RADIAL SYMMETRY AND MONOTONICITY FOR AN
INTEGRAL EQUATION
LI MA AND DEZHONG CHEN
Abstract. In this paper, we study radial symmetry and monotonicity
of positive solutions of an integral equation arising from some higherorder semilinear elliptic equations in the whole space Rn . Instead of the
usual method of moving planes, we use a new Hardy-Littlewood-Sobolev
(HLS) type inequality for the Bessel potentials to establish the radial
symmetry and monotonicity results.
1. Introduction
The radial symmetry and monotonicity of nonnegative solutions of
elliptic equations in the unit ball or the whole Euclidean space have
been studied by Gidas, Ni and Nirenberg using the technique of moving
planes ([7], [8]). However, their method of moving planes is based on
the maximum principle, and hence one is not able to apply it in the
absence of the maximum principle. Recently, it was first observed by
Chen, Li and Ou ([4], [5], [12]) that instead of the maximum principle,
one can use a Hardy-Littlewood-Sobolev (HLS) type inequality to obtain the radial symmetry and monotonicity of nonnegative solutions of
(systems of) elliptic equations of Yamabe type.
Motivated by the works mentioned above, we consider in this paper the positive solutions of the following semilinear partial differential
equation in Rn :
α
(I − ∆) 2 (u) = uβ ,
P
∂2
where α > 0, β > 1, and ∆ = ni=1 ∂x
2 denotes the usual Laplace opi
n
erator in R . We note that under some appropriate decay assumptions
of the solutions at infinity, (1) is equivalent to the following integral
(1)
Date: Aug 3rd, 2006.
1991 Mathematics Subject Classification. 35Q40, 35Q55.
Key words and phrases. Bessel potential, radial symmetry, monotonicity.
The research is partially supported by the National Natural Science Foundation of
China 10631020 and SRFDP 20060003002. The first named author would like to thank
Prof. W.Chen for a helpful discussion in the summer of 2004 in Tsinghua University,
Beijing.
1
2
LI MA AND DEZHONG CHEN
equation
u = gα ∗ uβ ,
(2)
where ∗ denotes the convolution of functions and gα is the Bessel kernel
whose precise definition will be given in section 2.
An important case of (1) corresponding to α = 2 appears in the
study of standing waves of the nonlinear Klein-Gordon equations
−∆u + u = u|u|β−1 ,
(3)
in Rn .
Such a kind of equation also arises from the ground states of the
Schrödinger equation ([2]). Here we just want to mention that it has
been shown that under some decay assumptions at the infinity, the
smooth positive solutions of (3) are unique and radially symmetric.
For more details, we refer to the paper [9] and the references therein.
Our main theorem is
Theorem 1. For any q > max{β, n(β−1)
}, if u ∈ Lq (Rn ) is a positive
α
solution of (2), then u must be radially symmetric and monotonely
decrease about some point.
The proof of the main theorem is based on the method of moving
planes introduced by Chen, Li and Ou. Here the new ingredient is a
HLS type inequality for the Bessel potentials, which is established in
Theorem 9.
2. Preliminaries on Bessel Potentials
In this section, we recall some basic properties of the Bessel potentials. For more details, one may refer to [14], [17].
Definition 2. The Bessel kernel gα , α ≥ 0, is given by
Z ∞
1
π
δ α−n
gα (x) =
exp(− |x|2 ) exp(− )δ 2 −1 dδ,
γ(α) 0
δ
4π
α
where γ(α) = (4π) 2 Γ( α2 ).
Here are some elementary facts about the Bessel kernel gα .
Proposition 3. (1) The Fourier transform of gα is given by
1
gbα (x) =
α ,
(1 + 4π 2 |x|2 ) 2
where the Fourier transform of f is defined by
Z
fb(x) =
f (t) exp(−2πixt)dt.
Rn
(2) For each α > 0, gα (x) ∈ L1 (Rn ).
RADIAL SYMMETRY
3
(3) The following Bessel composition formula holds
(4)
gα ∗ gβ = gα+β ,
α, β ≥ 0.
Definition 4. The Bessel potentials Bα (f ), f ∈ Lp (Rn ) with 1 ≤ p ≤
∞, is defined by
gα ∗ f, α > 0,
Bα (f ) =
f,
α = 0.
For later use, we mention the following two properties of the Bessel
potentials.
Proposition 5. (1) kBα (f )kLp (Rn ) ≤ kf kLp (Rn ) , 1 ≤ p ≤ ∞.
(2) Bα ◦ Bβ = Bα+β , α, β ≥ 0.
Proof. (1) follows from the Young inequality and the fact that kgα kL1 (Rn ) =
1.
(2) follows immediately from the Bessel composition formula (4). The most interesting fact concerning Bessel potentials is that they
can be employed to characterize the Sobolev spaces W k,p (Rn ). This
is expressed in the following theorem where we employ the notation
Lα,p (Rn ), α > 0 and 1 ≤ p ≤ ∞, to denote all functions u such that
u = gα ∗ f for some f ∈ Lp (Rn ). We may define a norm on Lα,p (Rn )
as follows
kukLα,p (Rn ) = kf kLp (Rn ) .
With respect to this norm, Lα,p (Rn ) becomes a Banach space.
Theorem 6. If k is a nonnegative integer and 1 < p < ∞, then
Lk,p (Rn ) = W k,p (Rn ). Moreover, if u ∈ Lk,p (Rn ) with u = gα ∗ f , then
C −1 kf kLp (Rn ) ≤ kukW k,p (Rn ) ≤ Ckf kLp (Rn )
where C = C(α, n, p).
Before ending this section, we would like to point out that on Sobolev
α
spaces W k,p (Rn ), Bα = (I − 4)− 2 .
3. A HLS type inequality
In this section, we prove a new HLS type inequality which will play
an important role in the proof of Theorem 1. To that purpose, we
recall the classical Sobolev imbedding theorem of the spaces W k,p (Rn )
([1]).
4
LI MA AND DEZHONG CHEN
Theorem 7. (The Sobolev imbedding theorem) For 1 ≤ p < ∞, there
exist the following imbeddings:
 q n
np
 L (R ), kp < n and p ≤ q ≤ n−kp
W k,p (Rn ) ,→
Lq (Rn ), kp = n and p ≤ q < ∞

Cb (Rn ), kp > n
where Cb (Rn ) = {u ∈ C(Rn )|u is bounded on Rn }.
The following lemma comes from [1] (Theorem 7.63, (d), (e)).
np
Lemma 8. (1) If t ≤ s and 1 < p ≤ q ≤ n−(s−t)p
< ∞, then
s,p
n
t,q
n
L (R ) ,→ L (R ).
(2) If 0 ≤ µ ≤ s − n/p < 1, then Ls,p (Rn ) ,→ C 0,µ (Rn ).
Now we are ready to prove the following HLS type inequality.
q
Theorem 9. Let q > max{β, n(β−1)
}. If f ∈ L β (Rn ), then Bα (f ) ∈
α
Lq (Rn ). Moreover, we have the estimate kBα (f )kLq (Rn ) ≤ Ckf k βq n ,
L (R )
where C = C(α, β, n, q).
q
q
Proof. By definition, Bα (f ) ∈ Lα, β (Rn ) provided that f ∈ L β (Rn ).
We divide the proof into several cases.
Case 1 α is an integer.
q
It follows from Theorem 6 that Bα (f ) ∈ W α, β (Rn ). Note that
n·
q
β
n−α·
q
β
=
n
n
·q >
nβ − αq
nβ − α ·
n(β−1)
α
q
.
β
·q =q >
So if α · βq ≤ n, then by the Sobolev imbedding theorem, we have
Bα (f ) ∈ Lq (Rn ). Now it follows again from Theorem 6 that
kBα (f )kLp (Rn ) ≤ CkBα (f )k
α,
L
q
β
(Rn )
≤ Ckf k
q
L β (Rn )
.
If α · βq > n, then it follows from the Sobolev imbedding theorem that
Bα (f ) ∈ Cb (Rn ) and kBα (f )kCb (Rn ) ≤ CkBα (f )k α, βq n ≤ Ckf k βq n .
L
(R )
L (R )
RADIAL SYMMETRY
5
Now a straightforward computation shows that
Z
1
kBα (f )kLq (Rn ) = (
|Bα (f )|q ) q
n
ZR
q
1
q− βq
q
≤ (
|Bα (f )| β · kBα (f )kCb (R
n))
Rn
1
1− 1
= kBα (f )k β q
L β (Rn )
≤ kf k
1
β
1− β1
q
Lβ
= Ckf k
β
· kBα (f )kCb (R
n)
(Rn )
· Ckf k
q
L β (Rn )
q
L β (Rn )
.
Case 2 α is a fraction.
In this case, α − [α] > 0, where [α] denotes the integer satisfying
nβ
[α] ≤ α < [α] + 1. We divide this case into two subcases, ie. q < α−[α]
nβ
.
and q ≥ α−[α]
nβ
Case 2.1 q < α−[α]
Let r0 =
n βq
n−(α−[α]) βq
. Since [α] ≤ α and
8, (1) that for any r satisfying
q
≤ r
β
α, βq
n
q
β
> 1, it follows from Lemma
q
≤ r0 , we have Lα, β (Rn ) ,→
L[α],r (Rn ). Note that Bα (f ) ∈ L (R ) and [α] is an integer. Thus
Bα (f ) ∈ W [α],r (Rn ). There are two possibilities.
Case 2.1.1 q ≤ r0
This is equivalent to q ≥ n(β−1)
. In this situation, q lies in the
α−[α]
q
[α],q
interval [ β , r0 ]. Hence Bα (f ) ∈ W
(Rn ). Moreover,
kBα (f )kLq (Rn ) ≤ kBα (f )kW [α],q (Rn )
≤ CkBα (f )kL[α],q (Rn )
≤ CkBα (f )k
= Ckf k
α,
L
q
L β (Rn )
q
β
(Rn )
.
Case 2.1.2 q > r0
This is equivalent to q < n(β−1)
. In this situation, we have Bα (f ) ∈
α−[α]
W [α],r0 (Rn ).
Case 2.1.2.1 q < nβ
α
We have [α]r0 < n. By assumption, q ≥ n(β−1)
. Hence q actually lies
α
nr0
].
Now
it
follows
from
the
Sobolev imbedding
in the interval [r0 , n−[α]r
0
q
n
theorem that Bα (f ) ∈ L (R ).
Case 2.1.2.2 q = nβ
α
6
LI MA AND DEZHONG CHEN
We have [α]r0 = n. Again it follows from the Sobolev imbedding
theorem that Bα (f ) ∈ Lq (Rn ).
Case 2.1.2.3 q > nβ
α
We have [α]r0 > n. Again it follows from the Sobolev imbedding
theorem that Bα (f ) ∈ Cb (Rn ). A straightforward calculation shows
that
1− 1
1
β
β
kBα (f )kLq (Rn ) ≤ kBα (f )kCb (R
n ) kBα (f )k q
L β (Rn )
≤ CkBα (f )k
= CkBα (f )k
= Ckf k
Case 2.2 q ≥
1− β1
q
L β (Rn )
1
kBα (f )k β q
L β (Rn )
q
L β (Rn )
q
L β (Rn )
.
nβ
α−[α]
q
First, we observe that Bα−[α] (f ) ∈ Lα−[α], β (Rn ). Note that q ≥
nβ
⇔ 0 ≤ µ0 < 1, where µ0 = α − [α] − nβ
. It follows from
α−[α]
q
0,µ0
n
Lemma 8, (2) that Bα−[α] (f ) ∈ C (R ). Then as before, we obtain
the following estimate.
kBα (f )kLq (Rn ) = kB[α] ◦ Bα−[α] (f )kLq (Rn )
≤ kBα−[α] (f )kLq (Rn )
1− 1
1
≤ kBα−[α] (f )kC 0,µβ 0 (Rn ) kBα−[α] (f )k β q
L β (Rn )
≤ CkBα−[α] (f )k
≤ Ckf k
= Ckf k
1− β1
q
Lβ
(Rn )
q
L β (Rn )
1− β1
q
α−[α],
β
L
1
(Rn )
kBα−[α] (f )k β q
L β (Rn )
1
kf k β q
L β (Rn )
.
This completes the proof.
4. Proof of Theorem 1
For a given real number λ, we may define
Σλ = {x = (x1 , · · · , xn )|x1 ≥ λ}.
Let xλ = (2λ − x1 , · · · , xn ) and define uλ (x) = u(xλ ).
Lemma 10. For any solution u(x) of (2), we have
Z
u(x) − uλ (x) =
(gα (x − y) − gα (xλ − y))(u(y)β − uλ (y)β )dy.
Σλ
RADIAL SYMMETRY
7
Proof. Let Σcλ = {x = (x1 , · · · , xn )|x1 < λ}. It is easy to see that
Z
Z
β
gα (x − y)u(y)β dy
gα (x − y)u(y) dy +
u(x) =
Σcλ
Σλ
Z
Z
β
gα (x − y λ )u(y λ )β dy
gα (x − y)u(y) dy +
=
ZΣλ
=
ZΣλ
gα (x − y)u(y)β dy +
Σλ
gα (xλ − y)uλ (y)β dy.
Σλ
Here we have used the fact that |x − y λ | = |xλ − y|. Substituting x by
xλ , we get
λ
Z
λ
β
Z
gα (x − y)u(y) dy +
u(x ) =
Σλ
gα (x − y)uλ (y)β dy.
Σλ
Thus
Z
λ
gα (x − y)(u(y)β − uλ (y)β )dy
u(x) − u(x ) =
Σλ
Z
−
Z
gα (xλ − y)(u(y)β − uλ (y)β )dy
Σλ
(gα (x − y) − gα (xλ − y))(u(y)β − uλ (y)β )dy.
=
Σλ
This completes the proof.
We may define Σ−
λ = {x|x ∈ Σλ , u(x) < uλ (x)}.
Lemma 11. For λ 0, Σ−
λ must be measure zero.
Proof. Whenever x, y ∈ Σλ , we have |x − y| ≤ |xλ − y|. Moreover, since
λ < 0, |y λ | ≥ |y| for any y ∈ Σλ . Then it follows from Lemma 10 that
for any x ∈ Σ−
λ,
Z
uλ (x) − u(x) ≤
Σ−
λ
gα (x − y)(uλ (y)β − u(y)β )dy
Z
≤ β
Σ−
λ
gα (x − y)[uβ−1
λ (uλ − u)](y)dy.
8
LI MA AND DEZHONG CHEN
It follows first from the HLS type inequality (Theorem 9) and then the
Hölder inequality that for q as in Theorem 1, we have
kuλ − ukLq (Σ− ) ≤ βkBα (uβ−1
λ (uλ − u))kLq (Σ− )
λ
λ
≤
Ckuβ−1
λ (uλ
− u)k
q
L β (Σ−
λ)
kuλ − ukLq (Σ− )
≤ Ckuλ kβ−1
Lq (Σ− )
λ
λ
≤
Ckuλ kβ−1
Lq (Σλ ) kuλ
=
Ckukβ−1
Lq (Σcλ ) kuλ
− ukLq (Σ− )
λ
− ukLq (Σ− ) .
λ
Since u ∈ Lq (Rn ), we can choose N 0, st. for λ ≤ −N , Ckukβ−1
Lq (Σcλ ) ≤
1
. Then it follows from the last inequality that kuλ − ukLq (Σ− ) = 0,
2
λ
and therefore Σ−
λ must be measure zero.
Now we fix an N 0 st. for λ ≤ −N ,
(5)
u(x) ≥ uλ (x),
∀x ∈ Σλ .
We start moving the planes continuously from λ ≤ −N to the right as
long as (5) holds. Suppose that at a λ0 < 0, we have u(x) ≥ uλ0 (x) and
m({x ∈ Σλ0 |u(x) > uλ0 (x)}) > 0. By Lemma 10, we know that u(x) >
uλ0 (x) in the interior of Σλ0 . Let Σ−
λ0 = {x ∈ Σλ0 |u(x) ≤ uλ0 (x)}. It
−
−
is easy to see that Σλ0 has measure zero and limλ→λ0 Σ−
λ ⊂ Σλ0 . Let
−
?
(Σ−
λ ) be the reflection of Σλ about the planes x1 = λ. It follows from
the above arguments that
(6)
kuλ − ukLq (Σ− ) ≤ Ckukβ−1
kuλ − ukLq (Σ− ) .
Lq (Σ− )?
λ
λ
λ
As before, the integrability condition on u guarantees that there exists
an 0 st. ∀λ ∈ [λ0 , λ0 + ), Ckukβ−1
≤ 21 . Now by (6), we have
Lq (Σ− )?
λ
kuλ − ukLq (Σ− ) = 0, and hence Σ−
λ must be measure zero. Therefore
λ
the planes can be moved further to the right., i.e., there exists an depending on n, α, β, q and the solution u such that u(x) ≥ uλ (x) on
Σλ , ∀λ ∈ [λ0 , λ0 + ).
The planes either stops at some λ < 0 or can be moved until λ = 0.
In the former case, it turns out that uλ (x) = u(x), ∀x ∈ Σλ . In the
latter case, we have uλ (x) ≤ u(x), ∀x ∈ Σ0 . However, if we move the
planes from the right to the left, then we may see that uλ (x) ≥ u(x),
∀x ∈ Σ0 . Thus uλ (x) = u(x), ∀x ∈ Σ0 . Since we can choose any
direction to start the process, we conclude that u is actually radial
symmetric and monotone decreases about some point in Rn . This
completes the proof of Theorem 1.
RADIAL SYMMETRY
9
Acknowledgement: The authors would like to thank the referee
for helpful suggestions.
References
[1] R.Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press,
New York-London, 1975.
[2] J.Bourgain, Global solutions of nonlinear Schrödinger equations, AMS Colloquium
Publications 46, Providence, RI, 1999.
[3] L.Caffarelli, B.Gidas and J.Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math.,
42(1989), 271–297.
[4] W.Chen, C.Li and B. Ou, Classification of solutions for a system of integral equations,
Comm. Partial Differential Equations, 30(2005), 59–65.
[5] W.Chen, C.Li and B.Ou, Classification of solutions for an integral equation, Comm.
Pure Appl. Math., 59(2006), 330–343.
[6] Y.Du and L.Ma, Some remarks related to De Giorgi’s conjecture, Proc. Amer. Math.
Soc., 131(2003), 2415–2422.
[7] B.Gidas, W.Ni and L.Nirenberg, Symmetry and related properties via the maximum
principle, Comm. Math. Phys., 68(1979), 209–243.
[8] B.Gidas, W.Ni and L.Nirenberg, Symmetry of positive solutions of nonlinear elliptic
equations in Rn , Mathematical analysis and applications, Part A, pp. 369–402, Adv.
in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.
[9] M.Kwong, Uniqueness of positive solutions of ∆u − u + up = 0 in Rn , Arch. Rational
Mech. Anal., 105(1989), 243–266.
[10] Y.Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6(2004), 153–180.
[11] E.Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,
Ann. of Math., 118(1983), 349–374.
[12] L.Ma and D.Chen, A Liouville type theorem for an integral system, Commun. Pure
Appl. Anal., 5(2006), 855–859.
[13] W.Ni and I.Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70(1993), 247–281.
[14] E.Stein, Singular integrals and differentiability properties of functions, Princeton
Mathematical Series 30, Princeton University Press, Princeton, NJ, 1970.
[15] E.Stein and G.Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton
Mathematical Series 32, Princeton University Press, Princeton, NJ, 1971.
[16] J.Wei and X.Xu, Classification of solutions of higher order conformally invariant
equations, Math. Ann., 313(1999), 207–228.
[17] W.Ziemer, Weakly differentiable functions: Sobolev spaces and functions of bounded
variation, GTM 120, Springer-Verlag, NY, 1989.
Li Ma, Department of mathematical sciences, Tsinghua university, Beijing,
100084, China
E-mail address: [email protected]
Dezhong Chen, Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada
E-mail address: [email protected]