1 , for ∣ x ∣ ⩾ R 0 + 1 , 0 , for 0 ⩽ ∣ x ∣ ⩽ R 0 , 0 ⩽ ϕ ⩽ 1 , ∣ ∇ ϕ ( x ) ∣ ⩽ C .
Then for R > R 0 + 1 , we have by (2.3), (2.4), (2.8), and the CKN inequality (1.15) that

(2.9) ∫ R N ⧹ B R Q ( x ) ∣ u k ∣ p d x ⩽ C 0 R b + p − ℓ ∫ R N ∣ x ∣ ℓ − p ∣ ϕ u k ∣ p d x ⩽ C R b + p − ℓ ∫ R N ∣ x ∣ ℓ ∣ ∇ ( ϕ u k ) ∣ p d x ⩽ C R b + p − ℓ ∫ R N [ A ( x ) ∣ ∇ u k ∣ p + V ( x ) ∣ u k ∣ p ] d x ⩽ C M R b + p − ℓ .
In above we have used a fact that ∣ x ∣ ℓ ∣ ∇ ϕ ∣ p ⩽ C V ( x ) for R 0 ⩽ ∣ x ∣ ⩽ R 0 + 1 .

Next we consider the case in a small ball centered at the origin. It follows from b 0 > min { a 0 , ℓ 0 − p } that either b 0 > a 0 or b 0 > ℓ 0 − p .
Let b 0 > a 0 . For 0 < r < r 0 , we have by (2.3) and (2.4) that
(2.10) ∫ B r Q ( x ) ∣ u k ∣ p d x ⩽ r b 0 − a 0 ∫ R N V ( x ) ∣ u k ∣ p d x ⩽ M r b 0 − a 0 .
Let b 0 > ℓ 0 − p . We choose a smooth cutoff function ψ ∈ C 0 ∞ ( R N ) satisfying
(2.11) ψ ( x ) = 0 , for ∣ x ∣ ⩾ r 0 , 1 , for 0 ⩽ ∣ x ∣ ⩽ r 0 2 , 0 ⩽ ψ ⩽ 1 , ∣ ∇ ψ ( x ) ∣ ⩽ C .

Then for 0 < r < r 0 2 , we have by (2.3), (2.4), (2.11), and (1.15) that

(2.12) ∫ B r Q ( x ) ∣ u k ∣ p d x ⩽ C 0 r b 0 + p − ℓ 0 ∫ R N ∣ x ∣ ℓ 0 − p ∣ ψ u k ∣ p d x ⩽ C r b 0 + p − ℓ 0 ∫ R N ∣ x ∣ ℓ 0 ∣ ∇ ( ψ u k ) ∣ p d x ⩽ C r b 0 + p − ℓ 0 ∫ R N [ A ( x ) ∣ ∇ u k ∣ p + V ( x ) ∣ u k ∣ p ] d x ⩽ C M r b 0 + p − ℓ 0 .

Since b < max { a , ℓ − p } and b 0 > min { a 0 , ℓ 0 − p } , it follows from (2.7) or (2.9) and (2.10) or (2.12) that for any ε > 0 , there exist R > R 0 and r 0 > r > 0 such that (2.5) and (2.6) hold.

For R > r > 0 given above, we consider the integral over B R ⧹ B r . We have by ( Q ) that

(2.13) ∫ B R ⧹ B r Q ( x ) ∣ u k ∣ p d x ⩽ C ∫ B R ⧹ B r ∣ x ∣ b 0 ∣ u k ∣ p d x ⩽ C ∫ B R ⧹ B r ∣ u k ∣ p d x .

By the classical Sobolev embedding theorem [1], we have that

(2.14) W 1 , p ( B R ⧹ B r ; A , V ) ↪ W 1 , p ( B R ⧹ B r ) ↪ ↪ L p ( B R ⧹ B r ) ,

where ↪ ↪ indicates compact embedding. It follows from (2.13), (2.14), and (2.1) that

(2.15) lim k → ∞ ∫ B R ⧹ B r Q ( x ) ∣ u k ∣ p d x = 0 .

Since ε > 0 is arbitrary, by (2.5), (2.6), and (2.15) we arrive at (2.2). The proof is complete.□

We remark that in Theorem 1.1, the functions A , V , and Q are not needed to be radial.

3 The weighted critical exponents

In this section, we give the proofs of those theorems about the weighted critical exponents. We first prove Theorem 1.2. Denote B R = { x ∈ R N : ∣ x ∣ < R } .

Proof of Theorem 1.2

To prove the first conclusion of this theorem, we only need to construct three weight radial functions A ˜ , V ˜ , and Q ˜ , satisfying the assumptions ( A ) , ( V ) , and ( Q ) , respectively, and to construct a sequence { u k } such that for any q > q ∗ , { u k } is bounded in W r 1 , p ( R N , A ˜ , V ˜ ) but { u k } is unbounded in L q ( R N ; Q ˜ ) . Define a smooth cutoff function φ ∈ C 0 , r ∞ ( R N ) satisfying φ ( x ) = 1 , for ∣ x ∣ ⩽ 1 0 , for ∣ x ∣ ⩾ 2 and 0 ⩽ φ ⩽ 1 . Define

A ˜ ( x ) = φ ( x ) ∣ x ∣ ℓ 0 + ( 1 − φ ( x ) ) ∣ x ∣ ℓ , V ˜ ( x ) = φ ( x ) ∣ x ∣ a 0 + ( 1 − φ ( x ) ) ∣ x ∣ a , Q ˜ ( x ) = φ ( x ) ∣ x ∣ b 0 + ( 1 − φ ( x ) ) ∣ x ∣ b .

Then A ˜ , V ˜ , and Q ˜ satisfy assumptions ( A ) , ( V ) , and ( Q ) , respectively. Construct a sequence { u k } k = 1 ∞ as follows:

(3.1) u k ( x ) = k N + ℓ 0 − p p 2 e − k ∣ x ∣ p p ⋅ q ∗ , k ∈ N .

For the integral term containing gradient, we have

(3.2) ∫ R N A ˜ ( x ) ∣ ∇ u k ∣ p d x ⩽ ∫ B 2 ∣ x ∣ ℓ 0 ∣ ∇ u k ∣ p d x + ∫ R N ⧹ B 1 ∣ x ∣ ℓ ∣ ∇ u k ∣ p d x = k N + ℓ 0 − p p + p ( q ∗ ) p ω N ∫ 0 2 r N + ℓ 0 + p ( p − 1 ) − 1 e − k r p q ∗ d r + k N + ℓ 0 − p p + p ( q ∗ ) p ω N ∫ 1 ∞ r N + ℓ + p ( p − 1 ) − 1 e − k r p q ∗ d r = ( q ∗ ) N + ℓ 0 − p p ω N p ∫ 0 k 2 p q ∗ s N + ℓ 0 + p ( p − 1 ) p − 1 e − s d s s = k q ∗ r p + k ℓ 0 − ℓ p ( q ∗ ) N + ℓ − p p ω N p ∫ k q ∗ ∞ s N + ℓ + p ( p − 1 ) p − 1 e − s d s ⩽ ( q ∗ ) N + ℓ 0 − p p ω N p Γ N + ℓ 0 + p 2 − p p + k ℓ 0 − ℓ p ( q ∗ ) N + ℓ − p p ω N p ∫ k q ∗ ∞ s N + ℓ + p ( p − 1 ) p − 1 e − s d s ,

where Γ is the Gamma function. It is easily seen that

(3.3) k ℓ 0 − ℓ p ∫ k q ∗ ∞ s N + ℓ + p ( p − 1 ) p − 1 e − s d s → 0 as k → ∞ .

It follows from (3.2) and (3.3) that for some C 1 > 0

(3.4) ∫ R N A ˜ ( x ) ∣ ∇ u k ∣ p d x ⩽ C 1 , ∀ k ∈ N .

Next we have

(3.5) ∫ R N V ˜ ( x ) ∣ u k ∣ p d x ⩽ ∫ B 2 ∣ x ∣ a 0 ∣ u k ∣ p d x + ∫ R N ⧹ B 1 ∣ x ∣ a ∣ u k ∣ p d x = k N + ℓ 0 − p p ω N ∫ 0 2 r N − 1 + a 0 e − k r p q ∗ d r + k N + ℓ 0 − p p ω N ∫ 1 ∞ r N − 1 + a e − k r p q ∗ d r ⩽ ( q ∗ ) N + a 0 p k − p + a 0 − ℓ 0 p ω N p Γ N + a 0 p + ( q ∗ ) N + a p k − p + a − ℓ 0 p ω N p ∫ k q ∗ ∞ s N + a p − 1 e − s d s .

Combining (3.5) with

k − p + a − ℓ 0 p ∫ k q ∗ ∞ s N + a p − 1 e − s d s → 0 , as k → ∞ ,

we have for some C 2 > 0

(3.6) ∫ R N V ˜ ( x ) ∣ u k ∣ p d x ⩽ C 2 , ∀ k ∈ N .

By (3.4) and (3.6), we see that the sequence { u k } is bounded in W r 1 , p ( R N ; A ˜ , V ˜ ) .

Let q ⩾ q ∗ . Then we have

(3.7) ∫ R N Q ˜ ( x ) ∣ u k ∣ q d x ⩾ ∫ B 1 ∣ x ∣ b 0 ∣ u k ∣ q d x + ∫ R N ⧹ B 2 ∣ x ∣ b ∣ u k ∣ q d x ⩾ k q ( N + ℓ 0 − p ) p 2 ω N ∫ B 1 ∣ x ∣ b 0 e − k q ∣ x ∣ p p ⋅ q ∗ d x = ω N p p q ∗ q N + b 0 p k q ( N + ℓ 0 − p ) − p ( N + b 0 ) p 2 ∫ 0 k q p q ∗ s N + b 0 p − 1 e − s d s .

Since q > q ∗ implies that q ( N + ℓ 0 − p ) − p ( N + b 0 ) > 0 and

(3.8) lim k → ∞ ∫ 0 k q p q ∗ s N + b 0 p − 1 e − s d s = Γ N + b 0 p ,

it follows from (3.7) that

(3.9) ∫ R N Q ˜ ( x ) ∣ u k ∣ q d x → ∞ as k → ∞ .

Therefore, there is no embedding from W r 1 , p ( R N ; A , V ) into L q ( R N ; Q ) for any q > q ∗ .

Now we prove the second conclusion of this theorem. Let q = q ∗ in (3.7). Then we have

(3.10) ∫ R N Q ˜ ( x ) ∣ u k ∣ q ∗ d x ⩾ p N + b 0 − p p ω N ∫ 0 k p s N + b 0 p − 1 e − s d s → p N + b 0 − p p ω N Γ N + b 0 p > 0 ,

as k → ∞ . From the definition in (3.1) we see that

(3.11) u k ( x ) → 0 a.e. in R N as k → ∞ .

Therefore, { u k } has no convergent subsequence in L q ∗ ( R N ; Q ˜ ) .□

Remark 3.1

From the arguments in the proof of Theorem 1.2 we have some remarks. Since W r 1 , p ( R N ; A , V ) is a subspace of W 1 , p ( R N ; A , V ) , under the assumptions of Theorem 1.2, there is no embedding from W 1 , p ( R N ; A , V ) into L q ( R N ; Q ) for any q > q ∗ . In particular, there is no embedding from W 1 , p ( R N ; ∣ x ∣ σ , ∣ x ∣ β ) into L q ( R N ; ∣ x ∣ α ) for any q > q ∗ ( σ , α , β ) when 1 < p < N + σ , β ⩾ σ − p , and α ⩾ σ − p . However, we do not know whether or not there is a continuous embedding from W 1 , p ( R N ; A , V ) into L q ∗ ( R N ; Q ) .

Now we turn our attention to the lower critical exponent and give the proof of Theorem 1.3.

Proof of Theorem 1.3

Define

A ^ ( x ) = ψ ( x ) ∣ x ∣ ℓ 0 + ( 1 − ψ ( x ) ) ∣ x ∣ ℓ , V ^ ( x ) = ψ ( x ) ∣ x ∣ a 0 + ( 1 − ψ ( x ) ) ∣ x ∣ a , Q ^ ( x ) = ψ ( x ) ∣ x ∣ b 0 + ( 1 − ψ ( x ) ) ∣ x ∣ b ,

where ψ ∈ C 0 , r ∞ ( R N ) satisfying ψ ( x ) = 1 , for ∣ x ∣ ⩽ 1 0 , for ∣ x ∣ ⩾ 2 and 0 ⩽ ψ ⩽ 1 . Then A ^ , V ^ , and Q ^ satisfy assumptions ( A ) , ( V ) , and ( Q ) , respectively. Define a sequence { u k } k = 1 ∞ as follows:

(3.12) u k ( x ) = k − N + b p 2 e − ∣ x ∣ p p k , k ∈ N .

Then we have

(3.13) ∫ R N A ^ ( x ) ∣ ∇ u k ∣ p d x ⩽ ∫ B 2 ∣ x ∣ ℓ 0 ∣ ∇ u k ∣ p d x + ∫ R N ⧹ B 1 ∣ x ∣ ℓ ∣ ∇ u k ∣ p d x = ω N k − N + b p − p ∫ 0 2 r N + ℓ 0 + p ( p − 1 ) − 1 e − r p k d r + ω N k − N + b p − p ∫ 1 ∞ r N + ℓ + p ( p − 1 ) − 1 e − r p k d r ⩽ ω N k − N + b p − p ∫ 0 2 r N + ℓ 0 + p ( p − 1 ) − 1 d r + k ℓ − b − p p ω N p ∫ 1 k ∞ s N + ℓ + p ( p − 1 ) p − 1 e − s d s ⩽ 2 N + ℓ 0 + p ( p − 1 ) ω N N + ℓ 0 + p ( p − 1 ) + ω N k ℓ − b − p p p Γ N + ℓ + p ( p − 1 ) p ,

(3.14) ∫ R N V ^ ( x ) ∣ u k ∣ p d x ⩽ ∫ B 2 ∣ x ∣ a 0 ∣ u k ∣ p d x + ∫ R N ⧹ B 1 ∣ x ∣ a ∣ u k ∣ p d x = k − N + b p ω N ∫ 0 2 r N − 1 + a 0 e − r p k d r + k − N + b p ω N ∫ 1 ∞ r N − 1 + a e − r p k d r ⩽ k − N + b p ω N ∫ 0 2 r N − 1 + a 0 d r + ω N p ∫ 1 k ∞ s N + a p − 1 e − s d s ⩽ ω N 2 N + a 0 N + a 0 + ω N p Γ N + a p .

It follows from (3.13) and (3.14) that the sequence { u k } is bounded in W r 1 , p ( R N ; A ^ , V ^ ) .

For q < p , we have

(3.15) ∫ R N Q ^ ( x ) ∣ u k ∣ q d x ⩾ ∫ R N ⧹ B 2 ∣ x ∣ b ∣ u k ∣ q d x = 1 p p q N + b p k ( p − q ) ( N + b ) p 2 ω N ∫ q 2 p p k ∞ s N + b p − 1 e − s d s .

Since N + b > 0 , we have that

(3.16) lim k → ∞ ∫ q 2 p p k ∞ s N + b p − 1 e − s d s = Γ N + b p .

It follows from (3.15), (3.16), and q < p that

(3.17) ∫ R N Q ^ ( x ) ∣ u k ∣ q d x → ∞ as k → ∞ .

Therefore, there are no embeddings from W r 1 , p ( R N ; A , V ) ↪ L q ( R N ; Q ) for any q < p .

Now on one hand, we have

(3.18) ∫ R N Q ^ ( x ) ∣ u k ∣ p d x ⩾ ∫ R N ⧹ B 2 ∣ x ∣ b ∣ u k ∣ p d x = ω N p ∫ 2 p k ∞ s N + b p − 1 e − s d s → ω N p Γ N + b p > 0 , as k → ∞ .

It follows that { ‖ u k ‖ L p ( R N ; Q ^ ) } is bounded from below by a positive constant. On the other hand, by the definition of (3.12), we see easily that

(3.19) u k ( x ) → 0 a.e. in R N as k → ∞ .

Thus, { u k } has no convergent subsequence in L p ( R N ; Q ^ ) .□

Remark 3.2

From the arguments in the proof of Theorem 1.3, we have a new observation. The assumptions of Theorem 1.3 plus Q ( x ) ⩽ V ( x ) for all x ∈ R N imply that there is no embedding from W 1 , p ( R N ; A , V ) into L p ( R N ; Q ) for any q < p and the embedding W 1 , p ( R N ; A , V ) ↪ L p ( R N ; Q ) is not compact.

Next we give the proof of Theorem 1.4. We follow some ideas from [8] by Gladiali et al. After finishing this article we found a paper by Horiuchi [10]. By a careful comparison we saw that (1.21) and (1.22) had been given in [10] without a detailed argument. We would like to keep the arguments of the proof for the readers’ convenience.

Proof of Theorem 1.4

For u ∈ D r 1 , p ( R N ) , we have

(3.20) ∫ R N ∣ x ∣ σ ∣ ∇ u ∣ p d x = ω N ∫ 0 ∞ r N + σ − 1 ∣ u ′ ( r ) ∣ p d r .

Set

(3.21) r = s p p + α − σ , v ( s ) = u ( s p p + α − σ ) = u ( r ) .

Then

(3.22) ∫ 0 ∞ r N + σ − 1 ∣ u ′ ( r ) ∣ p d r = p + α − σ p p − 1 ∫ 0 ∞ s κ − 1 ∣ v ′ ( s ) ∣ p d s ,

where

(3.23) κ = p ( N + α ) p + α − σ > p .

For 1 < p < κ and τ = κ p κ − p , we obtain by applying [18, Lemma 2] that

(3.24) ∫ 0 ∞ s κ − 1 ∣ v ′ ( s ) ∣ p d s ⩾ S κ ( R + ) ∫ 0 ∞ s κ − 1 ∣ v ( s ) ∣ τ d s p τ ,

where

(3.25) S κ ( R + ) = κ κ − p p − 1 p − 1 p − 1 p B κ p , κ ( p − 1 ) p p κ .

Now it follows from (3.20) to (3.25) that

(3.26) ∫ R N ∣ x ∣ σ ∣ ∇ u ∣ p d x ⩾ ω N p + α − σ p p − 1 S κ ( R + ) ∫ 0 ∞ s κ − 1 ∣ v ( s ) ∣ τ d s p τ = ω N p κ p + α − σ p p ( κ − 1 ) κ S κ ( R + ) ω N ∫ 0 ∞ r N + α − 1 ∣ u ( r ) ∣ p κ κ − p d r κ − p κ = S σ , α ∫ R N ∣ x ∣ α ∣ u ∣ q ∗ ( σ , α ) d x p q ∗ ( σ , α ) ,

where

(3.27) S σ , α = ω N p κ p + α − σ p p ( κ − 1 ) κ S κ ( R + ) = κ p + α − σ p p ( κ − 1 ) κ κ − p p − 1 p − 1 ω N p − 1 p B κ p , κ ( p − 1 ) p p κ = ( N + α ) 1 p + α − σ p + α − σ N + α N + σ − p p − 1 p − 1 ( p − 1 ) ω N B N + α p + α − σ , ( p − 1 ) ( N + α ) p + α − σ p + α − σ N + α .

By the density, we arrive at (1.20) and (1.21).

It follows from (3.24) and (3.21) that an extremal function u for S σ , α is exactly an extremal function v for S κ ( R + ) , i.e., the function v satisfies

(3.28) ∫ 0 ∞ s κ − 1 ∣ v ′ ( s ) ∣ p d s = S κ ( R + ) ∫ 0 ∞ s κ − 1 ∣ v ( s ) ∣ κ p κ − p d s κ − p κ .

By [18, Lemma 2], there is a unique extremal function for S κ ( R + ) taking the form

(3.29) v ˜ ( s ) = m + n s p p − 1 p − κ p , m , n > 0 .

Changing variable back to ∣ x ∣ by (3.21), we see that the extremal function for S σ , α ( R N ) takes the form

(3.30) u ˜ ( x ) = m + n ∣ x ∣ p + α − β p − 1 − N + σ − p p + α − σ .

Precisely, the function u ˜ satisfies

(3.31) ∫ R N ∣ x ∣ σ ∣ ∇ u ˜ ∣ p d x = S σ , α ∫ R N ∣ x ∣ α ∣ u ˜ ∣ q ∗ ( σ , α ) d x p q ∗ ( σ , α ) .

A complicated calculation shows that

(3.32) S σ , α ∫ R N ∣ x ∣ α ∣ u ˜ ∣ q ∗ ( σ , α ) d x p − q ∗ ( σ , α ) q ∗ ( σ , α ) = ( N + σ − p ) p − 1 ( N + α ) ( p − 1 ) p − 1 m n p − 1 .

By the Lagrange multiplier theorem and (3.32) we have

(3.33) − div ( ∣ x ∣ σ ∣ ∇ u ˜ ∣ p − 2 ∇ u ˜ ) = ( N + σ − p ) p − 1 ( N + α ) ( p − 1 ) p − 1 m n p − 1 ∣ x ∣ α u ˜ q ∗ ( σ , α ) − 1 .

After a first scaling, we have that the function

(3.34) U ( x ) = ( N + σ − p ) p − 1 ( N + α ) ( p − 1 ) p − 1 N + σ − p p ( p + α − σ ) n m ( p − 1 ) ( N + σ − p ) p ( p + α − σ ) 1 + n m ∣ x ∣ p + α − σ p − 1 N + σ − p p + α − σ

is the extremal function for S σ , α . By scaling invariance, for λ > 0 , we obtain U λ ( x ) ≔ λ N + σ − p p U ( λ x ) is the extremal function of S σ , α and a solution of equation (1.23). Taking λ 0 = m n p − 1 p + α − σ , then

(3.35) U λ 0 ( x ) = ( N + σ − p ) p − 1 ( N + α ) ( p − 1 ) p − 1 N + σ − p p ( p + α − σ ) 1 + ∣ x ∣ p + α − σ p − 1 N + σ − p p + α − σ .

By a second scaling, we obtain the functions

U ε , σ , α ( x ) = ε N + σ − p p ( p − 1 ) ( N + σ − p ) p − 1 ( N + α ) ( p − 1 ) p − 1 N + σ − p p ( p + α − σ ) ε p + α − σ p − 1 + ∣ x ∣ p + α − σ p − 1 N + σ − p p + α − σ ,

which are the extremal functions for S σ , α and the solutions of (1.23).□

Proof of Theorem 1.6

The proof is similar to Theorem 1.2. Define A ¯ ( x ) = ∣ x ∣ ℓ 0 , Q ¯ ( x ) = ∣ x ∣ b 0 . They satisfy ( A ′ ) and ( Q ′ ) , respectively. Define a sequence { u k } k = 1 ∞ as follows

(3.36) u k ( ∣ x ∣ ) = k N + ℓ 0 − p p 2 e − k ∣ x ∣ p p ⋅ q ⋆ − e − k p ⋅ q ⋆ , ∣ x ∣ < 1 , k ∈ N .

Since

(3.37) ∫ B A ¯ ( x ) ∣ ∇ u k ∣ p d x = k N + ℓ 0 − p p + p ( q ⋆ ) p ∫ B ∣ x ∣ ℓ 0 + p ( p − 1 ) e − k ∣ x ∣ p q ⋆ d x ⩽ ( q ⋆ ) N + ℓ 0 − p p ω N p Γ N + ℓ 0 + p ( p − 1 ) p .

It follows that { u k } is bounded in W 0 , r 1 , p ( B , A ¯ ) . Let q > q ⋆ . By a direct calculation, we have

(3.38) ∫ B Q ¯ ( x ) ∣ u k ∣ q d x = ω N p k q ( N + ℓ 0 − p ) p 2 − N + b 0 p ∫ 0 k s N + b 0 p − 1 e − s p ⋅ q ⋆ − e − k p ⋅ q ⋆ q d s ⩾ 1 p k q ( N + ℓ 0 − p ) p 2 − N + b 0 p e − 1 2 p ⋅ q ⋆ − e − 1 p ⋅ q ⋆ q ω N ∫ 0 1 2 s N + b 0 p − 1 d s = ω N ( N + b 0 ) 2 − N + b 0 p p 2 k q ( N + ℓ 0 − p ) p 2 − N + b 0 p e − 1 2 p ⋅ q ⋆ − e − 1 p ⋅ q ⋆ q .

Since q > q ⋆ implies q ( N + ℓ 0 − p ) − p ( N + b 0 ) > 0 , we have by (3.38) that

(3.39) ∫ B Q ¯ ( ∣ x ∣ ) ∣ u k ∣ q d x → ∞ as k → ∞ .

Therefore, there is no embedding from W r 1 , p ( B ; A ¯ ) into L q ( B ; Q ¯ ) for any q > q ⋆ . On one hand, we have

(3.40) ∫ B Q ¯ ( x ) ∣ u k ∣ q ⋆ d x = ω N p ∫ 0 k s N + b 0 p − 1 e − s p ⋅ q ⋆ − e − k p ⋅ q ⋆ q ⋆ d s ⩾ 1 p e − 1 2 p ⋅ q ⋆ − e − 1 p ⋅ q ⋆ q ⋆ ω N ∫ 0 1 2 s N + b 0 p − 1 d s = ω N ( N + b 0 ) 2 N + b 0 p p 2 e − 1 2 p ⋅ q ⋆ − e − 1 p ⋅ q ⋆ q ⋆ > 0 .

On the other hand, by the definition of (3.36), we have

(3.41) u k ( x ) → 0 a.e. in B as k → ∞ .

Thus, { u k } has no convergent subsequence in L q ⋆ ( B ; Q ¯ ) .□

Remark 3.3

We finally remark that under the assumptions of Theorem 1.6 there is no embedding from W 0 1 , p ( B ; A , V ) into L q ( B ; Q ) for any q > q ⋆ and there is no embedding from W 0 1 , p ( B ; ∣ x ∣ σ ) into L q ( B ; ∣ x ∣ α ) for any q > q ∗ ( σ , α ) as 1 < p < N + σ , α ⩾ σ − p .

Acknowledgements

The authors appreciate Professor Zhi-Qiang Wang for giving valuable suggestions and providing references [5,12]. We thank editors and the anonymous referees for a careful reading of the results and for making suggestions to improve the original manuscript.

Funding information: This work is supported by KZ202010028048 and NSFC(11771302, 12171326).
Conflict of interest: The authors state no conflict of interest.

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Received: 2021-08-27

Accepted: 2022-03-22

Published Online: 2022-04-20

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