Abstract
Using the nonsmooth critical point theory we investigate the existence and multiplicity of solutions for a differential inclusion problem with singular coefficients involving the p(x)Laplacian.
Mathematics Subject Classification 2000: 35D05; 35J20; 35J60; 35J70.
Keywords:
p(x)Laplacian; differential inclusion; singularity1 Introduction
In this article, we study the existence and multiplicity of solutions for the differential inclusion problem with singular coefficients involving the p(x)Laplacian of the form
where the following conditions are satisfied:
(P) Ω is a bounded open domain in ℝ^{N}, N ≥ 2, , 1 < p^{ }:= inf_{Ω }p(x) ≤ p^{+ }:= sup_{Ω }p(x) < +∞, λ, μ ∈ ℝ.
(A) For i = 1, 2, for x ∈ Ω, G_{i}(x, u) is measurable with respect to x (for every u ∈ ℝ) and locally Lipschitz with respect to u (for a.e. x ∈ Ω), ∂G_{i }: Ω × ℝ → ℝ is the Clarke subdifferential of G_{i }and for x ∈ Ω, t ∈ ℝ and ξ_{i }∈ ∂G_{i}, where c_{i }is a positive constant, , r_{i}(x) > q_{i}(x) for all x ∈ Ω, and
here
A typical example of (1.1) is the following problem involving subcritical SobolevHardy exponents of the form
and in this case the assumption corresponding to (A) is the following
, for i = 1, 2, ∂G_{i }: Ω × ℝ → ℝ is the Clarke subdifferential of G_{i }and for x ∈ Ω, t ∈ ℝ and ξ_{i }∈ ∂G_{i}, where c_{i }is a positive constant, , and
The operator div(∇u^{p(x)2 }∇u) is said to be the p(x)Laplacian, and becomes pLaplacian when p(x) ≡ p (a constant). The p(x)Laplacian possesses more complicated nonlinearities than the pLaplacian; for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth condition has been received considerable attention in recent years. These problems are interesting in applications and raise many difficult mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electrorheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electromagnetic field [1,2]. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermorheological viscous flows of nonNewtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium [3,4]. Another field of application of equations with variable exponent growth conditions is image processing [5]. The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to [611] for an overview of and references on this subject, and to [1221] for the study of the p(x)Laplacian equations and the corresponding variational problems.
Since many free boundary problems and obstacle problems may be reduced to partial differential equations with discontinuous nonlinearities, the existence of multiple solutions for Dirichlet boundary value problems with discontinuous nonlinearities has been widely investigated in recent years. Chang [22] extended the variational methods to a class of nondifferentiable functionals, and directly applied the variational methods for nondifferentiable functionals to prove some existence theorems for PDE with discontinuous nonlinearities. Later Kourogenis and Papageorgiou [23] obtained some nonsmooth critical point theories and applied these to nonlinear elliptic equations at resonance, involving the pLaplacian with discontinuous nonlinearities. In the celebrated work [24,25], Ricceri elaborated a Ricceritype variational principle and a three critical points theorem for the Gâteaux differentiable functional, respectively. Later, Marano and Motreanu [26,27] extended Ricceri's results to a large class of nondifferentiable functionals and gave some applications to differential inclusion problems involving the pLaplacian with discontinuous nonlinearities.
In [21], by means of the critical point theory, Fan obtain the existence and multiplicity of solutions for (1.1) under the condition of satisfying the Carathéodory condition for i = 1, 2, x ∈ Ω. The aim of the present article is to generalize the main results of [21] to the case of the functional of problem (1.1) is nonsmooth.
This article is organized as follows: In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces and the generalized gradient of the locally Lipschitz function; In Section 3, we give the variational principle which is needed in the sequel; In Section 4, using the critical point theory, we prove the existence and multiplicity results for problem (1.1).
2 Preliminaries
2.1 Variable exponent Sobolev spaces
Let Ω be a bounded open subset of ℝ^{N}, denote .
On the basic properties of the space W^{1,p(x)}(Ω) we refer to [7,2830]. Here we display some facts which will be used later.
Denote by S(Ω) the set of all measurable real functions defined on Ω. Two functions in S(Ω) are considered as the same element of S(Ω) when they are equal almost everywhere. For , define the spaces L^{p(x) }(Ω) and W^{1,p(x) }(Ω) by
with the norm
and
with the norm
Denote by the closure of in W^{1,p(x) }(Ω) . Hereafter, we always assume that p^{ }> 1.
Proposition 2.1. [7,31]The spaces L^{p(x) }(Ω) , W^{1,p(x) }(Ω) and are separable and reflexive Banach spaces.
Proposition 2.2. [7,31]The conjugate space of L^{p(x) }(Ω) is , where . For any u ∈ L^{p(x) }(Ω) and v ∈ , .
Proposition 2.3. [7,31]In the Poincaré inequality holds, that is, there exists a positive constant c such that
Proposition 2.4. [7,28,29,31]Assume that the boundary of Ω possesses the cone property and . If and , then there is a compact embedding W^{1,p(x)}(Ω) → L^{q(x) }(Ω).
Let us now consider the weighted variable exponent Lebesgue space.
Let a ∈ S(Ω) and a(x) > 0 for x ∈ Ω. Define
with the norm
then is a Banach space. The following proposition follows easily from the definition of .
Proposition 2.5. (see [7,31]) Set ρ(u) = ∫_{Ω }a(x)u(x)^{p(x) }dx. For , we have
Proposition 2.6. (see [21]) Assume that the boundary of Ω possesses the cone property and . Suppose that a ∈ L^{r}(^{x})(Ω), a(x) > 0 for x ∈ Ω, and r^{ }> 1. If and
then there is a compact embedding .
The following proposition plays an important role in the present article.
Proposition 2.7. Assume that the boundary of Ω possesses the cone property and . Suppose that a ∈ L^{r}(^{x})(Ω), a(x) > 0 for x ∈ Ω, and r(x) > q(x) for all x ∈ Ω. If and
then there is a compact embedding .
Proof. Set , then and . Moreover, from (2.2) we can get
Using Proposition 2.6, we see that the embedding is compact.
■
2.2 Generalized gradient of the locally Lipschitz function
Let (X,  · ) be a real Banach space and X* be its topological dual. A function f : X → ℝ is called locally Lipschitz if each point u ∈ X possesses a neighborhood Ω_{u }such that f(u_{1})  f(u_{2}) ≤ Lu_{1 } u_{2} for all u_{1}, u_{2 }∈ Ω_{u}, for a constant L > 0 depending on Ω_{u}. The generalized directional derivative of f at the point u ∈ X in the direction v ∈ X is
The generalized gradient of f at u ∈ X is defined by
which is a nonempty, convex and w*compact subset of X, where 〈·,·〉 is the duality pairing between X* and X. We say that u ∈ X is a critical point of f if 0 ∈ ∂f(u). For further details, we refer the reader to Chang [22].
We list some fundamental properties of the generalized directional derivative and gradient that will be used throughout the article.
Proposition 2.8. (see [22,32]) (1) Let j : X → ℝ be a continuously differentiable function. Then ∂j(u) = {j'(u)}, j^{0}(u; z) coincides with 〈j' (u), z〉_{X }and (f + j)^{0}(u, z) = f^{0}(u; z) + 〈j' (u), z〉_{X }for all u, z ∈ X.
(2) The setvalued mapping u → ∂f(u) is upper semicontinuous in the sense that for each u_{0 }∈ X, ε > 0, v ∈ X, there is a δ > 0, such that for each w ∈ ∂f (u) with w  u_{0} < δ, there is w_{0 }∈ ∂f (u_{0})
(3) (Lebourg's mean value theorem) Let u and v be two points in X. Then there exists a point w in the open segment joining u and v and such that
(4) The function
exists, and is lower semi continuous; i.e., .
In the following we need the nonsmooth version of PalaisSmale condition.
Definition 2.1. We say that φ satisfies the (PS)_{c}condition if any sequence {u_{n}} ⊂ X such that φ(u_{n}) → c and m(u_{n}) → 0, as n → +∞, has a strongly convergent subsequence, where m(u_{n}) = inf{u*_{X* }: u* ∈ ∂φ (u_{n})}.
In what follows we write the (PS)_{c}condition as simply the PScondition if it holds for every level c ∈ ℝ for the PalaisSmale condition at level c.
3 Variational principle
In this section we assume that Ω and p(x) satisfy the assumption (P). For simplicity we write and u = ∇u_{p}(_{x}) for u ∈ X. Denote by u_{n }⇀ u and u_{n }→ u the weak convergence and strong convergence of sequence {u_{n}} in X, respectively, denote by c and c_{i }the generic positive constants, B_{ρ }= {u ∈ X : u < ρ}, S_{ρ }= {u ∈ X : u = ρ}.
Set
where a_{i }and G_{i }(i = 1, 2) are as in (A).
Define the integral functional
We write
then it is easy to see that J ∈ C^{1}(X, ℝ) and φ = J  Ψ.
Below we give several propositions that will be used later.
Proposition 3.1. (see [19]) The functional J : X → ℝ is convex. The mapping J' : X → X* is a strictly monotone, bounded homeomorphism, and is of (S_{+}) type, namely
Proposition 3.2. Ψ is weaklystrongly continuous, i.e., u_{n }⇀ u implies Ψ(u_{n}) → Ψ(u).
Proof. Define ϒ_{1 }= ∫_{Ω }G_{1}(x, u) dx and ϒ_{2 }= ∫_{Ω }G_{2}(x, u) dx. In order to prove Ψ is weaklystrongly continuous, we only need to prove ϒ_{1 }and ϒ_{2 }are weaklystrongly continuous. Since the proofs of ϒ_{1 }and ϒ_{2 }are identical, we will just prove ϒ_{1}.
We assume u_{n }⇀ u in X. Then by Proposition 2.8.3, we have
where ξ_{n }∈ ∂G_{1}(,τ_{n}(x)) for some τ_{n}(x) in the open segment joining u and u_{n}. From Chang [22] we know that . So using Proposition 2.5, we have
■
Proposition 3.3. Assume (A) holds and F satisfies the following condition:
(B) for a.e.x ∈ Ω, all u ∈ X and ξ_{1 }∈ ∂G_{1}, ξ_{2 }∈ ∂G_{2}, where θ is a constant, , .
Then φ satisfies the nonsmooth (PS) condition on X.
Proof. Let {u_{n}} be a nonsmooth (PS) sequence, then by (B) we have
and consequently {u_{n}} is bounded.
Thus by passing to a subsequence if necessary, we may assume that u_{n }⇀ u in X as n → ∞. We have
with ε_{n }↓ 0, where ξ_{in}(x) ∈ ∂G_{i}(x, u_{n}) for a.e. x ∈ Ω, i = 1, 2. From Chang [22] or Theorem 1.3.10 of [33], we know that . Since X is embedded compactly in , we have that u_{n }→ u as n → ∞ in . So using Proposition 2.2, we have
Therefore we obtain . But we know that J' is a mapping of type (S_{+}). Thus we have
Remark 3.1. Note that our condition (1.2) is stronger than (1.2) of [21]. Because Ψ' is weaklystrongly continuous in [21], to verify that φ satisfies (PS) condition on X, it is enough to verify that any (PS) sequence is bounded. However, in this paper we do not know whether ξ(u) is weaklystrongly continuous, where ξ(u) ∈ ⇀Ψ. Therefore, it will be very useful to consider this problem.
Below we denote
We shall use the following conditions.
(B_{1}) ∃ c_{0 }> 0 such that G_{2}(x, t) ≥  c_{0 }for x ∈ Ω and t ∈ ℝ.
(B_{2}) and M > 0 such that 0 < G_{2}(x, u) ≤ θ 〈u, ξ_{2}〉 for x∈ Ω, u ∈ X and u ≥ M, ξ_{2 }∈ ⇀G_{2}.
Corollary 3.1. Assume (P), (A) and (A_{1}) hold. Then φ satisfies nonsmooth (PS) condition on X provided either one of the following conditions is satisfied.
(1). λ ∈ ℝ and μ = 0.
(2). λ ∈ ℝ, μ = 0 and (B_{1}) holds.
(3). λ ∈ ℝ, μ ∈ ℝ and (B_{2}) holds.
Proof. In case (1) or (2), we have, for x ∈ Ω and t ∈ ℝ,
which shows that the condition (B) with θ = 0 is satisfied.
In case (3), noting that (B_{2}) and (A) imply (B_{1}), by the conclusion (1) and (2) we know φ satisfies (PS) condition if μ ≤ 0. Below assume μ > 0. The conditions (B_{2}) and (A) imply that, for x ∈ Ω and u ∈ X,
so we have
which shows (B) holds. The proof is complete. ■
As X is a separable and reflexive Banach space, there exist (see [[34], Section 17]) such that
For k = 1, 2, . . . , denote
Proposition 3.5. [35]Assume that Ψ : X → ℝ is weaklystrongly continuous and Ψ (0) = 0. Let γ > 0 be given. Set
Then β_{k }→ 0 as k → ∞.
Proposition 3.6. (Nonsmooth Mountain pass theorem, see [23,33]) If X is a reflexive Banach space, φ : X → ℝ is a locally Lipschitz function which satisfies the nonsmooth (PS)_{c}condition, and for some r > 0 and e_{1 }∈ X with e_{1} > r, max{φ(0), φ(e_{1})} ≤·inf{φ(u) : u = r}. Then φ has a nontrivial critical u ∈ X such that the critical value c = φ(u) is characterized by the following minimax principle
where Γ = {γ ∈ C([0, 1], X) : γ(0) = 0, γ(1) = e_{1}}.
Proposition 3.7. (Nonsmooth Fountain theorem, see [36]) Assume (F_{1}) X is a Banach space, φ : X → ℝ be an invariant locally Lipschitz functional, the subspaces X_{k}, Y_{k }and Z_{k }are defined by (3.3).
If, for every k ∈ ℕ, there exist ρ_{k }> r_{k }> 0 such that
(F_{4}) φ satisfies the nonsmooth (PS)_{c }condition for every c > 0, then φ has an unbounded sequence of critical values.
Proposition 3.8. (Nonsmooth dual Fountain theorem, see [37]) Assume (F_{1}) is satisfied and there is a k_{0 }> 0 such that, for each k ≥ k_{0}, there exists ρ_{k }> γ_{k }> 0 such that
(D_{4}) φ satisfies the nonsmooth condition for every , then φ has a sequence of negative critical values converging to 0.
Remark 3.2. We say φ that satisfies the nonsmooth condition at level c ∈ ℝ (with respect to (Y_{n})) if any sequence {u_{n}} ⊂ X such that
contains a subsequence converging to a critical point of φ.
4 Existence and multiplicity of solutions
In this section, using the critical point theory, we give the existence and multiplicity results for problem (1.1). We shall use the following assumptions:
(S) For i = 1, 2, G_{i}(x, t) = G_{i}(x, t), ∀x ∈ Ω, ∀t ∈ ℝ.
Remark 4.1.
(1) It follows from (A), (A_{2}) and (O_{2}) that
(2)It follows from (A) and (B_{2}) that (see [33, p. 298])
The following is the main result of this article.
Theorem 4.1. Assume (P), (A), (A_{1}) hold.
(1) If (B_{1}) holds, then for every λ ∈ ℝ and μ ≤ 0, problem (1.1) has a solution which is a minimizer of the corresponding functional φ.
(2) If (B_{1}), (A_{2}), (O_{1}), (O_{2}) hold, then for every λ > 0 and μ ≤ 0, problem (1.1) has a nontrivial solution v_{1 }such that v_{1 }is a minimizer of φ and φ(v_{1}) < 0.
(3) If (A_{2}), (B_{2}), (O_{2}) hold, then for every μ > 0, there exists λ_{0}(μ) > 0 such that when λ ≤ λ_{0}(μ), problem (1.1) has a nontrivial solution u_{1 }such that φ(u_{1}) > 0.
(4) If (A_{2}), (B_{2}), (O_{1}), (O_{2}) holds, then for every μ > 0, there exists λ_{0}(μ) > 0 such that when 0 < λ ≤ λ_{0}(μ), problem (1.1) has two nontrivial solutions u_{1 }and v_{1 }such that φ(u_{1}) > 0 and φ(v_{1}) < 0.
(5) If (A_{2}), (B_{2}), (O_{1}), (O_{2}) and (S) holds, then for every μ > 0 and λ ∈ ℝ, problem (1.1) has a sequence of solutions {±u_{k}} such that φ(±u_{k}) → ∞ as k → ∞.
(6) If (A_{2}), (B_{2}), (O_{1}), (O_{2}) and (S) holds, then for every λ > 0 and μ ∈ ℝ, problem (1.1) has a sequence of solutions {±v_{k}} such that φ(±v_{k}) < 0 and φ(±v_{k}) → 0 as k → ∞.
Proof. We will use c, c' and c_{i }as a generic positive constant. By Corollary 3.1, under the assumptions of Theorem 4.1, φ satisfies nonsmooth (PS) condition. We write
then Ψ = Ψ_{1 }+ Ψ_{2}, φ(u) = J(u)  Ψ (u) = J(u)  Ψ_{1}(u)  Ψ_{2}(u). Firstly, we use to denote its extension to , where i = 1, 2. From (A) and Theorem 1.3.10 of [33] (or Chang [22]), we see that (u) is locally Lipschitz on and for a.e. x ∈ Ω and i = 1, 2. In view of Proposition 2.4 and Theorem 2.2 of [22], we have that is also locally Lipschitz, and ∂Ψ_{1}(u) ⊆ λ ∫_{Ω }a_{1}(x) ∂G_{1}(x, u) dx, ∂Ψ_{2}(u) ⊆ μ ∫_{Ω }a_{2}(x) ∂G_{1}(x, u) dx, (see [38]), where stands for the restriction of to X for i = 1, 2. Therefore, φ is a locally Lipschitz functional on X.
(1) Let λ ∈ ℝ and μ ≤ 0. By (A),
By (B_{1}), Ψ_{2}(u) ≤  μc_{0 }∫_{Ω }a_{2}(x) dx = c_{5}. Hence . By (A_{1}), , so φ is coercive, that is, φ(u) → ∞ as u → ∞. Thus φ has a minimize which is a solution of (1.1).
(2) Let λ > 0, μ ≤ 0 and the assumptions of (2) hold. By the above conclusion (1), φ has a minimize v_{1}. Take such that 0 ≤ v_{0}(x) ≤ min{δ_{1}, δ_{2}}, and . By (O_{1}) and (O_{2}) we have, for t ∈ (0, 1) small enough,
Since , we can find t_{0 }∈ (0, 1) such that φ(t_{0}v_{0}) < 0, and this shows φ(v_{1}) = inf_{u}_{∈}_{X }φ(u) < 0. So v_{1 }≠ 0 because φ(0) = 0. The conclusion (2) is proved.
(3) Let μ > 0 and the assumptions of (3) hold. By Remark 4.1.(1), for sufficiently small u
Since and , there exists γ > 0 and α > 0 such that J(u)  Ψ_{2}(u) ≥ α for u ∈ S_{γ}. We can find λ_{0}(μ) > 0 such that when λ ≤ λ_{0}(μ), Ψ_{1}(u) ≤ α/2 for u ∈ S_{γ}. So when λ ≤ λ_{0}(μ), φ(u) ≥ α/2 > 0 for u ∈ S_{γ}. By Remark 4.1.(2), noting that , we can find a u_{0 }∈ X such that u_{0} > γ and φ(u_{0}) < 0. By Proposition 3.6 problem (1.1) has a nontrivial solution u_{1 }such that φ(u_{1}) > 0.
(4) Let μ > 0 and the assumptions of (4) hold. By the conclusion (3), we know that, there exists λ_{0}(μ) > 0 such that when 0 < λ ≤ λ_{0}(μ), problem (1.1) has a nontrivial solution u_{1 }such that φ(u_{1}) > 0. Let γ and α be as in the proof of (3), that is, φ(u) ≥ α/2 > 0 for u ∈ S_{γ}. By (O_{1}), (O_{2}) and the proof of (2), there exists w ∈ X such that w < γ and φ(w) < 0. It is clear that there is v_{1 }∈ B_{γ}, a minimizer of φ on B_{γ}. Thus v_{1 }is a nontrivial solution of (1.1) and φ(v_{1}) < 0.
(5) Let μ > 0, λ ∈ ℝ and the assumptions of (5) hold. By (S), we can use the nonsmooth version Fountain theorem with the antipodal action of ℤ_{2 }to prove (5) (see Proposition 3.7). Denote
Let β_{k}(γ) be as in Proposition 3.5. By Proposition 3.5, for each positive integer n, there exists a positive integer k_{0}(n) such that β_{k}(n) ≤ 1 for all k ≥ k_{0}(n). We may assume k_{0}(n) < k_{0}(n + 1) for each n. We define {γ_{k }: k = 1, 2, . . . , } by
Note that γ_{k }→ ∞ as k → ∞. Then for u ∈ Z_{k }with u = γ_{k }we have
and consequently
i.e., the condition (F_{2}) of Proposition 3.7 is satisfied.
By (A), (A_{1}), (B_{2}) and Remark 4.1.(2), we have
Noting that and all norms on a finite dimensional vector space are equivalent each other, we can see that, for each Y_{k}, φ(u) →  ∞ as u ∈ Y_{k }and u → ∞. Thus for each k there exists ρ_{k }> γ_{k }such that φ(u) < 0 for u ∈ Y_{k }∩ S_{ρk}, so the condition (F_{3}) of Proposition 3.7 is satisfied. As was noted earlier, φ satisfies nonsmooth (PS) condition. By Proposition 3.7 the conclusion (5) is true.
(6) Let λ > 0, μ ∈ ℝ and the assumptions of (5) hold. Let us verify the conditions of the Nonsmooth dual Fountain theorem (see Proposition 3.8). By (S), φ is invariant on the antipodal action of ℤ_{2}. For Ψ(u) = ∫_{Ω }F(x, u)dx = Ψ_{1}(u)+ Ψ_{2}(u) let β_{k}(1) be as in Proposition 3.5, that is
By Proposition 3.5, there exists a positive integer k_{0 }such that for all k ≥ k_{0}. Setting ρ_{k }= 1, then for k ≥ k_{0 }and u ∈ Z_{k }∩ S_{1}, we have
which shows that the condition (D_{1}) of Proposition 3.8 is satisfied.
Since is the closure of in , we may choose {Y_{k }: k = 1, 2, . . . , }, a sequence of finite dimensional vector subspaces of X defined by (3.5), such that for all k. For each Y_{k}, because all norms on Y_{k }are equivalent each other, there is ε ∈ (0, 1) such that for every and By (O_{1}) and (O_{2}), for u ∈ Y_{k }∩ B_{ε }we have
Because there exists γ_{k }∈ (0, ε) such that
thus the condition (D_{2}) of Proposition 3.8 is satisfied.
Because Y_{k }∩ Z_{k }≠ ∅ and γ_{k }< ρ_{k}, we have
On the other hand, for any u ∈ Z_{k }with u ≤ 1 = ρ_{k}, we have φ(u) = J(u)  Ψ(u) ≥ Ψ(u) ≥ β_{k}(1). Noting that β_{k }→ 0 as k → ∞, we obtain d_{k }→ 0, i.e., (D_{3}) of Proposition 3.8 is satisfied.
Finally let us prove that φ satisfies nonsmooth condition for every c ∈ R. Suppose . Similar to the process of verifying the (PS) condition in the proof of Proposition 3.3, we can get in X. Let us prove 0 ∈ ∂φ(u) below. Notice that
Using Proposition 2.8.4, Going to limit in the right side of above equation, we have
so m(u) ≡ 0, i.e., 0 ∈ ∂φ(u), this shows that φ satisfies the nonsmooth condition for every c ∈ ℝ. So all conditions of Proposition 3.8 are satisfied and the conclusion (6) follows from Proposition 3.8. The proof of Theorem 4.1 is complete. ■
Remark 4.2
Theorem 4.1 includes several important special cases. In particular, in the case of the problem (1.4), i.e., in the case that
all conditions of Theorem 4.1 are satisfied provided (P), (A*), (A_{1}), and (A_{2}) hold.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
GD conceived of the study, and participated in its design and coordination and helped to draft the manuscript. RM participated in the design of the study. All authors read and approved the final manuscript.
Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions. Research supported by the NSFC (Nos. 11061030, 10971087), 1107RJZA223 and the Fundamental Research Funds for the Gansu Universities.
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