A Liouville-type theorem for higher order elliptic
systems
Mingfeng Zhao
University of Connecticut
February 21, 2014
Joint work with Frank Arthur and Xiaodong Yan
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
1 / 27
Lane-Emden Equation
(
−∆u = u p
u>0
Mingfeng Zhao(UConn)
in RN .
A Liouville-type theorem for higher order elliptic systems
(LE)
2 / 27
Lane-Emden Equation
(
−∆u = u p
u>0
in RN .
(LE)
Theorem (Gidas-Spruck[1981])
There is NO solution to (LE) if p is sub-critical, i.e.,

if n ≤ 2,
 ∞,
0 < p < pS :=
 n + 2 , if n ≥ 3.
n−2
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
2 / 27
Lane-Emden Equation
(
−∆u = u p
u>0
in RN .
(LE)
Theorem (Gidas-Spruck[1981])
There is NO solution to (LE) if p is sub-critical, i.e.,

if n ≤ 2,
 ∞,
0 < p < pS :=
 n + 2 , if n ≥ 3.
n−2
Remark
If p ≥ pS , the equation (LES) has bounded radial solutions.
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
2 / 27
Critical Case
When N ≥ 3 and p = pS , Caffarelli-Gidas-Spruck[1989] classified
all solutions to (LE) that all of them must be radial symmetric and
can be written as
u(x) =
[N(N − 2)λ2 ]
N−2
4
[λ2 + |x − x 0 |2 ]
N−2
2
,
where λ > 0 and x 0 ∈ RN . Chen-Li[1991] simplified the proof by
using a Kelvin transform and the method of moving planes.
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
3 / 27
Lane-Emden System

 −∆u = v p
−∆v = u q

u > 0, v > 0
Mingfeng Zhao(UConn)
in RN .
A Liouville-type theorem for higher order elliptic systems
(LES)
4 / 27
Lane-Emden System

 −∆u = v p
−∆v = u q

u > 0, v > 0
in RN .
(LES)
Remark
Recall that any super-harmonic function which is bounded below in
R2 is a constant function. Therefore, there is no solutions for the
system (LES) if N ≤ 2.
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
4 / 27
Lane-Emden System

 −∆u = v p
−∆v = u q

u > 0, v > 0
in RN .
(LES)
Remark
Recall that any super-harmonic function which is bounded below in
R2 is a constant function. Therefore, there is no solutions for the
system (LES) if N ≤ 2.
Conjecture (Lane-Emden Conjecture)
Let p, q > 0, if the pair (p, q) is sub-critical, i.e., if
1
1
2
+
>1− ,
p+1 q+1
N
then the system (LES) has NO solutions.
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
4 / 27
Radial Case
If we assume that both u and v in (LES) are radial symmetric, the
Lane-Emden conjecture is completely solved.
Theorem (Mitidieri[1996], Serrin-Zou[1994, 1998])
The Lane-Emden system (LES) has radial solutions if and only if
the pair (p, q) is critical or super-critical, i.e.,
1
1
2
+
≤1− ,
p+1 q+1
N
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
5 / 27
Higher Order Lane-Emden System
For any p, q > 0, m, N ∈ N and N ≥ 2, a higher order Lane-Emden
system is given by:

 (−∆)m u = v p
(−∆)m v = u q
in RN .
(HLES)

u > 0, v > 0
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
6 / 27
Higher Order Lane-Emden System
For any p, q > 0, m, N ∈ N and N ≥ 2, a higher order Lane-Emden
system is given by:

 (−∆)m u = v p
(−∆)m v = u q
in RN .
(HLES)

u > 0, v > 0
Conjecture (Higher Order Lane-Emden Conjecture)
Let p, q > 0, if the pair (p, q) is sub-critical, i.e., if
1
1
2m
+
>1−
,
p+1 q+1
N
then the system (HLES) has NO solutions.
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
6 / 27
Radial Case
Theorem (Liu-Guo-Zhang[2006], Yan[2012])
The high order Lane-Emden system (HLES) has NO radial
solutions if N > 2m, p, q ≥ 1 and the pair (p, q) is sub-critical,
i.e.,
1
1
2m
+
>1−
,
p+1 q+1
N
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
7 / 27
In the following, if pq > 1, let
α=
2m(p + 1)
,
pq − 1
β=
2m(q + 1)
.
pq − 1
Then the conjecture can be written in the following way:
Conjecture
Let p, q > 0, the system (HLES) has NO solutions whenever
pq ≤ 1,
Mingfeng Zhao(UConn)
or
pq > 1 and α + β > N − 2m.
A Liouville-type theorem for higher order elliptic systems
8 / 27
General Solutions for LES
The Lane-Emden system (LES), i.e., m = 1 in (HLES):
1
1
1
I
+
>1−
, Souto[1992]
p+1 q+1
N −1
I pq ≤ 1, Serrin-Zou[1996]
N +2 N +2
N +2
I 0 < p, q ≤
and (p, q) 6=
,
, Felmer-de
N −2
N −2 N −2
Figureiredo[1994]
I
I
pq > 1, α+β > N −2 and max{α, β} > N −3, Mitidieri[1996],
Serrin-Zou[1996], Souplet[2009]
N −2
pq > 1, α+β > N−2 and
≤ min{α, β} ≤ max{α, β} <
2
N − 2, Busca-Manasevich[2002]
I
min{p, q} = 1, Lin[1998], Xu[2000]
I
N = 3, Polacik-Quittner-Souplet[2007]
I
N = 4, Souplet[2009]
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
9 / 27
General Solutions for HLES
The Higher Order Lane-Emden system (HLES):
I
pq > 1 and N ≤ 2m, Yan[2012]
I
pq ≤ 1, N > 2m and (−∆)k u, (−∆)k v > 0 for k = 1, · · · , m−
1, Yan[2012]
p, q ≥ 1, (p, q) 6= (1, 1) and max{α, β} ≥ N − 2m, Mitidieri[1993], Liu-Guo-Zhang[2006], Yan[2012]
N + 2m
, (p, q) 6= (1, 1) and (p, q) 6=
1 ≤ p, q ≤
N − 2m
N+2m N+2m
N−2m , N−2m , Liu-Guo-Zhang[2006]
I
I
I
N + 2m
, all solutions to (HLES) must be radial
N − 2m
symmetric, Liu-Guo-Zhang[2006]
p = q =
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
10 / 27
General Solutions for HLES
The Higher Order Lane-Emden system (HLES):
I
pq > 1 and N ≤ 2m, Yan[2012]
I
pq ≤ 1, N > 2m and (−∆)k u, (−∆)k v > 0 for k = 1, · · · , m−
1, Yan[2012]
p, q ≥ 1, (p, q) 6= (1, 1) and max{α, β} ≥ N − 2m, Mitidieri[1993], Liu-Guo-Zhang[2006], Yan[2012]
N + 2m
, (p, q) 6= (1, 1) and (p, q) 6=
1 ≤ p, q ≤
N − 2m
N+2m N+2m
N−2m , N−2m , Liu-Guo-Zhang[2006]
I
I
I
N + 2m
, all solutions to (HLES) must be radial
N − 2m
symmetric, Liu-Guo-Zhang[2006]
p = q =
Remark: The condition p, q ≥ 1 can be used to prove that
(−∆)k u, (−∆)k v > 0 for k = 1, · · · , m − 1, Wei-Xu[1999],
Yan[2012]
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
10 / 27
Main Theorem
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
11 / 27
Main Theorem
Theorem (Arthur-Yan-Zhao[2013])
Assume N > 2m, p, q ≥ 1 and (p, q) 6= (1, 1) satisfies
1
2m
1
+
>1−
.
p+1 q+1
N
If N ≤ 2m + 2 or N ≥ 2m + 3 and (p, q) satisfies
2m(p + 1) 2m(q + 1)
max
,
> N − 2m − 1,
pq − 1
pq − 1
Then the system (HLES) has NO solutions.
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
12 / 27
Some Notations
I
kw kLp denotes the Lp norm on S N−1 .
I
uk = (−∆)k u and vk = (−∆)k v , for all k = 1, · · · , m − 1.
Z
m
X
N
G1 (R) =
R
|uk ||vm−k |dσ.
I
S N−1
k=0
I
G2 (R) =
m
X
Z
R
I
F (R) =
BR
Mingfeng Zhao(UConn)
0
0
|uk |+R −1 |uk | |vm−k−1
|+R −1 |vm−k−1 | dσ.
S N−1
k=0
Z
N
v p+1
u q+1
+
dx.
p+1 q+1
A Liouville-type theorem for higher order elliptic systems
13 / 27
Outline of Proofs
Eventually, we will prove that there exist constants C , a > 0, b < 1
such that
F (R) ≤ CR −a F b (4R), R ≥ 1.
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
14 / 27
Rellich-Pohozaev Identity
For any
> 0, we have Z
a1 + a2 =NZ − 2m and R N
N
p+1
− a1
v
dx +
− a2
u q+1 dx
p+1
q
+
1
BR
BR
p+1
Z
Z
m−2
q+1
X
u
v
+
dσ − R N
= RN
uk+1 vm−k−1 dσ
q+1
N−1
S
S N−1 p + 1
k=0
(
m−1
XZ
∂uk ∂vm−k−1
N
− R
2
·
− ∇uk · ∇vm−k−1 dσ
∂n
∂n
N−1
k=0 S
Z
m−1
X
∂uk
N−1
+R
(2m − 2k − 2 + a1 )
vm−k−1
dσ
∂n
N−1
S
k=0
)
Z
m−1
X
∂v
m−k−1
+R N−1
(a2 + 2k)
uk
dσ .
∂n
N−1
S
k=0
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
15 / 27
Rellich-Pohozaev Identity
For any
> 0, we have Z
a1 + a2 =NZ − 2m and R N
N
p+1
− a1
v
dx +
− a2
u q+1 dx
p+1
q
+
1
BR
BR
p+1
Z
Z
m−2
q+1
X
u
v
+
dσ − R N
= RN
uk+1 vm−k−1 dσ
q+1
N−1
S
S N−1 p + 1
k=0
(
m−1
XZ
∂uk ∂vm−k−1
N
− R
2
·
− ∇uk · ∇vm−k−1 dσ
∂n
∂n
N−1
k=0 S
Z
m−1
X
∂uk
N−1
+R
(2m − 2k − 2 + a1 )
vm−k−1
dσ
∂n
N−1
S
k=0
)
Z
m−1
X
∂v
m−k−1
+R N−1
(a2 + 2k)
uk
dσ .
∂n
N−1
S
k=0
Hence we have
F (R) ≤ C [G1 (R) + G2 (R)],
Mingfeng Zhao(UConn)
∀R > 0.
A Liouville-type theorem for higher order elliptic systems
15 / 27
Reduce to the Bounded Case
By using a similar argument to Polaciak-Quittner-Souplet[2007] ,
we can reduce our problem to the case of bounded solutions. That
is, we can prove that assume (HLES) does not admit any bounded
solutions in Rn , then it does not admit any solutions in Rn .
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
16 / 27
Estimates of G1 (R) and G2 (R)
Use the condition N ≥ 2m + 1, the Holder and Sobolev inequalities
to estimate G1 (R) and G2 (R), in terms of Dx2k u and Dx2k v , k =
1, · · · , m.
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
17 / 27
Estimates of G1 (R) and G2 (R)
Use the condition N ≥ 2m + 1, the Holder and Sobolev inequalities
to estimate G1 (R) and G2 (R), in terms of Dx2k u and Dx2k v , k =
1, · · · , m.
Lemma (Sobolev Inequalities on S N−1 )
Let N ≥ 2, k ≥ 1 and 1 < µ < λ ≤ ∞, we have
i
h
kw kLλ ≤ C kDθk w kLµ + kw kL1 ,
where

1
1
N −1

 (N − 1)
−
, if µ <
,
µ λ
k
λ=
N −1

 ∞,
if µ >
.
k
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
17 / 27
Estimate of G1 (R)
There exist constants C , > 0 and ν1k , ν2k ∈ [0, 1] such that
G1 (R) ≤ CR
N+2m
m nh
X
kDx2m−2k uk kL1+ + R −2m+2k kuk kL1
iν1k
k=0
i1−ν1k
· kDx2m−2k uk k p+1 + R −2m+2k kuk kL1
L p
h
iν2k
· kDx2k vm−k kL1+ + R −2k kvm−k kL1
h
i1−ν2k 2k
−2k
· kDx vm−k k q+1 + R
kvm−k kL1
,
h
L
Mingfeng Zhao(UConn)
q
A Liouville-type theorem for higher order elliptic systems
∀R > 0.
18 / 27
Estimate of G2 (R)
There exist constants C , > 0 and τ1k , τ2k ∈ [0, 1] such that
m−1
X nh
G2 (R) ≤ CR N+2m
kDx2k+1 Dx um−k−1 kL1+
k=1
+R
−2k−1
kDx um−k−1 kL1 + R −2k−2 kum−k−1 kL1
h
· kDx2k+1 Dx um−k−1 k
L
p+1
p
iτ1k
+ R −2k−1 kDx um−k−1 kL1
i1−τ1k
+R −2k−2 kum−k−1 kL1
h
· kDx2m−2k−1 Dx vk kL1+ + R −2m+2k+1 kDx vk kL1
iτ2k
+R −2m+2k kvk kL1
h
· kDx2m−2k−1 Dx vk k q+1 + R −2m+2k+1 kDx vk kL1
L q
i1−τ2k +R −2m+2k kvk kL1
.
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
19 / 27
Estimates of Norms
For k = 1, · · · , m − 1, γ = 0, 1 and R > 1, we have
Z R
I
kukqLq r N−1 dr ≤ CR N−2m−β
0
Z R
I
kv kpLp r N−1 dr ≤ CR N−2m−α
0
Z R
I
kDxγ uk kL1 r N−1 dr ≤ CR N−α−2k−γ
0
Z R
I
kDxγ vk kL1 r N−1 dr ≤ CR N−β−2k−γ
0
Z R
I
kDx2m uk1+
r N−1 dr ≤ CR N−2m−α
L1+
0
Z R
I
kDx2m v k1+
r N−1 dr ≤ CR N−2m−β
L1+
0
Z R
q+1
p+1
p
q
2m
2m
I
kDx uk p+1 + kDx v k q+1 r N−1 dr ≤ CF (2R)
0
Mingfeng Zhao(UConn)
L
p
L
q
A Liouville-type theorem for higher order elliptic systems
20 / 27
Estimates of Norms
The above estimates follow directly from the following three lemmas:
Lemma (Lemma 3.3 in Yan[2012])
For any k = 0, · · · , m − 1, r > 0, we have
uk (r ) ≤ Mr −α−2k ,
and vk (r ) ≤ Mr −β−2k ,
where w denotes the spherical average of the function w .
Lemma (Interpolation Inequalities)
For any smooth function w and R > 0, we have
Z
Z
Z
−1
|∇w |dx ≤ C R
|∆w |dx + R
BR
B2R
|w |dx .
B2R
Lemma (Interior Lp Estimates)
For any smooth function w , k > 1 and R > 0, we have
Z
Z
Z
2m
k
m
k
−2mk
k
|D w | dx ≤ C
|∆ w | dx + R
|w | dx .
BR
Mingfeng Zhao(UConn)
B2R
B2R
A Liouville-type theorem for higher order elliptic systems
21 / 27
Measure and Feedback Argument
For k = 1, · · · , m − 1 and larger K 1, we define
n
o
q
−qα
I Γ1
0 (R) = r ∈ (R, 2R) : ku(r )kLq > KR
n
o
p
−pβ
I Γ2
(R)
=
r
∈
(R,
2R)
:
kv
(r
)k
>
KR
0
Lp
p+1
I Γ1 (R) = r ∈ (R, 2R) : kDx2m u(r )k pp+1 > KR −N F (4R)
L p
q+1
I Γ2 (R) = r ∈ (R, 2R) : kDx2m u(r )k qq+1 > KR −N F (4R)
L
I
I
q
o
n
−pβ
Γ3 (R) = r ∈ (R, 2R) : kDx2m u(r )k1+
>
KR
L1+
Γ4 (R) = r ∈ (R, 2R) : kDx2m v (r )k1+
> KR −qα
L1+
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
22 / 27
Measure and Feedback Argument
I
n
o
Γ5k (R) = r ∈ (R, 2R) : kuk (r )kL1 > KR −α−2k
I
n
o
−β−2k
Γ6k (R) = r ∈ (R, 2R) : kvk (r )kL1 > KR
I
n
o
Γ7k (R) = r ∈ (R, 2R) : kDx uk (r )kL1 > KR −α−2k−1
I
n
o
Γ8k (R) = r ∈ (R, 2R) : kDx vk (r )kL1 > KR −β−2k−1
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
23 / 27
Measure and Feedback Argument
By the estimates of norms, we have
Z 2R
CR N−2m−β ≥
kukqLq r N−1 dr > K |Γ10 (R)|R −qα R N−1 .
R
Since 2m + β = qα, then
|Γ10 (R)| <
R
,
4m + 2
∀K 1.
Similarly, we can prove that when K 1, all measures of the above
R
sets are less than
. In particular, we have there exists some
4m + 2
R ∈ (R, 2R) such that
!
!
!
2
4
8
[
[ [
[ m−1
[ [
R∈
/
Γi0 (R)
Γi (R)
Γik (R) .
i=1
Mingfeng Zhao(UConn)
i=1
k=1 i=5
A Liouville-type theorem for higher order elliptic systems
24 / 27
Measure and Feedback Argument
By the estimates of G1 (R) and G2 (R), we have
Gi (R) ≤ CR −ai F bi (4R),
for i = 1, 2,
where
qαν2k
Np
pβν1k
+
+
(1 − ν1k )
−N − 2m +
k=1,··· ,m
1+
1+
p+1
Nq
+
(1 − ν2k )
q+1
p
q
b1 = max
(1 − ν1k +
(1 − ν2k )
k=1,··· ,m p + 1
q+1
pβτ1k
qατ2k
Np
a2 = min
−N − 2m +
+
+
(1 − ν1k )
k=1,··· ,m
1+
1+
p+1
p
q
b2 = max
(1 − τ1k +
(1 − τ2k ) .
k=1,··· ,m p + 1
q+1
a1 =
min
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
25 / 27
Measure and Feedback Argument
Take a = min{a1 , a2 } and b = max{b1 , b2 }, use the conditions
N ≤ 2m + 2 or N ≥ 2m + 3 and max{α, β} > N − 2m − 1, we can
deduce that a > 0 and b < 1. Then
F (R) ≤ CR −a F b (4R),
∀R ≥ 1.
Since u, v ∈ L∞ , then there exist constants M > 0 and a sequence
{Ri } such that lim Ri = ∞ and
i→∞
F (4Ri ) ≤ MF (Ri ),
for i = 1, 2, · · · .
Hence
F (Ri ) ≤ CRi−a F b (Ri ),
for i = 1, 2, · · · .
By taking i → ∞, we have
Z
p+1
v
+ u q+1 dx = 0.
Rn
Therefore, u ≡ v ≡ 0 in Rn .
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
26 / 27
Thank You!
Mingfeng Zhao(UConn)
A Liouville-type theorem for higher order elliptic systems
27 / 27