Local and global sharp gradient estimates for weighted pharmonic functions

Local and global sharp gradient estimates for weighted
p-harmonic functions
Nguyen Thac Dung and Nguyen Duy Dat
May 28, 2015
Abstract
n

−f

Let (M , g, e dv) be a smooth metric measure space of dimensional n. Suppose that v is
a positive weighted p-harmonic functions on M , namely
ef div(e−f |∇v|p−2 ∇v) = 0.
We first give a local gradient estimate for v, as a consequence we show that if Ricm
f ≥ 0 then
v is constant provided that v is of sublinear growth. At the same time, we prove a Harnack
inequality for such a weighted p-harmonic function v. Moreover, we show a global sharp gradient
estimate for weighted p-eigenfunctions. Then we use this estimate to study geometric structure
at infinity when the first eigenvalue λ1,p obtains its maximal value.
2000 Mathematics Subject Classification: 53C23, 53C24
Keywords and Phrases: Gradient estimates, weighted p-harmonic functions, smooth metric measure spaces, Liouville property, Harnack inequality

1.

Introduction

The local Cheng-Yau gradient estimate is a standard result in Riemannian geometry,
see [6], also see [22]. It asserts that if M be an n dimensional complete Riemannian
manifold with Ric ≥ −(n − 1)κ for some κ ≥ 0, for u : B(o, R) ⊂ M → R harmonic
and positive then there is a constant cn depending only on n such that
√
1 + κR

|∇u|
≤ cn
.
(1.1)
sup
R
B(o,R/2) u
Here B(o, R) stands for the geodesic centered at a fixed point o ∈ M . Notice that
when κ = 0, this implies that a harmonic function with sublinear growth on a manifold
with non-negative Ricci curvature is constant. This result is clearly sharp since on
Rn there exist harmonic functions which are linear.
Cheng-Yau’s method is then extended and generalized by many mathematicians.
For example, Li-Yau (see [9]) obtained a gradient estimate for heat equations. Cheng

1


(see [5]) and H. I. Choi (see [7]) proved gradient estimates for harmonic mappings,
etc. We refer the reader to survey [8] for an overview of the subject.
When (M n , g, e−f dv) is a smooth metric measure space, it is very natural to find
similar results. Recall that the triple (M n , g, e−f dµ) is called a smooth metric measure
space if (M, g) is a Riemannian manifold, f is a smooth function on M and dµ is the
volume element induced by the metric g. On M , we consider the differential operator
∆f , which is called f −Laplacian and given by
∆f · := ∆ · −

f,

· .

It is symmetric with repect to the measure e−f dµ. That is,
(∆f ϕ)ψe−f ,

ϕ, ψ e−f = −
M

M

for any ϕ, ψ ∈ C0∞ (M ). Smooth metric measure spaces are also called manifolds with
´
density. By m-dimensional Bakry-Emery
Ricci tensor we mean
Ricm
f = Ric + Hessf −

∇f ⊗ ∇f
,
m−n

´
for m ≥ n. Here m = n iff f is constant. The ∞−Barky-Emery
tensor is refered as
Ricf = Ric + Hessf.
Brighton (see [2]) gave a gradient estimate of positive weighted harmonic function,
as a consequence, he proved that any bounded weighted harmonic function on a
smooth metric measure space with Ricf ≥ 0 has to be constant. Later, Munteanu and
Wang refined Brighton’s argument and proved that positive f -harmonic function of
sub-exponential growth on smooth metric measure space with nonnegative Ricf must
be a constant function. Moreover, Munteanu and Wang also applied the De GiorgiNash-Moser theory to get a sharp gradient estimate for any positive f -harmonic
function provided that the weighted function f is at most linear growth (see [18, 19]

for further results). On the other hand, Wu derived a Li-Yau type estimate for
parabolic equations. He also made some results for heat kernel (see [28, 30] for the
details.).
From a variational point of view, p-harmonic function, or more general weighted pharmonic functions are natural extensions of harmonic functions, or weight harmonic
functions, respectively. Compared with the theory for (weighted) harmonic functions,
the study of (weighted) p-harmonic functions is generally harder, even though elliptic,
is degenerate and the regularity results are far weaker. We refer the reader to [17, 11]
for the connection between p-harmonic functions and the inverse mean curvature flow.
For the weighted p-harmonic function, Wang (see [24]) estimated eigenvalues of this
2


operator. On the other hand, Wang, Yang and Chen (see [27]) shown gradient estimates and entropy formulae for weighted p-heat equations. Their works generalized
Li’s and Kotschwar-Ni’s results (see [16, 11]).
In this paper, motived by Wang-Zhang’s gradient estimate for the p-harmonic
function, we give the following result.
Theorem 1.1. Let (M n , g, e−f ) be a smooth metric measure space of dimension n
with Ricm
f ≥ −(m − 1)κ. Suppose that v is a positive smooth weighted p-harmonic
function on the ball BR = B(o, R) ⊂ M . Then there exists a constant C = C(p, m, n)
such that
√
|∇v|
C(1 + κR)
≤
on B(o, R/2).
(1.2)
v
R
We note that this theorem is very much motivated by and follows the ideas of

Wang and Zhang in the paper [26]. Our argument is close to the one used in [26].
In the gerenal case, we obtain a sharp gradient estimate for weighted p-eigenfunction
as follows.
Theorem 1.2. Let (M n , g, e−f dv) be an n-dimensional complete noncompact manifold with Ricm
f ≥ −(m − 1). If v is a positive weighted p-eigenfunction with respect
to the first eigenvalue λ1,p , that is,
ef div(e−f |∇v|p−2 ∇v) = −λ1,p v p−1
then
|∇ ln v| ≤ y.
Here y is the unique positive root of the equation
(p − 1)y p − (m − 1)y p−1 + λ1,p = 0.
A directly consequence of the theorem 1.2 is a sharp gradient estimate for positive
weighted p-harmonic function.
Corollary 1.3. Let (M n , g, e−f dv) be an n-dimensional complete noncompact manifold with Ricm
f ≥ −(m − 1). If v is a positive weighted p-harmonic function then
|∇ ln v| ≤

m−1
.
p−1

The sharpness of the estimate is demonstrated by the below example.
Example 1.4. Let M n = R × N n−1 with a warped product metric
ds2 = dt2 + e2t ds2N ,

3


where N is a complete manifold with non-negative Ricci curvature. Then it can be
directly checked that RicM ≥ −(n−1) (See [14] for details of computation). Moreover,

we have
∂2
∂
∆ = 2 + (n − 1) + e−2t ∆N .
∂t
∂t
´
Choose weighted function f = (m − n)t, then the m-dimensional Bakry-Emery
curvature is bounded from below by −(m − 1).
Let v(t, x) = e−at , where m−1
≤ a ≤ m−1
, we have
p
p−1
|∇v|
= a.
v
It is easy to show that
|∇v|p−2 ∇v, ∇f = −(m − n)ap−1 v p−1
∇|∇v|p−2 , ∇v = (p − 2)ap v p−1
|∇v|p−2 ∆v = (1 − n + a)ap−1 v p−1 .
Hence,
ef div(e−f |∇v|p−2 ∇v) = ((p − 1)a − (m − 1))ap−1 v p−1 .
This implies that
λ1,p = (m − 1 − (p − 1)a)ap−1 ,
or equivalently,
(p − 1)ap − (m − 1)ap−1 + λ1,p = 0.
It is also very interesting to ask what is geometric structure of manifolds with λ1,p
acheiving its maximal value. When f is constant, this problem has been studied by
Li-Wang, Sung-Wang in [12, 13, 23]. In this paper, we prove a generalization of their

result.
Theorem 1.5. Let (M n , g, e−f dv) be a smooth metric measure space of dimension
n ≥ 2. Suppose that Ricm
f ≥ −(m − 1) and λ1,p =

m−1
p

p

. Then either M has no

p-parabolic ends or M = R × N n−1 for some compact manifold N . Here the definition
of p-parabolic ends is given in the section 3.
This paper is organized as follows. In the section 2, we give a proof of the main
theorem 1.1 by using the Moser’s iteration. As its applications, we show a Liouville
property and a Harnack inequality for weighted p-harmonic functions. In the section
3, we prove the theorem 1.2. The proof the theorem 1.5 is given in the section 4.

4


2.

Local gradient estimates for weighted p-harmonic functions on (M, g, e−f dµ)

Let v be a positive weighted p-eigenfunction function with respect to the first eigenvalue λ1,p , namely, v is a smooth solution of weighted p-Laplacian equation,
∆p,f v := ef div(e−f |∇v|p−2 ∇v) = −λ1,p v p−1 .

(2.1)

Note that, when λ1,p = 0 then v is a weighted p-harmonic function. Let u = −(p −
1) log v, then v = e−u/(p−1) . It is easy to see that u satisfies
ef div(e−f |∇u|p−2 ∇u) = |∇u|p + λ1,p (p − 1)p−1 .
Put h := | u|2 , the above equation can be rewritten as follows.
p
− 1 hp/2−2 ∇h, ∇u + hp/2−1 ∆f u = hp/2 + λ1,p (p − 1)p−1 .
2

(2.2)

Assume that h > 0. As in [11], [17], we consider the below operator
Lf (ψ) := ef div e−f hp/2−1 A(∇ψ) − php/2−1 ∇u, ∇ψ ,
where
A = id + (p − 2)

∇u ⊗ ∇u
.
|∇u|2

We have the following lemma and the proof is by direct computation.
Lemma 2.1.
Lf (h) = 2hp/2−1 (u2ij + Ricf (∇u, ∇u)) +

p
− 1 hp/2−2 |∇h|2 .
2

(2.3)

Proof. By the definition of Lf , we have
p

Lf (h) =ef div e−f h 2 −1 ∇h + (p − 2)
p

∇u, ∇h
∇u
h

p

− ph 2 −1 ∇u, ∇h

p

=h 2 −1 ∆h + (p − 2)h 2 −1 div h−1 ∇u, ∇h ∇u
p

p

+ ef ∇ e−f h 2 −1 , ∇h + (p − 2)h−1 ∇u, ∇h ∇u − ph 2 −1 ∇u, ∇h
p

p

p

=h 2 −1 ∆h + (p − 2)h 2 −2 ∇u, ∇h ∆u − (p − 2)h 2 −3 ∇u, ∇h 2
p

p
p
p
p
+ (p − 2)h 2 −2 (uij hi uj + hij ui uj ) +
− 1 h 2 −2 |∇h|2 + (p − 2)
− 1 h 2 −3 ∇u, ∇h
2
2
p
p
p
−1
−2
−1
2
2
2
−h
∇f, ∇h − (p − 2)h
∇u, ∇h ∇f, ∇u − ph
∇u, ∇h .

5

2


Hence,
p

p
− 1 h 2 −2 |∇h|2
2
p
p
p
−2
2
− 2 h 2 −3 ∇h, ∇u 2
+ (p − 2)h
∇h, ∇u ∆f u + (p − 2)
2
p
p
−2
2
2
+ (p − 2)h (uij hi uj + hij ui uj ) − ph −1 ∇u, ∇h
p
p
p
− 1 h 2 −2 |∇h|2
=2h 2 −1 u2ij + Ricf (∇u, ∇u) +
2
p
p
p
p
+ (p − 2)h 2 −2 ∇h, ∇u ∆f u + 2h 2 −1 ∇∆f u, ∇u + (p − 2)
− 2 h 2 −3 ∇h, ∇u

2
p
p
−2
−1
2
2
+ (p − 2)h (uij hi uj + hij ui uj ) − ph
∇u, ∇h .
p

Lf (h) =h 2 −1 ∆f h +

Here we used the Bochner identity
∆f h = ∆f |∇u|2 = 2 u2ij + Ricf (∇u, ∇u) + 2 ∇∆f u, ∇u
in the last equation.
On the other hand, differentiating both side of (2.2) then multiplying the obtained
results by ∇u, we have
p
p
p
p p −1
h2
∇h, ∇u =
− 1 h 2 −2 ∇h, ∇u ∆f u + h 2 −1 ∇∆f u, ∇u
2
2
p
p
p

p
p
+
−2
− 1 h 2 −3 ∇h, ∇u 2 +
− 1 h 2 −2 (uij hi uj + hijui uj )
2
2
2

Combining this equation and the above equation, we are done.
Now, suppose that v is a weighted p-harmonic function, so we can assume λ1,p = 0.
We choose a local orthonormal frame {ei } with e1 = ∇u/|∇u| then
n

u21i =

2hu11 = ∇u, ∇h ,
i=1

1 |∇h|2
.
4 h

Hence, (2.2) takes the following form
n

(p − 1)u11 +

uii = h + ∇f, ∇u .

i=2

6

2


Therefore
n

u2ij ≥ u211 + 2

n

u2ii

u21i +
i=2

i=2
n

≥ u211 + 2
i=2
n

= u211 + 2

1
u21i +

n−1

i=2

uii
i=2

u21i +

1
(h − (p − 1)u11 + ∇f, ∇u )2
n−1

u21i +

1
n−1

i=2
n

≥ u211 + 2

2

n

∇u, ∇f
(h − (p − 1)u11 )2
−

m−n
m−n
1 + n−1
n−1

1
2(p − 1)
(p − 1)2
≥
h2 −
hu11 + 1 +
m−1
m−1
m−1
≥

2(p − 1)
1
h2 −
hu11 + a0
m−1
m−1

n

2

n

u211

u21i −

+2
i=2

∇f, ∇u
m−n

2

2

u21i −
i=1

∇f, ∇u
,
m−n

(p−1)2

where a0 = max 1 + m−1 , 2 > 1. Note that we used (a − b)2 ≥
in the fourth inequality. Again, by using the identities
n

u21i =

2hu11 = ∇u, ∇h ,
i=1

a2
1+δ

−

b2
δ

for δ > 0

1 |∇h|2
4 h

we conclude that
1
p−1
a0 |∇h|2
∇f, ∇u 2
2
≥
h −
∇u, ∇h +
−
.
m−1
m−1
4 h
m−n
Assume that Ricm

f ≥ −(m − 1)κ, we infer
p
− 1 hp/2−2 |∇h|2
Lf (h) = 2hp/2−1 (u2ij + Ricf (∇u, ∇u)) +
2
1
p−1
a0 |∇h|2
df ⊗ df
≥ 2hp/2−1
h2 −
∇u, ∇h +
+ Ricf −
m−1
m−1
4 h
m−n
p
+
− 1 hp/2−2 |∇h|2
2
p + a0
≥ −2(m − 1)κhp/2 +
− 1 |∇h|2 hp/2−2
2
2
2(p − 1) p/2−1
+
hp/2+1 −
h

∇u, ∇h
m−1
m−1
2
2(p − 1) p/2−1
≥ −2(m − 1)κhp/2 +
hp/2+1 −
h
∇u, ∇h
m−1
m−1
u2ij

7

(∇u, ∇u)


The above equation holds wherever h is strictly positive. Let K = {x ∈ M, h(x) = 0}.
Then for any non-negative function ψ with compact support in Ω \ K, we have
hp/2−1 ∇h + (p − 2)hp/2−2 ∇u, ∇h ∇u, ∇ψ e−f
Ω

hp/2−1 ∇u, ∇h ψe−f +

+p
Ω

hp/2 ψe−f

≤ 2(m − 1)κ
Ω

2(p − 1)
+
m−1

2
m−1

hp/2+1 ψe−f
Ω

hp/2−1 ∇u, ∇h ψe−f

(2.4)

Ω

In order to consider the cases h = 0, for ε > 0, b > 2 we choose ψ = hbε η 2 where
hε = (h − ε)+ , η ∈ C0∞ (BR ) is non-negative, 0 ≤ η ≤ 1, b is to be determined later.
Then direct computation shows that
2
b
∇ψ = bhb−1
ε ∇hη + 2hε η∇η.

Plugging this identity into (2.4), we have
p

2
h 2 −1 hb−1
ε |∇h| + (p − 2)h

b

p−2
−2
2

hb−1
∇u, ∇h
ε

2

η 2 e−f

Ω
p

p

h 2 −1 hbε ∇η, ∇h ηe−f + 2(p − 2)

+2
Ω

+p

h 2 −2 hbε ∇u, ∇h ∇u, ∇η ηf −f
Ω

h

p
−1
2

p
2
h 2 +1 hbε η 2 e−f
m−1 Ω
p
2(p − 1)
h 2 −1 hbε η 2 e−f .
+
m−1 Ω

hbε ∇u, ∇h η 2 e−f +

Ω
p

h 2 hbε η 2 e−f

≤ 2(m − 1)κ
Ω

Let

if p ≥ 2
,
if 1 < p < 2

1
p−1

a1 =
it is easy to see that
p

h 2 −1 hεb−1 |∇h|2 + (p − 2)h

p−2
−2
2

hb−1
∇u, ∇h
ε

2

p

2
≥ a1 h 2 −1 hb−1
ε |∇h| .

Hence, by passing ε to 0, we obtain

p

h 2 +b−2 |∇h|2 η 2 e−f

ba1
Ω

p

Ω

+p

h

p

h 2 +b−2 ∇u, ∇h ∇u, ∇η ηe−f + 2

+ 2(p − 2)
p
+b−1
2

Ω

p

h 2 +b η 2 e−f
Ω

2
h
η 2 e−f
m−1 Ω
p
2(p − 1)
+
h 2 +b−1 ∇u, ∇h η 2 e−f .
m−1 Ω
p
+b+1
2

∇u, ∇h η 2 e−f +

Ω

≤ 2(m − 1)κ

h 2 +b−1 ∇h, ∇η ηe−f

8

(2.5)


From now on, we use a1 , a2 , . . . to denote constants depending only on m, p, n. Combining terms in (2.5) and using the definition of h, we infer
p

h 2 +b−2 |∇h|2 η 2 e−f +

a1 b
Ω

≤ 2(m − 1)κ

h

p
+b
2

2
m−1

+ a3

h

Ω

η 2 e−f + a2

Ω
p
+b−1
2

p

h 2 +b+1 η 2 e−f
h

p−1
+b
2

|∇h|η 2 e−f

Ω

|∇h||∇η|ηe−f .

(2.6)

Ω

For i, j = 1, 2, 3, . . ., again, from now on we use Ri , Lj to denote the i-th term on
the right hand side, the j-th term in the left hand side. For the third term on the
2
right hand side, using the Cauchy’s inequality 2αβ ≤ α4c + cβ 2 for c > 0, we have the
following estimate.
ba1
4

|R3 | ≤

p

h 2 +b−2 |∇h|2 η 2 e−f +
Ω

a4
b

p

|∇η|2 h 2 +b e−f .
Ω

By Cauchy’s inequality again, R2 can be estimated by
ba1
4

|R2 | ≤

p

h 2 +b−2 |∇h|2 η 2 e−f +
Ω

a5
b

p

h 2 +b+1 η 2 e−f .
Ω

Combining these two inequalities togather with (2.6), we obtain
ba1
2

p
2
h 2 +b+1 η 2 e−f
m−1 Ω
Ω
p
p
a4
a5
|∇η|2 h 2 +b e−f +
≤ 2(m − 1)κ h 2 +b η 2 e−f +
b Ω
b
Ω
p

h 2 +b−2 |∇h|2 η 2 e−f +

p

h 2 +b+1 η 2 e−f .

(2.7)

Ω

By requiring
a5
1
≤
,
b
m−1

(2.8)

we have that (2.7) implies
ba1
2

p
1
2
h 2 +b+1 η e−f
m−1 Ω
Ω
p
p
a4
≤ 2(m − 1)κ h 2 +b η 2 e−f +
|∇η|2 h 2 +b e−f .
b Ω
Ω
p

h 2 +b−2 |∇h|2 η 2 e−f +

By Schwarz inequality, it is easy to see that
p

b

∇(h 4 + 2 η)

2

≤

1 p
+b
2 2

2

p

p

h 2 +b−2 |∇h|2 η 2 + 2h 2 +b |∇η|2 .

From (2.9), we have following lemma.
9

(2.9)

Lemma 2.2. Let M be a smooth metric measure space with Ricm
f ≥ −(m − 1)κ,
for some κ ≥ 0, and Ω ⊂ M is an open set and v is a smooth weighted p-harmonic
function on M . Let u = −(p − 1) log v and h = |∇u|2 . Then for any b > 2, there
exist c1 , c2 , c3 depending on b, m, n such that
p

b

2

p

∇(h 4 + 2 η) e−f +c1

h 2 +b+1 η 2 e−f
Ω

Ω

p

p

h 2 +b η 2 e−f + c3

≤ κc2
Ω

h 2 +b |∇η|2 e−f ,

(2.10)

Ω

for any η ∈ C0∞ (BR ), where BR is a geodesic ball centered at a fixed point o ∈ M .
Moreover, we have c1 ∼ b, c2 ∼ b (Here c1 ∼ b means c1 is comparable to b, c2 ∼ b is
understood the same way).
In [1], Bakry and Qian proved the following generalized Laplacian comparison
theorem (also see Remark 3.2 in [16])
√
√
(2.11)
∆f ρ := ∆ − ∇f, ∇ρ ≤ (m − 1) κ coth( κρ)
provided that Ricm
f ≥ −(m − 1)κ. Here ρ(x) := dist(o, x) stands for the distance
between x ∈ M and a fixed point o ∈ M . This implies the volume comparison
VHm (r2 )
Vf (Bx (r2 ))
≤
Vf (Bx (r1 ))
VHm (r1 )

(2.12)

(see [29] for details). It turns out that we have the local f -volume doubling property.
Then we follow the Buser’s proof [4] or the Saloff-Coste’s alternate proof (Theorem
5.6.5 in [20]), we can easily get a local Neumann Poincar´e inequality in the setting
of smooth metric measure spaces. Using the volume comparison theorem, the local
Neumann Poincar´e inequality and following the argument in [21], we obtain a local

Sobolev inequality as belows.
Theorem 2.3. Let (M, g, e−f dµ) be an n-dimensional complete noncompact smooth
metric measure space. If Ricm
f ≥ −(m − 1)κ for some nonnegative constants κ, then
for any p > 2, there exists a constant c = c(n, p, m) > 0 depending only on p, n, m
such that
|ϕ|

2p
p−2

−f

e

BR

√

p−2
p

≤

R2 .ec(1+

κR)

Vf (BR )

2
p

|∇ϕ|2 + R−2 ϕ2 e−f
BR

for any ϕ ∈ C0∞ (Bp (R)).
Proof. We refer the reader to [28] for the details of the argument.

10


From now on, we suppose Ω = BR . Theorem 2.3 implies
h

n(p/2+b)
n−2

2n
n−2

η

e

n−2
n

−f

BR

≤ ec(1+

√

κR)

2

Vf (BR )− n

p

b

2

p

∇(h 4 + 2 ) e−f +

R2

h 2 +b η 2 e−f

BR

(2.13)

BR

√
where c(n, p, m) > 0 depends only on n, p. Let b0 = C(n, p, m)(1 + κR) with
C(n, p, m) ≥ c(n, p, m) large enough to make b0 satisfy (2.8) and b0 > 2, then from
(2.10) and (2.13) we infer
h

n(p/2+b)
n−2

η

2n
n−2

n−2
n

−f

e

p

+ a6 beb0 R2 Vf (BR )−2/n

h 2 +b+1 η 2 e−f
BR

BR

p

p

h 2 +b η 2 e−f + a8 eb0 Vf (BR )−2/n R2

≤ a7 b20 beb0 Vf (BR )−2/n

h 2 +b |∇η|2 e−f .
BR

BR

(2.14)
Lemma 2.4. Let b1 = b0 +

p
2

n
.
n−2

Then there exists d = d(n, p, m) > 0 such that

||h||Lb1 (B3R/4 ) ≤ d

b20

Vf (BR )1/b1 .
R2

Proof. First let us choose b = b0 . We observe that if h >

(2.15)

2a7 b2
a6 R2

then

p
p
1
a7 b3 h 2 +b < a6 bR2 h 2 +b+1 .
2

Hence, in (2.14), R1 can be estimated by
p

R1 = a7 b3 eb Vf (BR )−2/n
BR ∩ h≤

≤

p+2b

b
R

ab9 b3 eb

2a7 b2
a6 R2
2

Vf (BR )1− n +

p

h 2 +b η 2 e−f + a7 b3 eb Vf (BR )−2/n
BR ∩ h>

2a7 b2
a6 R 2

h 2 +b η 2 e−f

L2
.
2

Now, (2.14) can be read as follows.
h

n(p/2+b)
n−2

η

2n
n−2

e

−f

n−2
n

+

BR

≤ ab9 b3 eb

b
R

p+2b

a6 b 2
be R Vf (BR )−2/n
2

p

h 2 +b+1 η 2 e−f
BR

2

p

Vf (BR )1− n + a8 eb Vf (BR )−2/n R2

h 2 +b |∇η|2 e−f .
BR

11

(2.16)


Now, we want to control R2 in terms of the left hand side. We choose η1 ∈ C0∞ (BR )
such that
C(n)
0 ≤ η1 ≤ 1, η1 ≡ 1, in B3R/4 , |∇η1 | ≤
.
R
p

Let η = η12

+b+1

. Direct computation shows
2b+p

R2 |∇η|2 ≤ a10 b2 η b+p/2+1 .
By the above inequality and Young’s inequality, the second term of the right hand
side of (2.16) can be estimated by
p

e−b Vf (BR )2/n R2 = a8 R2

h 2 +b |∇η|e−f
BR
2b+p

p

h 2 +b η b+p/2+1 e−f

≤ a8 a10 b2
BR

≤ a8 a10 b

2

h

p
+b+1
2

b+p/2
b+p/2+1

2 −f

η e

1

Vf (BR ) b+p/2+1

BR

ba6 2
R
2

≤

b+p/2 b

p

h 2 +b+1 η 2 e−f + a11
BR

b+p/2+2

R2b+p

Vf (BR ).

Plugging this inequality into (2.16), and note that b > 2, we obtain
n−2
n

h

n(p/2+b)
n−2

≤ ab12 b3 eb

B3R/4
n
Recall that b = b0 , b1 = b0 + p2 n−2
= b + p2
1
root on both sides of (2.17), we have
the b+p/2

||h||Lb1 (B3R/4 ) ≤ d

b
R

p+2b

n
n−2

and b0 = C(1 +

Vf (BR )1−2/n .

(2.17)
√

κR). Taking

1
b20
b1
V
(B
)
f
R
R2

here we used the boundedness of b1/b . The proof is complete.
Now, we give a proof of the main theorem.
Proof of Theorem 1.1. By (2.14), we have
h

n(p/2+b)
n−2

η

2n
n−2

e

−f

n−2
n

≤ a13

BR

eb0

p

2/n

Vf (BR )

b20 bη 2 + R2 |∇η|2 h 2 +b e−f . (2.18)
BR

In order to apply the Moser iteration, let us put
bk+1 = bk

n
,
n−2

Bk = B o,
12

R R
+ k
2
4

,

k = 1, 2, . . .


and choose ηk ∈ C0∞ (BR ) such that
ηk ≡ 1 in Bk+1 ,

η ≡ 0 in BR \ Bk ,

|∇ηk | ≤

C1 4k
,
R

0 ≤ ηk ≤ 1,

where C1 is a certain constant. Hence in (2.18), by letting b +
obtain
1
bk+1

bk+1 −f

h

≤

e

a13

Bk+1

p
2

= bk , η = ηk , we

1
bk

eb0

b20 bk

Vf (BR )2/n

2

2

+ R |∇ηk |

bk −f

h e

1
bk

.

Bk

By assumption of |∇ηk |, this implies
1
bk

b0

||h||Lbk+1 (Bk+1 ) ≤
∞

It is easy to see that
k=1

||h||L∞ (BR/2 ) ≤

a13

1
bk

=

e

n
.
2b1

e

Vf (BR )
e 2b1

1

(b20 bk + 16k ) bk ||h||Lbk (Bk ) .

Vf (BR )2/n

The above inequality leads to
∞
1
b
k=1 k

b0

nb0

≤ a14

a13

∞

b30

2/n
k=1

n
n−2

∞
k=1

k
bk

+ 16k

||h||Lb1 (B3R/4 )

3n

b 2b1 ||h||Lb1 (B3R/4 )
1/b1 0

Vf (BR )

here we used that

1
bk

k

(2.19)

converges. Now, by Lemma 2.4 and (2.19), we conclude
||h||L∞ (BR/2 ) ≤ a15

b20
.
R2

The proof is complete.
As a consequence, we obtain an important folowing theorem ralating to Liouvilleproperty for weighted p-harmonic functions.
Theorem 2.5. Assume that (M, g) is a smooth metric measure space with Ricm
f ≥ 0.
If u is a weighted p-harmonic function bounded from below on M and u is of sublinear
growth then u is constant.
Moreover, let x, y ∈ M be arbitrary points. There is a minimal geodesic γ(s)
joining x and y, γ : [0, 1] → M, γ(0) = x, γ(1) = y. By integrating (1.2) over this
geodesic, we obtain the below Harnack inequality.
13

Theorem 2.6. Let (M, g, e−f dv) be a complete smooth metric measure space of dimension n ≥ 2 with Ricm
f ≥ −(m − 1)κ. Suppose that v is a positive weighted pharmonic function on the geodesic ball B(o, R) ⊂ M . There exists a constant Cp,n,m
depending only on p, n, m such that
v(x) ≤ eCp,n,m (1+

√

κR)

v(y),

∀x, y ∈ B(o, R/2).

If κ = 0, we have a uniform constant cp,n,m (independent of R) such that
sup v ≤ cp,n
B(o,R/2)

3.

inf

v.

B(o,R/2)

Global sharp gradient estimate for weighted p-Laplacian

Recall that a function v is an eigenfunction of p-Laplacian with corresponding eigenvalue λ1,p ≥ 0 if
ef div(e−f |∇v|p−2 ∇v) = −λ1,p |v|p−2 v.
(3.1)

In this section, we only consider positive solution v. Set u = −(p−1) ln v, the equation
(3.1) can be rewritten as follows
ef div(e−f |∇u|p−2 ∇u) = |∇u|p + λ1,p (p − 1)p−1 .

(3.2)

Put h := | u|2 , assume that h > 0. As in [23], we consider
L(ψ) := ef div e−f hp/2−1 A(∇ψ)
which is a slight modification of L(ψ). By Lemma 2.1, we have that
L(h) = 2hp/2−1 (u2ij + Ricf (∇u, ∇u)) +

p
− 1 hp/2−2 |∇h|2
2

p

+ ph 2 −1 ∇u, ∇h
Let {e1 , e2 , . . . , en } be an orthonormal frame on M with |∇u|e1 = ∇u. Then (3.2)
can be read as
n

uii = h + ∇f, ∇u + |∇u|2−p λ1,p (p − 1)p−1 .

(p − 1)u11 +
i=2

14

Therefore,
n

u2ij ≥
i=1
n

≥

1
u21i +
n−1

(h + λ1,p (p − 1)p−1 |∇u|2−p + ∇f, ∇u − (p − 1)u11 )2
n−1

u21i +

1
n−1

u21i +

∇f, ∇u
(h + λ1,p (p − 1)p−1 |∇u|2−p )2
−
m−1
m−n

i=1

n

≥

uii
i=2

u21i +
i=1
n

≥

2

n

i=1

−

(h + λ1,p (p − 1)p−1 |∇u|2−p − (p − 1)u11 )2
∇u, ∇f
−
m−n
m−n
1 + n−1
n−1
2

2(p − 1)
2λ1,p (p − 1)p
hu11 −
|∇u|2−p u11 .
m−1
m−1

Note that we used (a − b)2 ≥
using the identities

a2
1+δ

−

b2
δ

for δ > 0 in the fourth inequality. Again, by
n

u21i =

2hu11 = ∇u, ∇h ,
i=1

1 |∇h|2
4 h

we conclude that

u2ij

2

1 |∇h|2 (h + λ1,p (p − 1)p−1 |∇u|2−p )2
∇f, ∇u 2
≥
+
−
4 h
m−1
m−n
2λ1,p (p − 1)p
∇h, ∇u
2(p − 1)
∇h, ∇u −
|∇u|2−p
.
−
m−1
m−1
h

15


Assume that Ricm
f ≥ −(m − 1)κ, we infer
L(h) =2hp/2−1 (u2ij + Ricf (∇u, ∇u)) +

p
− 1 hp/2−2 |∇h|2
2

p

+ ph 2 −1 ∇u, ∇h
1 |∇h|2 (h + λ1,p (p − 1)p−1 |∇u|2−p )2
df ⊗ df
≥2hp/2−1
+
+ Ricf −
4 h
m−1
m−n
p
p
+
− 1 hp/2−2 |∇h|2 + ph 2 −1 ∇u, ∇h
2
4(p − 1) p −1
2λ1,p (p − 1)p ∇h, ∇u
h2
−
∇h, ∇u −
m−1
m−1
h
p−1
2−p 2

(h + λ1,p (p − 1) |∇u| )
≥2hp/2−1
− (m − 1)h
m−1
p
p − 1 p/2−2
h
|∇h|2 + ph 2 −1 ∇u, ∇h
+
2
4(p − 1) p −1
2λ1,p (p − 1)p ∇h, ∇u
−
h2
.
∇h, ∇u −
m−1
m−1
h

(∇u, ∇u)

(3.3)

To show the sharp estimate, let x be the unique positive root of the equation
p

x 2 − (m − 1)x

p−1

2

+ λ1,p (p − 1)p−1 = 0.

For any δ > 0, we consider
ω=

h − (x + δ), h > x + δ
.
0,
otherwise

To show global sharp estimate of weighted p-eigenfunction, we need to have a upper
bound of λ1,p as follows
Lemma 3.1. Let (M n , g, e−f dv) be an n-dimensional complete noncompact manifold
with Ricm
f ≥ −(m − 1). Then
λ1,p ≤

m−1
p

p

.

In order to prove lemma 3.1, let us recall a definition.
Definition 3.2. (see [3]) Let (M n , g, e−f dv) be a smooth metric measure space. For
a fixed point o ∈ M , let B o (r) = {q ∈ M : dist(o, q) ≤ r}. An end E is an unbounded
component E of M \ B o (r0 ) for some r0 ≥ 0. For any 1 ≤ p < ∞. The end E is said

to be p-parabolic if for each K M and ε > 0 there exists a Lipschitz function φ with

16


compact support, φ ≥ 1 on K, such that
Here gφ (x) is defined as

E

gφp < ε. Otherwise, E is p-nonparabolic.

dist(φ(y), φ(x))
.
dist(y, x)
y∈Bx (r)

gφ (x) = lim inf
sup
+
r→0

Proof of lemma 3.1. Without loss of generality, we may assume that λ1,p is positive.
By the variational characterization of λ1,p , we know that M has infinite f -volume.
The theorem 0.1 in [3] implies M is p-nonparabolic, moreover
1/p

Vf (B(r)) ≥ Cepλ1,p r ,
for all sufficiently large r and C is a constant dependent on r. On the other hand,
the volume comparison theorem (2.12) infers

Vf (B(r)) ≤ C1 e(m−1)r .
Here C1 is a constant dependent on r. Therefore, we obtain
1/p

Cepλ1,p r ≤ C1 e(m−1)r ,
or equivalently,
1/p

λ1,p ≤

1
ln
pr

C1
C

+

m−1
p

for all sufficiently large r. Lettting r → ∞, we have
λ1,p ≤

m−1
p

p

.

The proof is complete.
We have following key lemma.
Lemma 3.3. Let (M n , g, e−f dv) be an n-dimensional complete noncompact manifold
with Ricm
f ≥ −(m − 1). Then there are some positive constants a and b depending on
p, n, m and δ such that
L(ω) ≥ aω − b|∇ω|,
(3.4)
in the weak sense, namely
L(φ)ωe−f ≥
M

φ(aω − b|∇ω|)e−f
M

for any non-negative function φ with compact support on M .

17

(3.5)


Proof. Since h = |∇u|2 , by the theorem 1.1, (see also the proof of Lemma 2.3 in [24])
we have
x + δ ≤ h ≤ c(n, p, m).
Denote by Ω = {h ≥ x + δ}, then (3.3) implies that there are positive constants c1 , c2
depending only on n, p, m such that on Ω
p

L(ω) ≥ c1 h 2 − (m − 1)h

p−1
2

+ λ1,p (p − 1)p−1 − c2 |∇ω|.

(3.6)

Now, we can follow an argument in [23] to prove that on Ω
p

h 2 − (m − 1)h

p−1
2

+ λ1,p (p − 1)p−1 ≥ c3 ω

(3.7)

for some positive constant c3 depending only on n, p, m, δ. Indeed, we consider both
sides of (3.7) as a function of h. It is easy to see that (3.7) is valid when h = x for
any choice of c3 . Now, we view the left hand side of (2.10) as a function of h, its
derivative is
1
p p −1 (m − 1)(p − 1) p−3
1 p−3
h2 −

h 2 = h 2 ph 2 − (p − 1)(n − 1) .
2
2
2

m−1
p

By lemma 3.1, we have λ1,p ≤
h ≥ x + δ, we conclude that

p

, this implies x ≥ (p − 1)2

m−1
p

2

. Since

1

ph 2 − (p − 1)(m − 1) ≥ c(n, p, m, δ) > 0.
Hence, (3.7) holds true on Ω for some 0 < c3 < c(n, p, m, δ).
From (3.6), (3.7), we obtain
L(ω) ≥ aω − b|∇ω|

(3.8)

on Ω. Here a, b are positive constants depending only on p, n, m, δ.
Using the integration by parts, we have
L(φ)ωe−f =
M

L(φ)ωe−f
Ω
p

φLωe−f +

=
M

p

h 2 −1 A(∇φ), ν ωe−f −
∂Ω

h 2 −1 A(∇ω), ν φe−f .
∂Ω

∇h
∇ω
Here ν is the outward unit normal vector of ∂Ω. Since ν = − |∇h|
= − |∇ω|
and ω = 0

18

on ∂Ω, this implies
L(φ)ωe−f =

p

φL(ω)e−f +

M

h 2 −1

Ω

∂Ω

φ A(∇ω), ∇ω −f
e
|∇ω|

φL(ω)e−f .

≥
Ω

φ(aω − b|∇ω|)e−f

≥
Ω

φ(aω − b|∇ω|)e−f .

=
M

where we used (3.8) in the third inequality. The proof is complete.
Now, we give a proof of the theorem 1.2.
Proof of Theorem 1.2. We follow the proof in [23]. First, we will prove that ω ≡ 0.
Indeed, for any cut-off function φ on M , and for any q > 0, by using (3.5), we have
(aφ2 ω q+1 − bφ2 ω q |∇ω|)e−f .

ωL(φ2 ω q )e−f ≥
M

M

Integration by parts implies
p

ωL(φ2 ω q )e−f = −

A(∇(φ2 ω q )), ∇ω h 2 −1 e−f .
Ω

M

Therefore,
φ2 ω q |∇ω|e−f + C(n, p, mδ)

φ2 ω q+1 e−f ≤ b

a

Ω

M

φ|∇φ||∇ω|ω q e−f
Ω

qφ2 ω q−1 A(∇ω), ∇ω e−f .

−
Ω

It is easy to see that
A(∇ω), ∇ω = |∇ω|2 + (p − 2)

∇u, ∇ω
|∇u|2

≥ (p − 1)|∇ω|2 .
Hence, for any ε > 0, we have
φ2 ω q+1 e−f ≤ bε

a
M

φ2 ω q+1 e−f +

Ω

c
+
4ε

b
4ε

φ2 ω q−1 |∇ω|2 e−f + ε
Ω

Ω

φ2 ω q−1 |∇ω|2 e−f − c
Ω

φ2 ω q−1 |∇ω|2 e−f ,
Ω

19

|∇φ|2 ω q+1 e−f


where c and c are constants depending only on n, p, m, δ. Choose q such that b + c =
4εc then
(a − bε)
φ2 ω q+1 e−f ≤ ε
|∇φ|2 ω q+1 e−f .

M

M

Now a standard argument implies either ω ≡ 0 or for all R ≥ 1,
ω q+1 e−f ≥ c1 eR ln

c2
ε

(3.9)

B(R)

for some positive constants c1 and c2 independent of ε.
Since ω is bounded and the f -volume of the ball B(R) satisfies Vf (B(R)) ≤
(m−1)R
ce
(by (2.12)), if ε > 0 is chosen sufficiently small, (3.9) can not hold. Hence,
ω ≡ 0. This implies h ≤ x since δ is arbitrary. Thus, |∇ ln v| ≤ y. The proof is
complete.
Let λ1,p =

m−1
p

p

, it is easy to see that the equation
(p − 1)y p − (m − 1)y p−1 + λ1,p = 0

has the unique positive solution y =

m−1
.
p

Hence, we have the following corollary.

Corollary 3.4. Let (M n , g, e−f ) be an n-dimensional complete noncompact manifold
with Ricm
f ≥ −(m − 1). Suppose that u is a positive solution of
f

e div e
Then

4.

−f

p−2

|∇u|

∇u = −

m−1
p

p

up−1 .

|∇u|
m−1
≤
.
u
p

Rigidity of manifolds with maximal λ1,p

In this section, we study structure at infinity of manifolds with maximal λ1,p . Our
main purpose is to prove the theorem 1.5 stated in the introduction part.
Theorem 4.1. Let (M n , g, e−f dv) be a smooth metric measure space of dimension
n ≥ 2. Suppose that Ricm
f ≥ −(m − 1) and λ1,p =

m−1
p

p

. Then either M has no

p-parabolic ends or M = R × N n−1 for some compact manifold N .
Proof. Our argument is close to the argument in [23]. Suppose that M has a pparabolic end E. Let β be the Busemann function associated with a geodesic ray γ
containd in E, namely,
β(x) = lim (t − dist(x, γ(t))).

t→∞

20


Using the Laplacian comparison theorem (2.11), we have
∆f β ≥ −(m − 1).
Hence,
∆p,f e

m−1
β
p

m−1
p

−f

f

= e div e
=

p−1

m−1
p

e

p−2

m−1
(p−1)β
p

e

m−1
(p−2)β
p

p−1

m−1
β
p

m−1
β
p

m−1
p

∆f β +

p−1

m−1
p

(p − 1)e

m−1
(p−1)β
p

p−1
−1
p

m−1
m−1
e p (p−1)β
≥ (m − 1)
p
p
m−1
m−1
=−
e p (p−1)β .
p

Therefore, let ω := e

∇e

, we obtain

∆p,f (ω) ≥ −λ1,p ω p−1 .

Suppose that φ is a nonnegative compactly supported smooth function on M . Then
by the variational principle,
|∇(φω)|p e−f .

(φω)p e−f ≤

λ1,p

M

M

Noting that, integration by parts implies

M

M

M

φp−1 ω ∇φ, ∇ω |∇ω|p−2 e−f

φp |∇ω|e−f − p

φp ω∆p,f (ω)e−f = −
and

|∇(φω)|p = |∇φ|2 ω 2 + 2φω ∇φ, ∇ω + φ2 |∇ω|p−2

p
2

≤ φp |∇ω|p + pφω ∇φ, ∇ω φp−2 |∇ω|p−2 + c|∇φ|2 ω p
for some constant c depending only on p, we infer
φp ω(∆p,f (ω) + λ1,p ω p−1 )e−f
M

(φω)p e−f −

= λ1,p
M

φp |∇ω|p e−f − p
M

|∇(φω)|p e−f −

≤
M

M

φp |∇ω|p e−f − p
M

φp−1 ω ∇φ, ∇ω |∇ω|p−2 e−f
M

|∇φ|2 ω p e−f .

≤c

φp−1 ω ∇φ, ∇ω |∇ω|p−2 e−f

(4.1)

M

21

|∇β|2


Now, we choose
φ=
such that |∇φ| ≤

2
.
R

in B(R)
on M \ B(2R)

Then we conclude

|∇φ|2 ω p e−f =
M

1
0,

|∇φ|2 e(m−1)β e−f
M

4
≤ 2
R
4
= 2
R

e(m−1)β e−f
B(2R)\B(R)

e(m−1)β e−f +
E∩(B(2R)\B(R))

4
R2

e(m−1)β e−f .
(M \E)∩(B(2R)\B(R))

(4.2)
Since λ1,p =

m−1

p

p

, by theorem 0.1 in [3], it turns out that
Vf (E \ B(R)) ≤ ce−(m−1)R .

Hence, the first term of (4.2) tends to 0 as R goes to ∞. On the other hand, by [15]
we have
β(x) ≤ −r(x) + c
on M \ E. The volume comparison (2.12) implies Vf (B(R)) ≤ c e(m−1)R . It turns out
that the second term of (4.2) also goes to 0 as R → ∞. Therefore, (4.1) infers
∆p,f (ω) + λ1,p ω p−1 ≡ 0.
This implies
∆f β = −(m − 1).
Moreover, all the inequalities used to prove ∆f β ≥ −(m − 1) become equalities (see
Theorem 1.1 in [16]). By the proof of the Theorem 1.1 in [16] and the argument in
[15], we conclude that M = R × N n−1 for some compact manifold N of dimension n.
The proof is complete.
For p-nonparabolic end, we have the following theorem.
Theorem 4.2. Let (M n , g, e−f dv) be a smooth metric measure space of dimension
n ≥ 3. Suppose that Ricm
f ≥ −(m − 1) and λ1,p =

m−1
p

p

for some 2 ≤ p ≤

(m−1)2
.
2(m−2)

Then either M has only one p-nonparabolic end or M = R × N n−1 for some compact
manifold N .
To prove theorem 4.2, let us recall a fact on weighted Poincar´e inequality in [14].
22


Proposition 4.3 ([14], Proposition 1.1). Let M be a complete Riemannian manifold. If there exists a nonnegative function h defined on M , that is not identically 0,
satisfying
∆h(x) + ∇g, ∇h (x) ≤ −ρ(x)h(x),
for some nonnegative function ρ, then the weighted Poincar´e inequality
ρ(x)φ2 (x)eg(x) ≤

|∇φ|2 (x)eg(x)

M

M

must be hold true for all compactly supported smooth function φ ∈ C0∞ (M ).
Now we give a proof of theorem 4.2.
Proof of theorem 4.2. We follow Sung-Wang’s argument in [23]. For the completeness,
2
we give a detail of the proof here. If p = 2 then we have λ1 (M ) = λ1,2 = (m−1)
.
4

Hence, by Theorem 1.5 in [29], we are done. Therefore, we may assume p > 2.
Now let u be a positive weighted p-eigenfunction of the weighted p-Laplacian satisfying
∆p,f u = ef div(e−f |∇u|p−2 ∇u) = −λ1,p up−1 .
This implies
∆u + −∇f + (p − 2)

= −λ1,p

up−2
u.
|∇u|p−2

ρ = λ1,p

up−2
.
|∇u|p−2

∇|∇u|
, ∇u
|∇u|

Let
g = −f + (p − 2) ln |∇u|,

and

By proposition 4.3, we obtain
λ1,p
M

up−2 2
φ |∇u|p−2 e−f ≤
p−2
|∇u|

|∇φ|2 |∇u|p−2 e−f

or equivalently,
up−2 φ2 e−f ≤

λ1,p
M

|∇u|p−2 |∇φ|2 e−f ,
M

for any compactly supported smooth function φ on M . Therefore
φ2 e−f = λ1,p

λ1,p
M

φu−

p−2
2

2

up−2 e−f

M

∇ φu−

≤
M

23

p−2
2

2

|∇u|p−2 e−f


By a direct computation, we have
∇ φu−

2

p−2
2

|∇u|p−2 e−f

M

|∇φ|2

=
M

p−2
|∇u|p−2 −f
e + φ2 ∇ u− 2
p−2
u

φu−

+2

p−2
2

∇φ, ∇ u−

p−2
2

2

|∇u|p−2 e−f

|∇u|p−2 e−f

M

|∇φ|2

=
M

|∇u|p−2 −f
e +
up−2

p−2
2

2
M

|∇u|p 2 −f 1
φe +
up
2

∇φ2 , ∇u−(p−2) |∇u|p−2 e−f .
M

(4.3)
We claim that
1
2

∇φ2 , ∇v −2 |∇u|p−2 e−f = −

p−2
2

λ1,p + (p − 1)
M

|∇u|p
up

φ2 e−f .

(4.4)

Suppose that the claim (4.4) is verified. By (4.3), (4.4), we obtain
2 −f

λ1,p

φe

≤

M

M

2

p−2
|∇u|p 2 −f
+
φe
2
up
M
|∇u|p
λ1,p + (p − 1) p
φ2 e−f .
u

|∇u|p−2
|∇φ|2 p−2 e−f
u

−

p−2
2

M

This means
p
λ1,p
2
Since λ1,p =

φ2 e−f +

M

m−1
p

p(p − 2)
4

M

|∇u|p −f
e ≤
up

p

, the corollary 3.4 implies
(m − 1)2
2p

Hence, λ1 (M ) ≥

φ2

(m−1)2
2p

∇u
u

≤

M

m−1
.
p

|∇u|p−2
|∇φ|2 e−f .
up−2
Therefore,

|∇φ|2 e−f .

φ2 e−f ≤
M

M

. Here λ1 (M ) is the first eigenvalue of the weighted Laplacian.

(m−1)2
,
2(m−2)

Since p ≤
we have λ1 (M ) ≥ (m − 2). By a Wang’s theorem (see [25]), we are
done.
The rest of the proof is to verify the claim (4.4). Indeed, we have

1
2

∇φ2 , ∇u−(p−2) |∇u|p−2 e−f
M

=−

1
2

∆f (u−(p−2) )|∇u|p−2 φ2 e−f −
M

1
2

24

∇u−(p−2) , ∇|∇u|p−2 φ2 e−f .
M

(4.5)


Note that
ef div e−f |∇u|p−2 ∇u = −λ1,p up−1
we have
|∇u|p−2 ∆f u + ∇|∇u|p−2 , ∇u = −λ1,p up−1 .
Therefore,

|∇u|p−2 ∆f (u2−p ) + ∇(u2−p , ∇(|∇p|p−2 ))
=(2 − p)|∇u|p−2 u1−p ∆f u + (2 − p)(1 − p)|∇u|p−2 u−p |∇u|2
+ (2 − p)u1−p ∇u, ∇(|∇u|p−2 )
= (2 − p)u1−p (−λ1,p up−1 ) + (2 − p)(1 − p)

|∇u|p
|∇u|p
=
(p
−
2)
λ
+
(p
−
1)
1,p
up
up
(4.6)

By (4.5) and (4.6), the claim 4.4 is verified.
Since the above-mentioned result of Wang plays a critical role in the proof of
theorem 4.2, we reformulate it here for reader’s convenience.
Theorem 4.4 ([25]). Let (M n , g, e−f dv) be a smooth metric measure space of dimension n ≥ 3. Suppose that the lower bound of the spectrum λ1 (M ) of the weighted
Laplacian is positive and
m−1
Ricm
λ1 (M ).
f ≥ −

m−2
Then either M has only one p-nonparabolic end or M = R × N n−1 for some compact
manifold N of dimension n − 1 with the product metric
ds2M = dt2 + cosh

λ1 (M ) 2
tdsN .
m−2

Acknowledgment
A part of this paper was done during a visit of the first author to Vietnam Institute
for Advanced Study in Mathematics (VIASM). He would like to express his thanks
to staffs there for the excellent working conditions, and financial support. He also
would like to express his deep gratitude to his advisor Prof. Chiung Jue Sung for her
constant support. The authors also thank Munteanu and J. Y. Wu for their usefull
comments on the earlier manuscript of this paper.

25

.