Orbital stability of bound states of nonlinear Schrödinger
equations with linear and nonlinear lattices
Tai-Chia Lin ∗†, Juncheng Wei ‡and Wei Yao
§
We study the orbital stability and instability of single-spike bound states of critical
semi-classical nonlinear Schrödinger equations (NLS) with linear and nonlinear lattices.
These equations may model an inhomogeneous Bose-Einstein condensate and an optical beam in a nonlinear lattice. When the linear lattice is switched off, we derive the
asymptotic expansion formulas and obtain necessary conditions for the orbital stability
and instability of single-spike bound states, respectively. When the linear lattice is turned
on, we consider three different cases and obtain the most general theorem on the orbital
stability problem for NLS with linear and nonlinear lattices.
1
Introduction
Nonlinear Schrödinger equations in the presence of the Kerr nonlinearity may describe an
optical beam in a nonlinear lattice and an inhomogeneous Bose-Einstein condensate (BEC)
given by
∂ψ
= D∆ψ − Vtrap ψ − g|ψ|2 ψ ,
(1.1)
−i
∂t
for x ∈ RN , N = 2 and t > 0. Here ψ = ψ(x, t) ∈ C is the wavefunction, D is the diffraction (or
dispersion) coefficient, and Vtrap is the potential of the linear lattice. Besides, g = µm(x) ∼ a
characterizes the nonlinear lattice, where a denotes the spatially modulated s-wave scattering
length, µ is a nonzero constant and m(x) = m(x1 , · · · , xN ) > 0 is a function depending on
spatial variables (transverse coordinates) x1 , · · · , xN (cf. [1], [6]).
Linear and nonlinear lattices, such as photonic structures for laser beams or optical lattices
for atomic BECs, can support stable bright solitons. For instance, Vtrap the potential of the
linear lattice varying along three spatial variables may stabilize bright solitons in BEC experiments (cf. [7]). When the coefficient g varies along two spatial variables, two-dimensional bright
solitons can also be observed experimentally in two-dimensional nonlinear lattices (cf. [13]).
∗
Department of Mathematics, National Taiwan University, Taipei, Taiwan 106.
tclin@math.ntu.edu.tw
†
Taida Institute of Mathematical Sciences (TIMS), Taipei, Taiwan
‡
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong.
wei@math.cuhk.edu.hk
§
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong.
wyao@math.cuhk.edu.hk
1
Email:
Email:
Email:
Consequently, under the effect of linear and nonlinear lattices, two-dimensional bright solitons must have suitable stability for experimental observations. However, most theoretical
results, e.g., [10] and [11] only focus on orbital (dynamical) stability of one-dimensional bound
states, i.e., steady state bright solitons in one-dimensional nonlinear lattices fulfilling specifically asymptotic behaviors. Here we study the orbital stability of two-dimensional single-spike
bound states of (1.1) in two-dimensional nonlinear lattices when the coefficient g varies along
two spatial variables. Basically, we shall provide rigorous arguments to show the orbital stability and instability of single-spike bound states of (1.1) without using any hypothesis on the
asymptotic behavior of nonlinear lattices.
To get single-spike bound states in nonlinear lattices, we may assume the coefficient D > 0
and the s-wave scattering length a, i.e., µ is negative and large due to the Feshbach resonance
(cf. [9]). Setting h2 = D/(−µ), V (x) = Vtrap (x)/(−µ) and suitable time scale, the equation (1.1) with negative and large µ can be equivalent to a semi-classical nonlinear Schrödinger
equation (NLS) given by
−ih
∂ψ
= h2 ∆ψ − V ψ + m |ψ|2 ψ,
∂t
x ∈ R2 , t > 0 ,
(1.2)
where 0 < h ¿ 1 is a small parameter, V = V (x) is a smooth nonnegative function and
m = m(x) is a smooth positive function. For the spatial dimension N ≥ 1, we may generalize
the equation (1.2) to a NLS having the following form
−ih
∂ψ
= h2 ∆ψ − V ψ + m |ψ|p−1 ψ,
∂t
x ∈ RN , t > 0 ,
(1.3)
with critical exponent
p=1+
4
,
N
N ≥ 1.
(1.4)
In particular, when N = 2, the equation (1.3) with (1.4) is exactly same as (1.2).
Single-spike bound states of (1.3) are of the form ψ(x, t) = eiλ t/h u(x), where λ is a positive
constant and u = u(x) is a positive solution of the following nonlinear elliptic equation
h2 ∆u − (V + λ) u + m up = 0 ,
u ∈ H 1 (RN ) ,
(1.5)
with zero Dirichlet boundary condition, i.e., u(x) → 0 as |x| → ∞. When V ≡ 0 and m ≡ 1,
problem (1.5) admits a unique radially symmetric ground state which is stable for any λ > 0 if
p < 1 + N4 , and unstable for any λ > 0 if p ≥ 1 + N4 (cf. [4], [8] and [41]). For V 6≡ 0 or m 6≡ 1,
there exists uh a single-spike solution of (1.5), provided both V and m are bounded and satisfy
another conditions (cf. [20]). Hereafter, we set ψh (x, t) := eiλt/h uh (x) as a single-spike bound
state of (1.3), where uh is the single-spike solution of (1.5).
In this paper, we want to study the orbital stability of the bound state ψh for the equation (1.3) with critical exponent (1.4). One may regard the bound state ψh as an orbit of (1.3).
From [17], the orbital stability of ψh is defined as follows: For all ² > 0, there exists δ > 0 such
that if kψ0 − uh kH 1 < δ and ψ is a solution of (1.3) in some interval [0, t0 ) with ψ|t=0 = ψ0 , then
ψ(·, t) can be extended to a solution in 0 ≤ t < ∞ and sup0<t<∞ inf s∈R kψ(·, t)−ψh (·, s)kH 1 < ².
Otherwise, the orbit ψh is called orbital unstable.
2
The functions V = V (x) and m = m(x) may play a crucial role on the orbital stability of
ψh . When m ≡ 1 and V is of class (V )a and fulfills other conditions in [28]-[29], the orbital
stability and instability of ψh for the equation (1.3) was established by Lin and Wei [25] if V
has non-degenerate critical points. Under different conditions, e.g., h = 1 and λ is large, results
of the orbital stability problem can be found in [15]. One may also remark that the orbital
stability problem of NLS with inhomogeneous nonlinearity has been investigated in [5] but only
for the subcritical case, i.e., 1 < p < 1 + N4 .
To state our main results, we need to introduce some notations. It is well-known that the
positive solution of
(
∆w − w + wp = 0 in RN ,
(1.6)
w(0) = max w(y) , w(y) → 0 as |y| → +∞ .
y∈RN
is radial [16] and unique [24]. We denote the solution and its linearized operator as w = w(r)
and
L0 := ∆ − 1 + pwp−1 ,
(1.7)
respectively. For the orbital stability of ψh , we set
Lh := h2 ∆ − (V + λ) + m pup−1
h
as the linearized operator of (1.5) with respect to uh and
¸
Z · 2
h
1
1
p+1
2
2
d(λ) =
|∇uh | + (V + λ) uh −
m uh
dx.
2
2
p+1
RN
(1.8)
(1.9)
as the energy of uh . Observe that uh may depend on the variable λ. Assume that d(λ) is
non-degenerate, i.e., d 00 (λ) 6= 0. Let p(d 00 ) = 1 if d 00 > 0; p(d 00 ) = 0 if d 00 < 0, and n(Lh ) be the
number of positive eigenvalues of Lh . According to general theory of orbital stability of bound
states (cf. [17], [18]), ψh is orbital stable if n(Lh ) = p(d 00 ), and orbital unstable if n(Lh ) − p(d 00 )
is odd (see page 309 of [18]). It is remarkable that if both V and m are constant and p = 1 + N4 ,
then d 00 (λ) = 0. Consequently, from now on, we consider the critical exponent p = 1 + N4 and
assume the point x0 as a non-degenerate critical point of the function G defined by (cf. [20],
[37])
£
¤
G(x) := V (x) + λ m−N/2 (x) , ∀x ∈ RN ,
(1.10)
provided V 6≡ 0 and m > 0 in RN . When V ≡ 0 in RN , x0 is set as a non-degenerate critical
point of the function m.
For simplicity, we firstly switch off the potential V and obtain the following result.
Theorem 1.1. Let N ≥ 1 be a positive integer, p = 1 +
the function m = m(x) satisfies
m ∈ C 4 ∩ L∞ ;
4
N
and the potential V ≡ 0. Assume
|m(i) (x)| ≤ Cexp(γ|x|), ∀ x ∈ RN , i = 1, 2, 3, 4,
(1.11)
where γ and C are positive constants, and m(i) (x) are the i-th derivatives of m(x). Let
ψh (x, t) := eiλt/h uh (x) for x ∈ RN and t > 0, where uh is a single-spike solution of (1.5)
3
concentrating at a non-degenerate critical point x0 of m(x). Assume
h
i
m(x0 )∆2 m(x0 ) < CN,1 |∆m(x0 )|2 + CN,2 N k∇2 m(x0 )k22 − |∆m(x0 )|2
£
¤−1
+CN,3 m(x0 )∇(∆m)(x0 ) · ∇2 m(x0 ) ∇(∆m)(x0 ) ,
(1.12)
where
2(N + 2)2
R∞
¡ 2 p¢
rN +1 wp L−1
r w dr
0
0
R∞
N 2 rN +3 wp+1 dr
CN,1 =
,
(1.13)
0
4(N + 2)
CN,2 =
N2
R∞
R∞
rN +1 wp Φ0 dr
0
,
(1.14)
rN +3 wp+1 dr
0
¡ R∞
¢2
(N + 2) rN +1 wp+1 dr
CN,3 =
N
R∞
0
rN −1 wp+1 dr
0
R∞
,
(1.15)
rN +3 wp+1 dr
0
are constants depending only on N . Here Φ0 = Φ0 (r) satisfies

 Φ00 + N − 1 Φ0 − Φ + pwp−1 Φ − 2N Φ − r2 wp = 0, r = |x| ∈ (0, ∞),
0
0
0
0
0
r
r2

Φ0 (0) = Φ00 (0) = 0.
(1.16)
where L0 is defined in (1.7). Then for any λ > 0, ψh is orbitally stable if h is sufficiently small
and x0 is a non-degenerate local maximum point of m(x). Furthermore, for any λ > 0, ψh is
orbitally unstable if h is sufficiently small and the number of positive eigenvalues of the Hessian
matrix ∇2 m(x0 ) is odd.
Remark 1: Theorem 1.1 may include the case that the third order derivatives of the function
m at x0 can be nonzero. When N = 1, x0 = 0 and the function m satisfies m000 (x0 ) = 0 ,
(see (C.2) of [10]), the condition (1.12) of Theorem 1.1 is exactly same as the condition (4.14)
of [10]. For N ≥ 2, G.Fibich and X.-P.Wang considered the function m with radial symmetry,
i.e., m = m(r), r = |x| and m000 (0) = 0 , and studied the orbital stability problem only for
radial perturbations (cf. [12]). Here we study the orbital stability problem for general perturbations including the non-radial perturbations and the case that the function m is not radially
symmetric.
When the potential V is turned on, we may follow the argument of Theorem 1.1 to obtain
Theorem 1.2. Let N ≥ 1 be a positive integer, p = 1+ N4 . Assume both the potential V = V (x)
and the function m = m(x) satisfy the following conditions: there exist positive constants γ
and C such that
V, m ∈ C 2 ∩ L∞ ;
|V (i) (x)|, |m(i) (x)| ≤ Cexp(γ|x|),
4
∀ i = 1, 2 , x ∈ RN ,
(1.17)
where V (i) (x), m(i) (x) are the i-th derivatives of V (x), m(x), respectively. Let ψh (x, t) :=
eiλt/h uh (x) for x ∈ RN and t > 0, where uh is a single-spike solution of (1.5) concentrating at
a non-degenerate critical point x0 of the function G defined in (1.10). Then for any λ > 0, ψh
is orbitally unstable if h is sufficiently small and x0 is a non-degenerate local minimum point
of G such that ∇V (x0 ) 6= 0.
Theorem 1.3. Under the same hypotheses of Theorem 1.2, assume ∇V (x0 ) = 0 and ∆V (x0 ) 6=
0. Let n be the number of negative eigenvalues of the matrix ∇2 G(x0 ). Then for any λ, uh
is orbitally stable if h is sufficiently small and x0 is a non-degenerate local minimum point
of G. Furthermore, for any λ > 0, uh is orbitally unstable if h is sufficiently small and
∆V (x0 )
n − 21 (1 + |∆V
) is even.
(x0 )|
Theorem 1.4. Under the same hypotheses of Theorem 1.2, assume ∇V (x0 ) = 0, ∆V (x0 ) = 0
and (1.11) hold for both V and m. Let n be the number of negative eigenvalues of the matrix
∇2 G(x0 ). Suppose H(x0 ) > 0, where H(x0 ) defined in (4.34) involves the i-th derivatives (for
0 ≤ i ≤ 4) of V and m at x0 . Then for any λ > 0, uh is orbitally stable if h is sufficiently
small and x0 is a non-degenerate local minimum point of G. Furthermore, for any λ > 0, uh
is orbitally unstable if n is odd.
Remark 2: Theorem 1.2-1.4 may include all the cases of values ∇V (x0 ) and ∆V (x0 ) for the
orbital stability problem of (1.3) with critical exponent (1.4). Theorem 1.3 may generalize the
main result of [25] to the case that the function m is a positive and nonconstant function. As
V ≡ 0, Theorem 1.4 coincides with Theorem 1.1 since
i
h
¤
N
N£
∇2 G(x0 ) = m(x0 )− 2 −1 m(x0 )∇2 V (x0 ) −
V (x0 ) + λ ∇2 m(x0 ) ,
2
holds in Theorem 1.4.
The rest of this paper is organized as follows: In Section 2, we switch off the potential V
and study the properties of uh . Then we state the proof of Theorem 1.1 in Section 3. Finally,
the potential V is turned on and we briefly outline the proofs of Theorem 1.2-1.4 in Section 4.
Acknowledgments: The research of the first author is partially supported by a grant from
NCTS and NSC of Taiwan. The research of the second author is partially supported by an
Earmarked Grant from RGC of Hong Kong.
2
Preliminaries
In this section, we study the properties of u£h a single-spike
¤ −N/2bound state of (1.5) concentrated
at a non-degenerate critical point of G(x) := V (x) + λ m
(x) (cf. [20], [37]). Let xh be the
unique local maximum point of uh . So xh → x0 as h → 0.
Let vh (y) := uh (hy + xh ) for all y ∈ RN . Then by (1.5), vh is a positive solution of
£
¤
∆v − V (hy + xh ) + λ v + m(hy + xh )v p = 0.
(2.1)
For notation convenience, we still denote
£
¤
Lh := ∆ − V (hy + xh ) + λ + m(hy + xh )pvhp−1
5
(2.2)
as the linearized operator of the equation (2.1) with respect to the solution vh . As the result
of [37], vh can be written as vh = wxh + φh , where wxh is the unique positive solution of
£
¤
(
∆w − V (xh ) + λ w + m(xh )wp = 0 in RN ,
(2.3)
w(0) = max w(y) , w(y) → 0 as |y| → +∞ ,
y∈RN
and
kφh k∞ → 0
Moreover,
where V := inf RN
as h → 0 .
(2.4)
¡
1−N
1/2 ¢
vh (y) ≤ C|y| 2 exp − V y , ∀ y ∈ RN ,
£
¤
V (x) + λ . From (2.3), it is easy to check that
p
£
¤ 1
1
wxh (y) = V (xh ) + λ p−1 m(xh )− p−1 w( V (xh ) + λy) ,
(2.5)
(2.6)
where w is the positive solution of (1.6).
For the single-spike solution of (1.5), we recall the following result from [36] and [37]:
Lemma 2.1. Assume that there are positive constants γ and C such that
|∇V (x)|, |∇m(x)| ≤ Cexp(γ|x|) ,
Then
∀ x ∈ RN .
Z h
i
1
1
p+1
2
∇m(hy + xh )vh − ∇V (hy + xh )vh dy = 0
p+1
2
(2.7)
(2.8)
RN
for 0 < h < h0 , where h0 is a positive constant depending on γ and λ.
In the rest of this section, for simplicity, we switch off the potential V , i.e., set V ≡ 0. Then
by Lemma 2.1, we obtain the uniqueness of uh as follows:
Lemma 2.2. Suppose (2.7) holds, V ≡ 0 and x0 is a non-degenerate critical point of m. Then
uh is unique.
Proof. Suppose uh,1 and uh,2 are different single-spike solutions of (1.5) concentrating at the
same point x0 . Let v1 (y) := uh,1 (hy + x0 ) and v2 (y) := uh,2 (hy + x0 ). Then both v1 and v2
satisfy
∆v − λv + m(hy + x0 )v p = 0 , for y ∈ RN ,
and v1 , v2 → wx0 uniformly on RN as h → 0. Due to v1 6≡ v2 , we may set
veh :=
v1 − v2
,
kv1 − v2 k ∞
and then veh satisfies
∆e
vh − λe
vh + m(x0 )pwxp−1
veh + [m(hy + x0 ) − m(x0 )]pwxp−1
veh + N (e
vh ) = 0,
0
0
6
(2.9)
£
¤
where N (e
vh ) = m(hy + x0 ) v1p − v2p − pwxp−1
(v
−
v
)
/kv1 − v2 k∞ . Hence by the standard elliptic
1
2
0
PDE theorems on the equation (2.9), we may take a subsequence veh → ve0 , where ve0 solves
∆e
v0 − ve0 + m(x0 )pwxp−1
ve0 = 0.
0
Consequently, there exist constants cj ’s such that
ve0 =
N
X
cj ∂j wx0 .
(2.10)
j=1
Let yh be such that veh (yh ) = ke
vh k∞ = 1 (the same proof applies if veh (yh ) = −1). Then by
the Maximum Principle, we have |yh | ≤ C. On the other hand, as (2.8), we may obtain
Z
Z
p+1
∇m(hy + x0 )v2p+1 dy.
∇m(hy + x0 )v1 dy = 0 =
RN
RN
Thus
Z
RN
³ v p+1 − v p+1 ´
2
veh dy = 0.
∇m(hy + x0 ) 1
v1 − v2
(2.11)
Note that for all i = 1, · · · , N , as h → 0,
N
X
v1p+1 − v2p+1
∂ik m(x0 )yk + o(h) , and
∂i m(hy + x0 ) = h
= (p + 1)wxp0 + o(1) .
v1 − v2
k=1
Hence from (2.10) and (2.11), we may obtain
Z h X
N
N
i
³X
´
∂ik m(x0 )yk (p + 1)wxp0
cj ∂j wx0 dy + o(h)
0=
h
RN
= −h
j=1
k=1
N
X
j=1
Z
wxp+1
dy + o(h) .
0
∂ij m(x0 )cj
RN
Hence by the assumption that ∇2 m(x0 ) is non-degenerate, cj = 0 for j = 1, · · · , N , i.e., ve0 ≡ 0.
This may contradict to the fact that 1 = veh (yh ) → ve0 (y0 ) for some y0 ∈ RN . Therefore, we may
complete the proof of Lemma 2.2.
By Lemma 2.1, we may simplify the proof of [21] and get a shorter proof of the asymptotic
behavior of xh ’s as follows:
Lemma 2.3. Under the same hypotheses of Lemma 2.2,
xh = x0 + o(h)
7
as h → 0 .
(2.12)
Proof. Fix i ∈ {1, · · · , N } arbitrarily. By Taylor’s expansion of ∂i m(x) and ∇m(x0 ) = 0, we
obtain
N
X
∂i m(hy + xh ) =
∂ij m(x0 )(hyj + xh,j − x0,j ) + o(h) + o(|xh − x0 |).
j=1
Hence by Lemma 2.1 and vh = wx0 + o(1), we have
Z
0=
∂i m(hy + xh )vhp+1 dy
=
RN
N
X
Z
wxp+1
dy + o(h) + o(|xh − x0 |)
0
∂ij m(x0 )(xh,j − x0,j )
j=1
RN
Here we have used the fact that
R
RN
yj wxp+1
dy = 0 for j = 1, · · · , N . Using the assumption that
0
∇2 m(x0 ) is non-degenerate, we obtain (2.12).
Following the idea of [25], we may use Lemma 2.3 to show the asymptotic behavior of vh as
follows:
Lemma 2.4. Under the same hypotheses of Lemma 2.2,
vh = wxh + h2 φ2 + o(h2 ) ,
as h → 0 ,
(2.13)
where φ2 satisfies
∆φ2 − λφ2 +
m(xh )pwxp−1
φ2
h
N
1X
∂ij m(x0 )yi yj wxph = 0 , and ∇φ2 (0) = 0.
+
2 i,j=1
(2.14)
Proof. Let φh = vh − wxh . Then it is easy to check that |φh | → 0 uniformly, and φh satisfies
∆φh − λφh + m(hy + xh )pwxp−1
φh + N (φh ) + R(φh ) = 0 , and ∇φh (0) = 0,
h
where
and
(2.15)
h
i
p
p
p−1
N (φh ) = m(hy + xh ) (wxh + φh ) − wxh − pwxh φh ,
h
i
R(φh ) = m(hy + xh ) − m(xh ) wxph .
Note that by Lemma 2.3 and ∇m(x0 ) = 0,
N
h2 X
∂ij m(xh )yi yj + o(h2 )
m(hy + xh ) − m(xh ) =hy · ∇m(xh ) +
2 i,j=1
N
h2 X
∂ij m(x0 )yi yj + o(h2 ).
=
2 i,j=1
8
(2.16)
Now we claim that |φh | ≤ c h2 by contradiction. Suppose that h−2 kφh k∞ → ∞. Let
φeh = φh /kφh k∞ . Then φeh satisfies
N (φh )
R(φh )
∆φeh − λφeh + m(hy + xh )pwxp−1
φeh +
+
= 0.
h
kφh k∞ kφh k∞
(2.17)
Note that by (2.16),
R(φh )
h2
≤C
.
kφh k∞
k φh k∞
(2.18)
Let yh be such that φeh (yh ) = kφeh k∞ = 1 (the same proof applies if φeh (yh ) = −1). Then by
(2.17)−(2.18) and the Maximum Principle, we have |yh | ≤ C. On the other hand, by the usual
elliptic regularity theory, we may take a subsequence φeh → φe0 , where φe0 satisfies
∆φe0 − φe0 + m(x0 )pwxp−1
φe0 = 0 , and ∇φe0 (0) = 0.
0
Hence φe0 ≡ 0. This may contradict to the fact that 1 = φeh (yh ) → φe0 (y0 ) for some y0 . Therefore,
we may complete the claim that | φh | ≤ c h2 .
Now we set φh,2 = φh − h2 φ2 . Then φh,2 = O(h2 ) and satisfies
∆φh,2 − λφh,2 + m(hy + xh )pwxp−1
φh,2 + N (φh,2 ) + R(φh,2 ) = 0 , and ∇φh,2 (0) = 0
h
where
h
i
2
N (φh,2 ) = m(hy + xh ) (wxh + h2 φ2 + φh,2 )p − wxph − pwxp−1
(h
φ
+
φ
)
,
2
h,2
h
and
h
N
i
h
i
h2 X
p
2
R(φh,2 ) = m(hy + xh ) − m(xh ) −
∂ij m(x0 )yi yj wxh + h m(hy + xh ) − m(xh ) pwxp−1
φ2 .
h
2 i,j=1
Thus as for previous argument, we may have φh,2 = o(h2 ) and complete the proof of Lemma 2.4.
As for Proposition 3.1 of [23], one may get two lemmas as follows:
Lemma 2.5. For h small enough, the maps
£
¤
Lxh φ := ∆φ − V (xh ) + λ φ + m(xh )pwxp−1
φ
h
are uniformly invertible from Kx⊥h to Cx⊥h , where
¯Z
½
¾
¯
2
N ¯
⊥
φ∂j wxh dy = 0 , j = 1, · · · , N ⊂ H 2 (RN ),
Kxh = φ ∈ H (R ) ¯
RN
¯Z
½
¾
¯
2
N ¯
⊥
φ∂j wxh dy = 0 , j = 1, · · · , N ⊂ L2 (RN ).
Cxh = φ ∈ L (R ) ¯
RN
9
Lemma 2.6. The map
£
¤
Lx0 φ := ∆φ − V (x0 ) + λ φ + m(x0 )pwxp−1
φ
0
has eigenvalues µj , j = 1, · · · , N + 2 satisfying
µ1 > 0 = µ2 = · · · = µN +1 > µN +2 ≥ · · · ,
where the kernel of Lx0 is spanned by ∂j wx0 , j = 1, · · · , N .
In this section, our main result is the small eigenvalue estimates of Lh given by
Theorem 2.7. Under the same hypotheses of Lemma 2.2, for h small enough, the eigenvalue
problem
L h ϕ h = µh ϕ h
(2.19)
has exactly N eigenvalues µjh , j = 1, · · · , N satisfying
1
1
µ1 ≥ µ1h ≥ µ2h ≥ · · · ≥ µN
h ≥ µN +2 ,
2
2
and
µjh
→ c0 νj , (up to a subsequence)
h2
as h → 0 ,
for j = 1, · · · , N,
(2.20)
where µ1 and µN +2 are defined in Lemma 2.6, νj ’s are the eigenvalues of the Hessian matrix
N
is a positive constant. Furthermore, the corresponding eigenfunctions
∇2 m(x0 ) and c0 = 2m(x
0)
ϕjh ’s satisfy
ϕjh
N
X
£
¤
=
aij + o(1) ∂i wxh + O(h2 ) ,
j = 1, · · · , N ,
(2.21)
i=1
where aj = (a1j , · · · , aN j )T is the eigenvector associated with νj , namely,
∇2 m(x0 )aj = νj aj .
(2.22)
Here o(1) is a small quantity tending to zero and O(1) is a bounded quantity as h goes to zero.
Proof. We may follow the arguments given in Section 5 of [40]. Assume that kϕh kL2 = 1. It
is easy to see that µh → 0 as h → 0, where µh ∈ {µ1h , · · · , µN
h }. Then the corresponding
eigenfunctions ϕh ’s can be written as
ϕh =
N
X
ajh ∂j wxh + ϕ⊥
h,
(2.23)
j=1
⊥
⊥
where ϕ⊥
h ∈ Kxh . Hence by (2.19) and (2.23), ϕh satisfies
⊥
p−1 ⊥
⊥
∆ϕ⊥
h − λϕh + m(xh )pwxh ϕh + R(ϕh ) +
N
P
j=1
ajh Lh ∂j wxh = µh
à N
X
!
ajh ∂j wxh + ϕ⊥
h
,
j=1
(2.24)
10
where
h
i
p−1
p−1
⊥
p−1 ⊥
R(ϕ⊥
)
=
m(hy
+
x
)p(v
−
w
)ϕ
+
m(hy
+
x
)
−
m(x
)
h
h
h pwxh ϕh .
h
xh
h
h
Using (2.16) and Lemma 2.4, we have
Lh ∂j wxh = m(hy +
xh )p(vhp−1
−
wxp−1
)∂j wxh
h
h
i
+ m(hy + xh ) − m(xh ) pwxp−1
∂j wxh = O(h2 ).
h
(2.25)
From Lemma 2.5, the map Lxh = ∆ − λ + m(xh )pwxp−1
is uniformly invertible in the space Kx⊥h .
h
Thus by (2.25) and µh → 0, we have
2
kϕ⊥
h kH 2 ≤ c(h + |µh |)
N
X
|ajh | .
(2.26)
j=1
To estimate µh and ajh ’s, multiplying (2.24) by ∂k wxh and integrating over RN , we may
obtain
Z
Z
Z
N
N
X
X
¡
¢
j
j
⊥
ah
∂j wxh ∂k wxh dy . (2.27)
(Lh ∂j wxh ) ∂k wxh dy = µh
ah
Lh ϕh ∂k wxh dy +
RN
RN
j=1
j=1
RN
⊥
Here we have used the fact that ϕ⊥
h ∈ Kxh . Using (2.25), (2.26), µh = o(1) and integration by
parts, we obtain
Z
Z
¡
¢
⊥
2
Lh ϕh ∂k wxh dy =
ϕ⊥
(2.28)
h Lh ∂k wxh dy = o(h ) ,
RN
and
Z
RN
RN
h2
(Lh ∂j wxh ) ∂k wxh dy =
p+1
Z
RN
wxp+1
dy∂jk m(x0 ) + o(h2 ) ,
h
(2.29)
which we have proved in Appendix A. Substituting (2.28) and (2.29) into (2.27), we may obtain
Z
Z
N
X
¢2
1
µh k ¡
j
p+1
wxh dy
∂jk m(x0 )ah = 2 ah
∂k wxh dy + o(1).
p + 1 RN
h
j=1
RN
T
Since kϕh kL2 = 1, (2.23) implies that ah := (a1h , · · · , aN
h ) is bound. Moreover, by (2.26), ah
does not converge to 0. Thus
µjh
h2
→ c0 νj for j = 1, · · · , N and ah → aj , where
R
N RN wxp+1
dy
N
0
R
=
,
c0 =
2m(x0 )
(p + 1) RN |∇wx0 |2 dy
and aj is the eigenvector corresponding to νj . Here we have use the fact that
Z
Z
N
2
wxp+1
dy ,
m(x0 )
|∇wx0 | dy =
0
N
+
2
N
N
R
R
which can be proved by Pohozeve identity. The rest of the proof follows from a perturbation
result, similar to page 1473-1474 of [40]. We may omit the details here.
11
3
Proof of Theorem 1.1
In this Section, we firstly study the asymptotic expansion of d 00 (λ) as h → 0, and then
complete the proof of Theorem 1.1. To drive the O(h4 ) order terms of d 00 (λ)/hN , we need the
following lemma:
Lemma 3.1. Under the same hypotheses of Lemma 2.2,
xh = x0 + h2 x1 + O(h3 ) ,
where x1 ∈ RN satisfies
as h → 0 ,
(3.1)
R
|y|2 wp+1 dy
N
R
∇2 m(x0 )x1 = − R
∇(∆m)(x0 ).
2N λ wp+1 dy
(3.2)
RN
Proof. By Lemma 2.3 and ∇m(x0 ) = 0, for all i = 1, · · · , N , we have
N
X
∂i m(hy + xh ) =
¡
¢
∂ij m(x0 ) hyj + xh,j − x0,j + O(h2 ).
(3.3)
j=1
Then by (2.8), (3.3) and Lemma 2.4, we have
Z
∂i m(hy + xh )vhp+1 dy
0=
=
RN
N
X
Z
∂ij m(x0 )
j=1
=
N
X
¡
hyj + xh,j − x0,j
RN
¡
¢
∂ij m(x0 ) xh,j − x0,j
j=1
Here we have used the fact that
R
RN
¢h
i
wxp+1
+
O(h)
dy + O(h2 )
h
Z
wxp+1
dy + O(h2 ) .
0
RN
yj wxp+1
dy = 0 for j = 1, · · · , N . Thus xh = x0 + O(h2 ).
h
Consequently, we may set xh = x0 + h2 xh . Then xh = O(1) and by Taylor’s formula of ∂i m(x),
we have
N
X
N
¡
¢ h2 X
2
∂i m(hy + xh ) =
∂ij m(x0 ) hyj + h xh +
∂ijk m(x0 )yj yk + O(h3 ).
2
j=1
j,k=1
Hence by(2.8), (3.4) and Lemma 2.4, we may obtain
2
0 =h
N
X
Z
∂ij m(x0 )xh,j
j=1
2
=h
N
X
j=1
wxp+1
dy
h
RN
RN
Z
dy
wxp+1
0
∂ij m(x0 )xh,j
Z
N
h2 X
+
∂ijk m(x0 ) yj yk wxp+1
dy + O(h3 )
h
2 j,k=1
N
2 X
h
+
2N
RN
12
k=1
Z
∂ikk m(x0 )
RN
|y|2 wxp+1
dy + O(h3 ).
0
(3.4)
Here we have used the fact that
 R

yj wxp+1
dy = 0 , ∀j = 1, · · · , N,

0
N
RR
R
δ

yj yk wxp+1
= Njk
|y|2 wxp+1
dy , ∀j, k = 1, · · · , N.

0
0
RN
RN
Therefore, we may complete the proof because
√
wx0 (y) = λN/4 m(x0 )−N/4 w( λy) .
From Lemma 2.4 and 3.1, we may deduce that
Theorem 3.2. Under the same hypotheses of Lemma 2.2, for h small enough, uh is smooth
h
on λ. Let Rh := ∂u
(hy + xh ). Then
∂λ
Lh Rh − vh = 0.
(3.5)
and
Rh = R0 +
N
X
cjh ∂j wxh + h2 R1 + Rh⊥ ,
(3.6)
j=1
where R0 = λ−1
¡
1
v
p−1 h
¢
+ 12 y · ∇vh , cjh = O(h), Rh⊥ = O(h3 ) and R1 satisfies
∆R1 − λR1 +
m(xh )pwxp−1
R1
h
N
1 X
−
∂ij m(x0 )yi yj wxph = 0 .
2λ i,j=1
(3.7)
Furthermore,
R
¡
¢
∇2 m(x0 ) h−1 ch → −
|y|2 wp+1 dy
RN
R
∇(∆m)(x0 ) ,
2N λ2 wp+1 dy
as h → 0 ,
(3.8)
RN
T
where ch := (c1h , · · · , cN
h ) .
Proof. By Lemma 2.2 and Theorem 2.7, uh is unique and non-degenerate. Consequently, uh is
smooth on λ and Rh satisfies (3.5). Now we decompose Rh as
Rh = R0 +
N
X
cjh ∂j wxh + h2 R1 + Rh⊥ ,
j=1
where Rh⊥ ∈ Kx⊥h . Then Rh⊥ satisfies
Lh Rh⊥
N
£
¤ X
2
+ L h R 0 + h Lh R 1 − v h +
cjh Lh ∂j wxh = 0.
j=1
13
(3.9)
As for the proof of Theorem 2.7, we have
³
kRh⊥ kH 2
2
≤ c kLh R0 + h Lh R1 − vh kL2 +
N
X
´
|cjh |h2 .
(3.10)
j=1
It is easy to check
h
y · ∇m(hy + xh )vhp .
2λ
Hence by Lemma 2.4, 3.1, (3.7) and (3.11), we obtain
Lh R 0 = v h −
Lh R0 + h2 Lh R1 − vh
N
i
h3 h X
1 X
=−
∂ij m(x0 )x1,i yj +
∂ijk m(x0 )yi yj yk wxph + O(h4 ) .
2λ i,j=1
2 i,j,k=1
Consequently, by (3.10),
kRh⊥ kH 2
³
3
≤c h +
N
X
|cjh |h2
(3.11)
(3.12)
´
.
(3.13)
j=1
To estimate cjh ’s, we may multiply (3.9) by ∂k wxh and integrate over RN . Then
Z
Z h
i
⊥
(Lh Rh )∂k wxh dy +
Lh R0 + h2 Lh R1 − vh ∂k wxh dy
RN
N
X
+
j=1
RN
Z
cjh
RN
(Lh ∂j wxh )∂k wxh dy = 0.
Hence by (2.29), (3.14) may imply
· Z
¸
Z
¯ ¯
¯
£
¤
C ¯¯
j
⊥
2
|ch | ≤ 2
(Lh Rh )∂k wxh dy ¯ + ¯
Lh R0 + h Lh R1 − vh ∂k wxh dy ¯ .
h
RN
RN
Using integration by parts and (2.25), we have
Z
Z
⊥
Rh⊥ Lh ∂k wxh dy = kRh⊥ kL2 O(h2 ) .
(Lh Rh )∂k wxh dy =
RN
(3.14)
(3.15)
(3.16)
RN
Therefore, by (3.12), (3.13), (3.15) and (3.16), we may obtain |cjh | = O(h). Consequently,
by (3.13), Rh⊥ = O(h3 ). Thus by (3.16),
Z
(Lh Rh⊥ )∂k wxh dy = O(h5 ).
(3.17)
RN
Hence by (2.29), (3.12) and (3.17), (3.14) gives
Z
N
X
¡
¢
1
p+1
wx0 dy
∂jk m(x0 ) h−1 cjh
p + 1 RN
j=1
Z hX
N
N
i
1
1 X
∂ijl m(x0 )yi yj yl wxph ∂k wxh dy + o(1).
=
∂ij m(x0 )x1,i yj +
2λ RN i,j=1
2 i,j,l=1
14
(3.18)
Using integration by parts, we obtain
 R
δjk R

yj wxph ∂k wxh dy = − p+1
wxp+1
dy ,

h
RRN
RN
δ δ +δjk δil +δlk δij R

yi yj yl wxph ∂k wxh dy = − ik jl N (p+1)
|y|2 wxp+1
dy ,

h
RN
RN
where δ is the Kronecker symbol. Hence by (3.18), |cjh | = O(h) for j = 1, · · · , N . Moreover,
by (3.2), we obtain (3.8) and complete the proof.
Let us now compute d 00 (λ). From (1.9), it is easy to get
Z
1
0
u2h dx
d (λ) =
2
RN
and hence
Z
00
d (λ) =
Z
∂uh
uh
dx = hN
∂λ
RN
vh Rh dy .
(3.19)
RN
Using integration by parts and (3.5), we have
Z
Z
Z
¡ 1
¢
¡ 1
1
N¢
−1
−1
vh R0 dy =
vh λ
vh + y · ∇vh dy = λ
−
vh2 dy = 0 ,
p−1
2
p−1
4
RN
RN
4
.
N
since p = 1 +
d 00 (λ)
=
hN
RN
Hence, by (3.19) and Theorem 3.2, we have
Z
h
vh R0 +
Z
vh
N
hX
Z
Rh
RN
RN
cjh ∂j wxh
2
+ h R1 +
Rh⊥
i
dy
2
+ h R1 +
Rh⊥
i
¡
dy
Z
because
¢
vh R0 dy = 0
RN
N
hX
i
cjh Lh ∂j wxh + h2 Lh R1 + Lh Rh⊥ dy
¡
because Lh Rh = vh
¢
j=1
Z h
=
cjh ∂j wxh
j=1
RN
=
N
X
j=1
RN
=
(3.20)
R0 +
N
X
cjh ∂j wxh
2
+ h R1 +
Rh⊥
j=1
N
ih X
cjh Lh ∂j wxh
2
+ h Lh R1 +
Lh Rh⊥
i
dy .
j=1
Therefore, by (2.25), (3.9) and cjh = O(h),
d 00 (λ)
=
hN
Z
h
i
N
X
R0 vh − Lh R0 dy +
RN
Z
+ h4
Z
cjh ckh
j,k=1
¡
¢
R1 Lh R1 dy + O(h5 ) .
RN
15
¢
¡
∂k wxh Lh ∂j wxh dy
RN
(3.21)
For the integral
R
h
i
R0 vh − Lh R0 dy, by (3.11)and using integration by parts, we have
RN
Z
h
Z
i
λ−1
R0 vh − Lh R0 dy =
RN
RN
1
= 2
2λ
Z
RN
i
¡ 1
¢h h
1
vh + y · ∇vh
y · ∇m(hy + xh )vhp dy
p−1
2
2λ
N
h
i
X
N
2
hy · ∇m(hy + xh ) − h
∂ij m(hy + xh )yi yj vhp+1 dy.
4(N + 2)
i,j=1
Note that by Lemma 2.4, 3.1 and Theorem 3.2, we have
2
hy · ∇m(hy + xh ) − h
N
X
∂ij m(hy + xh )yi yj
i,j=1
N
N
h4 X
h3 X
∂ijk m(xh )yi yj yk −
∂ijkl m(xh )yi yj yk yl + o(h4 ) ,
=hy · ∇m(xh ) −
2 i,j,k=1
3 i,j,k,l=1
and
vhp = wxph + h2 pwxp−1
φ2 + O(h3 ) .
h
(3.22)
Hence
Z
h
i
RN
Z h
N
i
h4 X
−
∂ijkl m(xh )yi yj yk yl wxp+1
dy + o(h4 )
h
3 i,j,k,l=1
RN
Z
h4
−2
λ
|y|4 wxp+1
dy∆2 m(x0 ) + o(h4 )
=−
h
8(N + 2)2
RN
Z
4
h
−3
−N
−1
=−
λ m(x0 ) 2
|y|4 wp+1 dy∆2 m(x0 ) + o(h4 ) . (3.23)
8(N + 2)2
N
R0 vh − Lh R0 dy =
λ−2
8(N + 2)
RN
Here we have used the following identities:
 R
R

yi wxp+1
dy =
yi yj yk wxp+1
dy = 0 , for all i, j, k = 1, · · · , N ;

h
h

N
N

R
R


 R y y y y wp+1 dy = 0 , if y y y y is an odd function on one of its variate ;

i j k l xh
i j k l

RRN
R
3
4 p+1

y
w
dy
=
dy , for all i = 1, · · · , N ;
|y|4 wxp+1

i
x
N
(N
+2)
h
h
 N

RR
RN R




yi2 yj2 wxp+1
dy = N (N1+2) |y|4 wxp+1
dy , for all i 6= j ,

h
h
RN
RN
which can be proved by polar coordinates.
16
For the sum
N
P
j,k=1
N
X
cjh ckh
R
¢
¡
∂k wxh Lh ∂j wxh dy, we may use (2.29) and (3.8) to get
RN
Z
cjh ckh
RN
j,k=1
h4
=
p+1
∂k wxh (Lh ∂j wxh ) dy
Z
dy
wxp+1
h
RN
N
X
(h−1 cjh )(h−1 ckh )∂jk m(x0 ) + o(h4 )
j,k=1
¡R
4
=
N
h
λ−3 m(x0 )− 2 −1
8N (N + 2)
¢2
|y|2 wp+1 dy
£ 2
¤−1
RN
R
∇(∆m)(x
)
·
∇
m(x
)
∇(∆m)(x0 ) + o(h4 ).
0
0
p+1
w dy
RN
(3.24)
For the integral h4
R
¡
¢
1
R1 Lh R1 dy, by (3.7), it is obvious that R1 (λ− 2 y) satisfies
RN
∆R − R + pw
p−1
N
X
1 N −2
−N
−1
4
4
m(xh )
R− λ
∂ij m(x0 )yi yj wp = 0 .
2
i,j=1
(3.25)
Hence
Z
4
h
RN
¡
¢
Z
4
R1 Lh R1 dy = h
¡
¢
R1 Lxh R1 dy + O(h6 )
RN
Z
N
X
¡
¢
h4 −3
−2
−N
2
∂ij m(x0 )∂kl m(x0 ) yi yi wp L−1
yk yl wp dy + O(h6 )
= λ m(x0 )
0
4
i,j,k,l=1
RN
Z
4
N
h
2 p
= 2 λ−3 m(x0 )− 2 −2 |∆m(x0 )|2 r2 wp L−1
0 (r w )dy
4N
RN
Z
4
h
−3
−N
−2
2
2
+
λ m(x0 ) 2 k∇ m(x0 )k2 r2 wp Φ0 (r)dy
2N (N + 2)
RN
Z
N
h4
−
λ−3 m(x0 )− 2 −2 |∆m(x0 )|2 r2 wp Φ0 (r)dy + O(h6 ).
2
2N (N + 2)
RN
17
(3.26)
Here k∇2 m(x0 )k22 =
N
P
i,j=1
m2ij (x0 ) and we have used the following identities:
Z
2
2
p
yN
wp L−1
0 (yN w )dy
1
= 2
N
RN
Z
2
r w
RN
Z
2
p −1 2
p
yN
−1 w L0 (yN w )dy
RN
1
= 2
N
p
2 p
L−1
0 (r w )dy
Z
r2 wp Φ0 (r)dy ,
RN
Z
2
r w
p
2 p
L−1
0 (r w )dy
RN
Z
2(N − 1)
+ 2
N (N + 2)
p
yN −1 yN wp L−1
0 (yN −1 yN w )dy =
2
− 2
N (N + 2)
Z
r2 wp Φ0 (r)dy ,
(3.28)
RN
Z
1
N (N + 2)
(3.27)
r2 wp Φ0 (r)dy ,
(3.29)
RN
RN
where Φ0 satisfies (1.16), which we have proved in Appendix B.
Therefore, combining (3.21), (3.23), (3.24) and (3.26), we obtain
d 00 (λ)
+ o(h4 )
hN
Z
h4
−3
−N
−1
2
=−
λ m(x0 )
|y|4 wp+1 dy∆2 m(x0 )
2
8(N + 2)
RN
¢2
|y|2 wp+1 dy
£
¤−1
RN
R
∇(∆m)(x0 ) · ∇2 m(x0 ) ∇(∆m)(x0 )
p+1
w dy
¡R
+
N
h4
λ−3 m(x0 )− 2 −1
8N (N + 2)
RN
4
N
h −3
+
λ m(x0 )− 2 −2 |∆m(x0 )|2
2
4N
h
Z
¡ 2 p¢
|y|2 wp L−1
|y| w dy
0
RN
i
h
−2
2
2
2
−3
−N
2
+
N
k∇
m(x
)k
−
|∆m(x
)|
λ
m(x
)
0 2
0
0
2N 2 (N + 2)
4
Z
|y|2 wp Φ0 (|y|)dy.
RN
Consequently,
N
¡
¢
8(N + 2)2 m(x0 ) 2 +2 λ3 00
R
d (λ) =CN,1 |∆m(x0 )|2 + CN,2 N k∇2 m(x0 )k22 − |∆m(x0 )|2
4
p+1
N
+4
|y| w dy
h
RN
h
i
£
¤−1
+ CN,3 m(x0 ) ∇(∆m)(x0 ) · ∇2 m(x0 ) ∇(∆m)(x0 )
− m(x0 )∆2 m(x0 ) + o(h) ,
where CN,1 , CN,2 , CN,3 are constants given by (1.13), (1.14), (1.15), respectively.
Now we may prove Theorem 1.1 as follows: Suppose that x0 is a non-degenerate local
maximum point of the function m(x), then the Hessian matrix ∇2 m(x0 ) of m at x0 is negative
definite. By Theorem 2.7, we have n(Lh ) = 1. On the other hand, we have p(d 00 ) = 1. Thus ψh
is orbital stable by the orbital stability criteria of [17]-[18]. For orbital instability, we denote
the number of positive eigenvalues of the Hessian matrix ∇2 m(x0 ) by n. Then by Theorem 2.7,
we obtain n(Lh ) = n + 1. On the other hand, we have p(d 00 ) = 1. Thus by the instability
criteria of [18], we conclude that ψh is orbital unstable if n is odd. This may complete the proof
of Theorem 1.1.
18
4
Proof of Theorem 1.2-1.4
In this section, we may follow the argument in Section 2 and 3 to prove Theorem 1.2-1.4.
Let vh (y) := uh (hy + xh ), where uh is a single-spike bound state of (1.5) with a unique local
maximum point at xh . Then vh satisfies
h
i
∆ vh − V (hy + xh ) + λ vh + m(hy + xh )vhp = 0 in RN .
(4.1)
Suppose (2.7) hold. By (2.8) and [37], we have
m(x0 )∇V (x0 ) =
N
[V (x0 ) + λ] ∇m(x0 ) ,
2
(4.2)
so x0 may depend on λ. Note that by (4.2), ∇m(x0 ) = 0 if and only if ∇V (x0 ) = 0. By direct
computation on the function G,
h
N
N
∂ij G(x0 ) = m(x0 )− 2 −1 m(x0 )∂ij V (x0 ) + (1 − )∂i V (x0 )∂j m(x0 )
2
i
N
− [V (x0 ) + λ] ∂ij m(x0 ) .
2
In particular, if ∇m(x0 ) = 0, then
h
i
¤
N
N£
∇2 G(x0 ) = m(x0 )− 2 −1 m(x0 )∇2 V (x0 ) −
V (x0 ) + λ ∇2 m(x0 ) .
2
Using the identity (2.8), one may follow the arguments of Lemma 2.2-2.4 to get the uniqueness of uh and
xh =x0 + o(h) ;
vh =wxh + hφ1 + h2 φ2 + o(h2 ) ,
(4.3)
(4.4)
where φ1 and φ2 satisfy ∇φ1 (0) = ∇φ2 (0) = 0 ,
∆φ1 − [V (x0 ) + λ] φ1 + m(x0 )pwxp−1
φ1 − y · ∇V (x0 )wx0 + y · ∇m(x0 )wxp0 = 0 ,
0
(4.5)
and
N
£
¤
1X
∆φ2 − V (xh ) + λ φ2 + m(xh )pwxp−1
φ
−
y
·
∇V
(x
)φ
−
∂ij V (x0 )yi yj wxh
2
0
1
h
2 i,j=1
+y·
φ1
∇m(x0 )pwxp−1
h
N
1X
1
+
φ21 = 0 .
∂ij m(x0 )yi yj wxph + m(x0 )p(p − 1)wxp−2
h
2 i,j=1
2
(4.6)
Here we have used the hypothesis that x0 is a non-degenerate point of the function G. And the
only difference in the proof is that we need to estimate the term
Z
Z
1
1
p+1
∇m(x0 ) vh dy − ∇V (x0 ) vh2 dy ,
p+1
2
RN
RN
19
to estimate which one may use the following Pohozaev identity (cf. [33])
Z h
i
2
h
m(hy + xh ) +
y · ∇m(hy + xh ) vhp+1 dy
N +2
p+1
RN
Z h
i
h
=
V (hy + xh ) + λ + y · ∇V (hy + xh ) vh2 dy .
2
RN
For the small eigenvalue estimates of Lh , one may generalize the idea of Theorem 2.7 to get
Theorem 4.1. For h small enough, the eigenvalue problem
L h ϕ h = µh ϕ h
(4.7)
has exactly N eigenvalues µjh , j = 1, · · · N satisfying
1
1
µ1 ≥ µ1h ≥ µ2h ≥ · · · ≥ µN
h ≥ µN +2 ,
2
2
(4.8)
and
µjh
→ c0 νj ,
h2
as h → 0 ,
for j = 1, · · · N ,
(4.9)
where µ1 and µN +2 are defined Lemma 2.6, νj ’s are the eigenvalues of the Hessian matrix
N
2
0)
is a negative constant. Furthermore, the corresponding eigenfunc∇2 G(x0 ), and c0 = − Vm(x
(x0 )+λ
tions ϕjh ’s satisfy
ϕjh
N
X
£
¤¡
¢
=
aij + o(1) ∂i wxh + hψi + O(h2 ) ,
j = 1, · · · , N ,
(4.10)
i=1
where each ψi is the solution of
h
i
∆ψi − V (xh ) + λ ψi + m(xh )pwxp−1
ψi
h
h
i
p−2
+ − y · ∇V (xh ) + y · ∇m(xh )pwxp−1
+
m(x
)p(p
−
1)w
φ
h
1 ∂i wxh = 0 ,
xh
h
(4.11)
and aj = (a1j , · · · , aN j )T is the eigenvector corresponding to νj , namely,
∇2 G(x0 )aj = νj aj .
(4.12)
Remark: To prove it, one may follow the arguments in the proof of Theorem 2.7 and use the
following identity
Z
Z
¡
¢
h2
∂k wxh Lh ∂j wxh + hψj dy = −
wp+1 dy∂jk G(x0 ) + o(h2 ),
(4.13)
N +2
RN
RN
20
which replaces (2.29) (see Appendix C). The main difference between Theorem 2.7 and 4.1 is
the solution ψi of (4.11) which comes from
h
Lh ∂i wxh = h − y · ∇V (x0 ) + y · ∇m(x0 )pwxp−1
h
i
+ m(x0 )p(p − 1)wxp−2
φ1 ∂i wxh + O(h2 ).
(4.14)
h
Since the potential function V is nonzero, then x0 may depend on λ and the asymptotic
expansion of d 00 (λ) becomes more complicated. Indeed, when m ≡ 1 and ∆V (x0 ) 6= 0, the
result in [25] shows that the effect of potential function V on d 00 (λ) is O(h2 ). On the other
hand, when V ≡ 0 and condition (1.12) holds, the effect of m on d 00 (λ) is O(h4 ) (see Section 3).
Generally, when both m and V are not constant, we may show
(I) The effect of V and m on d 00 (λ) is O(1) if ∇V (x0 ) 6= 0 (see Theorem 1.2);
(II) The effect of V and m on d 00 (λ) is O(h2 ) if ∇V (x0 ) = 0 and ∆V (x0 ) 6= 0 (see Theorem 1.3);
(III)The effect of V and m on d 00 (λ) is O(h4 ) if ∇V (x0 ) = 0 , ∆V (x0 ) = 0 and some local
condition hold (see Theorem 1.4).
Now we divide three cases to prove these results.
Case I: ∇V (x0 ) 6= 0.
h
Let Rh := ∂u
(hy + xh ). Then (3.5) and (3.20) hold. Hence one may apply the idea of
∂λ
Theorem 3.2 to get
Rh =
N
X
¡
¢
cih ∂i wxh + hψi + R0 + Rh⊥ ,
(4.15)
i=1
where as h → 0, ch =
(c1h , · · ·
, cN
h )
satisfies
∇2 G(x0 )(hch ) → −
and
N
N
m(x0 )− 2 −1 ∇m(x0 ) ,
2
(4.16)
£
¤−1 ¡ 1
¢
1
R0 = V (xh ) + λ
vh + y · ∇vh , Rh⊥ = O(h) .
p−1
2
Thus
d 00 (λ)
=
hN
Z
Z
vh Rh dy =
RN
RN
Z
=
vh
N
X
Z
Rh
RN
=
N
X
¡
¡
¢
∂i wxh + hψi + R0 +
Rh⊥
i
dy
i=1
Z
³
¢
∂i wxh + hψi dy + O(h)
because
´
vh R0 dy = 0
RN
¡
¢
cih Lh ∂i wxh + hψi dy + O(h)
³
´
because Lh Rh = vh
i=1
Z hX
N
RN
cih
cih
i=1
RN
=
vh
N
hX
(4.17)
ckh
¡
¢
∂k wxh + hψk + R0 +
Rh⊥
N
iX
i=1
k=1
21
¡
¢
cih Lh ∂i wxh + hψi dy + O(h) .
Therefore, by (4.11), (4.14), (4.13), (4.16) and (4.17), we obtain
Z
£ 2
¤−1
N2
d 00 (λ)
−N −2
p+1
=
−
m(x
)
w
dy∇m(x
)
·
∇
G(x
)
∇m(x0 ) + O(h) .
0
0
0
hN
4(N + 2)
(4.18)
RN
Consequently, if x0 is a non-degenerate local minimum point of G, then the Hessian matrix
∇2 G(x0 ) is positive definite. By Theorem 4.1, we have n(Lh ) = 1. On the other hand,
by (4.18), we have p(d 00 ) = 0. Thus we complete the proof of Theorem 1.2 by the orbital
instability criteria of [17]-[18].
Case II: ∇V (x0 ) = 0 and ∆V (x0 ) 6= 0.
Firstly, note that in this case, φ1 ≡ 0 and ψi ≡ 0. Then one may apply the idea of Lemma 3.1
and Theorem 3.2 to obtain
xh =x0 + h2 x1 + O(h3 ) ;
Rh =R0 +
N
X
(4.19)
cjh ∂j wxh + h2 R1 + Rh⊥ ,
(4.20)
j=1
where x1 ∈ R
N
satisfies
R
∇2 G(x0 )x1 = −
|y|2 w2 dy

¤−1
N  N
N + 2£

V (x0 ) + λ m(x0 )− 2 R R p+1  ∇(∆V )(x0 )
4N
w dy
R
2

p+1
RN
|y| w dy
N
N
1


R
+ m(x0 )− 2 −1  R p+1
 ∇(∆m)(x0 ) ,
4
w dy
(4.21)
RN
R1 satisfies
£
¤
∆R1 − V (xh ) + λ R1 + m(xh )pwxp−1
R1
h
N
N
i
£
¤−1 h X
1X
+ V (xh ) + λ
∂ij V (x0 )yi yj wxh −
∂ij m(x0 )yi yj wxph = 0 ,
2 i,j=1
i,j=1
Rh⊥ = O(h3 ) and cjh = O(h) for j = 1, · · · , N . Moreover, ch := (c1h , · · · , cN
h ) satisfies
¡
¢
∇2 G(x0 ) h−1 ch = c0 + o(1) ,
where
(4.22)
(4.23)
£
¤−1
N
c0 = − V (x0 ) + λ m(x0 )− 2 ∇2 V (x0 )x1
R

|y|2 w2 dy
¤−2
N  N
N + 2£

V (x0 ) + λ m(x0 )− 2 R R p+1  ∇(∆V )(x0 )
−
2N
w dy
RN
2

|y| wp+1 dy
¤−1
N
1£

 N
+ V (x0 ) + λ m(x0 )− 2 −1 R R p+1
 ∇(∆m)(x0 ).
4
w dy
R
RN
22
(4.24)
Hence
d 00 (λ)
=
hN
Z
Z
h
vh Rh dy =
vh R0 +
Z
=
vh
N
hX
i
cjh ∂j wxh + h2 R1 + Rh⊥ dy
j=1
RN
RN
N
X
cjh ∂j wxh
2
+ h R1 +
Rh⊥
i
Z
³
dy
because
j=1
RN
Z
Rh
=
´
vh R0 dy = 0
RN
N
hX
i
cjh Lh ∂j wxh + h2 Lh R1 + Lh Rh⊥ dy
³
´
because Lh Rh = vh
j=1
RN
Z h
=
R0 +
N
X
ckh ∂k wxh
2
+ h R1 +
Rh⊥
N
ih X
j=1
k=1
RN
i
cjh Lh ∂j wxh + h2 Lh R1 + Lh Rh⊥ dy .
Therefore, by (4.11), (4.14) and (4.20), we obtain
d 00 (λ)
=
hN
Z
Z
N
X
£
¤
¡
¢
j k
R0 vh − Lh R0 dy +
ch ch ∂k wxh Lh ∂j wxh dy
RN
j,k=1
Z
¡
+ h4
RN
¢
R1 Lh R1 dy + O(h5 ) .
(4.25)
RN
For the integral
R
£
¤
R0 vh − Lh R0 dy, by direct computation, we have
RN
i
£
¤−1 h
h
vh − Lh R0 = − V (xh ) + λ
V (hy + xh ) − V (xh ) + y · ∇V (hy + xh )
2
£
¤
h
−1
+ V (xh ) + λ y · ∇m(hy + xh ).
2
Thus by (4.4), (4.19) and (2.6), we obtain
Z
Z
£
¤−3
¤
h2 £
−N
2
R0 vh − Lh R0 dy =
V (x0 ) + λ m(x0 )
|y|2 w2 dy∆V (x0 ) + O(h4 ) .
2N
RN
(4.26)
(4.27)
RN
N
P
For the sum
j,k=1
1, · · · , N , we have
cjh ckh
R
RN
¡
¢
∂k wxh Lh ∂j wxh dy, by (4.11), (4.14) and cjh = O(h) for j =
N
X
j,k=1
Z
cjh ckh
¢
¡
∂k wxh Lh ∂j wxh dy = O(h4 ) .
(4.28)
RN
Combining (4.27), (4.28) and (4.25), we obtain
¤−3
N
d 00 (λ)
h2 £
V (x0 ) + λ m(x0 )− 2
=
N
h
2N
Z
|y|2 w2 dy∆V (x0 ) + O(h4 ) .
RN
23
(4.29)
∆V (x0 )
Consequently, by (4.29), we have p(d 00 ) = 12 (1 + |∆V
). On the other hand, by Theorem 4.1,
(x0 )|
we have n(Lh ) = n + 1. Thus we complete the proof of Theorem 1.3 by the orbital stability
and instability criteria of [17]-[18].
Case III: ∇V (x0 ) = 0 , ∆V (x0 ) = 0.
In this case, we shall use
and (4.25) to compute the O(h4 ) term of d 00 (λ)/hN .
h (4.24), (4.21)
i
R
For the integral
R0 vh − Lh R0 dy, by (4.26) and integration by parts, we obtain
RN
Z
h
i
R0 vh − Lh R0 dy
RN
£
¤−2
= − V (xh ) + λ
Z
¡ 1
¢£
¤
1
h
vh + y · ∇vh V (hy + xh ) − V (xh ) + y · ∇V (hy + xh ) vh dy
p−1
2
2
RN
£
¤−2
+ V (xh ) + λ
Z
¡ 1
¢h
1
vh + y · ∇vh y · ∇m(hy + xh )vhp dy
p−1
2
2
RN
=
¤−2
1£
V (xh ) + λ
8
Z h
3hy · ∇V (hy + xh ) + h2
N
X
i
∂ij V (hy + xh )yi yj vh2 dy
i,j=1
RN
£
¤−2
N
+
V (xh ) + λ
8(N + 2)
Z h
2
hy · ∇m(hy + xh ) − h
i
N
X
∂ij m(hy + xh )yi yj vhp+1 dy .
i,j=1
RN
Hence by (4.19), (4.20) and Taylor’s formulas of V and m, we have
Z
i
h
R0 vh − Lh R0 dy
RN
¤−2
1£
= V (xh ) + λ
8
Z h
4h
4
+ 4h
N
X
∂ij V
(x0 )yi yj wx2h
Vijk (x0 )x1,i yj yk wx2h
4
+h
Z h
−
RN
N
P
j,k=1
N
X
j,k=1
Z
cjh ckh
∂ij V (x0 )yi yj wxh φ2
N
X
i
Vijkl (x0 )yi yj yk yl wx2h dy
i,j,k,l=1
£
¤−2
N
V (xh ) + λ
8(N + 2)
For the sum
N
X
i,j=1
i,j,k=1
+
4
+ 8h
i,j=1
RN
N
X
2
cjh ckh
R
N
i
h4 X
dy + o(h4 ).
∂ijkl m(x0 )yi yj yk yl wxp+1
h
3 i,j,k,l=1
(4.30)
¢
¡
∂k wxh Lh ∂j wxh dy, by (4.13) and (4.23), we obtain
RN
¡
∂k wxh Lh ∂j wxh
¢
h4
dy = −
N +2
RN
Z
wp+1 dy∇2 G(x0 )c0 · c0 + o(h4 ) .
RN
24
(4.31)
¡
¢
¡
R1 Lh R1 dy, by (4.22), R1 √
R
For the integral
RN
∆R − R + pw
p−1
y
V (xh )+λ
¢
satisfies
N
X
£
¤ N4 −3
−N
R + V (xh ) + λ
m(xh ) 4
∂ij V (x0 )yi yj w
i,j=1
−
Hence
Z
¡
N
X
¤ N −2
N
1£
V (xh ) + λ 4 m(xh )− 4 −1
∂ij m(x0 )yi yj wp = 0 .
2
i,j=1
(4.32)
Z
¢
R1 (Lxh R1 )dy + O(h2 )
R1 Lh R1 dy =
RN
RN
Z
N
X
£
¤−5
−N
= V (xh ) + λ m(xh ) 2
∂ij V (x0 )Vkl (x0 ) yi yj wL−1
0 (yk yl w)dy
i,j,k,l=1
RN
Z
N
X
£
¤−4
p
−N
−1
− V (xh ) + λ m(xh ) 2
∂ij V (x0 )mkl (x0 ) yi yj wL−1
0 (yk yl w )dy
i,j,k,l=1
RN
N
X
¤−3
N
1£
+ V (xh ) + λ m(xh )− 2 −2
∂ij m(x0 )mkl (x0 )
4
i,j,k,l=1
Z
p
2
yi yj wp L−1
0 (yk yl w )dy + O(h ).
RN
As in Section 3, we have used the following identities:
Z
Z
N
X
1
2
|y|2 wx2h dy∆V (x0 ) = 0 ,
∂ij V (x0 ) yi yj wxh dy =
N
i,j=1
RN
RN
N
X
Z
yi yj wxh φ2 dy
∂ij V (x0 )
i,j=1
RN
Z
N
X
¤−3
1£
−N
= V (xh ) + λ m(xh ) 2
∂ij V (x0 )Vkl (x0 ) yi yj wL−1
0 (yk yl w)dy
2
i,j,k,l=1
RN
Z
N
X
¤−2
1£
−1
−N
p
2
− V (xh ) + λ m(xh )
∂ij V (x0 )mkl (x0 ) yi yj wL−1
0 (yk yl w )dy ,
2
i,j,k,l=1
RN

















N
P
R
Vijk (x0 )x1,i
i,j,k=1
N
P
i,j,k,l=1
N
P
i,j,k,l=1
RN
Vijkl (x0 )
R
RN
∂ijkl m(x0 )
yj yk wx2h dy =
yi yj yk yl wx2h =
R
RN
1
N
RN
|y|2 wx2h dy∇(∆V )(x0 ) · x1 ,
3
N (N +2)
yi yj yk yl wxp+1
=
h
25
R
R
RN
3
N (N +2)
(4.33)
|y|4 wx2h dy∆2 V (x0 ) ,
R
RN
|y|4 wxp+1
dy∆2 m(x0 ) ,
h











R
2(N −1) R
2
r2 wL−1
(r
w)dy
+
r2 wΦ1 (r)dy ,
0
N 2 (N +2)
N
N
N
RR
R R
R R
−1 2
−1 2
1
2
2
2
yN
wL
(y
w)dy
=
r
wL
(r
w)dy
−
r2 wΦ1 (r)dy ,
0
0
−1
N
N2
N 2 (N +2)
RRN
RN
RN
R 2
1
yN −1 yN wL−1
(y
y
w)dy
=
r
wΦ
(r)dy
,
N −1 N
1
0
N (N +2)
2
2
yN
wL−1
0 (yN w)dy =
1
N2
R
RN
RN











R
2(N −1) R
2 p
r2 wL−1
r2 wΦ0 (r)dy ,
0 (r w )dy + N 2 (N +2)
N
N
N
RR
R R
R R
−1 2
−1 2 p
2
1
2
p
2
yN
(y
(r
w
)dy
−
wL
w
)dy
=
r
wL
r2 wΦ0 (r)dy ,
0
0
−1
N
N2
N 2 (N +2)
RRN
RN
RN
R 2
−1
1
p
yN −1 yN wL0 (yN −1 yN w )dy = N (N +2) r wΦ0 (r)dy ,
2
2
p
yN
wL−1
0 (yN w )dy =
1
N2
R
RN
RN
where Φ0 , Φ1 satisfy
(
Φ000 + N r−1 Φ00 − Φ0 + pwp−1 Φ0 −
Φ0 (0) = Φ00 (0) = 0 ,
2N
Φ
r2 0
− r2 wp = 0, r ∈ (0, ∞) ,
2N
Φ
r2 1
− r2 w = 0, r ∈ (0, ∞) ,
and
(
Φ001 + N r−1 Φ01 − Φ0 + pwp−1 Φ1 −
Φ1 (0) = Φ01 (0) = 0 ,
which can be proved as in Appendix B.
Therefore, combining (4.25), (4.30), (4.31) and (4.33), we obtain
d 00 (λ)
+ o(h4 ) = H2 (x0 ) + H3 (x0 ) + H4 (x0 ) ≡ H(x0 ) ,
hN +4
(4.34)
where
£
¤−5
N
3
H2 (x0 ) =
V (x0 ) + λ m(x0 )− 2
N (N + 2)
Z
|y|2 wΦ1 (|y|)dyk∇2 V (x0 )k22
N
−
+
+
−
R
Z
£
¤−4
3
−1
−N
V (x0 ) + λ m(x0 ) 2
|y|2 wΦ0 (|y|)dy∇2 V (x0 ) · ∇2 m(x0 )
N (N + 2)
RN
Z
¤−3
N
1 £
2 p
2
V (x0 ) + λ m(x0 )− 2 −2 |y|2 wp L−1
0 (|y| w )dy|∆m(x0 )|
2
4N
RN
Z
£
¤
1
−3
−N
−2
V (x0 ) + λ m(x0 ) 2
|y|2 wp Φ0 (|y|)dyk∇2 m(x0 )k22
2N (N + 2)
RN
Z
£
¤
1
−3
−N
−2
V (x0 ) + λ m(x0 ) 2
|y|2 wp Φ0 (|y|)dy|∆m(x0 )|2 ,
(4.35)
2
2N (N + 2)
RN
26
Z
¤−3
1 £
−N
2
|y|2 w2 dy∇(∆m)(x0 ) · x1
H3 (x0 ) =
V (x0 ) + λ m(x0 )
2N
RN
Z
£
¤−1
1
−
wp+1 dyc0 · ∇2 G(x0 ) c0 ,
N +2
(4.36)
RN
£
¤−4
N
3
H4 (x0 ) =
V (x0 ) + λ m(x0 )− 2
8N (N + 2)
Z
|y|4 w2 dy∆2 V (x0 )
RN
£
¤−3
N
1
−
V (x0 ) + λ m(x0 )− 2 −1
2
8(N + 2)
Z
|y|4 wp+1 dy∆2 m(x0 ) .
(4.37)
RN
Consequently, p(d 00 ) = 1 if H(x0 ) > 0, where H(x0 ) defined in (4.34) involves the i-th derivatives (for 0 ≤ i ≤ 4) of V and m at x0 . On the other hand, by Theorem 4.1, we have
n(Lh ) = n + 1. Thus we complete the proof of Theorem 1.4 by the orbital stability and
instability criteria of [17]-[18].
5
Appendix A
In this Appendix we will prove (2.29) of Section 2, i.e., we shall prove
Z
Z
h2
wxp+1
dy∂jk m(x0 ) + o(h2 ) .
(Lh ∂j wxh ) ∂k wxh dy =
h
p
+
1
N
N
R
R
(5.1)
Proof. Note that by Lemma 2.3 and 2.4, we obtain
h
i
∂j wxh + m(hy + xh )p(vhp−1 − wxp−1
)∂j wxh
Lh ∂j wxh = m(hy + xh ) − m(xh ) pwxp−1
h
h
N
h2 X
=
∂il m(x0 )yi yl pwxp−1
∂j wxh + h2 m(xh )p(p − 1)wxp−2
φ2 ∂j wxh + o(h2 ) .
h
h
2 i,l
R
Hence we may write the integral RN (Lh ∂j wxh ) ∂k wxh dy as follows:
Z
(Lh ∂j wxh ) ∂k wxh dy = I1 + I2 + o(h2 ),
(5.2)
RN
where
Z
N
h2 X
∂j wxh ∂k wxh dy ,
I1 =
∂il m(x0 )
yi yl pwxp−1
h
2 i,l=1
RN
Z
2
I2 =h
m(xh )p(p − 1)wxp−2
φ2 ∂j wxh ∂k wxh dy .
h
(5.3)
(5.4)
RN
Note that from (2.3), we have
h
i
∂j wxh ∂k wxh = 0 .
∆ − λ + m(xh )pwxp−1
∂jk wxh + m(xh )p(p − 1)wxp−2
h
h
27
(5.5)
Hence by (2.14), (5.4) and (5.5), we may use integration by parts to get
Z
h
i
2
I2 = − h
φ2 ∆ − λ + m(xh )pwxp−1
∂jk wxh dy
h
N
R
Z
h
i
2
p−1
=−h
∂jk wxh ∆ − λ + m(xh )pwxh φ2 dy
RN
N
2 X
h
=
2
Z
∂il m(x0 )
i,l=1
RN
yi yl wxph ∂jk wxh dy
Z
N
∂(yi yl wxph )
h2 X
=−
∂il m(x0 )
∂k wxh dy
2 i,l=1
∂yj
RN
Z
Z
N
h2 X
p−1
2
∂il m(x0 )
yi yl pwxh ∂j wxh ∂k wxh dy − h ∂jk m(x0 )
yk wxph ∂k wxh dy
=−
2 i,l=1
N
N
R
R
Z
Z
N
h2 X
h2
p−1
=−
∂il m(x0 )
yi yl pwxh ∂j wxh ∂k wxh dy +
∂jk m(x0 )
wxp+1
dy .
h
2 i,l=1
p+1
RN
RN
(5.6)
Combining (5.2), (5.3) and (5.6), we obtain (5.1).
6
Appendix B
In this Appendix, we shall prove (3.27), (3.28) and (3.29) of Section 3, i.e., we will prove
Z
Z
Z
1
2(N − 1)
2
p −1 2
p
2 p −1 2 p
yN w L0 (yN w )dy = 2
r2 wp Φ0 (r)dy ,
(6.1)
r w L0 (r w )dy + 2
N
N (N + 2)
RN
RN
RN
Z
Z
Z
2
1
2
2 p −1 2 p
p −1 2
p
yN −1 w L0 (yN w )dy = 2
r w L0 (r w )dy − 2
r2 wp Φ0 (r)dy ,
(6.2)
N
N (N + 2)
RN
RN
RN
Z
Z
1
p
yN −1 yN wp L−1
r2 wp Φ0 (r)dy ,
(6.3)
0 (yN −1 yN w )dy =
N (N + 2)
RN
RN
where r := |y| and Φ0 satisfies

 Φ00 + N − 1 Φ0 − Φ + pwp−1 Φ − 2N Φ − r2 wp = 0, r ∈ (0, ∞),
0
0
0
0
0
r
r2

Φ0 (0) = Φ00 (0) = 0.
Proof. From (6.4), it is easy to check that
¸
·
h y
i
2
1
1 −1 2 p
yN
N −1 yN
2
wp , and L0 Φ0
= yN −1 yN wp .
L0 Φ0 2 + L0 (r w ) − Φ0 = yN
r
N
N
r2
28
(6.4)
(6.5)
Then using the polar coordinate, we obtain
Z
2
2
p
wp L−1
yN
0 (yN w )dy
RN
·
¸
2
yN
1
1 −1 2 p
Φ0 (r) 2 − Φ0 (r) + L0 (r w ) dy
=
r
N
N
RN
·
¸
Z
r2 cos2 θN −1
1
1 −1 2 p
2
2
p
= r cos θN −1 w Φ0 (r)
− Φ0 (r) + L0 (r w ) dy
r2
N
N
Z
2
wxp0
yN
RN
Rπ
cos4 θN −1 sinN −2 θN −1 dθN −1 Z
=0
Rπ
r2 wp Φ0 (r)dy
sinN −2 θN −1 dθN −1
0
Rπ
+
RN
cos2 θN −1 sinN −2 θN −1 dθN −1 Z
0
Rπ
0
r
sin
Z
N −2
θN −1 dθN −1
2
wxp0
·
¸
1
1 −1 2 p
− Φ0 (r) + L0 (r w0 ) dy
N
N
RN
¸
·
Z
3
1
1 −1 2 p
1
2 p
2 p
=
r w Φ0 (r)dy +
r w − Φ0 (r) + L0 (r w ) dy
N (N + 2)
N
N
N
RN
RN
Z
Z
1
2(N − 1)
2 p −1 2 p
= 2
r w L0 (r w )dy + 2
r2 wp Φ0 (r)dy .
N
N (N + 2)
RN
RN
This completes the proof of (6.1). Similarly, one may obtain (6.2) and (6.3), respectively.
7
Appendix C
In this Appendix we will prove (4.13) of Section 4, i.e., we shall prove
Z
Z
¡
¢
h2
∂k wxh Lh ∂j wxh + hψj dy = −
wp+1 dy∂jk G(x0 ) + o(h2 ).
N +2
RN
RN
29
(7.1)
Proof. Note that by (4.3), (4.4) and (4.11), we obtain
h
i
Lh ∂j wxh =Lxh ∂j wxh + m(hy + xh ) − m(xh ) pwxp−1
∂j wxh
h
h
i
p−1
p−1
+ m(hy + xh )p(vh − wxh )∂j wxh − V (hy + xh ) − V (xh ) ∂j wxh
h
i
p−2
=h y · ∇m(xh )pwxp−1
+
m(x
)p(p
−
1)w
φ
−
y
·
∇V
(x
)
∂j wxh
h
1
h
xh
h
+ h2
N
h1 X
2
∂il m(xh )yi yl pwxp−1
+ y · ∇m(xh )p(p − 1)wxp−2
φ1 + m(xh )p(p − 1)wxp−2
φ2
h
h
h
i,l
i
1X
1
2
∂
V
(x
)y
y
φ
−
∂j wxh + o(h2 ) ,
+ m(xh )p(p − 1)(p − 2)wxp−3
il
h
i
l
1
h
2
2 i,l
N
and
h
i
Lh ψj =Lxh ψj + m(hy + xh ) − m(xh ) pwxp−1
ψj
h
h
i
+ m(hy + xh )p(vhp−1 − wxp−1
)ψ
−
V
(hy
+
x
)
−
V
(x
)
j
h
h ψj
h
h
i
p−1
p−2
= − y · ∇m(xh )pwxh + m(xh )p(p − 1)wxh φ1 − y · ∇V (xh ) ∂j wxh
h
i
p−2
2
+ h y · ∇m(xh )pwxp−1
+
m(x
)p(p
−
1)w
φ
−
y
·
∇V
(x
)
h
1
h ψj + O(h ).
xh
h
Hence we may write the integral
R
¡
¢
∂k wxh Lh ∂j wxh + hψj dy as follows:
RN
Z
¡
¢
∂k wxh Lh ∂j wxh + hψj dy = I0 + I1 + I2 + o(h2 ),
(7.2)
RN
where
Z
2
I0 =h
RN
Z
2
I1 =h
RN
h
i
p−2
y · ∇m(xh )pwxp−1
+
m(x
)p(p
−
1)w
φ
−
y
·
∇V
(x
)
ψj ∂k wxh dy ,
h
1
h
xh
h
N
h1 X
2
(7.3)
∂il m(xh )yi yl pwxp−1
+ y · ∇m(xh )p(p − 1)wxp−2
φ1
h
h
i,l
i
1X
1
p−3 2
∂il V (xh )yi yl ∂j wxh ∂k wxh dy ,
+ m(xh )p(p − 1)(p − 2)wxh φ1 −
2
2 i,l
(7.4)
m(xh )p(p − 1)wxp−2
φ2 ∂j wxh ∂k wxh dy .
h
(7.5)
N
Z
2
I2 =h
RN
Note that from (2.3), we have
i
h
¡
¢
p−1
∂j wxh ∂k wxh = 0 .
∆ − V (xh ) + λ + m(xh )pwxh ∂jk wxh + m(xh )p(p − 1)wxp−2
h
30
(7.6)
Hence by (4.6), (7.4) and (7.5), we may use integration by parts to get
Z
h
i
¡
¢
2
I2 = − h
φ2 ∆ − V (xh ) + λ + m(xh )pwxp−1
∂jk wxh dy
h
N
R
Z
h
i
¡
¢
2
p−1
=−h
∂jk wxh ∆ − V (xh ) + λ + m(xh )pwxh φ2 dy
Z
2
RN
h
=h
RN
N
1X
− y · ∇V (xh )φ1 −
∂il V (xh )yi yl wxh + y · ∇m(xh )pwxp−1
φ1
h
2 i,l=1
+
Z
2
=h
RN
h
N
i
1X
1
2
∂il m(xh )yi yl wxph + m(xh )p(p − 1)wxp−2
φ
1 ∂jk wxh dy
h
2 i,l=1
2
N
1X
∂j V (xh )φ1 + y · ∇V (xh )∂j φ1 +
∂il V (xh )yi yl ∂j wxh + ∂jk V (xh )yk wxh
2 i,l=1
− ∂j m(xh )pwxp−1
φ1 − y · ∇m(xh )p(p − 1)wxp−2
φ1 ∂j wxh − y · ∇m(xh )pwxp−1
∂j φ1
h
h
h
N
1X
∂j wxh − ∂jk m(xh )yk wxph
∂il m(xh )yi yl pwxp−1
−
h
2 i,l=1
i
1
2
p−2
− m(xh )p(p − 1)(p − 2)wxp−3
φ
∂
w
−
m(x
)p(p
−
1)w
φ
∂
φ
h
1 j 1 ∂k wxh dy
1 j xh
xh
h
2Z
h
i
p−2
= − I1 − h2
y · ∇m(xh )pwxp−1
+
m(x
)p(p
−
1)w
φ
−
y
·
∇V
(x
)
h
1
h ∂j φ1 ∂k wxh dy
xh
h
RN
Z h
i
2
p
+h
∂j V (xh )φ1 + ∂jk V (xh )yk wxh − ∂j m(xh )pwxp−1
φ
−
∂
m(x
)y
w
1
jk
h k xh ∂k wxh dy.
h
RN
(7.7)
Note that from (4.5), we have
∆(∂j φ1 ) − [V (x0 ) + λ] ∂j φ1 + m(x0 )pwxp−1
∂j φ1 + m(x0 )p(p − 1)wxp−2
φ1 ∂j wx0
0
0
p−1
− y · ∇V (x0 )∂j wx0 − ∂j V (x0 )wx0 + y · ∇m(x0 )pwx0 ∂j wx0 + ∂j m(x0 )wxp0 = 0 ,
and by direct computation,
(
Lx0 wx0 = (p − 1)m(x0 )wxp0 ,
1
Lx0 ( p−1
wx0 + 21 y · ∇wx0 ) = [V (x0 ) + λ] wx0 .
31
(7.8)
(7.9)
Thus we may use (7.3)-(7.9) and integration by parts to get
I0 + I1 + I2
Z h
´
i³
p−2
2
=h
y · ∇m(xh )pwxp−1
+
m(x
)p(p
−
1)w
φ
−
y
·
∇V
(x
)
ψ
−
∂
φ
h
1
h
j
j 1 ∂k wxh dy
xh
h
RNZ
h
i
p
+ h2
∂j V (xh )φ1 + ∂jk V (xh )yk wxh − ∂j m(xh )pwxp−1
φ
−
∂
m(x
)y
w
1
jk
h
k
xh ∂k wxh dy
h
RN
Z h
Z ³
i
´
p−1
2
2
∂j V (xh ) − ∂j m(xh )pwxh φ1 ∂k wxh dy
ψj − ∂j φ1 Lxh ψk dy + h
=−h
N
RN
ZR h
i
∂jk V (xh )yk wxh − ∂jk m(xh )yk wxph ∂k wxh dy
+ h2
N
Z h
Z Rh
i
i
p
2
2
∂j V (xh )wxh − ∂j m(xh )wxph ∂k φ1 dy
∂j V (x0 )wx0 − ∂j m(x0 )wx0 ψk dy − h
=h
RN
RNZ
h1
i
1
− h2
dy + o(h2 )
∂jk V (xh )wx2h −
∂jk m(xh )wxp+1
h
2
p
+
1
N
Z Rh
¡
¢−1 ¡ 1
¢
1
2
∂j V (x0 ) V (x0 ) + λ
=h
wx0 + y · ∇wx0
p−1
2
RN
´
i ³
1
m(x0 )−1 ∂j m(x0 )wx0 Lx0 ψk − ∂k φ1 dy
−
p−1
Z h
i
1
1
− h2
∂jk V (x0 )wx20 −
∂jk m(x0 )wxp+1
dy + o(h2 )
0
2
p
+
1
N
ZR h
¡
¢−1 ¡ 1
¢
1
= − h2
wx0 + y · ∇wx0
∂j V (x0 ) V (x0 ) + λ
p−1
2
RN
ih
i
1
m(x0 )−1 ∂j m(x0 )wx0 ∂k V (x0 )wx0 − ∂k m(x0 )wxp0 dy
−
p−1
Z h
i
1
1
∂jk V (x0 )wx20 −
∂jk m(x0 )wxp+1
− h2
dy + o(h2 )
0
2
p
+
1
N
R
h¡ 1
N¢
−
[V (x0 ) + λ]−1 ∂j V (x0 )∂k V (x0 )
= − h2
p−1
4
iZ
1
−1
wx20 dy
−
m(x0 ) ∂j m(x0 )∂k V (x0 )
p−1
RN
h¡ 1
¢
1
N
−
+ h2
[V (x0 ) + λ]−1 ∂j V (x0 )∂k m(x0 )
p−1 2p+1
iZ
1
−1
m(x0 ) ∂j m(x0 )∂k m(x0 )
wxp+1
dy
−
0
p−1
N
R
Z h
i
1
1
2
−h
∂jk V (x0 )wx20 −
∂jk m(x0 )wxp+1
dy + o(h2 ) .
(7.10)
0
2
p
+
1
N
R
Recall that

1
p
£
¤ p−1
1
− p−1

m(x
)
w(
V (x0 ) + λy) ,
w
(y)
=
V
(x
)
+
λ

0
x
0
 0
N
m(x0 )∇V (x0 ) = 2 [V (xh0 ) + λ] ∇m(x0 ) ,


∂ij G(x0 ) = m(x0 )− N2 −1 m(x0 )∂ij V (x0 ) + (1 − N )∂i V (x0 )∂j m(x0 ) −
2
32
N
2
i
[V (x0 ) + λ] ∂ij m(x0 ) ,
and the integral identity
Z
[V (x0 ) + λ]
RN
wx20 dy
2
=
m(x0 )
N +2
Z
RN
wxp+1
dy .
0
Combining (7.2) and (7.10), we obtain (7.1).
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