Báo cáo hóa học: "Research Article A Hardy Inequality with Remainder Terms in the Heisenberg Group and the Weighted Eigenvalue Problem" docx

Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2007, Article ID 32585, 24 pages
doi:10.1155/2007/32585
Research Article
A Hardy Inequality with Remainder Terms in the Heisenberg
Group and the Weighted Eigenvalue Problem
Jingbo Dou, Pengcheng Niu, and Zixia Yuan
Received 22 March 2007; Revised 26 May 2007; Accepted 20 October 2007
Recommended by L
´
aszl
´
oLosonczi
Based on properties of vector fields, we prove Hardy inequalities with remainder terms
in the Heisenberg group and a compact embedding in weighted Sobolev spaces. The best
constants in Hardy inequalities are determined. Then we discuss the existence of solutions
for the nonlinear eigenvalue problems in the Heisenberg group with weights for the p-
sub-Laplacian. The asymptotic behaviour, simplicity, and isolation of the first eigenvalue
are also considered.
Copyright © 2007 Jingbo Dou et al. This is an open access article distributed under the
Creative Commons Attribution License, w hich permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let
L
p,μ
u =−Δ
H,p
u −μψ
p

|u|
p−2
u
d
p
,0≤ μ ≤

Q − p
p

p
, (1.1)
be the Hardy operator on the Heisenberg group. We consider the following weighted
eigenvalue problem with a singular weight:
L
p,μ
u = λf(ξ)|u|
p−2
u,inΩ ⊂ H
n
,
u
= 0, on ∂Ω,
(1.2)
where 1 <p<Q
= 2n +2, λ ∈ R, f (ξ) ∈ Ᏺ
p
:={f : Ω→R
+
| lim

d(ξ)→0
(d
p
(ξ) f (ξ)/
(ψ
p
(ξ)) = 0, f (ξ) ∈ L
∞
loc
(Ω \{0})}, Ω is a bounded domain in the Heisenberg group, and
the definitions of d(ξ)andψ
p
(ξ); see below. We investigate the weak solution of (1.2)and
the asymptotic behavior of the first eigenvalue for different singular weights as μ increases
to ((Q
− p)/p)
p
. Furthermore, we show that the first eigenvalue is simple and isolated, as
2 Journal of Inequalities and Applications
well as the eigenfunctions corresponding to other eigenvalues change sign. Our proof is
mainly based on a Hardy inequality with remainder terms. It is established by the vec-
tor field method and an elementary integral inequality. In addition, we show that the
constants appearing in Hardy inequality are the best. Then we conclude a compact em-
bedding in the weighted Sobole v space.
Themaindifficulty to study the properties of the first eigenvalue is the lack of regu-
larit y of the weak solutions of the p-sub-Laplacian in the Heisenberg group. Let us note
that the C
α
regularity for the weak solutions of the p-subelliptic operators formed by
the vector field satisfy ing H

¨
ormander’s condition was given in [1] and the C
1,α
regularity
of the weak solutions of the p-sub-Laplacian Δ
H,p
in the Heisenberg group for p near 2
was proved in [2]. To obtain results here, we employ the Picone identity and Harnack
inequality to avoid effectively the use of the regularity.
The eigenvalue problems in the Euclidean space have been studied by many authors.
We refer to [3–11]. These results depend usually on Hardy inequalities or improved Hardy
inequalities (see [4, 12–14]).
Let us recall some elementary facts on the Heisenberg group (e.g., see [15]). Let
H
n
be
a Heisenberg group endowed with the group law
ξ
◦ ξ

=

x + x

, y + y

,t + t

+2
n


i=1

x
i
y

i
− x

i
y
i


, (1.3)
where ξ
= (z,t) = (x, y,t) = (x
1
,x
2
, ,x
n
, y
1
, , y
n
,t), z = (x, y), x ∈ R
n
, y ∈ R

n
, t ∈ R,
n
≥ 1; ξ

= ( x

, y

,t

) ∈ R
2n+1
. This group multiplication endows H
n
with a structure of
nilpotent Lie group. A family of dilations on
H
n
is defined as
δ
τ
(x, y,t) =

τx,τy,τ
2
t

, τ>0. (1.4)
The homogeneous dimension with respect to dilations is Q

= 2n +2. The left invariant
vector fields on the Heisenberg group have the form
X
i
=
∂
∂x
i
+2y
i
∂
∂t
, Y
i
=
∂
∂y
i
− 2x
i
∂
∂t
, i
= 1,2, ,n. (1.5)
We denote the horizontal gradient by
∇
H
= (X
1
, ,X

n
,Y
1
, ,Y
n
), and write
div
H
(v
1
,v
2
, ,v
2n
) =

n
i
=1
(X
i
v
i
+ Y
i
v
n+i
). Hence, the sub-Laplacian Δ
H
and the p-sub-

Laplacian Δ
H,p
are expressed by
Δ
H
=
n

i=1
X
2
i
+ Y
2
i
=∇
H
·∇
H
,
Δ
H,p
u =∇
H



∇
H
u



p−2
∇
H
u

=
div
H



∇
H
u


p−2
∇
H
u

, p>1,
(1.6)
respectively .
The distance function is
d(ξ, ξ

) =


(x − x

)
2
+(y − y

)
2

2
+

t − t

− 2(x·y

− x

·y)

2

1/4
,forξ, ξ

∈ H
n
.
(1.7)

Jingbo Dou et al. 3
If ξ

= 0, we denote
d(ξ) = d(ξ,0) =

|
z|
4
+ t
2

1/4
,with|z|=

x
2
+ y
2

1/2
. (1.8)
Note that d(ξ) is usually called the homogeneous norm.
For d
= d(ξ), it is easy to calculate
∇
H
d =
1
d

3

|
z|
2
x + yt
|z|
2
y − xt

,


∇
H
d


p
=
|
z|
p
d
p
= ψ
p
, Δ
H,p
d = ψ

p
Q − 1
d
. (1.9)
Denote by B
H
(R) ={ξ ∈ H
n
| d(ξ) <R} the ball of r adius R centered at the origin. Let
Ω
1
= B
H
(R
2
)\ B
H
(R
1
)with0≤ R
1
<R
2
≤∞ and u(ξ) = v(d(ξ)) ∈ C
2
(Ω
1
)bearadial
function with respect to d(ξ). Then
Δ

H,p
u = ψ
p


v



p−2

(p − 1)v

+
Q
− 1
d
v


. (1.10)
Let us recall the change of polar coordinates (x, y,t)
→(ρ,θ,θ
1
, ,θ
2n−1
)in[16]. If
u(ξ)
= ψ
p

(ξ)v(d(ξ)), then

Ω1
u(ξ)dξ
= s
H

R
2
R
1
ρ
Q−1
v(ρ)dρ,
(1.11)
where s
H
= ω
n

π
0
(sinθ)
n−1+p/2
dθ, ω
n
is the 2n-Lebesgue measure of the unitary Euclidean
sphere in
R
2n

.
TheSobolevspacein
H
n
is written by D
1,p
(Ω)={u : Ω→R;u,|∇
H
u|∈L
p
(Ω)}. D
1,p
0
(Ω)
is the closure of C
∞
0
(Ω) with respect to the norm u
D
1,p
0
(Ω)
= (

Ω
|∇
H
u|
p
dξ)

1/p
.
In the sequel, we denote by c, c
1
, C, and so forth some positive constants usually except
special narr ating.
This paper is organized as follows. In Section 2, we prove the Hardy inequality with
remainder terms by the vector field method in the Heisenberg group. In Section 3,we
discuss the optimality of the constants in the inequalities which is of its independent
interest. In Section 4, we show some useful properties concerning the Hardy operator
(1.1), and then check the existence of solutions of the eigenvalue problem (1.2)(1<p<
Q) and the asymptotic behavior of the first eigenvalue as μ increases to ((Q
− p)/p)
p
.In
Section 5, we study the simplicity and isolation of the first eigenvalue.
2. The Hardy inequality with remainder terms
D’Ambrosio in [17] has proved a Hardy inequality in the bounded domain Ω
⊂ H
n
:let
p>1andp
=Q.Foranyu ∈ D
1,p
0
(Ω,|z|
p
/d
2p
), it holds that

C
Q,p

Ω
ψ
p
|u|
p
d
p
dξ ≤

Ω


∇
H
u


p
dξ, (2.1)
where C
Q,p
=|(Q − p)/p|
p
.Moreover,if0∈ Ω, then the constant C
Q,p
is best. In this
section, we give the Hardy inequality with remainder terms on Ω, based on the careful

4 Journal of Inequalities and Applications
choice of a suitable vector field and an elementary integral inequality. Note that we also
require that 0
∈ Ω.
Theorem 2.1. Let u
∈ D
1,p
0
(Ω / {0}). Then
(1) if p
=Q and there exists a positive constant M
0
such that sup
ξ∈Ω
d(ξ)e
1/M
0
:= R
0
< ∞,
then for any R
≥ R
0
,

Ω


∇
H

u


p
dξ ≥




Q − p
p




p

Ω
ψ
p
|u|
p
d
p
dξ +
p
− 1
2p





Q − p
p




p−2

Ω
ψ
p
|u|
p
d
p

ln

R
d

−2
dξ;
(2.2)
moreover, if 2
≤ p<Q, then choose sup
ξ∈Ω
d(ξ) = R

0
;
(2) if p
= Q and there exits M
0
such that sup
ξ∈Ω
d(ξ)e
1/M
0
<R, then

Ω


∇
H
u


p
dξ ≥

p − 1
p

p

Ω
ψ

p
|u|
p

d ln(R/d)

p
dξ. (2.3)
Before we prove the theorem, let us recall that
Γ

d(ξ)

=
⎧
⎨
⎩
d(ξ)
(p−Q)/(p−1)
if p=Q,
−lnd(ξ)ifp = Q
(2.4)
is the solution of Δ
H,p
at the origin, that is, Δ
H,p
Γ(d(ξ)) = 0onΩ \{0}. Equation (2.4)
is useful in our proof. For convenience, write Ꮾ(s)
=−1/ ln(s), s ∈ (0,1), and A = (Q −
p)/p. Thus, for some positive constant M>0,

0
≤ Ꮾ

d(ξ)
R

≤
M,sup
ξ∈Ω
d(ξ) <R, ξ ∈ Ω. (2.5)
Furthermore,
∇
H
Ꮾ
γ

d
R

=
γ
Ꮾ
γ+1
(d/R)∇
H
d
d
,
dᏮ
γ

(ρ/R)
dρ
= γ
Ꮾ
γ+1
(ρ/R)
ρ
∀γ ∈ R, (2.6)

b
a
Ꮾ
γ+1
(s)
s
ds
=
1
γ

Ꮾ
γ
(b) − Ꮾ
γ
(a)

. (2.7)
Proof. Let T be a C
1
vector field on Ω and let it be specified later. For any u ∈ C

∞
0
(Ω \{0}),
we use H
¨
older’s inequality and Young’s inequality to get

Ω

div
H
T

|
u|
p
dξ =−p

Ω

T,∇
H
u

|
u|
p−2
udξ
≤ p



Ω


∇
H
u


p
dξ

1/p


Ω
|T|
p/(p−1)
|u|
p
dξ

(p−1)/p
≤

Ω


∇
H

u


p
dξ +(p − 1)

Ω
|T|
p/(p−1)
|u|
p
dξ.
(2.8)
Jingbo Dou et al. 5
Thus, the following elementary integral inequality:

Ω


∇
H
u


p
dξ ≥

Ω

div

H
T − (p − 1)|T|
p/(p−1)

|
u|
p
dξ (2.9)
holds.
(1) Let a be a free parameter to be chosen later. Denote
I
1
(Ꮾ) = 1+
p
− 1
pA
Ꮾ

d
R

+ aᏮ
2

d
R

, I
2
(Ꮾ) =

p − 1
pA
Ꮾ
2

d
R

+2aᏮ
3

d
R

,
(2.10)
and pick T(d)
= A|A|
p−2
(|∇
H
d|
p−2
∇
H
d/d
p−1
)I
1
. An immediate computation shows

div
H

A|A|
p−2


∇
H
d


p−2
∇
H
d
d
p−1

=
A|A|
p−2
dΔ
H,p
d − (p − 1)


∇
H
d



p
d
p
= A|A|
p−2
(Q − 1 − p +1)


∇
H
d


p
d
p
= p|A|
p


∇
H
d


p
d
p

.
(2.11)
By (2.6),
div
H
T = p|A|
p


∇
H
d


p
d
p
I
1
+ A|A|
p−2


∇
H
d


p−2
∇

H
d
d
p−1
×
∇
H
d
d

p − 1
pA
Ꮾ
2

d
R

+2aᏮ
3

d
R

=
p|A|
p


∇

H
d


p
d
p
I
1
+ A|A|
p−2


∇
H
d


p
d
p
I
2
,
div
H
T − (p − 1)|T|
p/(p−1)
= p|A|
p



∇
H
d


p
d
p
I
1
+ A|A|
p−2


∇
H
d


p
d
p
I
2
− (p − 1)|A|
p



∇
H
d


p
d
p
I
p/(p−1)
1
=|A|
p


∇
H
d


p
d
p

pI
1
+
1
A
I

2
− (p − 1)I
p/(p−1)
1

.
(2.12)
We claim
div
H
T − (p − 1)|T|
p/(p−1)
≥|A|
p


∇
H
d


p
d
p

1+
p
− 1
2pA
2

Ꮾ
2

d
R

. (2.13)
In fact, arguing as in the proof of [13, Theorem 4.1], we set
f (s):
= pI
1
(s)+
1
A
I
2
(s) − (p − 1)I
p/(p−1)
1
(s) (2.14)
6 Journal of Inequalities and Applications
and M
= M(R):= sup
ξ∈Ω
Ꮾ(d(ξ)/R), and distinguish three cases
(i)
1 <p<2 <Q, a>
(2
− p)(p − 1)
6p

2
A
2
,
(2.15)
(ii)
2
≤ p<Q, a = 0,
(2.16)
(iii)
p>Q, a<
(2
− p)(p − 1)
6p
2
A
2
< 0.
(2.17)
It yields that
f (s)
≥ 1+
p
− 1
2pA
2
s
2
,0≤ s ≤ M, (2.18)
(see [13]) and then fol lows (2.13). Hence (2.2)isproved.

(2) If p
= Q, then we choose the vector field T(d) = ((p − 1)/p)
p−1
(|∇
H
d|
p−2
∇
H
d/
d
p−1
)Ꮾ
p−1
(d/R). It gives
div
H
T =

p − 1
p

p−1


Q − 1 − (p − 1)



∇

H
d


p
d
p
Ꮾ
p−1

d
R

+(p − 1)Ꮾ
p

d
R



∇
H
d


p
d
p


=
p

p − 1
p

p
Ꮾ
p

d
R



∇
H
d


p
d
p
,
(2.19)
and hence
div
H
T − (p − 1)|T|
p/(p−1)

=

p − 1
p

p
Ꮾ
p

d
R



∇
H
d


p
d
p
.
(2.20)
Combining (2.20)with(2.9)follows(2.3).

Remark 2.2. The domain Ω in (2.9) may be bounded or unbounded. In addition, if we
select that T(d)
= A|A|
p−2

(|∇
H
d|
p−2
∇
H
d/d
p−1
), then
div
H
T − (p − 1)|T|
p/(p−1)
= p|A|
p


∇
H
d


p
d
p
− (p − 1)|A|
p


∇

H
d


p
d
p
=|A|
p


∇
H
d


p
d
p
.
(2.21)
Hence, from (2.9)weconclude(2.1) on the bounded domain Ω and on
H
n
(see [15]),
respectively .
We will prove in next section that the constants in (2.2)and(2.3)arebest.
Now, we state the Poincar
´
einequalityprovedin[17].

Jingbo Dou et al. 7
Lemma 2.3. Let Ω be a subset of
H
n
bounded in x
1
direction, that is, there ex ists R>0 such
that 0 <r
=|x
1
|≤R for ξ = (x
1
,x
2
, ,x
n
, y
1
, , y
n
,t) ∈ Ω. Then for any u ∈ D
1,p
0
(Ω),
then
c

Ω
|u|
p

dξ ≤

Ω


∇
H
u


p
dξ, (2.22)
where c
= ((p − 1)/pR)
p
.
Using (2.9) by choosing T
=−((p − 1)/p)
p−1
(∇
H
r/r
p−1
) immediately provides a dif-
ferent proof to (2.22).
In the following, we describe a compactness result by using (2.1)and(2.22).
Theorem 2.4. Suppose p
=Q and f (ξ) ∈ Ᏺ
p
. Then there exists a positive constant C

f ,Q,p
such that
C
f ,Q,p

Ω
f (ξ)|u|
p
dξ ≤

Ω


∇
H
u


p
dξ, (2.23)
and the embedding D
1,p
0
(Ω)  L
p
(Ω, fdξ) is compact.
Proof. Since f (ξ)
∈ Ᏺ
p
,wehavethatforany > 0, there exist δ>0andC

δ
> 0suchthat
sup
B
H
(δ)⊆Ω
d
p
ψ
p
f (ξ) ≤

, f (ξ)|
Ω\B
H
(δ)
≤ C
δ
. (2.24)
By (2.1)and(2.22), it follows

Ω
f (ξ)|u|
p
dξ =

B
H
(δ)
f |u|

p
dξ +

Ω\B
H
(δ)
f |u|
p
dξ
≤


B
H
(δ)
ψ
p
|u|
p
d
p
dξ + C
δ

Ω\B
H
(δ)
|u|
p
dξ ≤ C

−1
f ,Q,p

Ω


∇
H
u


p
dξ,
(2.25)
then (2.23)isobtained.
Now, we prove the compactness. Let
{u
m
}⊆D
1,p
0
(Ω) be a bounded sequence. By re-
flexivity of the space D
1,p
0
(Ω) and the Sobolev embedding for vector fields (see [18]), it
yields
u
m
j

 u weakly in D
1,p
0
(Ω),
u
m
j
−→ u strongly in L
p
(Ω)
(2.26)
for a subsequence
{u
m
j
} of {u
m
} as j→∞.WriteC
δ
=f 
L
∞
(Ω\B
H
(δ))
.From(2.1),

Ω
f



u
m
j
− u


p
dξ =

B
H
(δ)
f


u
m
j
− u


p
dξ +

Ω\B
H
(δ)
f



u
m
j
− u


p
dξ
≤ 

B
H
(δ)
ψ
p


u
m
j
− u


p
d
p
dξ + C
δ


Ω\B
H
(δ)


u
m
j
− u


p
dξ
≤ C
−1
Q,p

Ω


∇
H

u
m
j
− u




p
dξ + C
δ

Ω


u
m
j
− u


p
dξ.
(2.27)
8 Journal of Inequalities and Applications
Since
{u
m
}⊆D
1,p
0
(Ω) is bounded, we have

Ω
f


u

m
j
− u


p
dξ ≤

M + C
δ

Ω


u
m
j
− u


p
dξ,
(2.28)
where M>0 is a constant depending on Q and p.By(2.26),
lim
j→∞

Ω
f



u
m
j
− u


p
dξ ≤

M.
(2.29)
As
 is arbitrary, lim
j→∞

Ω
f |u
m
j
− u|
p
dξ = 0. Hence D
1,p
0
(Ω)  L
p
(Ω, fdξ)iscompact.

Remark 2.5. The class of the functions f (ξ) ∈ Ᏺ

p
has lower-order singularity than d
−p
(ξ)
at the origin. The examples of such functions are
(a) any bounded function,
(b) in a small neighborhood of 0, f (ξ)
= ψ
p
(ξ)/d
β
(ξ), 0 <β<p,
(c) f (ξ)
= ψ
p
(ξ)/d
p
(ξ)(ln(1/d(ξ)))
2
in a small neighborhood of 0.
3.Proofofbestconstantsin(2.2)and(2.3)
In this section, we prove that the constants appearing in Theorem 2.1 are the best. To do
this, we need two lemmas. First we introduce some notations.
For some fixed small δ>0, let the test function ϕ(ξ)
∈ C
∞
0
(Ω) satisfy 0 ≤ ϕ ≤ 1and
ϕ(ξ)
=

⎧
⎪
⎨
⎪
⎩
1ifξ ∈ B
H

0,
δ
2

,
0ifξ
∈ Ω \ B
H
(0,δ),
(3.1)
with
|∇
H
ϕ| < 2|∇
H
d|/d. Let  > 0 small enough, and define
V

(ξ) = ϕ(ξ)

,with


= d
−A+
Ꮾ
−κ

d
R

,
1
p
<κ<
2
p
,
J
γ
() =

Ω
ϕ
p
(ξ)


∇
H
d



p
d
Q−p
Ꮾ
−γ

d
R

dξ, γ ∈ R.
(3.2)
Lemma 3.1. For
 > 0 small, it holds
(i) c

−1−γ
≤ J
γ
() ≤ C
−1−γ
, γ>−1,
(ii) J
γ
() = (p/(γ +1))J
γ+1
()+O

(1), γ>−1,
(iii) J
γ

() = O

(1), γ<−1.
Proof. By the change of p olar coordinates (1.11)and0
≤ ϕ ≤ 1, we have
J
γ
() ≤

B
H
(δ)


∇
H
d


p
d
Q−p
Ꮾ
−γ

d
R

dξ = s
H


ρ<δ
ρ
−Q+p
Ꮾ
−γ

ρ
R

ρ
Q−1
dρ
= s
H

ρ<δ
ρ
−1+p
Ꮾ
−γ

ρ
R

dρ.
(3.3)
Jingbo Dou et al. 9
By (2.7)weknowthatforγ<
−1 the right-hand side of (3.3) has a finite limit, hence (iii)

follows from

→
0.
To show (i), we set ρ
= Rτ
1/
.Thus,dρ = (1/)Rτ
1/
−
1
dτ, Ꮾ
−γ
(τ
1/
) = 
−γ
Ꮾ
−γ
(τ), and
J
γ
() ≤ s
H

ρ<δ
ρ
−1+p
Ꮾ
−γ


ρ
R

dρ = s
H

(δ/R)

0

Rτ
1/

−1+p
Ꮾ
−γ

Rτ
1/
R

1

Rτ
1/
−
1
dτ
= s

H
R
p

−1−γ

(δ/R)

0
τ
p−1
Ꮾ
−γ
(τ)dτ.
(3.4)
It follows the right-hand side of (i). Using the fact that ϕ
= 1inB
H
(δ/2),
J
γ
() ≥

B
H
(δ/2)


∇
H

d


p
d
Q−p
Ꮾ
−γ

d
R

dξ = s
H
R
p

−1−γ

(δ/2R)

0
τ
p−1
Ꮾ
−γ
(τ)dτ,
(3.5)
and the left-hand side of (i) is proved.
Now we prove (ii). Let Ω

η
: ={ξ ∈ Ω | d(ξ) >η}, η>0, be small and note the boundary
term
−

d=η

ϕ
p


∇
H
d


p−2
∇
H
d
d
Q−1−p

Ꮾ
−γ−1

d
R

∇

H
d·

ndS
−→ 0asη −→ 0. (3.6)
From (2.6),

Ω
div
H

ϕ
p


∇
H
d


p−2
∇
H
d
d
Q−1−p

Ꮾ
−γ−1


d
R

dξ
=−

Ω
ϕ
p


∇
H
d


p−2
d
Q−1−p

∇
H
d,∇
H
Ꮾ
−γ−1

d
R


dξ
= (γ +1)

Ω
ϕ
p


∇
H
d


p
d
Q−p
Ꮾ
−γ

d
R

dξ = (γ +1)J
γ
().
(3.7)
On the other hand,

Ω
div

H

ϕ
p


∇
H
d


p−2
∇
H
d
d
Q−1−p

Ꮾ
−γ−1

d
R

dξ
= p

Ω
ϕ
p−1



∇
H
d


p−2

∇
H
d,∇
H
ϕ

d
Q−1−p
Ꮾ
−γ−1

d
R

dξ
+(1
− Q + p + Q − 1)

Ω
ϕ
p



∇
H
d


p
d
Q−p
Ꮾ
−γ−1

d
R

dξ
= p

Ω
ϕ
p−1


∇
H
d


p−2


∇
H
d,∇
H
ϕ

d
Q−1−p
Ꮾ
−γ−1

d
R

dξ + pJ
γ+1
().
(3.8)
10 Journal of Inequalities and Applications
We claim that p

Ω
ϕ
p−1
(|∇
H
d|
p−2
∇

H
d,∇
H
ϕ/d
Q−1−p
)Ꮾ
−γ−1
(d/R)dξ = O

(1). In fact,
by (3.1)and(1.11),

Ω
ϕ
p−1


∇
H
d


p−2

∇
H
d,∇
H
ϕ


d
Q−1−p
Ꮾ
−γ−1

d
R

dξ
≤ 2

B
H
(δ)


∇
H
d


p
d
Q−p
Ꮾ
−γ−1

d
R


dξ
≤ 2s
H

B
H
(δ)
ρ
−Q+p
Ꮾ
−γ−1

ρ
R

ρ
Q−1
dρ
= 2s
H

B
H
(δ)
ρ
p
−
1
Ꮾ
−γ−1


ρ
R

dρ.
(3.9)
Using the estimate (i) follows that

B
H
(δ)
ρ
p
−
1
Ꮾ
−γ−1
(ρ/R)dρ = O

(1). Combining (3.7)
with (3.8)gives
(γ +1)J
γ
() = pJ
γ+1
()+O

(1). (3.10)
This allows us to conclude (ii).


We next estimate the quantity
I[V

] =

Ω


∇
H
V



p
dξ −|A|
p

Ω


∇
H
d


p


V




p
d
p
dξ. (3.11)
Lemma 3.2. As

→
0,itholds
(i) I(V

) ≤ (κ(p − 1)/2)|A|
p−2
J
pκ−2
()+O

(1);
(ii)

B
H
(δ)
|∇
H
V

|

p
dξ ≤|A|
p
J
pκ
()+O

(
1−pκ
).
Proof. By the definition of V

(ξ), we see ∇
H
V

(ξ) = ϕ(ξ)∇
H


+ 

∇
H
ϕ. Using the ele-
mentary inequality
|a + b|
p
≤|a|
p

+ c
p

|
a|
p−1
|b| + |b|
p

, a,b ∈ R
2n
, p>1, (3.12)
one has


∇
H
V



p
≤ ϕ
p


∇
H





p
+ c
p



∇
H
ϕ




ϕ
p−1


∇
H




p−1
+


∇

H
ϕ


p






p

≤
ϕ
p


∇
H




p
+ c
p




2∇
H
d


d


ϕ
p−1


∇
H




p−1
+



2∇
H
d


d


p






p

.
(3.13)
Jingbo Dou et al. 11
Since
∇
H


=−d
−A+
−
1
Ꮾ
−κ
(d/R)(A −

+ κᏮ(d/R))∇
H
d,itfollows

Ω



∇
H
V



p
dξ ≤

B
H
(δ)


∇
H
d


p
ϕ
p
d
−Q+p
Ꮾ
−pκ

d

R





A −


− κᏮ

d
R





p
dξ
+2c
p

B
H
(δ)


∇
H

d


p
ϕ
p−1
d
−Q+p
Ꮾ
−pκ

d
R





A −


− κᏮ

d
R






p−1
dξ
+2
p
c
p

B
H
(δ)


∇
H
d


p
d
−Q+p
Ꮾ
−pκ

d
R

dξ := Π
A
+ Π
1

+ Π
2
.
(3.14)
We claim that
Π
1
,Π
2
= O

(1). (3.15)
Indeed, since
|A − (
−
κᏮ(d/R))| is bounded, using (3.1 )weget
Π
1
≤ C

B
H
(δ)


∇
H
d



p
ϕ
p−1
d
−Q+p
Ꮾ
−pκ

d
R





A −


− κᏮ

d
R





p−1
dξ
≤ C


B
H
(δ)


∇
H
d


p
d
−Q+p
Ꮾ
−pκ

d
R

dξ,
Π
2
≤ C

B
H
(δ)



∇
H
d


p
d
−Q+p
Ꮾ
−pκ

d
R

dξ.
(3.16)
By (i) of Lemma 3.1,itderivesΠ
1
, Π
2
= O

(1), as 
→
0.
From (3.14), (3.15) and the definition of J
γ
(), it clearly shows
I


V


=

B
H
(δ)


∇
H
V



p
dξ −|A|
p
J
pκ
() ≤ Π
A
−|A|
p
J
pκ
()+O

(1) = Π

3
+ O

(1),
(3.17)
where Π
3
=

B
H
(δ)
|∇
H
d|
p
ϕ
p
d
−Q+p
Ꮾ
−pκ
(d/R)(|A − (
−
κᏮ(d/R))|
p
−|A|
p
)dξ. For sim-
plicity, denote ζ

= 
−
κᏮ(d/R). Since ζ is small compared to A, we use Taylor’s expansion
to yield
|A − ζ|
p
−|A|
p
≤−pA|A|
p−2
ζ +
p(p
− 1)
2
|A|
p−2
ζ
2
+ C|ζ|
3
. (3.18)
Thus, we can estimate Π
3
by
Π
3
≤ Π
31
+ Π
32

+ Π
33
, (3.19)
where
Π
31
=−pA|A|
p−2

B
H
(δ)


∇
H
d


p
ϕ
p
d
−Q+p
Ꮾ
−pκ

d
R



−
κᏮ

d
R

dξ,
Π
32
=
p(p − 1)
2
|A|
p−2

B
H
(δ)


∇
H
d


p
ϕ
p
d

−Q+p
Ꮾ
−pκ

d
R

 −
κᏮ

d
R

2
dξ,
Π
33
= C

B
H
(δ)


∇
H
d


p

ϕ
p
d
−Q+p
Ꮾ
−pκ

d
R





 −
κᏮ

d
R





3
dξ.
(3.20)
12 Journal of Inequalities and Applications
We will show that
Π

31
, Π
33
= O

(1), as 
−→
0. (3.21)
In fact, using (ii) of Lemma 3.1 with γ
=−1+pκ yields
Π
31
=−pA|A|
p−2


J
pκ
() − κJ
pκ−1
()

=−
pA|A|
p−2


J
pκ
() −


J
pκ
()+O

(1)

=
O

(1).
(3.22)
Recalling (a
− b)
3
≤ (|a| + |b|)
3
≤ c(|a|
3
+ |b|
3
), we obtain
Π
33
≤ c
3
J
pκ
()+cJ
pκ−3

()for > 0. (3.23)
From (i) and (iii) in Lemma 3.1 and the fact that 1 <pκ<2itfollowsΠ
33
= O

(1). Using
(ii) of Lemma 3.1 twice (pick γ
= pκ − 1 > −1andγ = pκ − 2 > −1, resp.), we conclude
that
Π
32
=
p(p − 1)
2
|A|
p−2

B
H
(δ)


∇
H
d


p
ϕ
p

d
−Q+p
Ꮾ
−pκ

d
R


2
−2κᏮ

d
R

+κ
2
Ꮾ
2

d
R

dξ
=
p(p − 1)
2
|A|
p−2



2
J
pκ
() − 2κJ
pκ−1
()+κ
2

pκ − 1
pκ
+
1
pκ

J
pκ−2
()

=
p(p − 1)
2
|A|
p−2


2
J
pκ
() −


κ
p

pκ
J
pκ
() −

κJ
pκ−1
()
+
κ
2
(pκ − 1)
pκ
p

pκ − 1
J
pκ−1
()+
κ
p
J
pκ−2
()+O

(1)


=
κ(p − 1)
2
|A|
p−2
J
pκ−2
()+O

(1).
(3.24)
In virtue of (3.17), (3.19), (3.21), and (3.24) we deduce (i) of Lemma 3.2.By(3.17),
(3.24), and (i) of Lemma 3.2,

B
H
(δ)


∇
H
V



p
dξ = I

V



+ |A|
p
J
pκ
() ≤|A|
p
J
pκ
()+
κ(p
− 1)
2
|A|
p−2
J
pκ−2
()+O

(1).
(3.25)
Hence (ii) of Lemma 3.2 follows from (i) in Lemma 3.1. It completes the proof.

We are now ready to give the proof of the best constants in Theorem 2.1.
Theorem 3.3. Let 0
∈ Ω be a bounded domain in H
n
and p=Q.Supposethatforsome
constants B>0, D

≥ 0,andι>0, the following inequality holds for any u(ξ) ∈ C
∞
0
(Ω \{0}):

Ω


∇
H
u


p
dξ ≥ B

Ω
ψ
p
|u|
p
d
p
dξ + D

Ω
ψ
p
|u|
p

d
p
Ꮾ
ι

d
R

dξ. (3.26)
Jingbo Dou et al. 13
Then,
(i) B
≤|A|
p
;
(ii) if B
=|A|
p
and D>0, then ι ≥ 2;
(iii) if B
=|A|
p
and ι = 2, then D ≤ ((p − 1)/2p)|A|
p−2
.
Proof. Choose u(ξ)
= V

(ξ).
(i) By (ii) of Lemma 3.2,wehave

B
≤

B
H
(δ)


∇
H
V



p
dξ

B
H
(δ)
ψ
p
(|V

|
p
/d
p
)dξ
≤

|
A|
p
J
pκ
()+O



1−pκ


B
H
(δ)
ψ
p
(|ϕd
−A+
Ꮾ
−κ
(d/R)|
p
/d
p
)dξ
=
|A|
p


1+c
2

J
pκ
()+O

(1)
J
pκ
()
.
(3.27)
Note that J
pκ
()→∞,as
→
0, so B ≤|A|
p
.
(ii) Set B
=|A|
p
and assume by contradiction that ι<2. Since pκ − ι>−1, using (i)
of Lemma 3.2 and (i) of Lemma 3.1 leads to
0 <D
≤
I

V




Ω
ψ
p
(|V

|
p
/d
p
)Ꮾ
ι
(d/R)dξ
=
I

V


J
pκ−ι
()
≤

κ(p − 1)/2

|
A|

p−2
J
pκ−2
()+O

(1)
J
pκ−ι
()
≤
C
1−pκ
c
−1−pκ+ι
= C
2−ι
−→ 0, as 
−→
0,
(3.28)
which is a contradiction. Hence ι
≥ 2.
(iii) If B
=|A|
p
and ι = 2, then by (i) of Lemma 3.2,
D
≤
I


V


J
pκ−2
()
≤

κ(p − 1)/2

|
A|
p−2
J
pκ−2
()+O

(1)
J
pκ−2
()
.
(3.29)
The assumption κ>1/p implies J
pκ−2
()→∞,as
→
0. Hence, by (i) of Lemma 3.1 we
conclude that D
≤ (κ(p − 1)/2)|A|

p−2
,as
→
0. Then letting κ→1/p, the proof is finished.

Theorem 3.4. Set 0 ∈ Ω and p = Q. Suppose that there exist some constants D ≥ 0 and
ι>0 such that the following inequality holds for all u(ξ)
∈ C
∞
0
(Ω \{0}):

Ω


∇
H
u


p
dξ ≥ D

Ω
ψ
p
|u|
p
d
p

Ꮾ
ι

d
R

dξ. (3.30)
Then,
(i) if D>0, then ι
≥ p;
(ii) if ι
= p, then D ≤ ((p − 1)/p)
p
.
Proof. The proof is essentially similar to one of Theorem 3.3. Let the test function ϕ be
as before (see (3.1)). For
 > 0, κ>(p − 1)/p,defineV

= ϕ

with 

= d

(−ln(d/R))
κ
.
14 Journal of Inequalities and Applications
Using (3.12)yields


Ω


∇
H
V



p
dξ =

B
H
(δ)


ϕ∇
H


+ 

∇
H
ϕ


p
dξ

≤

B
H
(δ)
ϕ
p


∇
H




p
dξ + c
p

B
H
(δ)
ϕ
p−1




∇
H





p−1


∇
H
ϕ


dξ
+ c
p

B
H
(δ)

p



∇
H
ϕ


p

dξ := Π
4
+ Π
5
+ Π
6
.
(3.31)
Arguing as in the proof of previous theorem and letting

→
0, we have
Π
5
, Π
6
= O

(1). (3.32)
Denoting by c
p
i
the coefficients of binormial expansion, we get


∇
H





p
=


∇
H
d


p
d
p
−
p
Ꮾ
−pκ

d
R






−
κᏮ

d

R





p
≤


∇
H
d


p
d
p
−
p
Ꮾ
−pκ

d
R


+ κᏮ

d

R

p
=


∇
H
d


p
d
p
−
p
Ꮾ
−pκ

d
R

Σ
p
i
=0
c
p
i


p−i
κ
i
Ꮾ
i

d
R

.
(3.33)
Hence,
Π
4
≤ Σ
p
i
=0
c
p
i

p−i
κ
i
J
pκ−i
(), (3.34)
where J
γ

() =

Ω
|∇
H
d|
p
ϕ
p
d
p
−
p
Ꮾ
−γ
(d/R). By (ii) of Lemma 3.1 and the induction ar-
gument it holds

p−i
J
pκ−i
() =

κ −
i
p

κ −
i +1
p


···

κ−
p − 1
p

J
pκ−p
()+O

(1), i = 0,1, , p − 1.
(3.35)
Now (i) of Lemma 3.1 and the assumption κ>(p
− 1)/pimply that J
pκ−p
()→∞,as
→
0,
and
D
≤ limsup
→0

Ω


∇
H
V




p
dξ

Ω
ψ
p



V



p
/d
p

Ꮾ
p
(d/R)dξ
≤ limsup
→0

κ
p
+ Σ
p−1

i
=0
c
p
i
κ
i

κ − i/p

κ − (i +1)/p

···

κ − (p − 1)/p

J
pκ−p
()+O

(1)
J
pκ−p
()
=

κ
p
+ Σ
p−1

i
=0
c
p
i
κ
i

κ −
i
p

κ −
i +1
p

···

κ −
p − 1
p

.
(3.36)
The last expression converges to ((p
− 1)/p)
p
as κ→(p − 1)/p.Theproofisover. 
Jingbo Dou et al. 15
4.Theweightedeigenvalueproblem

This section is devoted to the problem (1.2) by using the Hardy inequality with remainder
terms.
We begin with some properties concerning the Hardy operator (1.1).
Lemma 4.1. Suppose that u(ξ)
∈ D
1,p
0
(Ω) and p=Q. The n
(1) L
p,μ
is a positive operator if μ ≤ C
Q,p
;inparticular,ifμ = C
Q,p
, then v(ξ) =
d
(p−Q)/p
(ξ) is a solution of L
p,μ
u = 0;
(2) L
p,μ
is unbounded from below if μ>C
Q,p
.
Proof. (1) It is obvious from (2.1)thatL
p,μ
is a positive operator.
We now suppose that μ
= C

Q,p
and verify that v = d
(p−Q)/p
satisfies L
p,μ
u = 0. For the
purpose, set v

= d
(p−Q)/p+
∈ D
1,p
0
(Ω)andA = (Q − p)/p.Since
v


= (
−
A)d
−A−1+
, v


= (
−
A)(
−
A − 1)d
−A−2+

,
(4.1)
it yields from (1.10)and(4.1)that
−Δ
H,p
v

=−ψ
p


v




p−2

(p − 1)v


+
Q
− 1
d
v



=−

ψ
p


(
−
A)d
−A−1+


p−2

(p − 1)(
−
A)(
−
A − 1)d
−A−2+
+
Q
− 1
d
(

−
A)d
−A−1+

=−
ψ

p
(
−
A)|

− A|
p−2
d
(−A−1+)(p−2)+(−A−2+)
[(p − 1)(
−
A − 1) + Q − 1]
=−ψ
p

(
−
A)(p − 1) + Q − p

(
−
A)|

− A|
p−2
d
(−A+)(p−1)−p
=−

(

−
A)(p − 1) + pA

(
−
A)|

− A|
p−2
ψ
p
v
p−1

d
p
.
(4.2)
Letting

→
0, the conclusion follows.
(2) By the density argument, we select φ(ξ)
∈ C
∞
0
(Ω), φ
L
p
= 1, such that C

Q,p
=

Ω
|∇
H
φ|
p
/(

Ω
(|z|
p
/d
p
)(|φ|
p
/d
p
)). Using the best constant of the inequality (2.1), one
has

L
p,μ
φ,φ

=

Ω



∇
H
φ


p
− μ

Ω
|z|
p
d
p
|φ|
p
d
p
<

Ω


∇
H
φ


p
− C

Q,p

Ω
|z|
p
d
p
|φ|
p
d
p
= 0.
(4.3)
Denote u
τ
(x, y,t) = τ
Q/p
φ(δ
τ
(x, y,t)) and ξ = (x, y,t). Thus,


u
τ
(x, y,t)


p
L
p

=

Ω


τ
Q/p
φ

δ
τ
(x, y,t)



p
dξ =

Ω


φ

δ
τ
(x, y,t)



p

τ
Q
dξ = 1,
(4.4)
and
L
p,μ
u
τ
,u
τ
=τ
p
L
p,μ
φ,φ < 0. This concludes the result. 
Inordertoprovethemainresult(Theorem 4.6 below) we need the following two
preliminary lemmas.
16 Journal of Inequalities and Applications
Lemma 4.2. Let
{g
m
}⊂L
p
(Ω)(1 ≤ p<∞) be such that as m→∞,
g
m
 g weakly in L
p
(Ω),

g
m
−→ g a.e. in Ω.
(4.5)
Then,
lim
m→∞



g
m


p
L
p
(Ω)
−


g
m
− g


p
L
p
(Ω)


=
g
p
L
p
(Ω)
. (4.6)
The proof is similar to one in the Euclidean space (see [19, Chapter 1, Section 4]. We
omit it here.
Lemma 4.3. Suppose that {u
m
}⊂D
1,p
0
(Ω)(1 ≤ p<∞) satis fies
u
m
 u weakly in D
1,p
0
(Ω),
u
m
−→ u strongly in L
p
loc
(Ω),
(4.7)
as m

→∞,and
−Δ
H,p
u
m
= f
m
+ g
m
, in Ᏸ

(Ω), (4.8)
where f
m
→ f strongly in D
−1,p

(Ω)(p

= p/(p − 1)), g
m
is bounded in M(Ω) (the space of
Radon measures), that is,



g
m
,ϕ




≤
C
K
ϕ
L
∞
(4.9)
for all ϕ
∈ Ᏸ(Ω) with supp(ϕ) ⊂ K, where C
K
is a constant which depends on the compact
set K. Then there exists a subsequence
{u
m
j
} of {u
m
} such that
u
m
j
−→ u strongly in D
1,q
0
(Ω), ∀q<p. (4.10)
Its proof is similar to one of [20, Theorem 2.1].
Lemma 4.4. Let p
=Q and

I
p
:=

f : Ω −→ R
+
| lim
d(ξ)→0
d
p
(ξ)
ψ
p
(ξ)
f (ξ)

ln
1
d(ξ)

2
< ∞, f (ξ) ∈ L
∞
loc

Ω \{0}


. (4.11)
(i) If f (ξ)

∈ I
p
, then there exists λ( f ) > 0 such that for all u ∈ D
1,p
0
(Ω \{0}),the
following holds:

Ω


∇
H
u


p
dξ ≥




Q − p
p




p


Ω
ψ
p
(ξ)
|u|
p
d
p
dξ + λ( f )

Ω
|u|
p
f (ξ)dξ. (4.12)
(ii) If f (ξ)
∈ I
p
and (d
p
(ξ)/ψ
p
(ξ)) f (ξ)(ln1/d(ξ))
2
→∞ as d(ξ)→0,then(4.12)isnot
true.
Jingbo Dou et al. 17
Proof. (i) If f (ξ)
∈ I
p
,then

lim

→
0
sup
ξ∈B
H
()
d
p
(ξ)
ψ
p
f (ξ)

ln
1
d(ξ)

2
< ∞. (4.13)
Without any loss of generality we assume that R
= 1in(2.2). For the sufficiently small
 > 0, we have
f (ξ) <
Cψ
p
d
p
(ln1/d)

2
,inB
H
(). (4.14)
Outside B
H
(), f (ξ) is also bounded. Hence there exists C( f ) > 0suchthat

Ω
|u|
p
f (ξ)dξ
≤ C( f )

Ω
ψ
p
|u|
p
d
p
(ln1/d)
2
dξ,onΩ. (4.15)
Taking λ( f )
= C( f )
−1
((p − 1)/2p)|A|
p−2
> 0, (4.12)followsfrom(2.2).

(ii) We write f (ξ)
= ψ
p
h(ξ)/d
p
(ξ)(ln 1/d(ξ))
2
,whereh(ξ)→∞ as d(ξ)→0. Then, for
the sufficiently small
 > 0, we select u(ξ) = V

(ξ) = ϕ(ξ)d
−A+
(ξ)Ꮾ
−κ
(d(ξ)/R)andget
from (i) of Lemma 3.2,
0 <λ( f )
≤
I

V



B
H
(δ)



V



p
f (ξ)dξ
≤
I

V



B
H
(δ)
ψ
p
(|V

|
p
h(ξ)/d
p
(ln 1/d)
2
)dξ
≤
I


V


K
δ

B
H
(δ)
ψ
p



V



p
/d
p
(ln 1/d)
2

dξ
≤

κ(p − 1)/2

|

A|
p−2
J
pκ−2
()+O

(1)
K
δ
J
pκ−2
()
≤
c
1−pκ
cK
δ

1−pκ
=
C
K
δ
−→ 0, as δ,
−→
0,
(4.16)
where K
δ
= inf

ξ∈B
H
(δ)
h(ξ). The impossibility shows that (4.12) cannot hold for f (ξ) ∈
I
p
. 
Definit ion 4.5. Let λ ∈ R, u ∈ D
1,p
0
(Ω), and u ≡ 0. Call that (λ,u) is a weak solution of
(1.2)if

Ω


∇
H
u


p−2

∇
H
u,∇
H
ϕ

dξ − μ


Ω
ψ
p
d
p
|u|
p−2
uϕ dξ = λ

Ω
f (ξ)|u|
p−2
uϕ dξ (4.17)
for any ϕ
∈ C
∞
0
(Ω). In this case, we cal l that u is the eigenfunction of problem (1.2)
associated to the eigenvalue λ.
Theorem 4.6. Suppose that 1 <p<Q,0
≤ μ<((Q − p)/p)
p
,and f (ξ) ∈ Ᏺ
p
. The problem
(1.2) admits a positive weak solution u
∈ D
1,p
0

(Ω), corresponding to the first eigenvalue λ =
λ
1
μ
( f ) > 0. Moreover, as μ increases to ((Q − p)/p)
p
, λ
1
μ
( f )→λ( f ) ≥ 0 for f (ξ) ∈ Ᏺ
p
and
the limit λ( f ) > 0 for f (ξ)
∈ I
p
.If f (ξ) ∈ I
p
and ψ
−1
p
(ξ)d
p
(ξ) f (ξ)(ln 1/d(ξ))
2
→∞, as
d(ξ)
→0, then the limit λ( f ) = 0.
18 Journal of Inequalities and Applications
Proof. We define
J

μ
(u): =

Ω


∇
H
u


p
− μ

Ω
ψ
p
|u|
p
d
p
. (4.18)
Obviously, J
μ
is continuous and G
ˆ
ateaux differentiable on D
1,p
0
(Ω). By (2.1),

J
μ
(u) ≥

Ω


∇
H
u


p
−
μ
C
Q,p

Ω


∇
H
u


p
=

1 −

μ
C
Q,p


Ω


∇
H
u


p
(4.19)
for 0 ≤ μ<((Q − p)/p)
p
.NotethatC
Q,p
= ((Q − p)/p)
p
for 1 <p<Q.HenceJ
μ
is coer-
cive in D
1,p
0
(Ω). We minimize the function J
μ
(u) over the mainfold ᏹ ={u ∈ D

1,p
0
(Ω) |

Ω
|u|
p
f (ξ)dξ = 1} and let λ
1
μ
be the infimum. It is clear that λ
1
μ
> 0fromLemma 4.1.
Now, we choose a special minimizing sequence
{u
m
}⊂ᏹ with

Ω
|u
m
|
p
f (ξ)dξ = 1, and
J
μ
(u
m
)→λ

1
μ
and J

μ
(u
m
)→0stronglyinD
−1,p

0
(Ω), when the component of J

μ
(u
m
)isre-
stricted to ᏹ.ThecoercivityofJ
μ
implies that {u
m
} is bounded and then there exists a
subsequence, still denoted by
{u
m
},suchthat
u
m
 u weakly in D
1,p

0
(Ω),
u
m
 u weakly in L
p

Ω,ψ
p
d
−p

,
u
m
−→ u strongly in L
p
(Ω),
(4.20)
as m
→∞.ByTheorem 2.4 in Section 2 we know that D
1,p
0
(Ω) is compactly embedded
in L
p
(Ω, fdξ), and it follows that ᏹ is weakly closed and hence u ∈ ᏹ.Moreover,u
m
satisfies
−Δ

H,p
u
m
− ψ
p
μ
d
p


u
m


p−2
u
m
= λ
m


u
m


p−2
u
m
f + f
m

,inᏰ

(Ω), (4.21)
where f
m
→0stronglyinD
−1,p

(Ω)andλ
m
→λ,asm→∞. Letting g
m
= ψ
p
(μ/d
p
)|u
m
|
p−2
u
m
+λ
m
|u
m
|
p−2
u
m

f , we check easily that g
m
is bounded in M(Ω) and conclude almost ev-
erywhere convergence of
∇
H
u
m
to ∇
H
u in Ω by Lemma 4.3,and
J
μ
(u
m
) =


∇
H
u
m


p
L
p
(Ω)
− μ



u
m


p
L
p
(Ω,ψ
p
d
−p
)
=


∇
H

u
m
− u



p
L
p
(Ω)
−μ



u
m
− u


p
L
p
(Ω,ψ
p
d
−p
)
+


∇
H
u


p
L
p
(Ω)
−μu
p
L

p
(Ω,ψ
p
d
−p
)
+o(1)
≥

C
Q,p
− μ



u
m
− u


p
L
p
(Ω,ψ
p
d
−p
)
+ λ
1

μ
+ o(1),
(4.22)
by applying Lemma 4.2 to u
m
and ∇
H
u
m
,whereo(1)→0asm→∞.ThusC
Q,p
>μ, u
m
−
u
p
L
p
(Ω,ψ
p
d
−p
)
→0, and ∇
H
(u
m
− u)
p
L

p
(Ω)
→0asm→∞. It shows that J
μ
(u) = λ
1
μ
and λ = λ
1
μ
.
Since J
μ
(|u|) = J
μ
(u), we can take u>0inΩ.ByLemma 4.3, u is a distribution solution
of (1.2) and since u
∈ D
1,p
0
(Ω), it is a weak solution to eigenvalue problem (1.2)corre-
sponding to λ
= λ
1
μ
.Moreover,if f (ξ) ∈ I
p
,thenbyLemma 4.4,
λ
1

μ
( f ) −→ λ( f ) = inf
u∈D
1,p
(Ω\{0})

Ω



∇
H
u


p
− C
Q,p
ψ
p

|
u|
p
/d
p

dξ

Ω

|u|
p
fdξ
> 0,
(4.23)
Jingbo Dou et al. 19
as μ increases to ((Q
− p)/p)
p
.When f (ξ) ∈ I
p
, using Lemma 4.4 again, it follows that
(4.12) is not true and hence λ( f )
= 0. This completes the proof. 
Remark 4.7. The set ᏹ is a C
1
manifold in D
1,p
0
(Ω). By Ljusternik-Schnirelman critical
point theory on C
1
manifold, there exists a sequence {λ
m
} of eigenvalues of (1.2), that
is, writing Γ
m
={A ⊂ Ᏻ | A is symmetric, compact, and γ(A) ≥ m},whereγ(A)isthe
Krasnoselski’s genus of A (see [19]), then for any integer m>0,
λ

m
= inf
A∈Γ
m
sup
u∈A
J
μ
(u) (4.24)
is an eigenvalue of (1.2). Moreover, lim
m→∞
λ
m
→∞.
5. Simplicity and isolation for the first eigenvalue
This section is to consider the simplicity and isolation for the first eigenvalue. We always
assume that f satisfies the conditions in Theorem 4.6 . From the previous results we know
clearly that the first eigenvalue is
λ
1
μ
= λ
1
μ
( f ) = inf

J
μ
(u) | u ∈ D
1,p

0

Ω \{0}

,

Ω
|u|
p
f (ξ)dξ = 1

. (5.1)
InwhatfollowsweneedthePiconeidentityprovedin[15].
Proposition 5.1 (Picone identity). For differentiable functions u
≥ 0, v>0 on Ω ⊂ H
n
,
with Ω a bounded or unbounded domain in
H
n
, then
L(u,v)
= R(u,v) ≥ 0, (5.2)
with
L(u,v)
=


∇
H

u


p
+(p − 1)
u
p
v
p


∇
H
v


p
− p
u
p−1
v
p−1


∇
H
v


p−2

∇
H
u·∇
H
v,
R(u,v)
=


∇
H
u


p
−


∇
H
v


p−2
∇
H

u
p
v

p−1

·∇
H
v
(5.3)
for p>1.Moreover,L(u,v)
= 0 a.e. on Ω if and only if ∇
H
(u/v) = 0 a.e. on Ω.
Theorem 5.2. (i) λ
1
μ
is simple, that is, the positive eige nfunction corresponding to λ
1
μ
is
unique up to a constant mult iple.
(ii) λ
1
μ
is unique, that is, if v ≥ 0 is an eigenfunction associated with an eigenvalue λ with

Ω
f (ξ)|v|
p
dξ = 1, then λ = λ
1
μ
.

(iii) Every eigenfunction corresponding to the eigenvalue λ (0 <λ
=λ
1
μ
) changes sign in Ω.
Proof. (i) Let u>0andv>0 be two eigenfunctions corresponding to λ
1
μ
in ᏹ.Forsuffi-
ciently small ε>0, set φ
= u
p
/(v + ε)
p−1
∈ D
1,p
0
(Ω). Then

Ω


∇
H
v


p−2

∇

H
v,∇
H
φ

dξ = μ

Ω
ψ
p
u
p−1
d
p
φdξ + λ
1
μ

Ω
f (ξ)v
p−1
φdξ. (5.4)
20 Journal of Inequalities and Applications
Using Proposition 5.1 and (5.4),
0
≤

Ω
L(u,v + ε) =


Ω
R(u,v + ε)
=

Ω


∇
H
u


p
dξ −

Ω


∇
H
v


p−2

∇
H

u
p

(v + ε)
p−1

,∇
H
v

dξ
=

Ω

μ
ψ
p
d
p
+ λ
1
μ
f (ξ)

u
p
dξ −

Ω

μ
ψ

p
d
p
+ λ
1
μ
f (ξ)

v
p−1
u
p
(v + ε)
p−1
dξ
=

Ω

μ
ψ
p
d
p
+ λ
1
μ
f (ξ)

u

p

1 −
v
p−1
(v + ε)
p−1

dξ.
(5.5)
The rig ht-hand side of (5.5) tends to zero when ε
→0. It follows that L(u,v) = 0andby
Proposition 5.1 there exists a constant c such that u
= cv.
(ii) Let u>0andv>0 be eigenfunctions corresponding to λ
1
μ
and λ, respectively. Sim-
ilarly to (5.5), we have

Ω

μ
ψ
p
d
p
+ λ
1
μ

f (ξ)

u
p
dξ −

Ω

μ
ψ
p
d
p
+ λf(ξ)

v
p−1
u
p
(v + ε)
p−1
dξ
=

Ω
μ
ψ
p
d
p


1 −
v
p−1
(v + ε)
p−1

u
p
dξ +

Ω
f (ξ)u
p

λ
1
μ
− λ
v
p−1
(v + ε)
p−1

dξ ≥ 0.
(5.6)
Letting ε
→0 shows that

λ

1
μ
− λ


Ω
f (ξ)u
p
dξ ≥ 0, which is impossible for λ>λ
1
μ
.Hence
λ
= λ
1
μ
.
(iii) With the same treatment as in (5.6)weget

λ
1
μ
− λ


Ω
f (ξ)u
p
dξ ≥ 0. (5.7)
Noting that


Ω
f (ξ)u
p
dξ > 0andλ>λ
1
μ
leads to a contradiction. So v must change sign
in Ω.

Lemma 5.3. If u ∈ D
1,p
0
(Ω) is a nonnegative weak solution of (1.2), then e ither u(ξ) ≡ 0 or
u(ξ) > 0 for all ξ
∈ Ω.
Proof. For any R>r, B
H
(0,R) ⊃ B
H
(0,r), let u ∈ D
1,p
0
(Ω) be a nonnegative weak solution
of (1.2). In virtue of Harnack’s inequalit y (see [1]), there exists a constant C
R
> 0suchthat
sup
B
H

(0,R)

u(ξ)

≤
C
R
inf
B
H
(0,R)

u(ξ)

. (5.8)
This implies u
≡ 0oru>0inΩ. 
Theorem 5.4. Every eigenfunction u
1
corresponding to λ
1
μ
does not change sign in Ω: either
u
1
> 0 or u
1
< 0.
Proof. From the proof of existence of the first eigenvalue we see that there exists a positive
eigenfunction, that is, if v is an eigenfunction, then u

1
=|v| is a solution of the minimiza-
tion problem and also an eigenfunction. Thus, from Lemma 5.3 it follows that
|v| > 0and
then u
1
has constant sign. 
Jingbo Dou et al. 21
Lemma 5.5. For u
∈ C(Ω \{0}) ∩ D
1,p
0
(Ω),letᏺ be a component of {ξ ∈ Ω | u(ξ) > 0}.
Then u
|
ᏺ
∈ D
1,p
0
(ᏺ).
Proof. Let u
m
∈ C(Ω \{0}) ∩ D
1,p
0
(Ω)besuchthatu
m
→u in D
1,p
0

(Ω). Therefore, u
+
m
→u
+
in D
1,p
0
(Ω). Set v
m
(ξ) = min{u
m
(ξ),u(ξ)},andletϕ
R
(ξ) ∈ C(Ω)beacutoff function such
that
ϕ
R
(ξ) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
0ifd(ξ) ≤
R
2
,

1ifd(ξ)
≥ R,
(5.9)
with
|∇
H
ϕ
R
|≤C|∇
H
d|/d(ξ), for some positive constant C. Now, consider the sequence
ω
m
(ξ) = ϕ
R
(ξ)v
m
(ξ)|
ᏺ
.Sinceϕ
R
(ξ)v
m
(ξ) ∈ C(
Ω
), we claim that ω
m
∈ C(
ᏺ
)andω

m
= 0
on the boundary ∂ᏺ.Infact,ifξ
∈ ∂ᏺ and ξ = 0, then ϕ
R
= 0, and so ω
m
= 0. If ξ ∈
∂ᏺ ∩ Ω and ξ=0, then u = 0 (since u is continuous except at {0}), and hence v
m
= 0. If
ξ
∈ ∂Ω,thenu
m
= 0andsov
m
= 0. Therefore, ω
m
= 0on∂ᏺ,andω
m
∈ D
1,p
0
(ᏺ). Noting

Ω


∇
H

ω
m
−∇
H

ϕ
R
u



p
dξ =

ᏺ


ϕ
R
∇
H
v
m
+ v
m
∇
H
ϕ
R
− ϕ

R
∇
H
u − u∇
H
ϕ
R


p
dξ
≤


ϕ
R

∇
H
v
m
−∇
H
u



p
L
p

(ᏺ)
+


∇
H
ϕ
R

v
m
− u



p
L
p
(ᏺ)
,
(5.10)
it is obvious that

Ω
|∇
H
ω
m
−∇
H

(ϕ
R
u)|
p
dξ→0, as m→∞.Thatisω
m
→ϕ
R
u|
ᏺ
in D
1,p
0
(ᏺ).
By (2.1),

ᏺ


u∇
H
ϕ
R
+ ϕ
R
∇
H
u −∇
H
u



p
dξ
≤

ᏺ


ϕ
R
∇
H
u −∇
H
u


p
dξ +

ᏺ∩{R/2<d<R}


u∇
H
ϕ
R



p
dξ
≤

ᏺ


ϕ
R
∇
H
u −∇
H
u


p
dξ + C
p

ᏺ∩{R/2<d<R}
ψ
p
|u|
p
d
p
dξ
≤


ᏺ


ϕ
R
∇
H
u −∇
H
u


p
dξ + C
1

ᏺ∩{R/2<d<R}


∇
H
u


p
dξ,
(5.11)
which approaches 0, as R
→0. Hence u|
ᏺ

∈ D
1,p
0
(ᏺ). 
Theorem 5.6. The eigenvalue λ
1
μ
is isolated in the spectrum, that is, there exists δ>0 such
that there is no other eigenvalues of (1.2)intheinterval(λ
1
μ
,λ
1
μ
+ δ).Moreover,ifv is an
eigenfunction corresponding to the eigenvalue λ
=λ
1
μ
and ᏺ is a nodal domain of v, then

Cλ f 
L
∞

−Q/p
≤|ᏺ|, (5.12)
where C is a constant depending only on Q and p.
22 Journal of Inequalities and Applications
Proof. Let u

1
be the eigenfunction corresponding to the eigenvalue λ
1
μ
.Let{λ
m
} be a
sequence of eigenvalues such that λ
m
>λ
1
μ
and λ
m
 λ
1
μ
, and the corresponding eigenfunc-
tions u
m
→u
1
with

Ω
f (ξ)|u
m
|
p
dξ = 1, that is, λ

m
and u
m
satisfy
L
p,μ
u
m
= λ
m
f (ξ)


u
m


p−2
u
m
. (5.13)
Since
0 <

Ω


∇
H
u

m


p
dξ − μ

Ω
ψ
p
d
p


u
m


p
dξ = λ
m

Ω
f (ξ)


u
m


p

dξ = λ
m
, (5.14)
it follows that u
m
is bounded. By Lemma 4.3, there exists a subsequence (still denoted
by
{u
m
})of{u
m
} such that u
m
 u weakly in D
1,p
0
(Ω), u
m
→u strongly in L
p
(Ω)and
∇
H
u
m
→∇
H
u a.e in Ω. Letting m→∞ in (5.13)yields
L
p,μ

u = λ
1
μ
f (ξ)|u|
p−2
u. (5.15)
Therefore, u
=±u
1
. Using (iii) of Theorem 5.2 we see that u
m
changes sign. For conve-
nience, we assume that u
= +u
1
.Then



ξ ∈ Ω | u
m
< 0



−→
0. (5.16)
Now, we check (5.13)withu
m
= u

−
m
,

Ω


∇
H
u
−
m


p
dξ − μ

Ω
ψ
p
d
p


u
−
m


p

dξ = λ
m

Ω
f (ξ)


u
−
m


p
dξ. (5.17)
Using the Hardy inequality and Sobolev inequality yields

1 −
μ
C
Q,p


Ω
−


∇
H
u
m



p
dξ
≤

Ω
−


∇
H
u
m


p
dξ − μ

Ω
−
ψ
p
d
p


u
m



p
dξ
= λ
m

Ω
−
f (ξ)


u
m


p
dξ
≤ λ
m
 f 
L
∞

Ω
−


u
m



p
dξ
≤ c
1
 f 
L
∞


u
m


p
D
1,p


Ω
−
m


p/Q
,


Ω
−

m


≥

c
2
 f 
L
∞

Q/p
, Ω
−
m
=

ξ ∈ Ω | u
m
< 0

.
(5.18)
It contradicts with (5.16). Hence, there is no other eigenvalue of (1.2)in(λ
1
μ
,λ
1
μ
+ δ)for

δ>0.
Next, we prove (5.12). Assume v>0inᏺ (the case v<0 being treated similarly). In
view of Lemma 5.5,wehavev
|
ᏺ
∈ D
1,p
0
(ᏺ). Define the function
ω(ξ)
=
⎧
⎨
⎩
v(ξ)ifξ ∈ ᏺ,
0ifξ
∈ Ω \ ᏺ.
(5.19)
Jingbo Dou et al. 23
Clearly, ω(ξ)
∈ D
1,p
0
(Ω). Taking ω as a test function in (4.17) satisfied by v and arguing as
in (5.18), we have

1 −
μ
C
Q,p



v
p
D
1,p
(ᏺ)
≤ λ f 
L
∞

ᏺ
|v|
p
dξ ≤ λ

C f 
L
∞
v
p
D
1,p
(ᏺ)
|ᏺ|
p/Q
(5.20)
for some constant

C = C(Q, p) and hence (5.12)holds. 

Corollary 5.7. Each eigenfunction has a finite number of nodal domains.
Proof. Let ᏺ
j
be a nodal domain of an eigenfunction associated to some positive eigen-
value λ.Itfollowsfrom(5.12)that
|Ω|≥

j


ᏺ
j


≥

Cλ f 
L
∞

−Q/p

j
1. (5.21)
The result is proved.

Acknowledgments
The authors would like to thank the referee for their careful reading of the manuscri pt and
many good suggestions. The project is supported by Natural Science Basic Research Plan
in Shaanxi Province of China, Program no. 2006A09. Pengcheng Niu is the corresponding

author (email address: ).
References
[1] L. Capogna, D. Danielli, and N. Garofalo, “An embedding theorem and the Harnack inequality
for nonlinear subelliptic equations,” Communications in Partial Differential Equations, vol. 18,
no. 9-10, pp. 1765–1794, 1993.
[2] A. Domokos and J. J. Manfredi, “C
1,α
-regularity for p-harmonic functions in the Heisenberg
group for p near 2,” in The p-Harmonic Equation and Recent Advances in Analysis, vol. 370 of
Contemporary Mathematics, pp. 17–23, American Mathematical Society, Providence, RI, USA,
2005.
[3] W. Allegretto, “Principal eigenvalues for indefinite-weight elliptic problems in
R
N
,” Proceedings
of the American Mathematical Society, vol. 116, no. 3, pp. 701–706, 1992.
[4] Adimurthi, N. Chaudhuri, and M. Ramaswamy, “An improved Hardy-Sobolev inequality and
its application,” Proceedings of the American Mathematical Society, vol. 130, no. 2, pp. 489–505,
2002.
[5] N. Chaudhuri and M. Ramaswamy, “Existence of positive solutions of some semilinear elliptic
equations with singular coefficients,” Proceedings of the Royal Society of Edinburgh: Section A,
vol. 131, no. 6, pp. 1275–1295, 2001.
[6] Adimurthi and M. J. Esteban, “An improved Hardy-Sobolev inequality in W
1,p
and its applica-
tion to Schr
¨
odinger operators,” Nonlinear Differential Equations and Applications, vol. 12, no. 2,
pp. 243–263, 2005.
[7] J. P. Garc

´
ıa Azorero and I. Peral Alonso, “Hardy inequalities and some critical elliptic and para-
bolic problems,” Journal of Differential Equations, vol. 144, no. 2, pp. 441–476, 1998.
[8] A. L
ˆ
e, “Eigenvalue problems for the p-Laplacian,” Nonlinear Analysis: Theory, Methods & Appli-
cations, vol. 64, no. 5, pp. 1057–1099, 2006.
24 Journal of Inequalities and Applications
[9] J.L.V
´
azquez and E. Zuazua, “The Hardy inequality and the asymptotic behaviour of the heat
equation with an inverse-square potential,” Journal of Functional Analysis, vol. 173, no. 1, pp.
103–153, 2000.
[10] P. L. De Napoli and J. P. Pinasco, “Eigenvalues of the p-Laplacian and disconjugacy criteria,”
Journal of Inequalities and Applications, vol. 2006, Article ID 37191, 8 pages, 2006.
[11] J. Jaro
ˇ
s, K. Taka
ˆ
si, and N. Yoshida, “Picone-type inequalities for nonlinear elliptic equations and
their applications,” Journal of Inequalities and Applications, vol. 6, no. 4, pp. 387–404, 2001.
[12] S. Yaotian and C. Zhihui, “General Hardy inequalities with optimal constants and remainder
terms,” Journal of Inequalities and Applications, no. 3, pp. 207–219, 2005.
[13] G. Barbatis, S. Filippas, and A. Tertikas, “A unified approach to improved L
p
Hardy inequalities
with best constants,” Transactions of the American Mathematical Society, vol. 356, no. 6, pp. 2169–
2196, 2004.
[14] G. Barbatis, S. Filippas, and A. Tertikas, “Series expansion for L
p

Hardy inequalities,” Indiana
University Mathematics Journal, vol. 52, no. 1, pp. 171–190, 2003.
[15] P. Niu, H. Zhang, and Y. Wang, “Hardy type and Rellich type inequalities on the Heisenberg
group,” Proceedings of the American Mathematical Society, vol. 129, no. 12, pp. 3623–3630, 2001.
[16] L. D’Ambrosio, “Critical degenerate inequalities on the Heisenberg group,” Manuscripta Math-
ematica, vol. 106, no. 4, pp. 519–536, 2001.
[17] L. D’Ambrosio, “Hardy-type inequalities related to degenerate elliptic differential operators,”
Annali della Scuola Normale Superiore di Pisa, vol. 4, no. 3, pp. 451–486, 2005.
[18] N. Garofalo and D M. Nhieu, “Isoperimetric and Sobolev inequalities for Carnot-Carath
´
eodory
spaces and the existence of mimimal surfaces,” Communications on Pure and Applied Mathemat-
ics, vol. 49, no. 10, pp. 1081–1144, 1996.
[19] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and
Hamiltonian Systems, vol. 34 of A Series of Modern Surveys in Mathematics, Springer, Berlin,
Germany, 3rd edition, 2000.
[20] L. Boccardo and F. Murat, “Almost everywhere convergence of the gradients of solutions to
elliptic and parabolic equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 19,
no. 6, pp. 581–597, 1992.
Jingbo Dou: Department of Applied Mathematics, Northwestern Polytechnical University,
Xi’an, Shaanxi 710072, China
Email address:
Pengcheng Niu: Department of Applied Mathematics, Northwestern Polytechnical University,
Xi’an, Shaanxi 710072, China
Email address:
Zixia Yuan: Department of Applied Mathematics, Northwestern Polytechnical University,
Xi’an, Shaanxi 710072, China
Email address: