Advances in Pure Mathematics, 2011, 1, 99-104
doi:10.4236/apm.2011.13022 Published Online May 2011 (http://www.scirp.org/journal/apm)
Copyright © 2011 SciRes. APM
Liouville Type Theor ems for Lichner owicz Equations
and Ginzburg-Landau Equation: Survey
Li Ma1
Department of Mat hematics, Henan Normal University, Xinxiang, China
Department of Mat he m at i c al Sci e nces, Tsinghua Universi ty, Beijing, China
E-mail: lma@math.tsinghua.edu.cn
Received April 11, 2011; revised May 10, 2011; accepted May 12, 2011
Abstract
In this survey paper, we firstly review some existence aspects of Lichnerowicz equation and Ginz-
burg-Landau equations. We then discuss the uniform bounds for both equations in Rn. In the last part of this
report, we consider the Liouville type theorems for Lichnerowicz equation and Ginzburg-Landau equations
in Rn via two approaches from the use of maximum principle and the monotonicity formula.
Keywords: Liouville Theorems, Ancient Solution, Ginzburg-Landau Equation, Lichenrowicz Equation
1. Introduction
This article is based on the lecture given at March 3rd,
2011 in the international conference “Recent Advances
in Nonlinear Partial Differential Equations: Part I” held
at the Chinese University of Hong Kong.
The initial-value problem of gen eral relativity consists
in the resolution of a coupled system of three linear equ-
ations and a quasilinear equation which determines the
conformity factor, on an initial Riemannian manifold

,
M
g. In the case where there are sources, the qua-
si-linear equation can be written as
735
8=0
guRuVu Quu

  (1.1)
where >0u is the unknown on the Riemannian mani-
fold

,
M
g, R is the scalar curvature, V, Q,
are functions derived from the Ricci curvature of

,
M
g. (1.1) is called the Lichnerowicz equation on

,
M
g. Let

2=2 2nn
and S be the best So-
bolev constant.
One interesting result derived from mountain pass
lemma is below.
Theorem 1.1 Assume that

,
n
M
g is compact,
3n. Consider the following Lichnerowicz equation
21 21
=u uBuAu


 
with >0A and >0
maxMB. Assume that there is a
positive function >0
and a constant

>0Cn such
that


1
222
1,>0
max n
HMM
M
Cn
AB
SB
 


Then there is a positive solution to Lichnerowicz equa-
tion.
Hebey-Pacard-Pollack [11] have applied the mountain
pass lemma to the perturbation functionals to get positive
approximation solutions and have proved the conver-
gence of a subsequence to a positive solution.
The Ginzburg-Landau (GL) model is proposed in 50’s
in the context of super-co nductivity th eory an d its ener gy
density is


2
22
11
=1.
24
eu uu 
Here :nk
uR R. The stationary E-L equation for GL
model is
2
=1.uuu
Yanyan Li and Z. C. Han (1995) have also studied
p-laplacian GL models. Other models with term
2
2
1m
u, 1m, is also of interesting.
The Lichnerowicz equation and the Ginzburg-Landau
equations are important models in mathematical physics.
The existence of solutions to both can be obtained via
variational methods, the monotone method (also called
sub-super solution method or barrier method), and per-
1The research is partially supported by the National Natural Science
Foundation of China 10631020 and SRFDP 2009 0 0 02110019.
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turbation methods.
There are huge literatures about G-L models [2,5,
10,14,15,20], but there is not much works about the
Lichnerowicz equation [4,12,18]. For the existence re-
sults of the Lichnerowicz equation, one may see the
works [7], [8], and [11]. For the existence results of the
Ginzburg-Landau equation, one may find more refer-
ences from the work [22].
Our main topic for both equations is about the Liou-
ville type results, which are closely related to compact-
ness theorems of the solution spaces. The heat flow me-
thod to both equations will be an interesting topic for
studying.
2. Classical Liouville Type The or e m s a n d
Keller-Osserman Theory
The famous Liouville theory says that any non-negative
harmonic function is constant. One can prove this by
Harnack inequality or differential gradient Harnack es-
timate.
People may extend this to non-negative p-harmonic
function or to non-negative solution to the elliptic equa-
tion
,
==0,in.
n
ij ij
ij
LuauR
We now recall the famous Keller-Osserman theory
obtained around 1957. Given a domain . Consider the
differential inequality

,in n
ufuR
where

t is a positive, continuous, and monotone
increasing function for 0
tt satisfying the Osgood
condition

12
00 dd<.
tfs st

Then any twice continuously differentiable function
u can not satisfy >0u on the whole space and

ufu outside of some ball.
As an application of above theory, J. B. Keller and R.
Osserman consider the non-existence result for the
Gaussian curvature equation on the plane

2
=,in.
u
uKxe R
Professor. Ni, Lin, W. Ding, W. Chen and C. Li, etc.,
have obtained a lot interesting existence results to this
problem. More results and references may be found in
my work [17].
It is also interesting to study the non-existence of non-
trivial non-negative solutions or energy solutions to

1
=,in.
pn
uKxu uR

where 2n and >1p.
Another application of the Keller and Osserman theory
(H. Brezis, 1984) we have that for
2
0n
vCR sat-
isfying
,>1,in
pn
vvpR
we have =0v.
In fact, H. Brezis getting upper bound of solutions by
using the following boundary blow-up super-solution on
the ball
R
Bp,


2
2
=
RCR
ux Rxp

with
=12p
and a suitable constant C inde-
pendent of >0R. Then for any fixed point n
pR,
sending R, we get =0v.
If the Laplacian is replaced by p-Laplacian, Du and
Guo (2002) can ex tend the Keller and Osserman theorem
to this case. For more, one may see the works of A. Fa-
rina, A. Ratto, M. Rigoli.
Parallel result to Keller and Osserman can be done for
non-negative ancient solutions to the parabolic inequal-
ity:

,>1,in ,0.
pn
tvvp R 
See also the interesting work of J. Serrin [24] for more
Liouville type theorems about elliptic and parabolic equ-
ations.
We have the following Liouville type Theorem.
Theorem 2.1 (L. Ma, 2010 [16]) Let >0u in (1.1)
with =n
M
R, =0Q, =1V, =1
. Then =1u.
For positive solutions to the general equation
2
=,
qq
uu u


on n
R with >1q, we have the same result.
However, H. Brezis [3] proves that for
0,1q, the
same result is not true, but we always have 1u.
Similar result is also true for
M
being a complete
Riemannian manifold
,
M
g with its Ricci curvature
bounded from blow.
Here is the argument of Theorem 2.1. Let
f
s
2
=qq
uu
for >0q. For any fixed n
x
R and
>0
, consider the new function
 
2
=.uy uyyx

Note that
uyu y
 as y. Then the
minimum of it can be achieved at some point z. Then,

=,uz ux ux

which implies that

uz ux.
Using the monotonicity of f, i .e. 0f, we have


f
uz fux
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At this point z, we have

220Du z

,
 

0=2= 2.uzuzn fuzn
 
Then we have




02 2fuzn fuxnfux


as 0
.
Recall that

=qp
fuuu
or some >0q and >0p. Then we have 1u.
Assume now that >1q. =10vu . Then

=1.
q
vfvv
Using the Keller-Osserman theory we then conclude that
=0,..=1.vieu
This completes the argument of Theorem 2.1.
We now give som e remark.
Set

3
=fuu u for 0u, which is the special
case of Ginzurg-Landau equation. We then conclude
from the argument above that =1u.
In general, we know the below.
Theorem 2.2 (Du, Ma [9], 2001) Assume that

2,
n
uCRR such that
3
=uuu
on n
R. Then we have 1u.
In the papers of Du and Ma (2001-2003), more general
logistic equations have been studied. It is there that we
are interested in a problem related to Di Giorgi conjec-
ture, which has been completely solved by Del Pino, J.
Wei, etc., Savin, C. Gui, Ambrosio and Cabre, etc.
Interestingly, Du and Guo can obtaine the below.
Theorem 2.3 (Du, Guo, 2002) Assume that

2,
n
uCRR such that
3
=,1<,
puu up
on n
R. Then we have 1u.
There is a similar De Giorgi conjecture related to the
equation above (see the work of L. Caffarelli, etc., A
Gradient Bound for Entire Solutions of Quasi-Linear
Equations and Its Consequences, Communications on
Pure and Applied Mathematics, Volume 47, Issue 11,
November 1994, Pages: 1457-1473 ).
It would be interesting to study the following evolu-
tion equation
3=0, in(0,)
tp
uuuuT 
with suitable initial and boundary conditions.
The results above can be extended to nonlinear heat
equations.
Set

=qp
fuu u
for some >0q and >0p.
Consider ancient solutions to the following parabolic
equation

=,>0,in ,0.
n
tufuu R  (2.1)
For any fixed

,,0
n
xR
 and >0
, con-
sider the new function
 

22
,= ,
,,
n
uytuyyxt
yt R

 

Note that
,uyt
 as yt
. Then the
minimum of it can be achieved at some point
,zt .
Then at this point
0
tu
 , which implies that


22
0,2=,2,
tuzt nfuztn

 
which is less than


2
,2 ,fux nfux
 

as 0
. Hence 1u (and one may also show that
=1u).
Theorem 2.4 Let >0u be an ancient solution to
(2.1). Then 1u.
3. Results for Ginzburg-Landau Equations
Our uniform bound result (due to H. Brezis) is
Theorem 3.1 Any smooth solution to GL model is
bounded in the sense that

1ux.
H. Brezis uses the Kato inequality to prove Theorem
3.1. We shall report here his argument. My argument is
different but it is also based on the maximum principle.
Before I give the proof, let’s recall the famous Kato
inequality. Assume that 1
loc
uL and 1
loc
uL . Firstly
we may assume that u is smooth. Note that for >0
,
we have
22
22
3
=.
()
u
uuu
uu

 

The latter is bigger than 2.
uu
u
Hence we have for any 2
0
0C
 we have
2
2.
u
uu
u

 

Sending 0
, we then get

sign ,uuu



i.e.,

sign .uuu

We have the parabolic version of the Kato inequality.
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Consider

=,uuxt with in addition 1
tloc
uL. Using
2
2
=,
t
t
uu
uu

we have
 
2
2.
tt
u
uuu
u
 
Then, letting 0
, we have


||sign .
tt
uuuu 
With the understanding above, we have that for any
non-negative ancient solution to

,>1,in ,0
pn
t
vvvpR 
is trivial.
The argument of this fact is almost the same as Bre-
zis’s argument (1984).
Therefore, we have the following extension of H. Bre-
zis’ theorem.
Theorem 3.2 Any ancient solution
:,
n
uR

0k
R to
2
=1
t
uu uu
must have 1u.
Going back to the parabolic version of Lichnerowicz
equation on manifold with non-negative Ricci curvature,
we have the following result.
Proposition 3.1 Any positive ancient solution to
=qp
t
uuuu

with >1q and >0p must be =1u.
We now give the proof o f T h eorem 3.1.
Brezis’s argument: Let

2
=1Wu
. Then we
have

22
sign1 .Wu u

Note that

2222
=2 221.uuu uuu 
Then we have


22 22
21sign12 12.WuuuWW W

Using the Keller-Osserman theory we then have
2
=0,..1.Wieu
In my proof of this bound, I have used the original
barrier functions use d by Kel ler -O sserman.
It is interesting to know if one can extend the result
above to p-Laplacian G-L model.
In the following, we present the monotnonicity for-
mula method to the Liouville type theorems [1,10,19].
It is our intention here to generalize the Liouville type
results to a large class of solutions of a more general
non-linear equations/systems

2
=0,in,,,
nnk
uWuRuC RR
 (3.1)
where W
is the gradient of the smooth function
Wu on k
R and
0Wu.
We shall use an idea from Professor Hesheng Hu
(1980) who introduced it for harmonic maps. Please see
the book of Y. L. Xin [25] for results of harmonic maps.
It is also interesting to know if one can extend this
kind of result to p-laplacian case.
Theorem 3.3 Assume that

0Wu is a non-trivial
smooth function. Let :nk
uR R, 2n, be a smooth
solution to the Ginzburg-Landau system (3.1). Assume
that there are a positive constant 0>0R and a positive
function
r on
0,R
such that
 
0
22<.
lim BB
RRR
ru Wu

 
and
0d= .
R
rr
r
Then u is a constant.
For any smooth mapping from the Riemannian mani-
fold
,
M
g to k
R, we define the Stress-Energy tensor
(see the paper of Baird-Eells) by
2
1
=dd,
2
u
Suguu
which is
1
=2
ijjj ikik
Suuguu

in local frames
j
e on
M
. By direct computation we
know that
div=,.
ug
Sudu
A consequence of this formula is that if u is harmonic,
then
div 0
u
S
.
Let
X
be a smooth vector field on
M
. Define the
tensor
X
by

,= ,.
ijej
i
X
ee g Xe
Then we hav e



2
1div=div,
2
,,
jj
u
uX duXue
du XuSX

Take any compact domain DM with smooth
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boundary D. Choose the local frame
j
e such that
=
n
e
be the unit outward normal to the boundary.
Then we hav e


2
1,d,
2
=div, .
D
uu
D
uX uXu
SXS X


(3.2)
Below we let =n
M
R.
To explain the main idea of the proof, we start with
the simple case when
=0Wu and

=1r. That is,
=0u and u has finite energy. We take in (3.2)

=0
R
DB , =r
X
r.
Note that =
ij ij
X
. Then we have
2
2
,= ==,
2
uijiju
n
SXSXtrS u

,=
X
R
and

2
d,=
r
uX uRu
on
0
R
B.
Hence

2
0
2
,=
2
u
BR
n
SX u


(3.3)
and by (3.2), we have
222
12
=.
22
r
BB
RR
n
Ruu u
 

Then we have the following Liouville theorem for
harmonic functions.
Theorem 3.4 Let :n
uR R be a harmonic fun ction
with slowly energy divergence, i.e., there are a positive
constant 0>0R and a positive function
r on
0,R such that

2
0
<.
lim BB
RRR
ru
 
and

0d= .
R
rr
r
Then u is a constant.
Here is the idea of proof:
Assume that u is not a constant. Then there are some
positive constants >0C and 0>0R such that

0
2
0
22>0.
2
BR
nuC

By this we know that
2
BR
RuC

(3.4)
for any 0
RR. Hence we have
00
0
22
d,
RR
BB RBR
RR R
r
uuC
r

 

as R, which gives a contradiction.
In fact, the proof of Theorem 3.4 goes below. As sume
u is not a constant. By (3.4) we know that
 

22
0
0
0d,
R
BBR B
RR r
R
R
rur u
r
Cr
r

 


as R, which gives a contradiction. This completes
the proof of Theorem 3.4.
We now turn to the proof of Theorem 3.3. Assume
that
Wu is non-trivial. In this case

div=, =,
u
Su duWudu
 
and
div=,=.
uXX
SXWu uWu

Again we take
=0
R
DB . Then by (3.2) we have



2
1,,
2
=,.
D
Xu
D
uX duXu
WuSX

 
(3.5)
Simplifying this identity we can derive th e following

2
1.
2
BR
RuWuC




Then using the argument above we obtain Theorem 3.3.
We remark that the results above can be extended to
complete Riemannian manif olds with bounded Ricci cu r-
vature.
We remark that for a large class of elliptic equations/
systems (also for parabolic equation/system), the Liou-
ville type theorems are equivalent to a local uniform
bound of solutions. For this direction, one may see the
recent works of P. Polacik, P. Souplet, P. Quittner, H.
Zou, etc. However, it is open for Lichnerowicz equation
on compact Riemannian manifolds (see also in [13]).
We also make a remark below. In the study of elliptic
systems, one may use the Pohozaev type identity (which
is a sister of monotonicity formula) and the interpolation
inequalities to derive a contraction mapping property
about
p
L norm of the solution. From the contraction
mapping property one then get the solution trivial (and
the Liouville type theorem). One may see the works of
Chen-Li [6], Souplet, etc. [21,23], for th is kind of results
for the Lane-Emden conjecture.
We would like to thank Professor H. Brezis, who (see
also [3]) has informed me that in the statements of
Theorems 1 and 2 in [16], the power >1p should be
>2p. Actually, we have used >2p in the proof of
Theorem 1.1 in [16].
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