# Multiplicity of high energy solutions for superlinear Kirchhoff equations

- First Online:

- Received:

DOI: 10.1007/s12190-012-0536-1

- Cite this article as:
- Liu, W. & He, X. J. Appl. Math. Comput. (2012) 39: 473. doi:10.1007/s12190-012-0536-1

## Abstract

*a*,

*b*>0 are constants,

*V*:ℝ

^{3}→ℝ is continuous and has a positive infimum.

*f*is a subcritical nonlinearity which needs not to satisfy the usual Ambrosetti-Rabinowitz-type growth conditions.

### Keywords

Nonlinear Kirchhoff equations High energy solutions Variational methods### Mathematics Subject Classification (2000)

35J60 35J25## 1 Introduction and preliminaries

*a*,

*b*>0 are constants,

*V*∈

*C*(ℝ

^{3},ℝ) and

*f*(

*x*,

*u*)∈

*C*(ℝ

^{3}×ℝ,ℝ) satisfy some further conditions.

*V*(

*x*)≡0 and replace ℝ

^{3}by a bounded domain Ω⊂ℝ

^{N}, then (1.1) reduces to the following Dirichlet problem of Kirchhoff type:

*f*is asymptotically

*t*

^{3}-growth at infinity. We refer to [16, 17, 18, 19] for more existence results of the Kirchhoff-type equations.

*a*=1,

*b*=0, problem (1.1) can be rewritten as the following well-known Schrödinger equation:

*V*and

*f*. For the potential

*V*we make the following assumption.

- (V)\(V(x)\in L^{\infty}_{loc}(\mathbb {R}^{3}),V_{0}:=\inf_{\mathbb {R}^{3}}V(x)>a_{1}>0\). For any
*M*,*r*>0,where$$m(\{x\in B_r(y):~V(x)\leq M\})\rightarrow 0\quad\mbox{as}\ |y|\rightarrow\infty,$$*B*_{r}(*y*) denotes the ball centered at*y*with radius*r*.

### Remark 1.1

*λ*

_{1}<

*λ*

_{2}<⋯<

*λ*

_{k}<⋯→∞ with finite multiplicity for each

*λ*

_{k}. The principal eigenvalue

*λ*

_{1}is simple with positive eigenfunction

*φ*

_{1}, and eigenfunction

*φ*

_{k}corresponding to

*λ*

_{k}(

*k*≥2) is sign-changing.

*f*, we suppose it satisfies the following conditions in this paper.

- (
*f*_{1}) \(\lim_{|s| \rightarrow0} \frac{f(x,s)}{s} = 0\) uniformly for

*x*∈ℝ^{3}.- (
*f*_{2}) *f*∈*C*(ℝ^{3}×ℝ,ℝ),*f*(*x*,*s*)*s*≥0, for*s*≥0; and there exists*a*_{2}>0,*q*∈(2,2^{∗}) such thatwhere 2$$|f(x,s)| \leq a_2 (1+|s|^{q-1}),\quad\mbox{for all}\ x \in \mathbb {R}^3, s\in \mathbb {R},$$^{∗}=6 is the critical exponent for the Sobolev embedding in dimension 3.- (
*f*_{3}) - There exists
*μ*>4 and*R*>0 such that$$\mu F(x,s)\leq sf(x,s),\quad\mbox{for all}\ (x,s)\in \mathbb {R}^3\times \mathbb {R}\ \mbox{and}\ |s|\geq R.$$ - (
*f*_{4}) There exists 4<

*α*<2^{∗}such that \(\lim\inf _{|s|\rightarrow\infty}\frac{F(x,s)}{|s|^{\alpha}}>0\), uniformly in*x*∈ℝ^{3}.- (
*f*_{5}) *f*(*x*,−*s*)=−*f*(*x*,*s*) for any*x*∈ℝ^{3},*s*∈ℝ.

*s*<∞,

*L*

^{s}(ℝ

^{3}) denotes the usual Lebesgue space with the norm \(\|u\|_{L^{s}} := (\int_{R^{3}} |u|^{s}dx)^{\frac{1}{s}}\).

*H*

^{1}(ℝ

^{3}) is the usual Sobolev space with the norm

*E*is defined by

*E*is a Hilbert space with the inner product \((u,v)_{E} :=\int_{R^{3}} (\nabla u \nabla v + V(x)uv)dx\), and its norm is \(\|u\|_{E} = (u,u)_{E} ^{\frac{1}{2}}\). Since

*V*(

*x*) is bounded form below, the embedding

*E*↪

*L*

^{s}(ℝ

^{3}) is continuous for any

*s*∈[2,2

^{∗}].

The main results are the following.

### Theorem 1.1

*Assume that conditions* (V), (*f*_{1})–(*f*_{5}) *hold*. *Then problem* (1.1) *has infinitely many solutions* {*u*_{n}} *satisfying* ∥*u*_{n}∥_{E}→∞ *and* Φ(*u*_{n})→∞ *as**n*→∞.

### Theorem 1.2

*In addition to conditions*(V), (

*f*

_{1})–(

*f*

_{2})

*and*(

*f*

_{5}),

*suppose that*

*f*

*satisfies the following conditions*

- (
*f*_{3})′ \(\lim_{|s| \rightarrow\infty} \frac{f(x,s)s}{s^{4}} = \infty\)

*uniformly for**x*∈ℝ^{3}.- (
*f*_{4})′ *For a*.*e*.*x*∈ℝ^{3},∀(*s*,*t*)∈ℝ^{+}×ℝ^{+},*s*≤*t*,*there holds*\(\mathcal{G}(x,s) \leq\mathcal{G}(x,t)\),*where*\(\mathcal{G}:\mathbb {R}^{3} \times \mathbb {R}^{+} \rightarrow \mathbb {R}\)*is defined by*\(\mathcal{G}(x,s):= \frac{1}{4} f(x,s)s -F(x,s)\).

*Then problem*(1.1)

*has a sequence of solutions*{

*u*

_{n}}

*in*

*E*

*with*∥

*u*

_{n}∥

_{E}→∞

*and*Φ(

*u*

_{n})→∞

*as*

*n*→∞.

Clearly, dealing with Φ, one has to face various difficulties: we mention that the competing effect of the non-local term with the nonlinear term gives rise to very different situations and that the lack of compactness of the embedding of *H*^{1}(*R*^{3}) in the Lebesgue spaces *L*^{p}(*R*^{3}),*p*∈(2,6), prevents from using the variational techniques in a standard way. This last difficulty can be avoided, if we restrict Φ to the subspace of *H*^{1}(*R*^{3}) consisting of radially symmetric functions, or, when one is searching for semi-classical states, by using perturbation methods or a reduction to a finite dimension by the projections method. As far as we know, except a perturbation result in [26], there are no existence results when the potential and the coefficient of the nonlinearity are not symmetric and a not singularly perturbed problem is considered.

In order to obtain infinitely many high energy solutions of problem (1.1), we shall apply Theorem 9.12 [27] and a variant version of fountain theorem [24], which will be stated later. The proofs of Theorems 1.1 and 1.2 are carried out in Sects. 2 and 3, respectively.

Hereafter, the letters *C*,*C*_{i},*i*=1,2,…, will be repeatedly used to denote various constants whose exact values are irrelevant.

## 2 Proof of Theorem 1.1

*X*

_{i}denote the eigenspace of

*λ*

_{i}; then dim

*X*

_{i}<∞. Let

*E*

_{1}=

*X*

_{1}⊕

*X*

_{2}⊕⋯⊕

*X*

_{k}and

*E*has a direct sum decomposition

*E*=

*E*

_{1}⊕

*E*

_{2}with dim

*E*

_{1}<∞. Under the conditions of Theorem 1.1, it is easy to see that Φ∈

*C*

^{1}(

*E*,ℝ).

### Definition 2.1

*u*

_{n}}⊂

*E*satisfying

In order to obtain infinitely many solutions of (1.1), we shall use the following critical point theorem [27] which was introduced by Rabinowitz.

### Lemma 2.1

([27], Theorem 9.12)

*Let*

*E*

*be an infinite dimensional real Banach space and*

*I*∈

*C*

^{1}(

*E*,ℝ)

*be even*,

*satisfy the*(PS)

*condition and*

*I*(0)=0.

*If*

*E*=

*E*

_{1}⊕

*E*

_{2}

*with*

*E*

_{1}

*being finite dimensional*,

*and*

*I*

*satisfies*

- (i)
*there exist constants**ρ*,*α*>0*such that*\(I|_{\partial B_{\rho}\cap E_{2}}\geq\alpha\),*where**∂B*_{ρ}={*u*∈*E*:∥*u*∥=*ρ*},*and* - (ii)
*for each finite dimensional subspace*\(\widehat{E}\subset E\),*there exists an*\(r=r_{\widehat{E}}>0\)*such that**I*≤0*on*\(\widehat{E}\backslash B_{r}\).

*Then*,

*I*

*possesses an unbounded sequence of critical values*.

Under condition (V), we have the following compactness result.

If we replace (V) by (V1) or (V2), then the conclusion of Lemma 2.2 is still true.

### Lemma 2.3

*If conditions* (*f*_{1}),(*f*_{2}),(*f*_{3}) *and* (*f*_{4}) *hold*. *Then any* (PS) *sequence of* Φ *is bounded in**E*.

### Proof

*u*

_{n}}⊂

*E*be a (PS)

_{c}sequence, that is Φ(

*u*

_{n})=

*c*+

*o*

_{n}(1) and Φ′(

*u*

_{n})→0. Suppose by contradiction that, {

*u*

_{n}} is unbounded in

*E*. Set \(v_{n}=\frac{u_{n}}{\|u_{n}\|_{E}}\), then ∥

*v*

_{n}∥

_{E}=1. Since In view of

*α*>4, we see that

*v*

_{n}∥

_{E}=1, we can find some

*v*∈

*E*such that

*v*

_{n}⇀

*v*in

*E*;

*v*

_{n}→

*v*in

*L*

^{s}(ℝ

^{3}) for

*s*∈[2,2

^{∗}) and

*v*

_{n}→

*v*a.e.

*x*∈ℝ

^{3}. Denote \(\Omega=\{x\in \mathbb {R}^{3}:~v(x)\not=0\}\). If meas(Ω)>0, then |

*v*

_{n}(

*x*)|>0 a.e.

*x*∈Ω. By (

*f*

_{1}),(

*f*

_{2}),(

*f*

_{3}) and (

*f*

_{4}), we have

*x*,

*u*)∈ℝ

^{3}×ℝ. So

*v*(

*x*)=0 a.e.

*x*∈ℝ

^{3}. By (

*f*

_{1}),(

*f*

_{2}) and (

*f*

_{3}), there is a constant

*C*

_{3}>0 such that

*x*,

*u*)∈ℝ×ℝ. Consequently,

*u*

_{n}} is bounded in

*E*. □

### Lemma 2.4

*Under the conditions of Theorem *1.1, *the functional* Φ *satisfies the Palais*-*Smale condition*.

### Proof

*u*

_{n}} is a (PS) sequence of Φ, then by Lemma 2.3 we see that {

*u*

_{n}} is bounded in

*E*. Therefore, we can extract a subsequence, still denoted by {

*u*

_{n}} such that

*u*

_{n}⇀

*u*in

*E*and

*u*

_{n}→

*u*in

*L*

^{s}(ℝ

^{3}),

*s*∈[2,2

^{∗}). We note that

*u*

_{n}} and

*u*

_{n}⇀

*u*in

*E*, one has

*f*

_{1}),(

*f*

_{2}), for any given

*ε*>0, there exists

*C*

_{ε}>0 such that

*C*are positive constants independent of

*ε*,

*n*. Since ∥

*u*

_{n}−

*u*∥

_{2}→0,∥

*u*

_{n}−

*u*∥

_{q}→0 as

*n*→∞, we see that

*u*

_{n})→0, we deduce from (2.4), (2.5) and (2.7) that ∥

*u*

_{n}−

*u*∥

_{E}→0, i.e., the (PS) condition holds. □

### Proof of Theorem 1.1

*Step 1.*- There exist constants
*ρ*>0 and*α*>0 such that \(\Phi|_{\partial B_{\rho}\cap E_{2}}\geq\alpha\). We fix a number*k*∈ℕ and choose*λ*∈[*λ*_{k},*λ*_{k+1}). Then by (*f*_{1}), there exists*δ*>0 such thatTherefore, by ($$|f(x,s)|\leq\lambda\min\{a,1\}|s|\quad\mbox{for all}\ x\in \mathbb {R}^3, |s|\leq\delta.$$*f*_{2}), we have$$F(x,s)\leq\frac{\min\{a,1\}}{2}\lambda s^2+C_4|s|^q,\quad\mbox{for all}\ (x,s)\in \mathbb {R}^3\times \mathbb {R}.$$For*u*∈*E*_{2}, we have Therefore, there exists some small*ρ*>0 such thatfor all$$\Phi(u)\geq\frac{\min\{a,1\}}{4}\left(1-\frac{\lambda}{\lambda _{k+1}}\right)\rho^2:=\alpha>0$$*u*∈*E*_{2}with ∥*u*∥_{E}=*ρ*. *Step 2.*- For any finite dimensional space \(\widehat{E}\subset E\), we have that Φ(
*u*)≤0 for all \(u\in\widehat{E}\backslash B_{r}\) with some \(r=r_{\widehat{E}}>0\). In fact, by (*f*_{1}),(*f*_{2}) and (*f*_{3}), there exist positive constants*C*_{6},*C*_{7}such thatfor all ($$F(x,u)\geq C_6|u|^{\mu}-C_7|u|^2$$*x*,*u*)∈ℝ^{3}×ℝ. Thus, for each \(u\in\widehat{E}\backslash B_{r}\), we have Since \(\widehat{E}\subset E\) is a finite dimensional space, all the norms on \(\widehat{E}\) are equivalent, we can choose \(r=r_{\widehat{E}}>0\) such that$$I(u)\leq0,\quad\mbox{for all}\ u\in\widehat{E}\backslash B_r.$$Now, by Lemma 2.4, Φ satisfies (PS) condition. Moreover, Φ is even due to (

*f*_{5}). Thus, the conclusion follows from Lemma 2.1.

## 3 Proof of Theorem 1.2

In order to prove Theorem 1.2, we shall use a variant version of fountain theorem due Zou [24], which can be stated as the following.

*X*be a Banach space with the norm ∥⋅∥ and let

*X*

_{i},

*i*∈ℕ be a sequence of subspace of

*X*with dim

*X*

_{j}<∞ for each

*j*∈ℕ. Further, \(X=\overline{\bigoplus_{j\in \mathbb {N}}X_{j}}\), the closure of the direct sum of all

*X*

_{j}. Set \(Y_{k} =\bigoplus_{j=0}^{k} X_{j}, Z_{k}=\overline{\bigoplus_{i=k}^{\infty}X_{i}}\) and

*B*

_{k}={

*u*∈

*Y*

_{k}:∥

*u*∥≤

*ρ*

_{k}},

*S*

_{k}={

*u*∈

*Z*

_{k}:∥

*u*∥≤

*γ*

_{k}}, for

*ρ*

_{k}>

*γ*

_{k}>0. Consider a family of \(\mathcal {C}^{1}\)-functional Φ

_{λ}(

*u*):

*X*→ℝ of the form:

- (
*H*_{1}) Φ

_{λ}(*u*) maps bounded sets into bounded sets uniformly for all*λ*∈[1,2]. Furthermore Φ_{λ}(−*u*)=Φ_{λ}(*u*) for all (*λ*,*u*)∈[1,2]×*X*.- (
*H*_{2}) *B*(*u*)≥0 for all*u*∈*X*;*A*(*u*)→∞ or*B*(*u*)→∞ as ∥*u*∥→∞, or- (
*H*_{3}) *B*(*u*)≤0 for all*u*∈*X*;*B*(*u*)→−∞ as ∥*u*∥→∞.

*K*≥2, let

### Lemma 3.1

([24], Theorem 3.1)

*Assume*(

*H*

_{1})

*and*(

*H*

_{2}) (

*or*(

*H*

_{3}))

*hold*.

*If*

*b*

_{k}(

*λ*)>

*a*

_{k}(

*λ*)

*for all*

*λ*∈[1,2],

*then*

*c*

_{k}(

*λ*)≥

*b*

_{k}(

*λ*)

*for all*

*λ*∈[1,2].

*Moreover*,

*for a*.

*e*.

*λ*∈[1,2],

*there exists a sequence*\(\{u_{n} ^{k} (\lambda)\}_{n} ^{\infty}\)

*such that*

*E*be the Banach space given in the previous section and

*X*

_{j}:=span{

*e*

_{j}}, where

*e*

_{j}is the eigenfunction corresponding to the eigenvalue

*λ*

_{j}. Denote \(Y_{k} = \bigoplus_{j=0} ^{k} X_{j}\), \(Z_{k} = \overline{\bigoplus _{j=k} ^{\infty}X_{j}}\). Consider the family of functional Φ

_{λ}:

*E*→ℝ defined by

*B*(

*u*)≥0 for all

*u*∈

*E*;

*A*(

*u*)→∞ as ∥

*u*∥

_{E}→∞, and Φ

_{λ}(−

*u*)=Φ

_{λ}(

*u*) for all (

*λ*,

*u*)∈[1,2]×

*E*due to (

*f*

_{5}). And it is easy to see that Φ

_{λ}maps bounded sets to bounded sets uniformly for

*λ*∈[1,2].

Now, we show that Φ_{λ} satisfies the geometric properties of Lemma 3.1.

### Lemma 3.2

*Suppose that*(

*f*

_{1}),(

*f*

_{2})

*and*(

*f*

_{3})′

*are satisfied*.

*Then there exists*

*ρ*

_{k}>

*γ*

_{k}>0

*such that*

- (i)
\(a_{k}(\lambda):= \max_{u \in Y_{k}, \|u\|_{E}= \rho_{k}} \Phi _{\lambda} (u) \leq 0\),

- (ii)
\(b_{k}(\lambda):= \inf_{u \in Z_{k}, \|u\|_{E}= \gamma_{k}}\Phi _{\lambda}(u)>0\).

### Proof

*f*

_{3})′, for any

*L*>0, there exists

*δ*=

*δ*(

*L*)>0, such that for all

*x*∈ℝ

^{3}, |

*u*|≥

*δ*, we have

*f*(

*x*,

*u*)

*u*≥

*Lu*

^{4}and

*f*

_{1}), we see that there exists

*β*>0 small, such that

*x*∈ℝ

^{3}, 0<|

*u*|≤

*β*. By (

*f*

_{2}), there exists some

*L*

_{1}=

*L*

_{1}(

*δ*,

*L*) such that for all

*x*∈ℝ

^{3},

*β*≤|

*u*|≤

*δ*,

*x*∈ℝ

^{3}, 0<|

*u*|≤

*δ*we have

*f*(

*x*,

*u*)

*u*≥−(

*L*

_{1}+1)|

*u*|

^{2}, and hence

*x*∈ℝ

^{3},

*u*∈ℝ,

*u*∈

*Y*

_{k}, we obtain where in the last inequality we have used the equivalence of all norms on the finite dimensional subspace

*Y*

_{k}.

*L*large enough such that

*u*∥

_{E}=

*ρ*

_{k}>0 large enough such that

*f*

_{1}) and (

*f*

_{2}) that, for any

*ξ*>0, there exists

*C*

_{ξ}>0 such that

*p*<2

^{∗}. Then

*β*

_{k}→0, as

*k*→∞ (cf. [22]). For simplicity, let

*a*

_{⋆}:=min(

*a*,1), then for any

*u*∈

*Z*

_{k}and

*ξ*>0 small enough, we have Denote \(\gamma_{k} = (\lambda C_{\xi}\frac {q}{a_{\star}}\beta_{k}^{q})^{\frac{1}{2-q}}\), then Since

*β*

_{k}→0 as

*k*→∞,

*q*>2 and

*ξ*can be small arbitrarily, we have

*λ*∈[1,2], there exists a sequence \(\{u_{n}^{k}(\lambda)\}_{n=1}^{\infty}\) such that where \(c_{k}(\lambda)=\inf_{\gamma\in\Gamma_{k}} \max_{u \in B_{k}}\Phi_{\lambda} (\gamma(u))\). Note that \(c_{k}(\lambda) \leq\sup_{u\in B_{k}} \Phi_{1} (u) := \widetilde{{c_{k}}}\) and the embedding

*E*↪

*L*

^{s}(ℝ

^{3}), 2≤

*s*<2

^{∗}is compact. By a similar argument as we have done in the proof of Theorem 1.1, \(\{u_{n}^{k}(\lambda )\}_{n=1}^{\infty}\) has a convergent subsequence. Suppose \(u_{n}^{k}(\lambda) \rightarrow u^{k}(\lambda)\) as

*n*→∞. Then we have

*λ*

_{n}}⊂[1,2] with

*λ*

_{n}→1, and set

*u*

_{n}:=

*u*

^{k}(

*λ*

_{n}) for simplicity. Then \(\Phi_{\lambda_{n}}'(u_{n})=0, \Phi_{\lambda_{n}} (u_{n})\in[\widetilde{{b_{k}}},\widetilde{{c_{k}}}]\).

### Lemma 3.3

*Under the assumptions of Theorem *1.2, *the sequence* {*u*_{n}} *is bounded in E*.

### Proof

*u*

_{n}∥

_{E}→∞ as

*n*→∞. Define \(\omega_{n}:=\frac{u_{n}}{\|u_{n}\|_{E}}\). Then ∥

*ω*

_{n}∥

_{E}=1,∀

*n*∈ℕ. Passing to a subsequence, and using the compact embedding

*E*↪

*L*

^{s}(ℝ

^{3}),2≤

*s*<2

^{∗}, we obtain

*v*≠0 and (ii)

*v*=0.

*x*∈ℝ

^{3}|

*v*(

*x*)≠0},

*M*>0, define \(\widetilde{{\omega_{n}}}=\sqrt{4M}~\frac{u_{n}}{\|u_{n}\|_{E}} =\sqrt{4M}~\omega_{n}\). From (3.6) we have

*n*→∞. Then, choosing

*n*sufficiently large we conclude which means that \(\lim_{n\rightarrow\infty}\Phi_{\lambda_{n}}(t_{n}u_{n}) = \infty\). By the definition of

*t*

_{n}, we see that \(\langle\Phi_{\lambda_{n}}' (t_{n}u_{n}),t_{n}u_{n}\rangle=0\). Consequently, by (

*f*

_{4})′ we have as

*n*→∞. This leads to a contradiction and the conclusion follows. □

### Proof of Theorem 1.2

*E*↪

*L*

^{s}(

*R*

^{3}), 2≤

*s*<2

^{∗}, standard arguments imply that there exists a convergent subsequence of {

*u*

_{n}} when

*λ*

_{n}→1. Notice that {

*u*

_{n}} is relevant to the choice of

*k*, we may assume that

*u*

_{n}→

*u*

^{k}for some

*u*

^{k}∈

*E*. Moreover, from the equality

*u*

_{n}} in

*E*, we conclude that \(|\int_{R^{3}}F(x,u_{n})dx|<\infty\) as

*n*→∞. Recalling that \(\Phi_{\lambda_{n}}(u_{n}) \in[\widetilde{{b_{k}}},\widetilde{{c_{k}}}],\lambda_{n}\rightarrow1\), we obtain

*v*∈

*E*. Since \(\Phi_{\lambda_{n}}'(u_{n})=0,~\{u_{n}\}\) is bounded in

*E*, the above equality yields that

*C*

^{1}(

*E*,ℝ), we have Φ′(

*u*

_{n})→Φ′(

*u*

^{k}) in

*E*

^{∗}. Therefore, for every

*v*∈

*E*,

*n*→∞. This means that 〈Φ′(

*u*

^{k}),

*v*〉=0 for all

*v*∈

*E*, i.e. Φ′(

*u*

^{k})=0 in

*E*

^{∗}. By (3.12) and \(\widetilde{b_{k}}\rightarrow+\infty\), we see that \(\{u^{k}\}_{k=1}^{\infty}\) is an unbounded sequence of critical points of functional Φ(

*u*). This completes the proof. □

### Remark 3.1

*f*which satisfies the conditions of Theorem 1.2, but does not satisfy the conditions of Theorem 1.1, especially the Ambrosetti-Rabinowitz-type growth condition (

*f*

_{3}). In fact, let

*f*(

*x*,

*s*)=

*s*

^{3}(4+ln(1+|

*s*|)). Simple computation yields that and it is easy to check that

*f*satisfies conditions (

*f*

_{1}),(

*f*

_{2}),(

*f*

_{3})′ and (

*f*

_{4})′.

*f*does not satisfy (

*f*

_{3}). Indeed, suppose by contradiction that there is some

*μ*>4 such that

*μF*(

*x*,

*s*)≤

*f*(

*x*,

*s*)

*s*for |

*s*| large. That is, the following inequality holds true for |

*s*| large. But this is impossible since

*μ*>4.

## Acknowledgements

The authors are grateful for the anonymous referees for very helpful suggestions and comments. This work was supported by NSFC Grants 10971238 and the Fundamental Research Funds for the Central Universities 0910KYZY51.