Applied Mathematics, 2010, 1, 293-300
doi:10.4236/am.2010.14038 Published Online October 2010 (http://www.SciRP.org/journal/am)
Entire Large Solutions of Quasilinear Elliptic
Equations of Mixed Type*
Hongxia Qin1, Zuodong Yang1,2
1Institute of Mathema tics, School of Mathematical Sciences, Nanjing Normal University,
Nanjing, China
2College of Zhongbei, Nanjing Normal University, Nanjing, China
Received March 10, 201 0; revised August 11, 2010; accepted August 15, 2010
Abstract
In this paper, the existence and nonexistence of nonnegative entire large solutions for the quasilinear elliptic
equation

2
|| =()()()()
m
divuup xfuq x gu

 are established, where2m,
f
and
g
are nondecreasing
and vanish at the origin. The locally Ho
 lder continuous functions
p
and qare nonnegative. We extend results
previously obtained for special cases of
f
and
g
.
Keywords: Entire Solutions, Large Solutions, Quasilinear Elliptic Equations
1. Introduction
In this paper, we consider the problem

2
||=()()()(),( 3)
(),| |
mN
divuup xfuq x guinRN
uxas x

 

 


(1)
where 2m, 1
,([0, ),[0, ))((0, ),[0, ))fg CC ,
the locally Ho
 lder continuous functions p and q are
nonnegative on N
R. In addition, we assume that
(0) =(0)=0;()0,()0,
() ()>0,>0
fg ftgt
ftgtfort



 (2)
We call nonnegative solutions of (1) entire large
solutions. In fact, this problem appears in the study of
non-Newtonian fluid s [1,2] and non -Newtonian filtration
[3,4], such problems also arise in the study of the
sub-sonic motion of a gas [5], the electric potential in
some bodies [6], and Riemannian geometry [7].
Large solutions of the problem
()=(()),,
|=,
uxfux x
u



 (3)
where  is a bounded domain in (1)
N
RN have
been extensively studied, see [8-20]. A problem with
()=u
f
ue and 2=N was first considered by Bieber-
bach [13] in 1916. Bieberbach showed that if

is a
bounded domain in 2 R such that  is a 2 C sub-
manifold of 2 R, then there exisSts a unique 2()uC


such that u eu = in

and 2 |()(()) |uxlndx   is
bounded on

. Here )(xd denotes the distance from a
point
x
to


. Rademacher [17], using the idea of
Bieberbach, extended the above result to a smooth
bounded domain in 3 R. In this case the problem plays
an important role, when 2=N, in the theory of
Riemann surfaces of constant negative curvature and in
the theory of automorphic functions, and when 3=N,
according to [17], in the study of the electric potential in
a glowing hollow metal body. Lazer and McKenna [6]
extended the results for a bounded domain

in
1)( NRN satisfying a uniform external sphere condi-
tion and the non-linearity u expuxff )(=),(= , where
)(xp is continuous and strictly positive on

. Lazer
and McKenna [12] obtained similar results when

is
replaced by the Monge-Ampere operator and

is a
smooth, strictly convex, bound ed domain. Similar results
were also obtained for a uxpf )(= with 1>a.
Posteraro [16], for u euf =)( and 2N, proved the
estimates for the solution )(xu of the problem (1,2) and
for the measure of

comparing with a problem of the
same type defined in a ball. In particular, when 2=N,
Posteraro [16] obtained an explicit estimate of the
minimum of )(xu in terms of the measure of

:
()(8/| |).
minux ln


*Project Supported by the National Natural Science Foundation o
f
China(Grant No.10871060). Project Supported by the Natural Science
Foundation of the Jiangsu Higher Education Institutions of China
(Grant No.08KJB110005)

H. X. QIN ET AL. 294
The existence, but not uniqueness, of solutions of the
problem (3) with f monotone was studied by Keller
[18]. For a
uuf =)( with 1>a, the problem (3) is of
interest in the study of the sub-sonic motion of a gas
when 2=a (see [15]) and is related to a problem
involving super-diffusion, particularly for 2<1

a
(see [21,22]). Pohozaev [15] proved the existence, but
not uniqueness, for the problem (1.2) when 2
=)(uuf .
For the case where (2)/(2)
()=( >2)
NN
fu uN

, Loewe-
ner and Nirenberg [20] proved that if  consists of a
disjoint union of finitely compact  C manifolds, each
having co-dimension less than 1/2N, then there exists
a unique solution of the problem (3). The uniqueness was
established for a
uuf =)( with 3>a, when


is
a 2
C-submanifold and  is replaced by a more
general second-order elliptic operator, by Kondrat'ev and
Nikishkin [19]. Marcus and Veron [14] proved the
uniqueness for a
uuf =)( with 1>a, when


is
compact and is locally the graph of a continuous function
defined on an 1)( N-dimensional space.
In [23], the authors considered the problem of
existence and nonexistence of positive entire large
solutions of the semilinear elliptic equation
=()() ,0<.upxu qxu


  
Recently [24], which is to extend some of these results
to a more general the problem
=()()()(),3,
()| |.
N
upxfuqxguinRN
uxas x
 
 

Quasilinear elliptic problems with boundary blowup

2
||=(()),,
|=,
m
divuufu xx
u


 




(4)
have been studied, see [9,25,26] and the references
therein. Diaz and Letelier [10] proved the existence and
uniqueness of large solu tions to the problem (4) both for
1>,=)( muuf 

(super-linear case) and

 being
of the class 2 C. Lu, Yang and E.H.Twizell [25] proved
the existence of Large solutions to the problem (4) both
for N
Rmuuf =1,>,=)(  

or  being a boun-
ded domain (super-linear case) and N
Rm =1, 
(sub-linear case) respectively.
Recently [27], which is to extend some results of [28]
to the following quasilinear elliptic problem

2
|| =()(),
() ,
m
divuup xfuin
ux on

 

 


(5)
where 
N
R, the non-negative function )()( Cxp ,
and the continuous functionfsatisfies (2) and the
Keller-Osserman condition
1/
10
[()]=,()=()
s
m
F
sdsFs ftdt


(6)
then the author also consider the nonexistence for the
non-negative non-trivial entire bounded radial solution
on N R of (5) when p satisfies
1/(1)
**
0||=
(( ))=,( )=().
min
m
xt
tp tdtp tpx

 (7)
On the other hand, if f does not satisfy (6), that is



<)]([ 1/
1dssF m, we can obtain from Lemma 2.4 in
[29] that
1/( 1)
1
1<
()
mds
fs


 (8)
which is also shown in [30]. In this paper, we will
require the above integral to be infinite, that is
1/( 1)
1
1=
()
mds
fs


 (9)
which is a very important condition in our main results.
Furthermore, motivated by the results of [24], we also
admit the following condition which is opposite to (7),
that is
*1/(1) *
0||=
(( ))<,( )=().
max
m
xt
tp tdtp tpx

 (10)
As far as the authors know, however, there are no
results which contain the existence criteria of positive
solutions to the problem (1). In this paper, we prove the
existence of the positive large solutions for the problem
(1). When 2=p, the related results have been obtained
by A.Lair and A.Mohammed [24]. The main results of
the present paper contain extension of the results in [24]
and complement of the results in [10,25,26].
The plan of the paper is as follows. In Section 2, for
the convenience of the reader we give some basic
lemmas that will be used in proving our results. In
Section 3 we state and prove the main results. Section 4
contains some consequences of the main theorems, and
other results. In Section 5 we present an Append ix wher e
we state and prove three lemmas needed for proofs in
previous sections.
2. Preliminary
In this section, we give some results that we shall use in
the rest of the paper.
Lemma 2.1.(Weak comparison principle)(see [25])
Let

be a bounded domain in
N
R(2)N with
smooth boundary


and )(0,)(0,: 

is con-
tinuous and non-decreasing. Let )(, 1,
21  m
Wuu satisfy
2
111
2
22 2
|| ()
|| ()
m
m
uudxudx
uudxudx












H. X. QIN ET AL. 295
for all non-negative )(
1,
0 m
W 
. Then the inequality
 onuu 21
implies that
.
21  inuu
Lemma 2.2. Let f verify (9), and )[0,)[0,:



be continuous. Then
11
((( )))=( )( ),>0
(0) =,(0) = 0
NN
m
rvrrrfvr
vv









 (11)
admits a non-negative solution v defined on )[0,

,
where sss m
m2
|=|)(  . If in addition f is nondecrea-
sing and

satisfies (7), then )( rv as


r
.
Proof. First we note that (11) has a solution )(0,
1RCv
for a maximal R<0 . As a consequence of (7) we
claim that =R. By way of contradiction, let us
suppose that <<0 R instead. Then we must have
)( rv as  Rr . Let
0>},<0:)({sup:=)( ttsst 


Then

is nondecreasing, and clearly )()( tt



for 0>t. Integrating Equation (11) from 0 to
r
yields
11
0
(())=()(())
r
NN
mvr rssfvsds



 (12)
From (12) we see that 0)( 

rv , therefore, v is a
non-decreasing function and we can obtain from (12)
that (()) ()(())
mr
vr rfvr
N


 . Then we can obtain
1/(1) 1/( 1)
1/(1)
00
() ()
(())
m
rr
m
m
vt t
dtt dt
N
fvt








That is
1/( 1)
() 1/( 1)
1/( 1)0
1()
()
m
vrr m
m
t
dss ds
N
fs









Letting Rr , and recalling that

)(rv , we
conclude that
1/( 1)1/( 1)
1/( 1)0
1()
()
mRm
m
R
dssds
N
fs









which is an obvious contradiction. Thus, indeed v is
defined on )(0,.
We now show that

)(rv as 
r
. For this we
will use (7) on

. Integrating the equation in (11) we
find

1/(1)
11
00
1/( 1)1/(1)
*
0
()=() (())
() (())
m
rt
NN
mrm
vrtss f vsdsdt
fttdt
N












That is
1/(1)
*
0
()( ,,)(()),>0
rm
vrCmNttdt r 


and as a consequence of (7) we conclude that

)(rv
as


r
.
3. Main Theorems
In this section, we will state the first of our main results.
Theorem 3.1. Under the following hypotheses
 
1
1
1, 0;
t
Ht dst
fs





 

 



 



 
1/ 11/ 1
**
0
1/ 1
*
0
2
, ,
mm
mt
Hptp t
tfptdtptsps ds





 


where

is the inverse of

;
  



1/ 1
*
0
3m Htqtgpt dt 



Let f and
g
satisfy (2). Furth ermore, assu me that (9)
and (10) hold. If p satisfies (7), then (1) admits a
solution.
Proof. Let v be an entire radial large solution of
)(|)(|=)|(| *
2vfxpvvdivm  such that

=(0)v for
some 1<<0

. This is possible by Lemma 2.2, since
f satisfies (9) and * p satisfies (7). Thus v is a
super-solution of (1). We proceed to construct a
sub-solution u of (1) such that vu  on N
R. Then
by the standard regularity argument for elliptic problems,
it is a straight forward argument to prove that (1) would
have a solution w such that vwu  on N
R. For
each positive integer n, let n
u be a solution of
2
**
(| |)
= (||)()(||)(),0.4,
()=,0.4 ,
m
n
n
div uu
pxfuqxgu cminB
ux vcmon B






(13)
where )(0,= nBBn is the ball of radius n centered at
the origin. That such a solution exists is shown in
Lemma 5.2 of Appendix. Then we note that each n u is
a radial solution and that
1
0<, .
nn n
uuvonB



Let
():= (),.
lim
N
n
n
uxuxxR
 
Since each n u is radial, it follows that u is radial as
well. By a standard argument we can show that u is a
solution of the differential equation in (1). Clearly vu

on N
R. So We only prove that u is nontrivial and that

)(xu as

|| x.
Recalling that n u and v are radial and that
)(=)( nvnun we see that

H. X. QIN ET AL. 296

11* *1/(1)
00
1/( 1)1/( 1)
11 11
**
00 00
(0)((( )()()()))
=(0) ()()(0) ()()
nt
NN m
nnn
mm
nt nt
NNNN n
utspsfuqsgudsdt
vtspsfvdsdtvtspsfudsdt
 

 




for 20,  mx , we can use the inequality 1)1/(1)1/( 1)(1   mm xx , then we obtain





1/( 1)1/(1)
11* 11*
00 00
1/( 1)
11
*
00
(0)()()( )()
() ()(0)=
mm
nt nt
NN NN
nn n
m
nt
NN n
u tspsfudsdttsqsgudsdt
tspsf udsdtv 

 




 

that is


(1 )/( 1)1*1/(1)11/(1)
*
00 0
1/( 1)
11*
00
(0)((()( ))(()( ) )
() ()(0)=
ntt
Nm NmNm
nnn
m
nt
NN n
utspsfudsspsfudsdt
tsqsg udsdtv 
 




 

Since )(
*sp is increasing and )(
*sp is decreasing, so




1/( 1)1/( 1)
(1)/(1)1*1 *
00 0
1/( 1)
(1)/( 1)*1/(1)11/( 1)11/( 1)
*
00 0
* 1/(1)1/(
*
0
() ()() ()
(()()(())()() (()))
() ()()
mm
ntt
Nm NN
nn
m
nt t
Nm mNmNm
nn
nmm
tspsfudsspsfudsdt
tptsfusdsptsfusds dt
ptp

 

  







 
 






1/( 1)
1)(1)/( 1)1
0
1/( 1)
* 1/(1)1/(1)
*
00
*1/(1)1/( 1)1/( 1)
1*
0
()(()
() ()() ()(())
()() ()() ()((()))
m
t
NmN n
m
nt
mm
n
nmm m
n
tts fusdsdt
ptptfusdsdt
Cmptpt tfutdt




 





and


1/( 1)1/( 1)
11* *
2
00 0
() ()()()(())
mm
nt n
NN nn
tsqsg udsdtCmtqtg utdt

 
 
Therefore we get
 
1/( 1)
*1/(1)1/(1)1/(1)*
1* 2
0 0
(0)(,)()( )()()(())(,)( )()m
n n
mm m
nnn
uCm NptpttfudtCm Ntqtgudt


 
 

(14)
Now, let )(t

be the inverse of the increasing
function defined in (9). We note that 1)( t

for all
0t. Furt h ermore, we have 0.>)),(())((=)()),((=)( ttftfttft







Let w be an entire large solution of |)(|= *xpw
such that 0=(0)w. Set ))((:=)( xwxa

. Then
)(|)(|
*afxpa. Since 0>(0)=>1(0))(=(0) vwa


, we invoke Lemma
2.1 in [24] to conclude that )()( xaxv  for all

x
N
R.
Moreover, ,)(:=)()( *
0dtttprprw r

 we have
|))(|()( xpxv

.
Now, recalling that vun for all Nn we see that
***
*1/( 1)1/( 1)1/( 1)
*
*1/(1)1/( 1)1/( 1)
*
*1/(1)1/( 1)1/( 1)
*
() (())()(())()((())),
(()( )()())((( )))
(()()()( ))((( )))
(()( )()())(((())))
n
mmm
n
mm m
mm m
tqtgu ttqtgvttq tgpt
ptpttfut
ptpttfvt
ptpttfpt


 
 
 




Take note of (9) and (10), we invoke the Lebesgue
dominated convergence theorem to infer from (14) that


*1/(1)1/( 1)1/( 1)
1*
0
1/( 1)
*
20
(0)
(,)()()()()(())
(, )()()>0.
nmm m
m
n
u
CmNptpt tfudt
CmN tqtgudt 
 





This show s that u is nontrivial. Now we note that



1/(1)
11* *
00
1/(1)
11*
00
()=(0)
()()() ()
() ().
nn
m
rt
NN nn
m
rt
NN n
uru
tspsfuqsg udsdt
tsps f udsdt








Recalling that vun

for all n, we invoke the
Lebesgue dominated convergence theorem again, on
letting

n


1/( 1)
11*
00
()() ().
m
rt
NN
u rtspsfu dsdt




H. X. QIN ET AL. 297
Since u is nontrivial we see that 0>)
2
(0
r
u for
some 0>
0
r. Thus for 0 >rr , we have
dtdsspst
r
ufru m
N
t
N
r
r
m1)1/( *1
0
1
0
0 1)1/( )()
2
()( 
 












then
dtttp
N
r
ufru m
r
r
m
m1)1/(
*
0
1)1/(
0 1)1/( ))((
1
)
2
()( 















Therefore, as a consequence of (7) we see that
)(xu as || x.
To show our next main result, now we set p is c-
positive on  (i.e., for any 

0
x satisfying 0=)(0 xp ,
there exists a domain 0  such that  000,x,
and 0>)(xp for all 0 x.) we know that p is
c-positive on
N
R if and only if there is a sequence
n
 of smooth bounded domains with 1

 nn for
each n such that n
n

1=
N
R and p is c-positive
on each . It is easy to see that if

is a non-negative
and locally Ho
 lder continuous function in
N
R that
satisfies (10), then the following problem admits a
positive solution.
2
(||) =(),
()0,||
mN
div wwxx
wx x





R (15)
In fact
.)(=)( 1)1/(
*1
0
1
||dtdssstxv m
N
t
N
x









is a super-solution of (15) such that 0)( xv as
|| x. On the other hand, 0 is a sub-solution of (15),
(See [31], Lemma 3) the assertion follows.
Theorem 3.2. Suppose f and
g
satisfy (2). If (1)
has a solution, f satisfies (8) and p is c-positive in
N
R, (3) admits a solution. Conversely, if gf

satisfies (8), (15) admits a non-negative solution with
)()(=)( xqxpx 

and )}(),({min:=)( xqxpx

is c-positive, then (1) has a solution.
Proof. Let }{ n  be a sequence of bounded smooth
domains in
N
R as provided in the definition of the
c-positivity of p.
Suppose (1) has a solution, say
v
is a solution. For
each n, the problem
2
(||)=( )( ),
()=,
mn
n
divuup xfux
ux x

 



 (16)
has a solution( see [29]). For each positive integer n, let
n
u be a solution of (16). Then by Lemma 2.1 it follows
that
.),()()( 1nnn xxuxuxv  
A standard procedure (for example, see [30]) can be
used to show that
,),(
lim
:=)( N
n
nxxuxu R 
is the desired solution of (3). For the converse, we let n
u
be a solution of the problem
2
(||)=( )( )()( ),
()= ,
mn
n
divuup xfuq x gux
ux x







 (17)
The existence of such a solution is demonstrated in
Lemma 5.3 of Appendix. It easily follows that the
sequence }{ n u is a non-in creasing sequence. Let
()= (),.
lim
N
n
n
uxu xx
 R
A standard argument shows that u is a solution of
the quasilinear equation in (17). Thus we need only show
that u is nontrivial and that )(xu as

|| x.
For this we consider the following function
1/( 1)
1
()=,>0,
()
m
t
tdst
hs



 (18)
where )()(:=)( tgtfth

. Obviously, (18) is finite for
all 0>t. We also notice that
0>
))(1)((
)(
=)(0,<
)(
1
=)( 1)1/(1)1/( mmm thm
th
t
th
t 





Now fix 0>

, and let
nnn xxuxv



),)((=)(


. Note the sequence n v is
nondecreasing. Moreover, a simple computation shows
that
212
2
1
(||)=|() |(||)
(1)|( )|( )||
|()| (()()()())
() ()()()
=()()
()
mmm
nn nnn
mm
nnn
m
nnn
nn
n
divvvudivuu
mu uu
upxfuqxgu
pxfu qxgupxqx
hu

 





  






We can also note that 0=
n
v on n  . Let w be a
solution of (15). Thus by Lemma 2.1 we see that wvn

on n

for all n, letting n, and then 0

we
see that wu

)(

on
N
R. Thus 0))(( xu

as

|| x, that is

)(xu as || x.
4. Consequences and Related Results
We can obtain some consequences of the main theorems,
and other results that are of independent interest.
Theorem 4.1. Let f and
g
be continuous, nonde-
creasing functions such that gf  satisfies (9), and
Suppose qp

is nontrivial. If there is a solution to
2
(||)() ()()(),,3
(),| |
mN
divuup xfuqx g uxRN
uxas x



 
 
(19)
then qp

satisfies (7).

H. X. QIN ET AL. 298
Proof. Let u be a solution of (19). Let v be a
solution of the initial value (11) with *
)(= qp  
, f
replaced by gf  and 0 =


where 0

is chosen
such that (0)>
0u

. Since gf  satisfies (9), we
recall from Lemma 2.2 that v is defined on )[0,

.
Then |)(|=)( xvxw is a solution of
)))(())((|)((|)(=)|(| *2xwgxwfxqpwwdiv m,
and hence )()()|(|2wqgwpfwwdiv m on N R.
Since 0>)(rv we see that Arv )( as


r
, for
some A<0 . Assume that <
A
so that Axw

)(
for all
N
xR. Since )(xu as || x, we see
that for some R, we have RxxuAxw  ||),()( .
Thus )()( xuxw  on Rx |=| and therefore by Lemma
2.1 we find that )()( xuxw  on )(0,RB . But this
contradicts the choice that (0)>(0) uw . So we have
=
A
, then  ||,)( xasxw . From Equation (11)
we find


1/(1)
11*
0
1/(1)
11*
0
()=() ( )()(())
()(())()()
m
r
NN
m
r
NN
vrrtp qt fgvtdt
rfgvrtpqtdt





 


(20)
Dividing (20) through by ))(()(1)1/( rvgfm
 and
integrating the resulting inequality on )(0,r we have

1/( 1)
0
1/( 1)
11*
00
()
()(())
()()
r
m
m
rt
NN
vt dt
fg vt
tspqsdsdt








That is
()
1/( 1)
0
1/( 1)*1/(1)
0
1
()()
1(( )())
vr
m
mrm
dt
fg t
tp qtdt
N











Letting 
r
and recalling that gf  satisfies
(9), the claim is proved.
As a consequence of Theorem 3.1 and Theorem 4.1
we also obtain the following corollaries.
Corolla ry 1. Suppose (2) and ( 9) hold for f. Further,
let p satisfy (10). (3) admits a solution if and only if
p satisfies (7).
Proof. If p satisfies (7) then Theorem 3.1, with
0=)(xq shows that (3) has a solution. The converse
follows from Theorem 4.1 on taking 0)( xq again.
The next corollary provides sufficient conditions for
the existence and nonexistence of solutions to (1) when
both p and q satisfy (7).
Corollary 2. Suppose f and
g
satisfy (2) and p
and q satisfy (7). If gf satisfies (9), then (1) has
no solution. On the other hand, (1) admits a solution if
gf  satisfies (8) and the function
)}(),({min)( xqxpx 

is c-positive on N
R.
Proof. By way of contradiction,we can obtain th e first
statement from Theorem 4.1. Since qp  satisfies (10)
and the remark noted ju st before Theorem 3.2 shows that
(15) admits a solution with qpb=. Thus the second
part of the corollary is an immediate consequence of
Theorem 3.2.
5. Appendix
In this appendix we state and prove results that have
been used in the proofs of the main results of the paper.
We start by proving the existence of a solution to the
following Dirichlet problem on a bounded smooth domain

in N
R.
2
(||)=( )( )( )( ),,
()= (),.
m
divuup xfuq x guin
uxx on






(21)
Lemma 5.1. Let 

N
R be a smooth bounded
domain and let f and
g
satisfy (2). Let )(
2C

be positive. If v is a positive super-solution of (21),
then the problem (21) has a solution u such that
vu

<0 on

.
Proof. Let )(
min
:= x
x


 . Obviously, 0>

.
Now we set
,))((=)( 1)1/(
0dsshtm
t


where )()(=)( sgsfsh

for all 0s. An application
of L'H o
ˆpital’s Rule shows that tt )(

for all

<<0 t and some 0>

. Without of generality we
can suppose that



)(<0 . Finally, let
z
be a
solution of the Dirichlet problem




 
.,=)(
,),()(=)|(|2
onxz
inxqxpzzdiv m

The n the maximu m pri ncip le sh ows th at


)(<0 xz
on

, we define ))((:=)( xzxw

for all x, we
note that)()( xzxw

for allx. A simple computation
shows that
0>
))(1)((
)(
=)(0,>)(=)( 21)1/(
1)1/(




 mm
m
thm
th
ttht

and
2
12 2
1
(| |)
=|| (||)(1)||||
||(( )( ))=(( )())(( )( ))
(()())(( )( ))( )()( )()
m
mm mm
m
div ww
'divzzm' ''z
' pxqx fzgzpxqx
f
wgwpxqx pxfwqxgw



 


 
 

and )()()( xxw







for x. Thus w is a
sub-solution of (21) and v is a super-solution of (21)
such that vw



on


. By the maximum principle
we note that vw

on

. Thus by lemma 1 in [31] we

H. X. QIN ET AL. 299
conclude that (21) has a solution usuch that vuw


which is what we want to show.
The following lemma was used in the proof of
Theorem 3.1.
Lemma 5.2. Let ))[0,),([0,, Cba and B be a
ball in N
R centered at the origin. If f and
g
are
nondecreasing on )[0,, then given a positive constant

, there exists a radial solution to the problem
2
(||)=(||)()(||)(),,
()= ,,
m
divuuaxfubxguin B
uxonB 

 

(22)
Proof. Let }{k a and }{ k b be decreasing sequences
of Ho
 lder continuous functions which converge uni-
formly on B to a and b respectively(Se e [32]). Then
by Lemma 5.1, for each k there exists a nonnegative
solution k u of
2
(| |)
=(||)()(||)(),,
()=,.
m
kk
kkkk
k
div uu
axfu bxguinB
uxon B 






Since the sequence }{ k a and }{ k b are decreasing,it
is easy to show that }{ k u is increasing. Of course,it is
also bounded above by

. Thus it converges, and
assume uuk. Since k u satisfies the integral equation


1/(1)
11
00
()= (0)
( )(( )( )(( )))
kk
m
rt
NN kk kk
ur u
tsasfusbsgusds dt





the function u will satisfy the integral equation


1/( 1)
11
00
()= (0)
()(())() (()).
m
rt
NN
ur u
tsa sfu sb s g usdsdt





Since

=)(Ruk for eachk, whereRis the radius of
the ballB, it is clear that

=)(Ru . Thusuis a non-
negative solution of problem (22) on Bas claimed.
Finally we state and prove a lemma that was used in
the proof of Theorem 3.2.
Lemma 5.3. Let 
N
R be smooth. Suppose f
and
g
satisfy (2). Iffsatisfies (6) andp is c-positive
on , then the problem
2
(||)=( )( )()( ),,
()= ,,
m
divuup xfuq xg ux
ux x

 

(23)
has a solution. Similarly, if instead of requiring f to
satisfy (6), we require only gf  to satisfy (6), and
require },{min:= qp

to to be c-positive on

, then
(23) has a solution.
Proof. Since p is c-positive and f satisfies (6), let
v
be a large solution of )()(=)|(| 2vfxpvvdivm
on  (see [29]). Now for each positive integer k, let
k
w be a solution (See Lemma 5.1) of
2
(| |)
=(()())(()()),,
()= ,,
m
div ww
pxqxfwgwx
wxkx








By Lemma 2.1 we see that


xxvxwxw kk),()()( 1
If )(
lim
=)( xwxw k
k, then by a standard procedure
we conclude that w is a solution of ))()())(()((=)|(| 2wgwfxqxpwwdivm  on

such that vw

. Since w is a sub-solution, and
v
is
a super-solution of the differential equation in (23), we
conclude that (23) has a solution u with vuw


(See [31]).
Similarly.We can obtain the second part by defining
v
in this case as a large solution of
))()()((=)|(|2vgvfxvvdivm   on  and the
argument is as the same as the previous process.
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