Applied Mathematics, 2010, 1, 351-356
doi:10.4236/am.2010.15046 Published Online November 2010 (http://www.SciRP.org/journal/am)
Existence and Non-Existence Result for Singular
Quasilinear Elliptic Equations*
Mingzhu Wu1, Zuodong Yang1,2
1Institute of Mathematics, School of Mathematical Science, Nanjing Normal University,
Nanjing, China
2College of Zhongbei, Nanjing Normal University, Nanjing, China
Received June 1, 2010; revised August 11, 2010; accepted August 14, 2010
Abstract
We prove the existence of a ground state solution for the qusilinear elliptic equation 2
(| |)
p
div uu

 
=(,)
f
xu in N
R, under suitable conditions on a locally Holder continuous non-linearity ),( txf , the
non-linearity may exhibit a singularity as 0t

. We also prove the non-existence of radially symmetric
solutions to the singular elliptic equation 2
(||)=( )[( )( )||],
pq
divuudxg ur uu

 ()>0ux in N R,
() 0ux  as || ,x where ()=(||)(,(0, )),
N
dxdxC

R 3, 0.Nq


10,
(0,),(0,), ([0,),
loc
gC rC


 
[0,)), 0<<1

.
Keywords: Quasilinear Elliptic Equations, Existence, Non-Existence, Singular
1. Introduction
In this paper, we are concerned with the existence of
ground state solution or positive solution for the follow-
ing problem
2
(||) =(,),
>0,
()=0,
p
divuufx ux
ux
ux x

 




(1)
in which
N
R, () 0ux  when ||x . Let
:[0,)[0,)f be a locally Holder continuous
function which may be singular at =0t.
The problem (1) appears in the study of non-Newto-
nian fluids [1,2] and non-Newtonian filtration [3]. The
quantity p is a characteristic of the medium. Media with
>2p are called dilatant fluids and those with <2p are
called pseudoplastics. If =2p, they are Newtonian fluid.
When 2p, the problem becomes more complicated
since certain nice properties inherent to the case =2p
seem to be lose or at least difficult to verify. The main
differences between =2p and 2p can be found in
[4-6].
In recent years the study of ground state solutions for
=2p has received a lot of interest and gets numerous
existence results (see [7-12]). For p-Laplacian equa-
tions, in most papers, the focus has been on separable
nonlinearities like (2).
2
(||)=( )( ),
>0,
()0,| |
pN
N
divuubx g ux
uxR
uxas x

 




R
(2)
We refer readers to the paper [13-19]. In this paper,
we consider the situation of 1>p.
In [20], the author extended results to the problem (1)
for =2p where
f
is not necessarily separable. For
2p we can see [1-6]. Motivated by papers [4-6,20],
we extend the results to >1p and get two theorems.
But we still have many difficulties to get entire ground
state solution of (1).
The second purpose is to give a result for nonexistence
of solution. To the best of our knowledge, there has been
very less result for nonexistence of solution about
singular elliptic equation. We solve an open problem in
[13] for >1p, for the case when u is a radially sym-
metric solution.
The paper is organized as follows. In Section 2 we
recall some facts and give many lemmas that will be
needed in the paper. In Section 3, we give the proof of
the main result of the paper.
*Project Supported by the National Natural Science Foundation o
f
China (No.10871060); the Natural Science Foundation of Educational
Department of Jiangsu Province (No.08KJB110005)

M. Z. WU ET AL. 352
2. Preliminaries
Firstly we list the following assumptions and results that
needed below.
2
(||) =(),
>0,
()0,| |
pN
N
divWWbxx
WxR
Wx asx

 




R
(3)
1
[]>0Bb is a locally Holder continuous function on
N
R.
1
11* 1
200
[](()) <
t
NN p
Btsbsdsdt
 
 , where
*()=max{ ():||=}btbx x t for >0t.
3
[]B Problem (3) has a solution b W in
N
R when
b satisfy some condition.
It can be proved that condition 2 []B implies 3 []B.
This follows from the observation that () 0vr  as
r where
1
11* 1
0
()=(())
t
NN p
r
vrtsb sdsdt
 

and satisfies the following equation
2*
(||) =().
p
divvvb x

 
Thus ()= (||)wxvx is a supersolution of (3). Since
0w, and zero is clearly a sub-solution it follows from
standard results that (3) admits a solution b W such that
0< b Ww.
The following eigenvalue problem
21
(||) =(),
0,| |
ppN
divc xx
as x



 


R (4)
where N  R is a bounded smooth domain, and
(,(0,))cC

 for some 0< <1

. The first eigen-
value of the problem (4) will be denoted by >0

. It is
easily noted from the variational characterization of
eigenvalues that 12



 where12  are smooth
bounded domains in
N
R.
Consider :(0, )(0, )g  satisfies the following
conditions:
1
[]Gg is 1 C.
2
()1
[] < limsupt
b
gt
GtW


, where
=()
max N
x
bb
WWx

R.
The nonlinearity
f
in problem (1) is assumed to be
a real function that satisfies the following conditions:
1
[](,)
F
fxt is locally Holder continuous on


0,
N

R
and continuously differentiable in the variable
t
.
2
[](,) ()()
F
fxsbxgs for all



,0,
N
xs R
and some functions


:0,
N
bR and
:(0, )(0, )g

.
][ 3 F There is a continuous function


:0,
N
aR
and some constants >0

and 1 >

 such that






1
,,,0,
pN
fxsaxs xs



R
where 1

is the first eigenvalue of the problem (4) on
the ball (0 , 1 )B of radius one and centered at the origin
and where ()= ()cxax.
Recall that the nonlinearity (,)fxt may exhibit sin-
gularity as 0t

. We will consider the following
Dirichlet problem for a given smooth bounded domain
N
 R.
2
(||) =(),
>0,
()=0,
p
divWWbxx
Wx
Wxx

 




(5)
To establish the main theorem, from reference [2] we
give the following lemmas.
Lemma 1. (Weak Comparison Principle) Let

be a
bounded domain in


2
NNR with smooth boundary

 and :(0, )(0,)


 is continuous and nonde-
creasing. Let 1,
12
,()()
p
loc
uu WC

  satisfy
2
11
2
1222
||
()||()
p
p
uudx
udxuudxu dx

 



 



 
for all non-negative 1, ()
p
loc
W 

. Then the inequality
12
sup( ()())0
lim
x
uxu x



implies that
12 .uuin


Before Lemma 2, we give the following equation
2
(||)(,) = 0,
p
N
divuufx ux

 R (6)
Lemma 2. Suppose that (,)
f
xu is defined on 1N

R
and is locally Holder continuous (with exponent
(0,1)


) in
x
. Suppose moreover that there exist
functions


1
,
N
loc
vw C
 
R such that
2
2
(||)( ,)0
(||)( ,)0
p
p
divvvfx v
divwwfx w


 




and
() ()vx wx

and that (,)
f
xu is locally Lipschitz continuous in u
on the set






,: ,
N
x
uxvx uwxR
Then Equation (6) possesses an entire solution )(xu
satisfying




,
N
vx uxwxx R
Lemma 3. Let b satisfy 1 []B, 3 []B and
g
sati-
sfy both 1 []G, 2 []G. Then there is

1
N
vCR such

M. Z. WU ET AL. 353
that
21
(||)( )(()),
>0,()0||.
pp
divvv bxgvx
vvx asx

 

Proof. Since
g
satisfies 2 []G, we define
()
ˆ():=sup:>, >0
gs
gtst t
s



(7)
Note that ˆ()
g
t is non-increasing, positive and
1
ˆ() ()
g
tgtt

. Furthermore, by 2 []G we have
1
ˆ()< b gtW 
 for sufficiently large t.
Let
2
2ˆ
()=( ),>0
t
t
htgsdst
t (8)
Then h is 1 C, non-increasing and
ˆˆ
()()( )
2
t
gt ht g for all )(0,

t. Since h is non-
increasing, we note that 1 ()<b ht W 


 as t
for some [0,)

. Now, let us set
0
1
():=, >0
()
t
tdst
hs

 (9)
On using 1
ˆ()< b
gtW 
 in (8) for sufficiently large
>0t, we see from (9) that
()> b W  
, (10)
for a sufficiently large 1

. Let 1 =


 be the
inverse function of

.
By direct calculation, we see that
()=( ())tht

, ()>0t

, for >0t and (0) = 0

.
Furthermore
()=(())(()), >0.thth tt


By condition 3 []B, we take a solution b W of (3)
with N
R. Let us now set ():= (())
b
vxW x

for
all x. We note from (10) that



()= ()<
bb
vxWxW
 

 (11)
A simple computation shows that
v
has the desired
properties.
Indeed, on recalling 2 (||) =
p
bb
div WWb

 , we
see that
2
2
11 2
(| |)
=(1)()| |
()(||)
p
pp p
b
pp p
bb
div vv
pW
div WW





 

 


111
11
=(1)(( ())( ())||()
()
pp ppp
b
pp
phthtW bhv
bh v
 




 
111 1
ˆ ()()().
ppp p
bvg vbgv
v

 

We have used (11) in the last inequality. Since
(0) = 0

, it is clear that () 0vx as ||x.
Since 1

, observe that the solution
v
constructed
in Lemma 3 also satisfies
21
ˆ (||)( ).
pp
divvv bgv

 



 
1
10,, 0,GgC

;




201=
limup
gu
Gu


;




31=0
limup
gu
Gu
 .
Lemma 4. Assume that
g
satisfies

13
[][]GG. Then
there exist functions 1
g
, 1
g
such that








1
1
1,0,,0,igg C

;
 


 
11
1
1p
gs
iig sg s
s

, >0s and


00
1
1==
lim lim
ss
gs gs


;


1
1,iiigg are non-increasing on (0, );


1
1()= ()=0
lim lim
ss
ivgsg s
.
Proof.





1
11
00
==,
limlim 1
1
p
pp
ss
gsgs s
s
ss















1
11
==0.
limlim 1
1
p
pp
ss
gsgs s
s
ss


 





We set
 

>0 1
=sup
1
ts p
g
t
gs
t

, then we have




1,>0
1p
gt
g
ssandts
t
 

and )(sg is non-increasing on )(0,. Moreover,


0
==0.
lim lim
s
s
gsand gs



Now we can assume

10,gC. If not, we can
replace it by
 
1
2
2
=,>0
s
s
g sgtdt s
s
Obviously,
 
1,>0;
2
s
gs gs gs

 


and

M. Z. WU ET AL. 354
  
12
2
21 2
'= 22
s
s
s
g
sgsg gtdt
ss



 



  
2
21 21
)=( 0.
22 22
ss s
gsggsgs g
ss s

 
 
  
 

1
1
. .,((0,),(0,)).ie gC
1
>0
()
()=inf (1)
p
st
g
t
gs t

Observe that
1
()
0<( ),>0;
(1)
p
gt
gs s
t


and )( sg is non-increasing on )(0,. Moreover,
0
()= ()=0.
lim lim
s
s
gsandgs



Now we can assume 1(0, )gC. If not, we can
replace it by
1
1
2
()=() , >0
s
s
gs gtdts
s


Obviously,
1
(1)() (),>0gsg sgss
and
1'()=(1)()0,> 0g sgsgss
1
1
. .,((0,),(0,)).ie gC
3. Proof of Main Theorems
In this section, we prove our main results.
Theorem 1. Let
N
 R be a bounded smooth
domain that contains (0 , 1 )B, the ball of radius one
centered at the origin, and let
f
satisfies 1 []
F
, 2 []
F
and 3 []
F
, where b satisfies 1 []B, 3 []B and
g
satisfies 1
[]G, 2 []G. Then the problem
2
(||) =(,),
()=0,
p
divuufx ux
ux x

 

 (12)
has a positive solution u in 1,() ()CC


 such that
u


 where 

is an eigenfunction of the eigenvalue
problem (4) on  with ()= ()cxax normalized such
that 0<


 on . Here

is the constant in
condition 3
[]
F
.
Proof. Let W be the solution of (5) and set
1
()= ()vx W 


where

and

are defined as in
(9) and (10) respectively. Then =0v on

, and
proceeding as in the proof of Lemma 3, we note that
21
(||)( )(),.
pp
divvvb x gvx

 
  (13)
Therefore, by condition ][ 2 F, we see that
2
(||)( ,),.
p
divvvfx vx

 

 (14)
We recall, by the above, that v also satisfies
21
ˆ (||)( )(),.
pp
divvvbx gvx

 

 (15)
Let

be a smooth bounded domain that contains
(0, 1 )B the unit ball centered at the origin. Now, let


be the first eigenfunction of the problem (4) with
()= ()cxax such that 0<


, where

is the
positive constant in 3 []
F
. Invoking conditions 2 []
F
and 3 []
F
, we get
2
11
1
(| |)
=()()(,),
p
pp
div
axaxf xx

 



 
 


(16)
Moreover, since 1<0



, we also note that,
2
11
(||)( ,)
ˆ
()()()(),
p
pp
divf x
bxg bxgx
 


 


 


Therefore, we get
21
ˆ (||)()(),
pp
divb x gx  

 

  (17)
Recalling that ˆ
g
is non-increasing, by Lemma 1 we
note, from (15) and (17), that v


. Then by the
elliptic regularity theory and Lemma 2, (14) and (16), we
conclude that (12) has a solution u such that
uv




 and 1,() ()uC C



.
Let W

be as in the proof of the above Theorem.
Then we note that b
WW


, and hence vv


where
v is as in Lemma 3. Then we deduce a non-singular
case.
Theorem 2. If f satisfies 1 []
F
, (,0)=0fx, where b
satisfies 1 []B, 3 []B and
g
satisfies 1 []G, 2 []G,
then problem (1) has a solution

1, N
loc
uC

R.
Proof. For each positive integer k, let )(0,=kBBk
be the ball of radius k centered at the origin. By
Theorem 1, for each positive integer k we let
)(
1,
kkBCu

 be a solution of
2
(||) =(,),
()=0,
p
k
k
divuufx uxB
uxx B

 



 (18)
Then 0()()
kk
ux vx

 in k B, where k v are as in
Theorem 1. Corresponding to the ball k B. It is easy to
see that 0 is a subsolution.
Recall the above, k vv on k B for all 1k, and
hence 0()()
k
ux vx

 for all 2 Bx.
By a standard procedure, we conclude that {}
kl
u

has
a subsequence that converges uniformly to a function in
2, ()C 

. By a diagonalization process it follows that
{}
k
u has a subsequence that converges uniformly on
open bounded subsets of N
R to


2,
N
loc
uC

R and
that u is a solution of (1). Since 0uv, it follows

M. Z. WU ET AL. 355
that () 0ux as ||x.
In the last part of the paper, we prove a nonexistence
result for the following problem,
2
(| |)
=()[()() ||],>0,
()0,| |
p
qN
div uu
dx guruuuin
uxas x

 




R(19)
The result solves an open problem in [4] for >1p,
0q for the case when u is a radially symmetric
solution. Before the proof, we state some conditions
which we needed at the below.


 
1
10,, 0,GgC;




201=
limup
gu
Gu
;



31=0
limup
gu
Gu
 .
Theorem 3. Suppose

13
()()GG are fulfilled and
d is a positive radial function, r is a nonnegative
radial function that is continuous on N R and satisfies
,=))(( 1
1
1
0
1
0
 




ddd p
NN
then the problem (19) has no positive radial solution that
decays to zero near infinity.
Proof. Suppose (19) has such a solution )(ru . Then
2
(|()|( ))
=( )[(( ))(( ))|( )|],
p
q
divu rur
dr gurrurur

 

or, equivalently, ()ur is a solution to the problem
12 1
(||)()[ (())(())]
Np N
ruurdr gurrur
 

  (20)
Integrating (20) from 0 to
r
, we have
12 1
0
| |()[(())(())].
r
Np N
ru udguru d

 
 
 

Hence ()<0ur
; i.e., ()ur is non-increasing. We
put
  
ln1=>0urur for all >0r. Then we
have
2
12
(|()|())
11
=()(|()|())(1)||,
1(1)
p
pp p
p
divu ru r
divu ru rpu
uu






and

ur
 satisfies
22
1
1
(|' |')|'|'
1(())(())
(1)| |()
(1) (1)
pp
p
pp
N
uu uu
r
g
ur rur
pudr
uu





 

 
(21)
Multiplying Equation (21) by 1N
r and integrating
on (0,)

yields
1
12
0
1
1
0
(1)
|'|'| |
(1)
(( ))(( ))
() (1)
N
Np p
p
N
p
p
uu ud
u
gu ru
dd
u






 








 



(22)
If p is even, we can deduce that




ddu
u
p
uru p
p
p
N
N
r1
1
1
0
1
0)||
1)(
1)(
((0)
~
)(
~




 
1
11 1
1
00
(( ))(( ))
(() )
(1)
rNN p
p
gu ru
ddd
u



 

 


 

(23)
We observe that ()< (0)ur u for all >0r and
()< (0)uru

for all >0r.
Since ()ur
 is positive, then (23) implies
1
11 1
1
00
(( ))(( ))
(() )(0)
(1)
rNN p
p
gu ru
dddu
u


 
 



 
(24)
for all 0>r. Now, using Lemma 4 in (24), we have
1
11 1
1 00
1
11 1
1
00
(()(()))
(( ))
(() )
(( )1)
rNN p
rNN p
p
dgu dd
gu
ddd
u


 


 

 
 




1
11 1
1
00
(( ))(( ))
(( ))(0).
(( )1)
rNN p
p
gu ru
dddu
u


 

 




 
If p is odd, we deduce that
1
1
11
00
(1)
() (0)(||)
(1)
N
rNp
p
p
p
uruu dd
u








 



1
11 1
1
00
(( ))(( ))
(() )
(1)
rNN p
p
gu ru
ddd
u



 

 




(25)
then (25) implies
1
11 1
1
00
(( ))(( ))
(() )
(1)
()< (0)
rNN p
p
gu ru
ddd
u
ur u



 
 






(26)
for all 0>r. As the same as the above we get
1
11 1
1
00
(()(()))<(0).
rNN p
dgudd u

 
 
 
But, since 1 g is non-increasing on )(0,, we have

M. Z. WU ET AL. 356
1
11 1
1
00
1
1
(0)
=(() )<<,
((0))
rNN p
p
u
dd d
gu

 
 


 
which is a contradiction.
4. References
[1] G. Astrita and G. Marrucci, “Principles of Non-New-
tonian Fluid Mechanics,” McGraw-Hill, Rochester, 1974.
[2] L. K. Martinson and K. B. Pavlov, “Unsteady Shear
Flows of a Conducting Fluid with a Rheological Power
Law, Magnit,” Gidrodinamika, Vol. 2, 1971, pp. 50-58.
[3] J. R. Esteban and J. L. Vazquez, “On the Equation of
Turbulent Filtration in One-Dimensional Porous Media,”
Nonlinear Analysis, Vol. 10, No. 11, 1982, pp. 1303-
1325.
[4] Z. M. Guo, “Existence and Uniqueness of the Positive
Radial Solutions for a Class of Quasilinear Elliptic
Equations,” Applied Analysis, Vol. 47, No. 3, 1992, pp.
173-190.
[5] Z. M. Guo, “Some Existence and Multiplicity Results for
a Class of Quasilinear Elliptic Equatons,” Nonlinear
Analysis, Vol. 18, No. 10, 1992, pp. 957-971.
[6] Z. M. Guo and J. R. L. Webb, “Uniqueness of Positive
Solutions for Quasilinear Elliptic Equations When a
Parameter is Large,” Proceedings of Royal Society of
Edinburgh, Edinburgh, Vol. 124, No. 1, 1994, pp. 189-
198.
[7] M. G. Crandall, P. H. Rabinowitz and L. Tartar, “On a
Dirichlet Problem with a Singular Nonlinearity,” Commu-
nications in Partial Differential Equations, Vol. 2, No. 2,
1977, pp. 193-222.
[8] S. Cui, “Existence and Nonexistence of Positive
Solutions for Singular Semilinear Elliptic Boundary
Value Problems,” Nonlinear Analysis, Vol. 41, No. 1-2,
2000, pp. 149-176.
[9] A. V. Lair and A. W. Shaker, “Classical and Weak
Solutions of a Singular Semilinear Elliptic Problem,”
Journal of Mathematical Analysis and Applications, Vol.
211, No. 2, 1997, pp. 371-385.
[10] A. C. Lazer and P. J. McKenna, “On a Singular Nonlinear
Elliptic Boundary-Value Problem,” Proceedings of the
American Mathematical Society, Vol. 111, No. 3, 1991,
pp. 721-730.
[11] H. Maagli and M. Zribi, “Existence and Estimates of
Solutions for Singular Nonlinearity Elliptic Problems,”
Journal of Mathematical Analysis and Applications, Vol.
263, No. 2, 2001, pp. 522-542.
[12] J. Shi and M. Yao, “Positive Solutions for Elliptic
Equations with Singular Nonlinearity,” Electronic
Journal of Qualitative Theory of Differential Equations,
Vol. 4, 2005, pp. 1-11.
[13] H. Xue and Z. Zhang, “A Remark on Ground State
Solutions for Lane-Emden-Fowler Equations with a
Convection Term,” Electronic Journal of Differential
Equations, Vol. 2007, No. 53, 2007, pp. 1-10.
[14] H. Brezis and S. Kamin, “Sublinear Elliptic Equations in
N
R,” Manuscripta Mathematica, Vol. 74, No. 1, 1992,
pp. 87-106.
[15] S. Wu and H. Yang, “The Existence Theorems for a Class
of Sublinear Elliptic Equations in
N
R,” Acta
Mathematica Sinica, Vol. 13, No. 3, 1997, pp. 259-304.
[16] Z. Zhang, “A Remark on the Existence of Entire
Solutions of a Singular Semilinear Elliptic Problem,”
Journal of Mathematical Analysis and Applications, Vol.
215, No. 2, 1997, pp. 579-582.
[17] Z. Zhang, “A Remark on the Existence of Positive Entire
Solutions of a Sublinear Elliptic Problem,” Nonlinear
Analysis, Vol. 67, 2007, pp. 147-153.
[18] K. E. Mabrouk, “Entire Bounded Solutions for a Class of
Sublinear Elliptic Equations,” Nonlinear Analysis, Vol.
58, No. 1-2, 2004, pp. 205-218.
[19] J. V. Goncalves and C. A. Santos, “Existence and Asy-
mptotic Behavior of Non-Radially Symmetric Ground
States of Semilinear Singular Elliptic Equations,” Non-
linear Analysis, Vol. 65, No. 4, 2006, pp. 719-727.
[20] A. Mohammed, “Ground State Solutions for Singular
Semilinear Elliptic Equations,” Nonlinear Analysis, Vol.
71, No. 3-4, 2009, pp. 1276-1280.