Acta Mathematicae Applicatae Sinica, English Series
Vol. 24, No. 3 (2008) 473–482
DOI: 10.1007/s10255-008-8030-0
www.Applmath.com.cn
Acta Mathemaca Applicatae
Sinica, English Series
The Editorial Office of AMAS
& Springer-Verlag 2008
Uniqueness and Radial Symmetry of Least Energy
Solution for a Semilinear Neumann Problem
Zheng-ping Wang1 , Huan-song Zhou2
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O.Box 71010, Wuhan 430071,
China (E-mail: 1 [email protected], 2 [email protected])
Abstract
Consider the following Neumann problem
dΔu − u + k(x)up = 0 and u > 0 in B1 ,
where d > 0, B1 is the unit ball in
∂u
= 0 on ∂B1 ,
∂ν
RN , k(x) = k(|x|) ≡ 0 is nonnegative and in C(B 1 ), 1 < p <
(∗)
N+2
N−2
with
N ≥ 3. It was shown in [2] that, for any d > 0, problem (∗) has no nonconstant radially symmetric least
energy solution if k(x) ≡ 1. By an implicit function theorem we prove that there is d0 > 0 such that (∗) has a
unique radially symmetric least energy solution if d > d0 , this solution is constant if k(x) ≡ 1 and nonconstant
if k(x) ≡ 1. In particular, for k(x) ≡ 1, d0 can be expressed explicitly.
Keywords Implicit function theorem, least energy solution, radial symmetry, Neumann problem, elliptic
2000 MR Subject Classification 35J70, 35J65
1
Introduction
In this paper, we consider the following semilinear elliptic
⎧
dΔu − u + k(x)up = 0
⎪
⎪
⎨
u>0
⎪
⎪
⎩ ∂u = 0
∂ν
Numann problem
in B1
in B1
(1.1)
on ∂B1 ,
+2
where B1 is the unit ball in RN , N ≥ 3, 1 < p < N
N −2 , k(x) = k(|x|) ∈ C(B 1 ), k(x) ≥ 0(≡ 0),
d > 0 is a parameter. This problem is related to the study of a mathematical model of biological
pattern formation, it also plays an important role in considering the shadow system of some
reaction-diffusion system in chemotaxis (see [4,9,10]) and the references therein.
ud is said to be a least energy solution of problem (1.1) if ud ∈ H 1 (B1 ) is a weak solution
of (1.1) such that the following energy functional on u ∈ H 1 (B1 )
d
1
1
Jd (u) =
|∇u|2 dx +
u2 dx −
k(x)(u+ )p+1 dx,
2 B1
2 B1
p + 1 B1
with u+ (x) = max{u(x), 0}, achieves the least positive value at ud among all positive solutions
of (1.1) . It turns out that a mountain pass solution is a least energy solution[8] .
In 1986, Ni and Takagi[10] established a priori estimate for solutions of (1.1) with k(x) ≡ 1,
upon which they proved that there exists d∗ > 0 such that u ≡ 1 is the only solution for
Manuscript received March 6, 2008.
Supported by the National Natural Science Foundation of China (No. 10571174, 10631030), Chinese Academy
of Sciences grant KJCX3-SYW-S03.
Z.P. Wang H.S. Zhou
474
problem (1.1) if d > d∗ . After that, there are many results on the existence, the shape and the
symmetry of the least energy solutions for problem (1.1) with k(x) ≡ 1 and d > 0 small, we
mention here only the papers[2,7,11,12] . In [5], Grossi proved that the least energy solution of
(1.1) with k(x) ≡ 1 is unique (up to rotation) if d > 0 is small enough, which is nonconstant
by [11]. So, if k(x) ≡ 1, by the results mentioned above we see that there exist d1 > 0, d2 > 0
such that problem (1.1) has only one nonconstant least energy solution for d ∈ (0, d1 ) and it
has only the constant solution u ≡ 1 for d > d2 . To the authors’ knowledge, it is still not clear
what would happen for problem (1.1) with d ∈ (d1 , d2 ), even if k(x) ≡ 1.
Recently, Chern and Lin in [2] showed that, for any d > 0, problem (1.1) with k(x) ≡ 1
has no any nonconstant, radially symmetric least energy solution. It is natural to ask whether
or not this fact is true for problem(1.1) with k(x) ≡ 1. In this paper, we try to get some
information about the radially symmetric least energy solution of problem (1.1) with k(x) ≡ 1.
Our method seems inapplicable for the case where d is small. Here, we prove only that there
exists d0 > 0 such that problem (1.1) has a unique radially symmetric least energy solution
(which may be a constant solution) if d > d0 and d0 can be given explicitly if k(x) ≡ 1. Even this
uniqueness result, it seems not trivial because the method used in [10] is no longer applicable
when k(x) ≡ 1. Motivated by a recent paper[1] we use a variant version of the implicit function
theorem for 1/d small to prove our results, this is why we require that d > 0 large. Our results
work also for the case of k(x) ≡ 1 and in this case the solution which we get is a constant.
To find a least energy solution of (1.1), we consider the following minimization problem
2 p+1
2
2
p+1
inf
(d|∇u|
+
u
)dx
k(x)|u|
dx
Sd =
u∈H 1 (B1 )\{0}
B1
B1
2
2
1
= inf
(d|∇u| + u )dx : u ∈ H (B1 ),
k(x)|u|p+1 dx = 1 .
(1.2)
B1
B1
Since the Sobolev embedding from H 1 (B1 ) to Lp+1 (B1 ) is compact,
by a standard procedure,
dx = 1 which
it is not difficult to see that there exists ud > 0 in H 1 (B1 ) with B1 k(x)|u|p+1
d
achieves Sd . Therefore, ud solves the following equation
⎧
dΔu − u + Sd k(x)up = 0
in B1
⎪
⎪
⎨
u>0
in B1
(1.3)
⎪
∂u
⎪
⎩
=0
on ∂B1 ,
∂ν
1
and then we see that vd = cd ud with cd = (Sd ) p−1 is a least energy solution of (1.1). Moreover,
any least energy solution of (1.1) achieves Sd . In what follows, we call the function which
achieves Sd as an extremal of Sd .
Remark 1.1.
The above conclusion implies that problem (1.1) has at least one least energy
solution ud for any d > 0. It is easy to see that ud → c (a constant) as d → ∞. If c = 0, it can
be expressed explicitly via k(x). Therefore, the uniqueness of least energy solution for d large
can be proved by slightly modifying the proof of Theorem 3 in [10] where k(x) ≡ 1. For general
k(x), it seems difficult to show c = 0 using only the priori estimate for solutions in [Theorem
2, 10]. However, c = 0 is essentially implied in our proof of Theorem 1.1., where an implicit
function theorem is used.
In the present paper, our main aim is to prove that the least energy solution is not only
unique but also radially symmetric if d > 0 is large enough.
Our main theorems are as follows.
Uniqueness and Radial Symmetry of Least Energy Solution for a Semilinear Neumann Problem
475
Theorem 1.1.
There exists d0 > 0 such that the least energy solution of (1.1) is unique
and radially symmetric if d > d0 . Moreover, this least energy solution is either nonconstant if
k(x) ≡ 1, or identically equal to 1 if k(x) ≡ 1.
Remark 1.2.
It follows from the proof of Theorem 2.1 in [1] (or refer to our proof of
Theorem 1.1) that if problem (1.1) has a unique solution, then this unique solution must be
radially symmetric.
Remark 1.3.
Our method of proving Theorem 1.1 also applies to the special case where
k(|x|) ≡ 1. In this case, Theorem 1.1 means that there exists d0 > 0 such that u ≡ 1 is the
only least energy solution of (1.1) if d > d0 , which is consistent with the result in [10].
Our next result is to establish an explicit expression for d0 when k(|x|) ≡ 1.
Theorem 1.2.
Let
Sd = inf
(d|∇u|2 + u2 )dx : u ∈ H 1 (B1 ),
B1
|u|p+1 dx = 1
B1
that is, taking k(x) ≡ 1 in (1.2), and define
d = inf{d0 : Sd has a unique extremal for any d > d0 }.
Then, 0 < d ≤ d∗ ≡
set
p−1
where C0 = |B1 | p+1 , |B1 | denotes the Lebesgue measure of B1 and
|∇u|2 dx
1
:
u
∈
H
(B
)
\
{0}
and
udx
=
0
.
(1.4)
C1 (p) = inf B1
1
2
B1
( B1 |u|p+1 dx) p+1
pC0
C1 (p) ,
Furthermore, if d > d∗ , problem (1.1) with k(x) ≡ 1 has only constant least energy solution
u ≡ 1.
Remark 1.4.
C1 (p) defined by (1.4) is positive. In fact, by the compactness of Sobolev
embedding it is easy to see that C1 (p) can be attained, then it can not be zero.
Throughout this paper, we denote the universal positive constant by C and set
· q = · Lq (B1 ) with 1 < q < +∞ and · = · H 1 (B1 ) .
2
(1.5)
An Implicit Function Theorem and Some Lemmas
To prove Theorem 1.1, the following version of implicit function theorem on Banach manifolds
is needed, this kind of theorem should be found in some references, but we could not get it. So
we give here a sketch of its proof.
Proposition 2.1 (Impicit Function Theorem).
Let X, Y be C 1 Banach manifolds,
1
W be a Banach space and, F : X × Y → W be a C map with F (x0 , y0 ) = 0 for some
(x0 , y0 ) ∈ X × Y . If ∂F
∂y (x0 , y0 ) : Ty0 Y → W is a bounded invertible linear transformation,
where Ty0 Y is the tangent space of Y at y0 , then there is a connected open neighborhood U of
x0 in X and a unique continuous map u : U → Y such that u(x0 ) = y0 and F (x, u(x)) = 0 for
every x ∈ U .
Proof. This Proposition can be proved by almost the same procedures as the proof of the
implicit function theorem on Banach space, Theorem 5.9 in [6]. Just here we need to use an
Z.P. Wang H.S. Zhou
476
inverse function theorem based on a local diffeomorphism theorem on C 1 Banach manifold,
Theorem 73.B in [13].
For applying Proposition 2.1 to our problem, it is convenient to change first the settings in
(1.2) as follows.
For d > 0, let μ = 1d and
−1
2
2
1
Cμ = inf
(μ |∇u| + u )dx : u ∈ H (B1 ),
k(x)|u|p+1 dx = 1 .
B1
B1
It follows from (1.2) that Cμ = Sd and for any μ > 0, Cμ has an extremal which is a solution
of the following problem:
⎧
Δu − μu + μCμ k(x)up = 0
in B1
⎪
⎪
⎨
u>0
in B1
(2.1)
⎪
⎪ ∂u
⎩
=0
on ∂B1 .
∂ν
Denote
E = v ∈ H 1 (B1 ) :
k(x)|v|p+1 dx = 1
B1
1
and E is a C Banach manifold.
Define a functional F : E × [0, 1] → (H 1 (B1 ))∗ by
F (v, μ)(φ) =
∇v · ∇φdx + μ
vφdx − μCμ
B1
B1
k(x)|v + |p φdx.
(2.2)
B1
Then F is C 1 (E × [0, 1]).
−1
Taking v0 ≡ ( B1 k(x)dx) p+1 , it is easy to see that
v0 ∈ E and F (v0 , 0) = 0,
(2.3)
2
since, for any μ > 0, Cμ ≤ C0 :≡ |B1 |{ B1 k(x)dx}− p+1 and Cμ is non-increasing with respect
to μ > 0.
Now, we turn to proving some lemmas to be used in verifying the conditions of the implicit
function theorem, Proposition 2.1. These lemmas are similar to those in section 2 of [1], we just
use a different constraint here. We give their proofs in Appendix for the readers’ convenient.
Lemma 2.1.
For v0 given in (2.3), the tangent space to E at v0 is given by
Tv0 E = z ∈ H 1 (B1 ) :
k(x)zdx = 0 .
B1
Lemma 2.2.
Let
A = {ϕ ∈ (H 1 (B1 ))∗ : ϕ, 1 = 0}.
Then, the derivative of F with respect to v at the point (v0 , 0) is given by
∂F
∂F
∇w∇φdx for all φ ∈ H 1 (B1 ).
(v0 , 0) : Tv0 E → A and
(v0 , 0)(w)(φ) =
∂v
∂v
B1
Lemma 2.3.
Let Ω be a bounded smooth domain in RN , k(x) is the function given in (1.1).
Then there exists a constant C > 0, depending only on N and Ω, such that
1
k(x)udx ≤ C
∇u
2
(2.4)
u −
|Ω| Ω
2
Uniqueness and Radial Symmetry of Least Energy Solution for a Semilinear Neumann Problem
477
for each function u ∈ H 1 (Ω).
∂F
Lemma 2.4.
∂v (v0 , 0) has a continuous inverse, that is, for any given ϕ ∈ A there exists a
unique w ∈ Tv0 E such that
∇w · ∇φdx = ϕ, φ, for all φ ∈ H 1 (B1 ).
B1
Moreover the map ϕ → w is continuous.
3
Proofs of Theorems 1.1 and 1.2
Proof of Theorem 1.1.
It follows from Lemma 2.4 that the functional F , defined by (2.2),
verifies all the hypotheses of Proposition 2.1, then there exists μ0 > 0 such that there is a
unique solution v = v(μ) satisfying F (v, μ) = 0 for any μ ∈ (0, μ0 ) and v near v0 . That is, for
μ > 0 small enough problem (2.1) has only one weak solution near v0 . This implies that the
extremal of Cμ is unique if μ > 0 is small. Recalling that d = 1/μ and by Remark 1.1, we see
that there is d0 = 1/μ0 such that problem (2.1) has a unique least energy solution for d > d0 .
Moreover, similar to the proof of Theroem 2.1 in [1] we take an extremal u1 for Cμ and let
R be any rotation. Noticing that
k(x)|u1 (Rx)|p+1 dx =
k(x)|u1 (x)|p+1 dx, since k(x) = k(|x|),
B1
B1
2
|∇u1 (Rx)| dx =
|∇u1 (x)|2 dx,
B1
B1
which means that u2 := u1 (Rx) is also an extremal. But the extremal of Cμ is unique for μ > 0
small, then u1 ≡ u2 and we conclude that the unique extremal must be a radial function if
μ > 0 is small.
Therefore, there exists d0 = μ10 such that if d > d0 the least energy solution of (1.1) is
unique and radially symmetric. Moreover, if k(x) ≡ 1 we see that the above procedures also
work, but it follows from the result of [2] that any nonconstant least energy solution of problem
(1.1) with k(x) ≡ 1 must be nonradially symmetric, so in this case u ≡ 1 is the only least
energy solution.
Proof of Theorem 1.2. Now, we consider only the special case: k(x) ≡ 1.
By Remark 1.1, Cμ has always an extremal for each μ > 0. If this extremal is unique, then
it is radially symmetric by Remark 1.2. But it follows from the results of Ni and Takagi[11] that
the extremal of Cμ must be nonradial if μ is large enough, which means that Cμ should have
at least two extremals if μ is large enough and it is reasonable to define
μ = sup{θ : Cμ has a unique extremal for any μ < θ},
(3.1)
and μ < ∞. But, μ = 1d for d given by Theorem 1.2, so d > 0. Moreover, by Theorem 1.1
we know that there exists μ0 > 0 such that if μ ∈ (0, μ0 ), Cμ has a unique extremal which is
radially symmetric, so μ0 ≤ μ.
1 (p)
. In fact, it follows from the
Hence, to prove Theorem 1.2 it is enough to show that μ ≥ CpC
0
definition of μ that Cμ must has an unique extemal uμ for any μ ∈ (0, μ). Hence, for any n ∈ N
large enough such that μn := μ − n1 ∈ (0, μ), Cμn has a unique extremal vn with vn p+1 = 1,
which is radially symmetric by Remark 1.2. But by the result of [2], that is, any radially
−1
symmetric least energy solution is constant, so vn must be constant and vn ≡ v = |B1 | p+1 .
Z.P. Wang H.S. Zhou
478
For v ∈ H 1 (B1 )\{0}, μ > 0, 1 < p <
−1
Q(v, μ, p) =
[μ
N +2
N −2 ,
2
let
2
|∇v| + v ]dx
B1
2
p+1
|v|p+1 dx
.
B1
Then, Cμ = inf{Q(v, μ, p) : v ∈ H 1 (B1 )\{0}}.
We claim that v is an extremal of Cμ . Indeed, if there exists v ∈ H 1 (B1 )\{0} such that
Q(
v , μ, p) < Q(v, μ, p), then there is a contradiction since vn achieves Cμn , vn ≡ v ≡ const and
Q(
v , μ, p) = lim Q(
v , μn , p) ≥ lim Q(vn , μn , p) = Q(v, μ, p).
n→+∞
n→+∞
Again by the definition of μ, we have that for any ε > 0 small enough, there exists ε1 ∈ [0, ε)
such that Cμ+ε1 has at least two extremals. Therefore, Proposition 2.1 implies that ∂F
∂v (v, μ),
the derivative of F with respect to v at the point (v, μ), is not regular.
By the proof of Lemma 2.1, the tangent space of E at v can be given by
zdx = 0 .
Tv E = z ∈ H 1 (B1 ) :
B1
Also by the similar procedure of the proof of Lemma 2.2, we have
∂F
(v, μ)(z)(φ) =
∇z · ∇φdx + μ
zφdx − μCμ
v p−1 zφdx.
∂v
B1
B1
B1
The map
∗
∂F
(v, μ) : Tv E → H 1 (B1 )
∂v
is not regular, so there exists z1 ∈ Tv E(≡ 0) such that
∂F
(v, μ)(z1 )(φ) =
∇z1 · ∇φdx + μ
z1 φdx − μCμ
v p−1 z1 φdx = 0.
∂v
B1
B1
B1
(3.2)
Take φ = z1 in (3.2), by Hölder inequality we have
|∇z1 |2 + μz12 dx =μCμ p
v p−1 z12 dx
B1
B1
≤μCμ p
p−1
p+1
v p+1 dx
B1
2
p+1
|z1 |p+1 dx
(3.3)
B1
1 (p)
, since Cμ ≤ C0 .
This proves that μ ≥ CpC
0
Moreover, the definition of d implies that problem (1.1) with k(x) ≡ 1 has a unique least
energy solution u for d > d∗ and, u is radially symmetric by Remark 1.2. So, the result of [2]
means that u ≡ 1.
4
Appendix: Proofs of Lemmas 2.1 to 2.4
Proof of Lemma 2.1. The proof is almost the same as that of Lemma 2.1 in [1].
First, we claim that
k(x)zdx = 0 E1 .
Tv0 E ⊂ z ∈ H 1 (B1 ) :
B1
Uniqueness and Radial Symmetry of Least Energy Solution for a Semilinear Neumann Problem
479
C1
In fact, forany z ∈ Tv0 E. there exits a curve γ : (−1, 1) −→ E with γ (0) = z and γ(0) = v0 .
Moreover, B1 k(x)|γ(t)|p+1 dx = 1 for all t ∈ (−1, 1), since γ(t) ∈ E. This implies that
(p + 1)
B1
k(x)γ(0)p γ (0)dx = (p + 1)v0p
k(x)zdx = 0, that is, z ∈ E1 .
B1
On the other hand, let z ∈ Tv0 E, that is, z ∈ H 1 (B1 ) and B1 k(x)zdx = 0. Define a curve
in (−1, 1) by
1
p+1
k(x)|v0 + tz|p+1 dx
.
γ(t) = {v0 + tz}/
B1
It is easy to see that γ(t) ∈ E with γ(0) = v0 . Furthermore, γ (0) = z. Indeed,
1
p+1
v
+
tz
−
v
(
dx) p+1
0
0
γ(t)
−
γ(0)
B1 k(x)|v0 + tz|
γ (0) = lim
= lim
1
t→0
t→0
t
t( B1 k(x)|v0 + tz|p+1 dx) p+1
1
v0 − v0 ( B1 k(x)|v0 + tz|p+1 dx) p+1
z
= lim + lim
1
1
t→0 (
t→0
k(x)|v0 + tz|p+1 dx) p+1
t( B1 k(x)|v0 + tz|p+1 dx) p+1
B1
1
p+1
v0
1 =z + lim · lim
k(x)v0p+1 dx
1
t→0 (
B1
k(x)|v0 + tz|p+1 dx) p+1 t→0 t
B1
1 p+1
, since
−
k(x)|v0 + tz|p+1 dx
k(x)v0p+1 dx = 1 by (2.3)
B1
d
1
p+1
k(x)|v0 + tz|p+1 dx
|t=0
B1
=z − v0
dt B1
=z + c
k(x)zdx = z.
B1
That is, z ∈ Tv0 E and E1 ⊂ Tv0 E. The proof is completed.
Proof of Lemma 2.2. Let F be defined by (2.2) and consider the mapping
∗
C1
F (·, 0) : E −→ F (E × {0}) ⊂ H 1 (B1 ) .
C1
It is well-known that for any given w ∈ Tv0 E, there exists a curve γ : (−1, 1) −→ E such that
γ(0) = v0 , γ (0) = w. Let
α(t) := F (·, 0) ◦ γ(t) = F (γ(t), 0) : (−1, 1) → F (E × {0}).
Then
∂F
∂v
(v0 , 0)(w) = α (0). Since α(0) = F (v0 , 0) = 0, we have
∗
α(t) − α(0)
α(t) 1
= lim
∈ H (B1 ) .
t→0
t→0 t
t
α (0) = lim
Since α(t) ∈ F (E × {0}), that is, there exists vt ∈ E such that α(t) = F (vt , 0). Therefore
α (0), 1 = lim
t→0
1
1
α(t), 1 = lim
F (vt , 0)(1) = 0.
t→0 t
t
Thus, for any φ ∈ H 1 (B1 ), by the definition of F in (2.2) and noting that γ(0) = v0 = const.,
Z.P. Wang H.S. Zhou
480
that is ∇γ(0) = 0, we have that
F (γ(t), 0)
∂F
(v0 , 0)(w)(φ) =α (0)(φ) = lim
(φ)
t→0
∂v
t
γ(t) − γ(0)
· ∇φdx
= lim
∇
t→0 B
t
1
=
∇γ (0) · ∇φdx =
∇w · ∇φdx.
B1
B1
Proof of Lemma 2.4. The proof of this lemma is similar to that of Theorem 1 of Section 5.8.1
in [3]. We argue by contradiction. If (2.4) is false, then for each k ∈ N, there exists a sequence
{uk } ⊂ H 1 (Ω) satisfying
1
k(x)uk dx > k
∇u
2 .
(4.1)
uk −
|Ω| Ω
2
Set
(u)Ω =
1
|Ω|
Ω
k(x)udx,
vk =
uk − (uk )Ω
.
uk − (uk )Ω 2
Then (vk )Ω = 0, vk 2 = 1 and (4.1) implies
Dvk 2 <
1
for all k = 1, 2, · · · .
k
(4.2)
Therefore, {vk } is bounded in H 1 (Ω) and, for some v ∈ L2 (Ω) we may assume that
vk → v strongly in L2 (Ω)
k
(4.3)
with v
2 = 1. Moreover, (v)Ω = 0, since (vk )Ω = 0 and
1 k
|(v)Ω − (vk )Ω | = k(x)(v − vk )dx ≤ C
v − vk 2 → 0, by (4.3).
|Ω| Ω
On the other hand, for i = 1, · · · , N , it follows from (4.2) that
vDi φdx = lim
vk Di φdx = − lim
(Di vk )φdx = 0, for any φ ∈ C0∞ (Ω).
Ω
k→+∞
Ω
k→+∞
Ω
1
k(x)vdx = 0.
This implies that Dv = 0 a.e in Ω and v = const. Thus, v ≡ 0 since (v)Ω = |Ω|
Ω
This contradicts that v
2 = 1. So (2.4) is proved.
Proof of Lemma 2.4.
First, we prove that Tv0 E is a Hilbert space with the inner product
given by
∇u · ∇vdx.
(u, v) =
B1
(4.4)
Indeed, for any u ∈ Tv0 E, if (u, u) = 0 then u ≡ constant. Since u ∈ Tv0 E, B1 k(x)udx = 0,
this implies that u ≡ 0. Thus, it is not difficult to verify that (·, ·) defined by (4.4) is an inner
product for Tv0 E. On the other hand, since (Tv0 E, .
) is a closed subspace of (H 1 (B1 ), .
),
(Tv0 E, .
) is a Banach space. By Lemma 2.3 there holds u
2 ≤ C
∇u
2 for all u ∈ Tv0 E.
1
This implies that, in Tv0 E, the norm · ∗ = (·, ·) 2 introduced by the inner product (4.4) is
equivalent to the norm · given by (1.5). Hence, (Tv0 E, .
∗ ) is also a Banach space, then
Tv0 E is a Hilbert space with the inner product given by (4.4) .
Uniqueness and Radial Symmetry of Least Energy Solution for a Semilinear Neumann Problem
481
For each ϕ ∈ A ⊂ (H 1 (B1 ))∗ ⊂ (Tv0 E)∗ , the Riesz representation theorem implies that
there is a unique w = w(ϕ) ∈ Tv0 E such that
ϕ, φ = (w, φ) =
∇w · ∇φdx, for all φ ∈ Tv0 E.
(4.5)
B1
Next, for any φ ∈ H 1 (B1 ), let
ψ =φ−
1
|B1 |
then ψ ∈ Tv0 E and (4.5) shows that
ϕ, ψ =
∇w · ∇ψdx =
B1
k(x)φdx,
B1
∇w · ∇φdx for all φ ∈ H 1 (B1 ).
(4.6)
B1
By Lemma 2.2, we have ϕ, 1 = 0. Hence ϕ, ψ = ϕ, φ and it follows from (4.6) that for any
ϕ ∈ A, there exists a unique w ∈ Tv0 E such that
∇w · ∇φdx = ϕ, φ,
∀φ ∈ H 1 (B1 ).
B1
n
Finally, we prove that the map ϕ → w is continuous. Assume that ϕn → ϕ in A, that is,
n
ϕn − ϕ
(H 1 (B1 ))∗ → 0, then there exist wn , w ∈ Tv0 E such that
∇w · ∇φdx = ϕ, φ and
∇wn · ∇φdx = ϕn , φ, for all φ ∈ H 1 (B1 ).
B1
B1
Since Tv0 E is a Hilbert space with the inner product given by (4.4) , we have
|(wn − w, φ)|
|(wn − w, φ)|
≤C
sup
φ
φ
∗
φ∈Tv0 E\{0}
φ∈Tv0 E\{0}
| B1 ∇(wn − w) · ∇φdx|
=C
sup
φ
φ∈Tv0 E\{0}
wn − w
∗ =
=C
sup
|ϕn − ϕ, φ|
φ
φ∈Tv0 E\{0}
sup
n
≤C
ϕn − ϕ
(H 1 (B1 ))∗ −→ 0.
Acknowledgements. . Thanks to Professors D.M. Cao, J.D. Rossi and S.J. Peng for some
helpful comments and also to Professor B.K. Driver for the references [6].
References
[1] Bonder, J.F., Dozo, E.L., Rossi, J. D. Symmetry properties for the extremals of the Sobolev trace embedding. Ann. Inst. H. Poincaré Anal. Nonlinéaire, 21: 795–805 (2004)
[2] Chern, J.L., Lin, C.S. The symmetry of least-energy solutions for semilinear equations. J. Differential
Equations, 187: 240–268 (2003)
[3] Evans, L.C. Partial Differential Equations. G.S.M. Vol.19. American Mathematical Society: Providence,
RI 1998
[4] Gierer, A., Meinhardt, H. A theory of biological pattern formation. Kybernetik, (Berlin) 12: 30–39 (1972)
[5] Grossi, M. Uniqueness of the least-energy solution for a semilinear Neumann problem. Proc.Amer. Math.
Soc., 128(6): 1665–16772 (2000)
482
Z.P. Wang H.S. Zhou
[6] Lang, S. Differential and Riemannian Manifolds. Springer-Verlag, Berlin, 1995
[7] Lin, C.S. Locating the peaks of solutions via the maximum principle: I. The Neumann problem. Comm.Pure
Appl. Math., LIV: 1065–1095 (2001)
[8] Lin, C.S., Ni, W.M. On the diffusion coefficient of a semilinear Neumann problem, in Calculus of Variations
and Partial Differential Equations, S.Hildebrandt, D.Kinderlchrer and M.Miranda, eds., Lecture Notes in
Math. 1340, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1988, pp.160–174
[9] Lin, C.S., Ni, W.M., Takagi, I. Large amplitude stationary solutions to a chemotaxis system. J. Differential
Equations, 72: 1–27 (1988)
[10] Ni, W.M., Takagi, I. On the Neumann problem for some semilinear elliptic equations and systems of
activator-inhibitor type. Trans. AMS, 297: 351–368 (1986)
[11] Ni, W.M., Takagi, I. On the shape of least-energy solutions to a semilinear Neumann problem. Comm.Pure
Appl. Math., XLIV: 819–851 (1991)
[12] Ni, W.M., Takagi, I. Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke
Math. J., 70: 247–281 (1993)
[13] Zeidler, E. Nonliner functional analysis and its applications, IV. Applications to Mathematical Physis,
Springer, 2nd edition, 1997