Hardy-type inequalities and Pohozaev-type identities for a class of p -degenerate subelliptic operators and applications

Nonlinear Analysis 54 (2003) 165 – 186

www.elsevier.com/locate/na

Hardy-type inequalities and Pohozaev-type identities for a class of p-degenerate subelliptic operators and applications
Huiqing Zhang∗ , Pengcheng Niu Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, People’s Republic of China Received 15 June 2002; accepted 1 September 2002

Abstract The purpose of this paper is to consider a class of p-degenerate subelliptic operators Lp constructed by generalizing Greiner’s vector 3elds. Their fundamental solutions at the origin are established with the aid of the properties of radial functions. A Picone-type identity and a Hardy-type inequality with respect to vector 3elds are proved. Some Pohozaev-type identities and applications to nonlinear equations are given. Finally, a Carleman-type estimate and uniqueness of the operator L2 are discussed. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Fundamental solution; Picone identity; Hardy inequality; Pohozaev identity; Nonexistence; Carleman estimate; Uniqueness

1. Introduction In this paper, we consider a class of degenerate subelliptic operators Lp u = divL (|∇L u|p−2 ∇L u);

(1.1)

where p ¿ 1, Xj = @=@xj + 2kyj |z|2k−2 @=@t, Yj = @=@yj − 2kxj |z|2k−2 @=@t, zj =
xj + √ n −1yj ∈ C, j = 1; : : : ; n, t ∈ R, ∇L = (X1 ; : : : ; Xn ; Y1 ; : : : ; Yn ), divL (u1 ; : : : ; u2n ) = j=1 (Xj uj + Yj un+j ), k ¿ 1. When p = 2 and k = 1, Lp becomes the sub-Laplacian >H n
This paper was supported by the Natural Science Foundation of Shaanxi Province and the National Natural Science Foundation of China. ∗

Corresponding author. E-mail address: [email protected] (H. Zhang).

0362-546X/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0362-546X(03)00062-2

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H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

on the Heisenberg group H n , see [5]. If p = 2 and k = 2; 3; : : :, Lp is a Greiner operator (see [9]). Operator (1.1) does not possess the translation invariance for k ¿ 1. We refer to [2,11] and the references therein. Since HGormander’s famous foundational paper [12], the study of partial diHerential operators constructed from noncommutative vector 3elds has received great attention. Folland [5] established an explicit fundamental solution for the sub-Laplacian >H n . Garofalo and Lanconelli [7] discussed the unique continuation of SchrGoding-type operators with a suitable potential and proved a Hardy-type inequality for p = 2. A Pohozaev-type identity for the sub-Laplacian was proved in [8] and used to nonexistence of semilinear sub-Laplace equations. Capogna et al. [3] studied the p-degenerate
m subelliptic operator j=1 Xj∗ (|Xj u|p−2 Xj u), where Xj are smooth vector 3elds satisfying HGormander’s hypoellipticity condition. We note that the vector 3elds X1 ; : : : ; Xn ; Y1 ; : : : ; Yn in (1.1) do not satisfy HGormander’s condition for k ¿ 1 (see [2,11]). Lp in (1.1) is the Euler-Lagrange equation associated to the functional 

p

|∇L u| =

  n

(|Xj u|2 + |Yj u|2 )p=2 ;

p¿1

j=1

for a function u satisfying u; ∇L u ∈ Lp . It is the purpose of this paper to present some important properties of Lp , including the fundamental solution with singularity at the origin; Hardy-type inequalities; Pohozaev-type identities; Carleman-type estimates and applications. We now state some preliminaries about the family of vector 3elds X1 ; : : : ; Xn ; Y1 ; : : : ; Yn and the operator Lp in (1.1). Let 

2ky1 |z|2k−2

        A =        

     2k−2  2kyn |z|   2k−2  −2kx1 |z|    ..  .   2k−2  −2kx n |z|  .. .

IR2n

2ky1 |z|2k−2 · · · 2kyn |z|2k−2 −2kx1 |z|2k−2 · · · −2kx n |z|2k−2  IR2n

 =  2ky|z|2k−2

−2kx|z|2k−2

2ky|z|2k−2





 −2kx|z|2k−2  ; 2 4k−2 4k |z|

4k 2 |z|4k−2

H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

167

where A is a (2n + 1) × (2n + 1) matrix, IR2n denotes the identity matrix in R2n , 

IR2n B= 2ky1 |z|2k−2 · · · 2kyn |z|2k−2 −2kx1 |z|2k−2 · · · −2kx n |z|2k−2 (2n+1)×2n

=



IR2n 2ky|z|2k−2

−2kx|z|2k−2

:

Then we have Lp u = div(B|∇L u|p−2 ∇L u);

(1.2)

B∇L u = A∇u;

(1.3)

where div and ∇ are taken with respect to the variable (x; y; t) ∈ R2n+1 . A natural family of anisotropic dilations attached to Lp is  (z; t) = (z; 2k t);

 ¿ 0; (z; t) = (x; y; t) ∈ R2n+1 :

(1.4)

One easily checks that Lp ◦  = p  ◦ Lp ; therefore Lp is a homogeneous partial diHerential operator of degree p with respect to { } in the sense of [4,13]. It is easy to verify that d (z; t) = Q d z dt; where Q = 2n + 2k

(1.5)

and d z dt denotes the Lebesgue measure on R2n+1 . The generator of the group dilations { }¿0 is the 3eld

n  @ @ @ xj + 2kt : + yj X= (1.6) @xj @yj @t j=1

Another smooth vector 3eld on R2n+1 used later is

n  @ @ yj : − xj T= @xj @yj

(1.7)

j=1

Consider the function d(z; t) = (|z|4k + t 2 )1=4k ;

(1.8)

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H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

where |z|2 =


n

j=1

Xd = d;

(xj2 + yj2 ). Since d ◦  = d for every  ¿ 0, we have

Td = 0:

(1.9)

2n+1

2n+1

| d(z; t) ¡ r}, @Br = {(z; t) ∈ R | d(z; t) = r} and call these We let Br = {(z; t) ∈ R sets, respectively, the generalized ball and the generalized sphere centered at the origin with radius r. Let p = |z|p(2k−1) =dp(2k−1) , ’p = t|z|p(2k−1)−2k =dp(2k−1) . Observing that p is homogeneous of degree zero with respect to (1.4), one has X

p

= 0:

(1.10)

An evident computation yields T’p = 0:

(1.11)

We obtain from (1.8) that

 x|z|4k−2 + ty|z|2k−2 1 ∇L d = 4k−1 ; d y|z|4k−2 − tx|z|2k−2 |z|2k−1 |∇L d| = 2k−1 d and |∇L d|p =

p:

(1.12)

Hence we have



x|z|4k−2 + ty|z|2k−2



   y|z|4k−2 − tx|z|2k−2   d(p−2)(2k−1)+4k−1  2kt|z|4k−2      x y p(2k−1)−2k    1  |z|p(2k−1)   y  + t|z|  −x  =       p(2k−1) p(2k−1) d d d 2kt 0

B|∇L d|p−2 ∇L d =

= Noting that div X = Q; we conclude

|z|(2k−1)(p−2)

1 [ p X + ’p T ]: d

div T = 0;

(1.13) (1.14)



 1 ( p X + ’p T ) d

1 1 1 1 ’ ’ = + div X + X div T + T p p p p d d d d

Lp d = div

=

p

d

(Q − 1):

(1.15)

H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

169

Let S01; p be the completion of C0∞ (R2n+1 ) in the norm

1=p  uS 1; p = |∇L u|p + |u|p : R2n+1

The plan of the paper is as follows. In Section 2 we deduce the fundamental solution of Lp according to the properties of the radial functions. Section 3 is devoted to a Picone-type identity associated to the family {Xj ; Yj } and a Hardy-type inequality. In Section 4, we establish the Pohozaev-type identities in our setting. A nonexistence result of the nonlinear degenerate subelliptic equation is given in Section 5. Finally, a Carleman-type estimate and uniqueness of the operator L2 are considered in Section 6 by using the result in Section 4.

2. A fundamental solution Theorem 2.1. The fundamental solution of Lp at the origin is −1 (p−Q)=(p−1) ; %p; k; Q = Cp; k; Q d

1 ¡ p ¡ Q;

(2.1)

where Cp; k; Q satis8es

p−1  |z|p(2k−1) d2k(p−2) Q−p −1 Cp; (Q + 4kp − p − 4k) : = − k; Q 4k (4kp−p+Q)=4k p−1 R2n+1 (1 + d ) Proof. Let d& = (d4k + &4k )1=4k ; We compute ∇ L d& =



1 d&4k−1

|∇L d& |p−2 =

& ¿ 0:

x|z|4k−2 + yt|z|2k−2 y|z|4k−2 − xt|z|2k−2



|z|2k−2 = 4k−1 d&



x|z|2k + yt



y|z|2k − xt

;

|z|(2k−1)(p−2) d2k(p−2)

(2.2)

d&(4k−1)(p−2)

and

 B|∇L d& |p−2 ∇L d& =

=

x|z|4k−2 + yt|z|2k−2



 |z|(2k−1)(p−2) d2k(p−2)   y|z|4k−2 − xt|z|2k−2  (4k−1)(p−1)   d& 4k−2 2kt|z| d2k(p−2) d&2k(p−2)+1

[

p& X

+ ’p& T ];

(2.3)

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H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

where p& = |z|p(2k−1) =dp(2k−1) , ’p& = t|z|p(2k−1)−2k =dp(2k−1) . Using the fact that Xd& = & & d4k =d&4k−1 , we have:

p(2k−1) |z|p(2k−1) dp(2k−1) d X p& = X p p(2k−1) = p(2k−1) X d d& dp(2k−1) &   p(2k−1) p(2k−1) d |z| dp(2k−1) d4k = p(2k−1) p(2k − 1) p(2k−1) − p(2k − 1) p(2k−1)+1 4k−1 d d& d& d&   d4k |z|p(2k−1) dp(2k−1) = p(2k − 1) p(2k−1) p(2k−1) 1 − 4k d d& d& = p(2k − 1) and

X

p&

&4k d4k &

(2.4)

d2k(p−2) d&2k(p−2)+1

= 2k(p − 2)

d2k(p−2) d&2k(p−2)+1

− [2k(p − 2) + 1]

d2kp d&2kp+1

:

(2.5)

Combining these equations, we obtain Lp d& = div(B|∇L d& |p−2 ∇L d& )  2k(p−2)  d = div 2k(p−2)+1 ( p& X + ’p& T ) d& 2k(p−2) d2k(p−2) d = p& 2k(p−2)+1 div X + X d& d&2k(p−2)+1 =Q =

p&

p&

d2k(p−2)

+

d&2k(p−2)+1  d2k(p−2)

d&2k(p−2)+1

d2k(p−2) d&2k(p−2)+1

X

p&

+

p&

p& X

Q + 2k(p − 2) + p(2k − 1)

− [2k(p − 2) + 1]

d4k d4k &

 ;

d2k(p−2)

d&2k(p−2)+1 &4k d4k & (2.6)

where we have used (1.14) and the fact T’p& = 0. Taking f(d& ) = d&(p−Q)=(p−1) and directly computing show f (d& ) =

p − Q (1−Q)=(p−1) : d p−1 &

We see after using (2.6) and (2.7) that Lp (f(d& )) = divL (|∇L f(d& )|p−2 ∇L f(d& )) = divL (|f (d& )∇L d& |p−2 f (d& )∇L d& )

(2.7)

H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

171



p−1 Q−p = divL − d&1−Q |∇L d& |p−2 ∇L d& p−1

Let now

p−1 Q−p p 1−Q =− Lp d& ] [(1 − Q)d−Q & |∇L d& | + d& p−1

p−1  Q−p |z|p(2k−1) d2kp =− (1 − Q)d−Q & p−1 dp(4k−1) &  d2k(p−2) + d&1−Q p& 2k(p−2)+1 Q + 2k(p − 2) d&

 &4k d4k + p(2k − 1) 4k − (2k(p − 2) + 1) 4k d& d&

p−1  2k(p−2) Q−p &4k −Q d =− Q + 2k(p − 2) + p(2k − 1) p& d& d4k p−1 d&2k(p−2) &  d4k d4k − (2k(p − 2) + 1) 4k + (1 − Q) 4k d& d&

p−1 2k(p−2) Q−p &4k −Q d =− (Q + 4kp − p − 4k) 4k p& d& 2k(p−2) p−1 d& d&

p−1 |z|p(2k−1) d2k(p−2) &4k Q−p (Q + 4kp − p − 4k) : (2.8) =− p−1 d4kp−p+Q &

Q−p p−1 Then (2.8) becomes K(z; t) = −

p−1 (Q + 4kp − p − 4k)

|z|p(2k−1) d2k(p−2) : (1 + d4k )(4kp−p+Q)=4k

(2.9)

Lp (d&(p−Q)=(p−1) ) = &−Q K(1=& (z; t)): For any u ∈ C0∞ (R2n+1 ), it follows:  (Lp (d(p−Q)=(p−1) ); u) = lim &→0

R2n+1

= lim &−Q &→0

= lim

&→0

 R2n+1

 R2n+1

= u(0; 0)

Lp (d&(p−Q)=(p−1) )u K(1=& (z; t))u(z; t)

K(z; t)u(&z; &2k t)



R2n+1

The conclusion is deduced from (2.10).

K(z; t):

(2.10)

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H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

Remark 2.1. A direct examination shows that the same formula hold for p ¿ Q. In the critical case p = Q, the fundamental solution is 1 Ck; Q log : d

(2.11)

In fact, let f(d& ) = log 1=d& , clearly f (d& ) = −1=d& and LQ (f(d& )) = divL (|∇L f(d& )|Q−2 ∇L f(d& )) = div(−d&1−Q |∇L d& |Q−2 ∇L d& ) Q 1−Q LQ d& ] = −[(1 − Q)d−Q & |∇L d& | + d&

=−

−Q Q& d&

d2k(Q−2)

(4kQ − 4k) 2k(Q−2)

d&

= −4k(Q − 1)

&4k d4k &

|z|Q(2k−1) d2k(Q−2) &4k d4kQ &

:

Denoting K(z; t) = −4k(Q − 1) and

|z|Q(2k−1) d2k(Q−2) (1 + d4k )Q

 Ck; Q =

R2n+1

−1 K(z; t) d z dt

;

then one has at (z; t) = (0; 0),

1 = : LQ Ck; Q log d 3. Hardy-type inequalities In this section, we establish a Picone-type identity for the family {Xj ; Yj }, which is an extension of the identity for the p-Laplacian in the Euclidean space in [1]. Using the identity, we prove a Hardy-type inequality to Lp . Note that for p = 2 and k = 1, it is the result obtained by Garofalo and Lanconelli [7]. Lemma 3.1. For di:erentiable functions v ¿ 0, u ¿ 0 on ) ⊂ R2n+1 , where ) is a bounded or unbounded domain in R2n+1 , it holds L(u; v) = R(u; v) ¿ 0;

(3.1)

where L(u; v) = |∇L u|p + (p − 1)

up up−1 p |∇ v| − p |∇L v|p−2 ∇L u · ∇L v; L vp vp−1

H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

p

p−2

R(u; v) = |∇L u| − |∇L v|

∇L

up

173

· ∇L v:

vp−1

Moreover, L(u; v) = 0 a.e. on ) i: ∇L (u=v) = 0 a.e. on ). Proof. It follows from straightforward calculations. It contains Lemma 2.1 in [14]. Lemma 3.2. Suppose that for some  ¿ 0, v ∈ C ∞ ()) satis8es − Lp v ¿ gvp−1 and v ¿ 0 in );

(3.2)

where g is some positive, weight function. Then we have for any u in S01; p ()),   (3.3) |∇L u|p ¿  g|u|p : )

)

Proof. Thanks to Lemma 3.1, we have   0 6 L(u; v) = R(u; v) )

)

 =

)

 =

)

 6

)

|∇L u|p − |∇L u|p +



 

|∇L u|p − 

∇L

)

up

)



vp−1

)

up vp−1

|∇L v|p−2 ∇L v

Lp v

gup :

Eq. (3.3) now follows. Theorem 3.1. Let u ∈ C0∞ (R2n+1 \{(0; 0)}); 1 ¡ p ¡ Q. Then

p  p(2k−1) p  |z| |u| Q−p p |∇L u| ¿ ; p d d R2n+1 R2n+1

(3.4)

where d is as in (1.8). Proof. Setting v = f(d) = d(p−Q)=p , one easily sees f (d) =

p − Q −Q=p : d p

Using (1.12), (1.15) and (3.5), we get Lp v = divL (|∇L v|p−2 ∇L v)



p−1 Q(p−1) Q−p − p−2 p = divL − d |∇L d| ∇L d p

(3.5)

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H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

=− + =− + =− =−

Q−p p Q−p p Q−p p Q−p p Q−p p Q−p p

p−1 d

p−1

d

p

p

pd

−

Q(p−1) −1 p

p (Q

− 1)

Q(p − 1) − Q(p−1) −1 p d p

pd

p

Q(p−1) p L

Q(p − 1) − Q(p−1) −1 p |∇L d|p d p

p−1

p−1

−

−

p

Qp−Q+p p

vp−1 : dp

This, combined with Lemma 3.2, yields (3.4). 4. Pohozaev-type identities In what follows we set Zj = Xj ; Zn+j = Yj ; j = 1; : : : ; n, and adopt the summation convention over repeated indices. The following Lemma 4.1 is a generalization of the Proposition 2.2 from [9]. Lemma 4.1. Let ) be a bounded, piecewise C 1 domain and G be a real vector 8eld in R2n+1 . Then, for any di:erentiable function u,   |Zu|p G · ˜n − p Gu|Zu|p−2 Zj uZj · ˜n @)



=

)

@)

div G|Zu|p − p

−p

 )

 )

div Zj Gu|Xu|p−2 Zj u + p

 )

[G; Zj ]u|Zu|p−2 Zj u

GuLp u;

(4.1)

where ˜n denotes the outward unit normal to @), |Zu| = (


2n

j=1

|Zj u|2 )1=2 .

Proof. Since

  p=2  p=2−1   2n 2n 2n    p   |Zj u|p   =  |Zj u|2  G |Zj u|2  G(|Zu|p ) = G  2 j=1

= p|Zu|p−2 GZj uZj u

j=1

j=1

H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

175

we deduce from the divergence theorem  |Zu|p G · ˜n @)



=

)

 =

)

 =

)

 =

)

 =

)

div(G|Zu|p ) div G|Zu|p +

div G|Zu|p + p div G|Zu|p + p 

p

 )

GZj u|Zu|p−2 Zj u

 )

 )

[G; Zj ]u|Zu|p−2 Zj u +



[G; Zj ]u|Zu|p−2 Zj u + p

)

Zj Gu|Zu|p−2 Zj u

 )





div G|Zu| + p

−p

G(|Zu|p )



Zj (Gu|Zu|p−2 Zj u)

GuZj (|Zu|p−2 Zj u)

)

 )

)

div G|Zu|p + p

− =



 )

p−2

)

[G; Zj ]u|Zu|

p−2

div Zj Gu|Zu|

 Zj u + p

 Zj u − p

)

@)

Gu|Zu|p−2 Zj uZj · ˜n

GuLp u:

Then (4.1) is valid. Lemma 4.2. For j = 1; 2; : : : ; 2n; it holds [Zj ; X ] = Zj ;

(4.2)

where X is given by (1.6). Proof. A direct calculation shows for j = 1; : : : ; n;   

 @ @ @ @ 2k−2 @ xi + 2kt + 2kyj |z| + yi Xj X = @xj @t @xi @yi @t i =

@ @2 @2 @2 @2 + 2kxi yj |z|2k−2 + xi + yi + 2kt @xj @xi @xj @xj @yi @xj @t @xi @t + 2kyi yj |z|2k−2

@2 @2 @ + 4k 2 yj t|z|2k−2 2 + 4k 2 yj |z|2k−2 @yi @t @t @t

176

H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

and

 XXj =

 i

= xi

@ @ xi + yi @xi @yi

@ + 2kt @t



@ @ + 2kyj |z|2k−2 @t @xj



@2 @2 @2 @ + 2kxi yj |z|2k−2 + 4k(k − 1)xi2 yj |z|2k−4 + yi @xj @yi @t @xi @t @xi @xj

+ 2kyj |z|2k−2 + 2kt Hence, we see [Xj ; X ] = =

@ @ @2 + 4k(k − 1)yi2 yj |z|2k−4 + 2kyi yj |z|2k−2 @t @t @yi @t

@2 @2 + 4k 2 yj t|z|2k−2 2 : @t @xj @t

@ @ @ @ + 4k 2 yj |z|2k−2 − 2kyj |z|2k−2 − 4k(k − 1)yj |z|2k−2 @xj @t @t @t @ @ + 2kyj |z|2k−2 = Xj : @t @xj

Similarly, [Yj ; X ] = Yj , j = 1; : : : ; n. Using Lemma 4.1 with G = X and Lemma 4.2 lead immediately to the following Pohozaev-type identity. Theorem 4.1. Let ) be a bounded, piecewise C 1 domain. Then     Xu|Zu|p−2 Zj uZj · ˜n − |Zu|p X · ˜n = (p − Q) |Zu|p + p XuLp u: p @)

@)

)

)

(4.3)

Remark 4.1. If p = 2 and k = 1, then (4.3) is the identity in [8]. Also see [10]. Letting G = T in Lemma 4.1 we will have from (1.7) and (1.14) that   p−2 Tu|Zu| Zj uZj · ˜n = TuLp u: @)

)

Theorem 4.2. Under the assumptions of Theorem 4.1, one has  2     Xu  Xu Tu s |Zu|p ds X · ˜n − p |Zu|p−2   ds X · ˜n − p |Zu|p−2 d ’X · ˜n d d d @Br @Br @Br  2    Xu  =(Q + s − p) ds |Zu|p − ps ds |Zu|p−2   d Br Br   Xu Tu −p −ps ds ’|Zu|p−2 ds XuLp u; (4.4) d d Br Br where Br = {(z; t) ∈ R2n+1 | d(z; t) 6 r}, @Br = {(z; t) ∈ R2n+1 | d(z; t) = r}, ’2 ; s ¿ 0.

=

2;

’=

H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

177

Proof. Set G = ds X in Lemma 4.1. Then [G; Zj ] = ds XZj − Zj (ds X ) = ds XZj − ds Zj X − sds−1 Zj dX = ds [X; Zj ] − sds−1 Zj dX = −ds Zj − sds−1 Zj dX;

(4.5)

div G = div(ds X ) = ds div X + X (ds ) = (Q + s)ds and div Zj = 0; where we have used (4.2) and (1.14), respectively. Substituting these equations in (4.1) yields   ds Xu|Zu|p−2 Zj uZj · ˜n |Zu|p ds X · ˜n − p @Br

@Br

 = (Q + s − p)  −p

Br

Br



ds |Zu|p − ps

Br

ds−1 |Zu|p−2 XuZj uZj d

ds XuLp u:

(4.6)

Noting the fact that ˜n = ∇d=|∇d| on @Br , we infer X · ˜n =

Xd d = |∇d| |∇d|

on @Br :

On the other hand, in view of the formula ZuZd = ∇L u · ∇L d = A∇d · ∇u = we have  @Br

1 [ Xu + ’Tu]; d

ds Xu|Zu|p−2 Zj uZj · ˜n 

=

@Br

 =

@Br

 =

@Br

ds Xu|Zu|p−2 Zj uZj d

1 |∇d|

ds−1 Xu|Zu|p−2 [ Xu + ’Tu] ds−2 |Zu|p−2 |Xu|2 X · ˜n +

1 |∇d|

 @Br

ds−2 ’|Zu|p−2 XuTuX · ˜n

(4.7)

178

and

H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

 Br

ds−1 |Zu|p−2 XuZj uZj d  =

Br

 =

Br

ds−2 |Zu|p−2 Xu[ Xu + ’Tu] ds−2 |Zu|p−2 |Xu|2 +

 Br

ds−2 ’|Zu|p−2 XuTu:

(4.8)

Replacing (4.7), (4.8) in (4.6) leads to (4.4). Let G = ds T in Lemma 4.1. We have the following: Theorem 4.3. Under the assumptions of Theorem 4.1, it holds  2    Tu  Xu Tu X · ˜n + p p ’ds |Zu|p−2   X · ˜n ds |Zu|p−2 d d d @Br @Br  2     Tu  Xu Tu =ps ds |Zu|p−2 ’ds |Zu|p−2   + p ds TuLp u: (4.9) + ps d d d Br Br Br 5. A nonexistence result As a consequence of theorems in Section 4, we give a simple nonexistence result for nonlinear equations. De'nition 5.1. Let ) be a C 1 open domain in R2n+1 , (0; 0) ∈ ). We say that it is strict generalized starshaped with respect to (0; 0) if X · ˜n ¿ 0 at every point of @). Theorem 5.1. Let ) ⊂ R2n+1 be connected, bounded and strict generalized starshaped with respect to (0; 0) ∈ ). Then the problem  Lp u + f(u) = 0 in ); (5.1) u=0 on @) L u = 0, if f is locally Lipschitz, has no nonnegative solution u ∈ S01; p ()) ∩ C 1 ()), f(0) = 0 and pQF(u) − (Q − p)uf(u) 6 0; u where F(u) = 0 f(s) ds.

(5.2)

H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

179

Proof. If u is a nonnegative solution of (5.1) which satis3es (5.2), we have from the fact ˜n = ∇u=|∇u| on @) (see [8]) that    Zj u Xu|Zu|p−2 Zj u |Zu|p X · ˜n: (5.3) Xu|Zu|p−2 Zj uZj · ˜n = = |∇u| @) @) @) Eq. (5.1) and an integration by parts conclude     p uf(u) = − uLp u = |∇L u| = |Zu|p )

and

)

)



 )

XuLp u = −

Xuf(u) = −

 =Q

)

)

)

X

X (F(u)) = −

@)

u

0



 =−





)

(5.4)

)

f(t) dt

F(u)X · ˜n +

 )

div XF(u)

F(u):

(5.5)

Inserting (5.3)–(5.5) into (4.3) implies   p (p − 1) |Zu| X · ˜n = [pQF(u) − (Q − p)uf(u)] ¿ 0; @)

)

which is a contradiction. The claim is proved.

6. A Carleman-type estimate There has been a considerable amount of work in the study of the unique continuation property for second-order elliptic diHerential equations, we refer the reader to the paper [15] and the references therein. The topic for generalized Baouendi–Grushin operator is considered by Garofalo [6]. In this section, we prove a Carleman-type estimate and then the unique continuation of the operator L2 . Theorem 6.1. Let R0 ¡ 1 be 8xed and u ∈ C0∞ (BR0 \{(0; 0)}), for each 2 ¿ 0 and some constants C ¿ 0, satisfying −2

e2d u2 = o(1) −1 2d−2

as d → 0;

e

(Lu)2 ∈ L1 (BR0 )

|Tu| 6 C

|z| 1=2 |u| = C d2

(6.1) (6.2)

and 2k=2(2k−1)

d

|u|:

(6.3)

180

H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

where d is as in (1.8), = 2 = |z|2(2k−1) =d2(2k−1) . Then there exist 20 ¿ 1; C0 ¿ 0 depending only on Q such that   −2 −1 2+2 2d−2 d e (Lu)2 ; e2d [ d−22−2 u2 + |∇L u|2 ] 6 C0 (2−2 + R20 ) R2n+1

R2n+1

(6.4)

where L = L2 . Theorem 6.2. Suppose that u satis8es |Lu| 6 C1

d

|u| + C2 2

1=2

|∇L u|

d

(6.5)

in BR for some constants C1 ; C2 ¿ 0; u; |∇L u|; Lu ∈ L2 (BR ) and for each 2 ¿ 1 −2

ed uL∞ (Br ) = o(1)

(r → 0)

(6.6)

and −2

ed |∇L u| ∈ L2 (BR ):

(6.7)

Then there exists r0 = r0 (Q; C1 ; C2 ) ¿ 0 such that u ≡ 0 in Br0 . −2

Proof of Theorem 6.1. Let u = f(d)w = e−d w and we have Lu = wL(f(d)) + f(d)Lw + 2∇L w · ∇L (f(d)): From the formula L(f(d)) =

(6.8)

  Q−1  f (d) + f (d) d

and the facts that −2

f (d) = 2d−2−1 e−d ; −2

f (d) = 2[(−2 − 1)d−2−2 + 2d−22−2 ]e−d ; we obtain L(f(d)) = e

−d−2



2 d2+2

 2 −2−2+Q : d2

(6.9)

On the other hand, it follows ∇L (f(d)) · ∇L w = f (d)∇L d · ∇L w 1 = f (d)A∇d · ∇w = f (d) [ Xw + ’Tw] d = e−d

−2

2 [ Xw + ’Tw]; d2+2

(6.10)

H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

181

where ’ = ’2 = t|z|2k−2 =d2(2k−1) . Applying (6.9), (6.10) to (6.8) yields     2 2 22 −d−2 − 2 − 2 + Q w + Lw + 2+2 [ Xw + ’Tw] Lu = e d2+2 d2 d and then d

2+2 2d−2

e





 2 −2−2+Q w (Lu) = d d2+2 d2 2 22 + Lw + 2+2 [ Xw + ’Tw] d     2 2 ¿ 42[ Xw + ’Tw] Lw + − 2 − 2 + Q w (6.11) d2+2 d2 2

2+2

2

where we have used the inequality (a + (b + c))2 ¿ 2a(b + c). Now we integrate (6.11) with respect to the measure −1 d z dt and obtain   −1 2+2 2d−2 d e (Lu)2 ¿ 42 XwLw R2n+1

R2n+1

+ 422

 + 42 + 422





R2n+1

d2+2

’

R2n+1



R2n+1

 2 − 2 − 2 + Q wXw d2

TwLw ’

d2+2



 2 − 2 − 2 + Q wTw: d2

Since u ∈ C0∞ (BR0 \{(0; 0)}), letting p = 2 in Theorem 4.1 leads to   Q−2 XwLw = |∇L w|2 ¿ 0: 2 R2n+1 R2n+1 By the divergence theorem we 3nd    2 422 − 2 − 2 + Q wXw 2+2 d2 R2n+1 d    2 2 =22 − 2 − 2 + Q Xw2 2+2 d2 R2n+1 d    2 2 = − 22 Q − 2 − 2 + Q w2 2+2 d2 R2n+1 d     2 −222 X − 2 − 2 + Q w2 d2+2 d2 R2n+1

(6.12)

(6.13)

182

H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186



 2 2 2 w2 = − 22 Q w + 22 Q(2 + 2 − Q) 2+2 22+2 R2n+1 d R2n+1 d   2 2 3 w2 w − 22 (2 + 2)(2 + 2 − Q) + 2(22 + 2)2 2+2 22+2 R2n+1 d R2n+1 d

  Q 2 2 2 3 w2 : = 42 2 + 1 − w − 22 (2 + 2 − Q) 2+2 22+2 2 R2n+1 d R2n+1 d (6.14) 2

Now we consider the third term on the right-hand side of (6.12). Since L = 6z + 4k 2 |z|4k−2 we have  42

’

R2n+1

@2 @ + 4k|z|2k−2 T; @t 2 @t

TwLw

@w @w t − x 6z w y 2 2 k @x @y R2n+1 (x + y )

2 @w @w t 2 4k−2 @ w − x · 4k y + 2 |z| @t 2 @x @y (x + y2 )k



@w @w t @w @w 2k−2 @ : · 4k|z| y y + 2 − x − x @x @y @x @y @t (x + y2 )k 



= 42

Denote that  I1 =

@w @w −x 6z w; @x @y R2n+1

 2 @w @w t 2 4k−2 @ w − x |z| I2 = 4k y ; 2 2 k @x @y @t 2 R2n+1 (x + y )



 @w @w t @w @w 2k−2 @ |z| y : I3 = 4k y −x −x 2 2 k @x @y @t @x @y R2n+1 (x + y ) t (x2 + y2 )k



y

It is clear that I1 = I2 = 0. From the fact −2

−2

−2

−2

Tw = T (ed u) = u(T ed ) + ed Tu = ed Tu and (6.3), one has |Tw|2 |z|−2 6 C

d4

|w|2 :

(6.15)

H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

Consequently, we see   −2 I3 = 4k |z| d x dy R2n

−∞



= −2k

∞

R2n+1

 tTwd(Tw) = 2k

|z|−2 |Tw|2 ¿ − 2kC

 R2n+1

d4

−2

R2n

|z|

183

 d x dy

∞

−∞

td(Tw)2

w2 :

Returning to (6.15) we have for 2 large enough,   ’ 42 w2 : TwLw ¿ − C1 2 22+2 d 2n+1 2n+1 R R

(6.16)

As for the fourth term on the right-hand side of (6.12), it follows    2 ’ − 2 − 2 + Q Tw2 2+2 d2 R2n+1 d    2 ’ =− div T 2+2 − 2 − 2 + Q w2 d2 d R2n+1     2 ’ T − 2 − 2 + Q w2 − d2+2 d2 R2n+1 = 0: Using (6.13), (6.14) and (6.16), we deduce from (6.12) that there exist C0 = C0 (Q) ¿ 0, 20 = 20 (Q; R) ¿ 0 such that when 2 ¿ 20 ,   −1 2+2 2d−2 w2 (Lu)2 ¿ C0 24 d e 22+2 d 2n+1 2n+1 R R  −2 = C0 2 4 e2d u2 ; 22+2 d 2n+1 R that is,  R2n+1

d22+2

e

1 u 6 C0 2 4

2d−2 2

 R2n+1

−1 2+2 2d−2

d

e

(Lu)2 :

(6.17)

On the other hand, for the smooth function g on R2n+1 ,   u∇g · A∇u = − u∇(Au∇g) R2n+1

R2n+1

 =−

R2n+1

u(∇u · A∇g + uLg)

 =− It shows  R2n+1

u∇g · A∇u = −

R2n+1

1 2

 u∇g · A∇u −

 R2n+1

u2 Lg

R2n+1

u2 Lg:

(6.18)

184

H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

and then  R2n+1

 guLu =

R2n+1

gu div(A∇u) 



=− =

1 2

R2n+1



R2n+1

u∇g · A∇u − u2 Lg −

 R2n+1

R2n+1

g∇u · A∇u

g|∇L u|2

(6.19)

−2

Letting g = e2d , we have from (6.9)

−2 22e2d 22 Lg = +2−Q+2 : d2+2 d2 Due to d ¡ R0 ¡ 1 we obtain |Lg| 6 C2 22

−2

d22+2

e2d :

(6.20)

Using (6.19) and HGolder’s inequality yields

1=2    2d−2 2 2d−2 2 2d−2 2 2 e u + e u e |∇L u| 6 C2 22+2 22+2 R2n+1 d R2n+1 R2n+1 d

1=2  2 −1 22+2 2d−2 d e (Lu) × R2n+1

6 C2

2

 R2n+1

d22+2

e

R2 u + 0 2

2d−2 2

 −2 1 + e2d u2 2 R2n+1 d22+2

 1 1 2  + 2 + R0 6C 24 2 R2n+1



−1 2+2 2d−2

R2n+1

−1 2+2 2d−2

d

e

d

e

(Lu)2 ;

(Lu)2

(6.21)

where we have used (6.17) in the last inequality. From this, together with (6.17), the result follows. Proof of Theorem 6.2. Without restriction, we assume that R ¿ 1 and pick a cut-oH function 7 ∈ C02 (B1 ) with 7 ≡ 1 on B1=2 . Applying Theorem 6.1 to 7u yields  −2 e2d [ d−22−2 u2 + |∇L u|2 ] B1=2

6

 B1

−2

e2d [ d−22−2 (7u)2 + |∇L (7u)|2 ]

H. Zhang, P. Niu / Nonlinear Analysis 54 (2003) 165 – 186

6 C0 (2

−2

+

R20 )



−1 2+2 2d−2

d

B1=2

+ C0 (2−2 + R20 )

e



According to (6.5) we have 2 2 2 |Lu| 6 max(C1 ; C2 ) 6 2 max(C12 ; C22 )

(Lu)2

−1 2+2 2d−2

d

B1 \B1=2

d2

1=2

|u| + 2

d4

d

2

|u| +

185

e

(L(7u))2 :

(6.22)

2 |∇L u|

d2

2

:

|∇L u|

Let 20 be large enough such that 2C0 (2−2 + R20 )max(C12 ; C22 ) 6 We have C0 (2

−2

and thus  B1=2

e

+

2d−2

R20 )



[ d

1 2

−1 2+2 2d−2

d

B1=2

−22−2 2

e

2

for 2 ¿ 20 : 1 (Lu) 6 2 2

u + |∇L u| ] 6 2C0 (2



−2

−2

B1=2

+

e2d [ d−22−2 u2 + |∇L u|2 ]

R20 )

(6.23)  B1 \B1=2

−1 2d−2

e

(L(7u))2 :

Letting 2 → ∞ and noting R0 ¡ 1, we have u ≡ 0 in B1=2 . This accomplishes the proof. Acknowledgements The authors are grateful to the referee for his/her invaluable comments and suggestions on the manuscript. We thank the referee for pointing [15] out to us. References [1] W. Allegretto, Y.X. Huang, A Picone’s identity for the p-Laplacian and applications, Nonlinear Anal. 32 (1998) 819–830. [2] R. Beals, B. Gaveau, P. Greiner, Uniforms hypoelliptic Green’s functions, J. Math. Pures Appl. 77 (1998) 209–248. [3] C. Capogna, D. Danielli, N. Garofalo, Capacity estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math. 118 (1997) 1153–1196. [4] Cui shangbin, Some necessary conditions for local solvability of linear partial diHerential operators, J. DiHerential Equations 106 (1993) 1–9. [5] G.B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc. 79 (1973) 373–376.

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[6] N. Garofalo, Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension, J. DiHerential Equations 104 (1993) 117–146. [7] N. Garofalo E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier Grenoble 40 (1990) 313–356. [8] N. Garofalo, E. Lanconelli, Existence and nonexistence results for semilinear equations on the Heisenberg group, Indian Univ. Math. J. 41 (1992) 71–98. [9] N. Garofalo, Z. Shen, Absence of positive eigenvalues for a class of subelliptic operators, Math. Ann. 304 (1996) 701–715. [10] N. Garofalo, D. Vassilev, Regularity near the characteristic set in the nonlinear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups, Math. Ann. 318 (2000) 453–516. [11] P.C. Greiner, A fundamental solution for a nonelliptic partial diHerential operator, Canad. J. Math. 31 (1979) 1107–1120. [12] L. HGormander, Hypoelliptic second order diHerential equations, Acta Math. Uppsala 119 (1967) 147–171. [13] Luo Xuebo, Global analyticity for quasi-homogeneous linear partial diHerential equations, Comm. Partial DiHerential Equation 23 (1998) 1171–1179. [14] P. Niu, H. Zhang, Y. Wang, Hardy type and Rellich type inequalities on the Heisenberg group, Proc. Amer. Math. Soc. 129 (2001) 3623–3630. [15] C.D. Sogge, Oscillatory integrals and unique continuation for second order elliptic diHerential equations, J. Amer. Math. Soc. 2 (1989) 491–516.