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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

含临界增长奇异椭圆问题的临界维现象

代晋军

，彭双阶

，彭艳芳

华中师范大学数学与统计学科学学院，武汉　 430079

贵州师范大学数学与计算机科学学院，贵阳 550001

摘要：本文研究一类奇异椭圆 Dirichlet 边值问题，这个问题涉及到著名的

Caﬀarelli-Kohn-Nirenberg 不等式。利用山路引理我们可以得到一个正解，而且，我们证明参

数 µ 和 d 在某些范围时，这个问题出现临界维现象。

关键词：正解，临界维，奇异性，Caﬀarelli-Kohn-Nirenberg 不等式

中图分类号： O175

Critical Dimensions for Singular Elliptic

Problems with Critical Growth

Dai jin-jun

, Peng Shuang-Jie

，Peng Yan-fang

School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079

Guizhou Normal Univercity, School of Mathematics and Computer Science, Guiyang, 550001

Abstract: In this paper, we investigate a class of elliptic equation with Dirichlet boundary

condition, which involves the well-known Caﬀarelli-Kohn-Nirenberg inequalities. We obtain a

positive solution to this problem by using a mountain pass lemma. Moreover, we prove that

in some ranges of the parameters µ and d, critical dimension phenomenon occurs for this

problem.

Key words: positive solutions, critical dimension, singularity, Caﬀarelli-Kohn-Nirenberg

inequality.

0 Introduction

In this paper, we consider the following elliptic problem with singular coeﬃcients











−div(|x|

−2a

∇u) − µ

|x|

2(1+a)

p−1

|x|

+ λ

|x|

, u > 0, x ∈ Ω,

u = 0 , x ∈ ∂Ω,

(1)

where Ω is a smooth bounded domain in R

(N ≥ 3), 0 ∈ Ω, 0 ≤ µ < (

√

¯µ − a)

, ¯µ =:

(

N−2

)

, 0 ≤ a <

√

¯µ, a ≤ b < a + 1, a ≤ d < a + 1, d ≤ b, λ > 0, p = p(a, b) =:

N−2(1+a−b)

, D = D(a, d) =:

N−2(1+a−d)

Problem (1) has the form

−div(a(x)∇u) = b(x)f (x, u), x ∈ Ω, (2)

Foundations: This work was supported by the PhD Specialized Grant of Ministry of Education of China (Grant

20110144110001) and NSFC(Great 11501143).

Author Introduction: Dai jin-jun（1978-），male，lecturer，major research direction：Partial diﬀerential equations E-

mail:jjdai@mail.ccnu.edu.cn. Correspondence author：Peng Shuang-Jie（1968-），male，professor，major research direction：

Partial diﬀerential equations. E-mail: sjpeng@mail.ccnu.edu.cn. Peng Yan-fang（1982-），famale，associate professor，major

research direction：Partial diﬀerential equations. E-mail:pyfang2005@sina.com.

- 1 -

，彭双阶

，彭艳芳

430079

550001

hlet 边值问题，这个问题涉及到著名的

ohn-Nirenberg 不等式

for Singular Elliptic

with Critical Growth

eng Shuang-Jie

，Peng Yan-fang

Central China Normal University, Wuhan, 430079

ol of Mathematics and Computer Science, Guiyang, 550001

a class of elliptic equation with Dirichlet boundary

wn Caﬀarelli-Kohn-Nirenberg inequalities. We obtain a

using a mountain pass lemma. Moreover, we prove that

and d, critical dimension phenomenon occurs for this

dimension, singularity, Caﬀarelli-Kohn-Nirenberg

following elliptic problem with singular coeﬃcients

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

where the weights a(x) and b(x) are non-negative measurable functions which are allowed to

be unbounded. We just remark that equations like (2) are introduced as models for several

physical phenomena related to equilibrium of anisotropic continuous media which possibly are

somewhere “perfect” insulators (see [1]). When a = 0, (1) has Hardy-type singular potential

|x|

and has been the object of a quite large interest in the recent literature, see e.g. [2, 3, 4, 5,

6, 7, 8, 9, 10, 11, 12]. The singularity of potential

|x|

2(1+a)

is critical both from the mathematical

and the physical point of view. As it does not belong to the Kato’s class, it cannot be regarded

as a lower order perturbation of the laplacian but strongly inﬂuences the properties of the

associated elliptic operator. Moreover singular potentials arise in many ﬁelds, such as quantum

mechanics, nuclear physics, molecular physics, and quantum cosmology; we refer to [13] for

further discussion and motivation.

Mathematically, problem (1) is related to the following well-known Caﬀarelli-Kohn-Nirenberg

inequalities [14]





|x|

−bp

|u|



≤ C

a,b



|x|

−2a

|∇u|

, ∀u ∈ C

∞

), (3)

where 0 ≤ a <

√

¯µ, a ≤ b < a + 1, p = p(a, b) =:

N−2(1+a−b)

. p = p(a, b) is called the critical

Sobolev-Hardy exponent since when a = b = 0 and a = 0, b = 1, (3) are classical Sobolev

and Hardy inequalities respectively. (3) has played an important role in many applications by

virtue of the complete knowledge about the best constant C

a,b

and the extremal functions (see

[15]).

Concerning (1) and the related problems, there are many results, we can refer to [2, 3,

5, 6, 8, 10, 11, 12, 15, 16, 17, 18, 19, 20] and references therein. For the quasilinear problems

involving critical Soblev-Hardy exponents, we can see [21] and [22]. We can also refer to [23]

and [24] for more recent results on this type of problems.

According to the deﬁnition proposed by Pucci and Serrin in [25] and [26], we say that a

dimension N is critical for a second order nonlinear elliptic equation



T u = f (x, u) + λh(x)u, u > 0, x ∈ Ω,

u = 0 , x ∈ ∂Ω

(4)

if there exists a bounded domain Ω ⊂ R

in which (1.4) has no solutions for some λ ∈ (0, λ

where T is a second order nonlinear elliptic operator, λ

is the ﬁrst eigenvalue related to the

operator T and f(x, u) is a nonlinear term, critical with respect to T . In the prominent literature

[27], N = 3 was proved to be critical for problem (1) in the case a = b = d = µ = 0. In [11], E.

Jannelli proved that when a = b = d = 0, any dimension N ≥ 3 is critical if ¯µ − 1 < µ < ¯µ.

In this paper, we are concerned with the existence and the critical dimension for problem

(1). For µ ∈ [0 , (

√

¯µ − a)

), deﬁne H = H

(Ω, |x|

−2a

) to be the completion of C

∞

(Ω) with

respect to the norm

∥u∥ = ∥u∥





Ω



|x|

−2a

|∇u|

− µ

|u|

|x|

2(1+a)





. (5)

Set b = a + 1 in (1.3), we have the following weighted Hardy inequality ([16], [15]),



|u|

|x|

2(1+a)

≤

(

√

¯µ − a)



|x|

−2a

|∇u|

, ∀ u ∈ C

∞

). (6)

Hence norm (5) is well deﬁned and equivalent to the usual norm (



Ω

|x|

−2a

|∇u|

dx)

- 2 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

Deﬁne the energy functional corresponding to problem (1)

J(u) =



Ω



|x|

−2a

|∇u|

− µ

|x|

2(1+a)

− λ

|x|



−



Ω

|x|

, ∀u ∈ H, (7)

where u

= max{0, u}. Due to the invariance of H−norm, L

(Ω, |x|

−bp

)−norm and



Ω

|x|

2(1+a)

with respect to rescaling u 7→ u

(·) = r

N−2−2a

u(r(·)), J(u) fails to satisfy the classical Palais-

Smale (P.S. in short) condition in H. Moreover, since the solutions to (1) may be singular and

the operator −div(|x|

−2a

∇·) is not elliptic uniformly in Ω, the strong maximum principle and

regularity theory cannot be applied directly. Meanwhile, concerning the critical dimensions for

(1), it is natural to investigate the nonexistence of solutions in a ball centered at the origin.

Hence we should study the singularity and radial symmetry of the solutions when Ω = B

(0).

Deﬁne β =



(

√

¯µ − a)

− µ, ν =

√

¯µ − a − β, γ =

√

¯µ − a + β. Set operator L(·) =

−div(|x|

−2a

∇·) −

|x|

2(1+a)

·. Deﬁne λ

∗

(µ) to be the ﬁrst eigenvalue of problem

L(u) = λ

|x|

, u ∈ H, (8)

and λ

(µ) the ﬁrst eigenvalue of problem

−div(|x|

−2(a+γ)

∇φ) = λ

|x|

2γ+dD

, φ ∈ H

(Ω, |x|

−2(a+γ)

), (9)

that is

∗

(µ) = inf

u∈H\{0}



Ω



|∇u|

|x|

− µ

|x|

2(1+a)







Ω

|x|

(µ) = inf

v∈H

(Ω,|x|

−2(a+γ)

)\{0}



Ω

|∇v|

|x|

2(a+γ)





Ω

|x|

2γ+dD

Our main result is as follows

Theorem 1. Suppose N ≥ 4 + 4a − dD, 0 ≤ a <

√

¯µ, a ≤ b < a + 1.

(i) If 0 ≤ µ ≤ (

√

¯µ − a)

− (1 + a −

)

, then problem (1) has a solution in H when

0 < λ < λ

∗

(µ).

(ii) If (

√

¯µ − a)

− (1 + a −

)

< µ < (

√

¯µ − a)

, then problem (1) has a solution in H

when λ

(µ) < λ < λ

∗

(µ).

(iii) If (

√

¯µ −a)

−(1 + a −

)

< µ < (

√

¯µ −a)

, and Ω = B

(0). Then problem (1) has

no solutions in H for λ ≤ λ

(µ).

Remark 2. In cases (ii) and (iii) of Theorem 1, (

√

¯µ −a)

−(1 + a −

)

< µ, we can easily

verify that N − 2(a + γ) > 0 and N − (2γ + dD) > 0, which guarantee that |x|

−2(a+γ)

and

|x|

−(2γ+dD)

are in L

(Ω). Hence the deﬁnition of λ

(µ) is reasonable.

Remark 3. From Theorem 1, the phenomenon of critical dimensions may occur for problem

(1).

To verify the theorem, we mainly employ the framework used in [11] and [27]. However,

compared with [11] and [27], in the present paper, the singularity of the solutions and the non-

uniform ellipticity of the operator −div(|x|

−2a

∇·) cause more diﬃculties and we need to ﬁnd

new arguments. First of all, to obtain positive solutions, a new maximum principle should be

established. Secondly, since −2(a + γ) < 2 − N in (9), Caﬀarelli-Kohn-Nirenberg inequalities

- 3 -

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