Advances in Pure Mathematics, 2011, 1, 235-237
doi:10.4236/apm.2011.14041 Published Online July 2011 (http://www.SciRP.org/journal/apm)
Extremum Principle for Very Weak Solutions of
A-Harmonic Equation with Weight*
Hong-Ya Gao, Chao Liu, Yu Zhang
College of Mathematics and Computer Science, Hebei University, Baoding, China
Received March 2, 2011; revised April 11, 2011; accepted April 20, 2011
Abstract
Extremum principle for very weak solutions of A-harmonic equation


,div Axu0

 is obtained, where the
operator :nn
A
RR satisfies some coercivity and controllable growth conditions with Muckenhoupt
weight.
Keywords: A-Harmonic Equation, Muckenhoupt Weight, Extremum Principle, Hodge Decomposition
1. Introduction
Throughout this paper will stands for a bounded
regular domain in , . By a regular domain we
understand any domain of finite measure for which the
estimates (1.6) and (1.7) for the Hodge decomposition
are justified, see [1]. A Lipschitz domain, for example, is
regular.

n
n
R2
Given a nonnegative locally integrable function ,
we say that belongs to the
w
w
p
A
class of Mucken-
houpt, , if
1< p<


1
11
11
dd= sup
p
p
p
QQ
Q
wxwxA w
QQ


 <

 
 
 
 (1)
where the supremum is taken over all cubes of .
When , replace the inequality (1.1) with
Qn R
=1p
 
M
wx cwx
for some fixed constant and a.e. cn
x
R, where
M
is the Hardy-Littlewood maximal operator.
It is well-known that 1
p
A
A whenever , see
[2]. We will denote by
>1p


Lw,
p, 1<<p

, the
Banach space of all measurable functions
f
defined on
for which


 

1
,d<
P
p
p
Lw
ffxwxx



The weighted Sobolev class consists of
all functions

1, , p
W

w
f
for which
f
and its first generalized
derivatives belong to


,Lw
p.
We will need the following definition. Given ,uv



1,r
W

, 1r<

, the function


1
==min
2uv uvuv

0,


also belong to


1,r
W

. The chain rule gives
=uv


 if




<ux vx and =0

 if


>ux


vx. We say




xux v on  in sobolev sense,
or symbolically, uv


 if the function

defined
above lies in


1,
0
r
W

.
Consider the following second order divergence type
elliptic equation (also called A-harmonic equation or
Leray-Lions equation)


div,=0Ax u (2)
where :nn
A
RR is a Carathéodory function and
satisfies
1)
 
,,
p
Ax wx

  ,
2)
 
1
,
p
Ax wx  

,
where 1< <p

, 0< <



 are fixed constants,
and


1
wx A

be a Muckenhoupt weight. The proto-
type of Equation (2) is the -harmonic equation with
weight
p



2
div= 0
p
wx uu


*Research supported by NSFC (10971224) and NSF of Hebei Province
(A2011201011).

236

H.-Y. GAO ET AL.
Definition: A function with
is called a very weak solution of
(2) if

1, , r
uW w

max 1,1p<rp

,,d=Ax ux 

0 (3)
for all


1,1 , rrp
W 



w

w
=

w
with compact support.
Recall that is a weak solution of (2)
if (3) holds for all with compact
support. The word very weak in the above definition
means that the Sobolev integrable exponent of u is
smaller than the natural exponent .

1, , p
uW w
 
1, (,) p
W
p
r
Extremum principle for weak and very weak solutions
of elliptic equations is an important and basic property. It
is closely related to the uniqueness results for some
boundary value problems of elliptic PDEs, see [4].
Motivated by this property, Gao, Li and Deng showed in
[3] the extremum principle for very weak solutions of (2)
with the weight . In the present paper, we
generalize the result obtained in [3] to weighted case,
and prove the extremum principle for very weak
solutions of (2). The main result of this paper is the
following theorem.

1wx
Theorem 1: Suppose that be a Muckenhoupt
weight. There exists 2
, such that if is a very
weak solution of the
1
wA
11
=,rr 
uW

,,,<<npw pr

1, , rw

2,,,,rnp

A
-harmonic Equation (1), and
on in the sobolev sense, then

mux M  m

almost everywhere in , provided that
.

ux M
12
<<rrr

With the extremum principle at hand, we can consider
the 0-Dirichlet problem


1,
0
,=
r
div Axu
uW





0
(4)
Theorem 2: Let and 2 be the exponents in
Theorem 1 and 12 . Then the 0-Dirichlet bound-
ary value problem (4) has only zero solution.
1
r
<<rr
r
r
We will need the following lemma in the proof of the
main theorem, which is a Hodge decomposition in
weighted spaces.
Lemma: [5] Let be a regular domain and



wx
be an 1
A
weight. If ,

1,
0,
p
uW w
 
1< <p

,
1<1< p

, then there exist


,Ww

1, (
0
p

)/(1 )  

and a divergence-free vector field

,
1p
H
Lw
 
such that
uu



h (5)
and




1
1
,
pp
p




1
1
,,
pp
p
L
wLw
hCAwu








 (7)
where


=p  and


=,,CCnpw depending only
on and , respectively.
p,,npw
2. Proof of Theorem 1 and Theorem 2
Proof of Throrem 1. If


ux is a very weak solution of
the Equation (2), then also is






1,
=
r
vxuxm Ww 

,. For every test function

1
1,
0,
p
rL
Ww





, we have


,, d=Ax ux 

0 (8)
Now let


=min0,v 
. It is easy to see that


w
1,
0,
r
W  . Consider the Hodge decomposition of
rp



,
=
rph


,
L
w
CA wu









Lw
(6)


By Lemma, we have the following estimate




1
1,,
rp
rrp r
p
L
wLw
hCAwrp



 
 (9)
The integral identity (8) with

as a test function of
class


1, 1
0,
rrp
W
 

w takes the form
 
,,d= ,,d
rp
A
xux Axuhx 


 

(10)
Let us put



=:<Xx vx 0
Since the gradient of

is equal to on X, while
it vanishes on
v
X

, then for this choice of

the
integral in (10) reduces to
 
,,d= ,,d
rp
XX
A
xv vvxAxvhx

 

By the conditions 1) and 2), the above equality yields
dd
rrp
XX
vwxv hwx 



Using Hölder’s inequality and (9) we obtain
  

1
1
,,
d
d
rp
rrp
r
Lw Lw
X
r
pX
vwxh
CA wprvwx










2
(11)
Taking sufficiently close to to satisfy
1
<<rprp

1
=p CA wpr



and

2
=p CA wrp



.
Thus, if , then
1
<<rrr
2

=<
p
CA wpr



1
(11)

H.-Y. GAO ET AL.
237
yields

,=0
r
Lw
v , from which we deduce


=0x 
almost everywhere in , and this simply means that
almost everywhere in .


mux
Similarly, by the same method we used above, we can
also derive almost everywhere in

ux M

. This
completes the proof of Theorem 1.
Proof of Throrem 2. By Theorem 1, we know that
and almost everywhere in

0ux

ux0

.
This simply means that almost everywhere in
. This completes the proof of Theorem 2.

=0ux

3. References
[1] T. Iwaniec and C. Sbordone, “Weak Minima of Varia-
tional Integrals,” Journal für die Reine und Angewandte
Mathematik, No. 454, 1994, pp. 143-162.
doi:10.1515/crll.1994.454.143
[2] J. Heinonen, T. Kilpeläinen and O. Martio, “Nonlinear
Potential Theory of Degenerate Elliptic Equations,” Clar-
endon Press, Oxford, 1993.
[3] H. Y. Gao, J. Li and Y. J. Deng, “Extremum Principle for
Very Weak Solutions of A-Harmonic Equation,” Journal
of Partial Differential Equations, Vol. 18, No. 3, 2005,
pp. 235-240.
[4] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differ-
ential Equations of Second Order,” Springer-Verlag, Ber-
lin, 1983.
[5] H. Y. Jia and L. Y. Jiang, “On Non-Linear Elliptic Equa-
tion with Weight,” Nonlinear Analysis: Theory , Methods
& Applications, Vol. 61, No. 3, 2005, pp. 477-483.