Nonlinear Differ. Equ. Appl.c© 2014 Springer BaselDOI 10.1007/s00030-014-0264-3

Nonlinear Differential Equationsand Applications NoDEA

Existence of solutions of sublinear ellipticsystems with quadratic growthin the gradient

Mohamed Benrhouma

Abstract. In this paper, we deal with the following sublinear elliptic sys-tem: {

−Δu+ u+ |∇u|2 = a(x)|v|p + f, x ∈ RN ,

−Δv + v + |∇v|2 = b(x)|u|q + g, x ∈ RN ,

where 0 < p < 1 and 0 < q < 1. Under suitable assumptions on the termsa, b, f and g and by using the Schauder fixed point theorem, we obtaina solution for an approximated system. The limit of the approximatedsolutions is a nonnegative solution.

Mathematics Subject Classification (2000). 35J55, 35J60.

Keywords. Sublinear elliptic systems, Quadratic growth in the gradient,Nonnegative solutions.

1. Introduction

In this paper, we are concerned with the existence of solutions of the followingsystem: {

−Δu+ u+ |∇u|2 = a(x)|v|p + f, x ∈ RN ,

−Δv + v + |∇v|2 = b(x)|u|q + g, x ∈ RN ,

(1.1)

Our main hypotheses are cited below:(H1) 0 < p < 1, 0 < q < 1, N > 2.(H2) a(x) , b(x) ≥ 0, a.e. in R

N , a ∈ Lr(RN ) ∩ L∞(RN ), b ∈ Ls(RN ) ∩L∞(RN ),

where r =2

1 − pand s =

21 − q

.

(H3) We suppose also that f, g ≥ 0, f, g �= 0 and f, g ∈ L2(RN ) ∩L∞(RN ).

M. Benrhouma NoDEA

The study of semilinear systems in the whole space RN has received

much attention recently. See, for example, [1–12]. Benrhouma [1] studied theexistence of solutions of the following system:{−Δu+ u = α

α+βa(x)|v|β |u|α−2u in RN

−Δv + v = βα+βa(x)|u|α|v|β−2v in R

N(1.2)

With the help of the Nehari manifold and the linking theorem, the authorproved the existence of at least two solutions of (1.2). Chen [3] studied theexistence of multiple positive solutions of the system:{

Δu = a(|x|)g(v), x ∈ RN ,

Δv = b(|x|)f(u), x ∈ RN ,

(1.3)

Costa [4] studied the existence of solutions of the system:{−Δu+ a(x)u = f(x, u, v), x ∈ RN ,

−Δv + b(x)v = g(x, u, v), x ∈ RN ,

(1.4)

By using a variant of the generalised Moutain-Pass theorem [13], the authorproved the existence of a nontrivial solution of (1.4). The supersolution andsubsolution method is used by Kawano and Kusano [10] to establish the exis-tence of infinitely positive entire solutions of the semilinear system:{

Δu+ f(x, u, v) = 0, x ∈ RN ,

Δv + g(x, u, v) = 0, x ∈ RN ,

(1.5)

Similar results have been proved for quasilinear or semilinear elliptic systemin bounded domains, we can for instance cite [14–20]. Other works for singleequation (see [21–28] and the references therein ). A good survey on the ex-istence results for quasilinear elliptic equations with quadratic growth in thegradient can be found in [26–28]. By contrast, it seems to us that very few re-sults are known on the quasilinear or semilinear elliptic system with gradientterm in R

N . We can quote [29–31].Miao and Yang [29] established the existence of infinitely positive bounded

entire solutions of the following system:{div

(|∇u|p−2∇u) + f(x, u, v) = 0, x ∈ RN ,

div(|∇v|p−2∇v) + g(x, u, v) = 0, x ∈ R

N. (1.6)

where the functions f and g are, among other assumptions, locally Lipschitzcontinuous in u and v which is not the case in system (1.1). Zhang and Liu[30] studied the existence and the nonexistence of entire positive solutions forthe system: {

Δu+ |∇u| = p(|x|)f(u, v), x ∈ RN ,

Δv + |∇v| = q(|x|)g(u, v), x ∈ RN ,

. (1.7)

with the imposed conditions on the potential functions p and q are not satisfiedin our case. Ghergu and Radulescu [31] studied the existence of explosivesolutions for the system:

Existence of solutions of sublinear

{Δu+ |∇u| = p(|x|)f(v), x ∈ R

N ,

Δv + |∇v| = q(|x|)g(u), x ∈ RN ,

. (1.8)

In the present work, we are interested in finding the existence of a nonnegativesolution of (1.1) [ie: (u, v) is nonnegative if u ≥ 0 and v ≥ 0 a.e.]. Using a prioriestimates technique and through an approximation process we will be able toshow the existence of a nonnegative and nontrivial solution of (1.1). Applyingthe Schauder fixed point theorem we prove the existence of a solution (un, vn)of an approximated system and we show that (un, vn) converges to a non-negative solution of (1.1). The main difficulty to deal with the approximatedsystems consists in the boundedness of (un) and (vn) in L∞(RN ) which cannot be solved by the arguments of Stampacchia [32]. In fact, these argumentsare only used for bounded domains. Moreover, to prove the almost everywhereconvergence of ∇un and of ∇vn, we give a technique which is different of theone often used [21,26,27,33,34]. Our main result is in the following

Theorem 1.1. Assume (H1) − (H3) hold. Then the system (1.1) has at leastone nonnegative solution.

We divide this paper into three sections. In Sect. 2, we establish theexistence of a solution of an approximated system and we give some usefulproperties of this solution. In Sect. 3, we prove the Theorem 1.1. We shouldproceed by steps.

We introduce the Banach space H = H1(RN )×H1(RN ) equipped by thenorm

‖(u, v)‖ =(‖u‖2

H1(RN ) + ‖v‖2H1(RN )

) 12

We say that (u, v) is a weak solution pair of the system (1.1), if (u, v) ∈ H and∫RN

(∇u∇ϕ+ uϕ+ ∇v∇ψ + vψ + |∇u|2ϕ+ |∇v|2ψ)dx

−∫

RN

((a(x)vp + f)ϕ+(b(x)uq + g)ψ) dx=0 ∀(ϕ,ψ)∈C∞c (RN )×C∞

c (RN ).

2. The approximated quasilinear systems

We devote this section to study a sequence of approximated systems to (1.1).We consider the following approximated system:⎧⎪⎨

⎪⎩−Δu+ u+ |∇u|2

1+ 1n |∇u|2 = a(x) |v|p

1+ 1n |v|p + f, x ∈ Bn

−Δv + v + |∇v|21+ 1

n |∇v|2 = b(x) |u|q1+ 1

n |u|q + g, x ∈ Bn

(2.1)

where Bn ={x ∈ R

N , |x| < n}.

Lemma 2.1. Assume (H1) − (H3) hold. Then, there exists (un, vn) ∈ Hn =H1

0 (Bn) ×H10 (Bn) which is a solution pair of (2.1).

M. Benrhouma NoDEA

Proof. To prove it, we define the function Ln : Hn → L2(Bn) × L2(Bn) by

Ln(u, v)

=(

− |∇u|21 + 1

n |∇u|2 + a(x)|v|p

1 + 1n |v|p + f,− |∇v|2

1 + 1n |∇v|2 + b(x)

|u|q1 + 1

n |u|q + g

)

First, observe that Ln is continuous on Hn. Indeed, let (uk, vk) be a se-quence in Hn which converges to (u, v) in Hn as k → ∞. Which implies,by a subsequence, that uk(x) → u(x), ∇uk(x) → ∇u(x), vk(x) → v(x) and∇vk(x) → ∇v(x) a.e. in Bn.

By virtue of Lebesgue’s dominated convergence theorem, we deduce that∫Bn

∣∣∣∣ |∇uk|21 + 1

n |∇uk|2 − |∇u|21 + 1

n |∇u|2 − a(x)( |vk|p

1 + 1n |vk|p − |v|p

1 + 1n |v|p

)∣∣∣∣2

dx → 0

as k → ∞and∫

Bn

∣∣∣∣ |∇vk|21 + 1

n |∇vk|2 − |∇v|21 + 1

n |∇v|2 − b(x)( |uk|q

1 + 1n |uk|q − |u|q

1 + 1n |u|q

)∣∣∣∣2

dx → 0

as k → ∞.

It follows that, Ln is continuous.Let E = L2(Bn) × L2(Bn). If we equipped E with the norm

‖(u, v)‖E = ‖u‖L2(Bn) + ‖v‖L2(Bn) ,

it becomes a Banach space. Define the following function

S : E → Hn

(u, v) �−→ ((−Δ + I)−1(u), (Δ + I)−1(v)

)where (−Δ + I)−1 is the inverse of the sum of the Laplacian operator andidentity function. We see that the solutions of (2.1) are just the fixed points ofthe composition S ◦Ln. By classical arguments, the operator S is compact andhence the composition of it with the continuous operator Ln is also compact.On the other hand, we have

‖Ln(u, v)‖E =∥∥∥∥− |∇u|2

1 + 1n |∇u|2 + a(x)

|v|p1 + 1

n |v|p + f

∥∥∥∥L2(Bn)

+∥∥∥∥− |∇v|2

1 + 1n |∇v|2 + b(x)

|u|p1 + 1

n |u|p + g

∥∥∥∥L2(Bn)

≤ (2n+ n|a|∞ + n|b|∞)√meas(Bn) + ‖f‖2 + ‖g‖2 .

where meas(Bn) is the Lebesgue’s measure of Bn.Therefore, by the continuity of S, there exits R > 0 such that

‖S ◦ Ln(u, v)‖n < R, ∀ (u, v) ∈ Hn.

In particular, the compact operator S◦Ln maps the ball in Hn centered at zeroand with radius R into itself. Which implies, by the Schauder fixed theorem,the existence of a fixed point (un, vn) ∈ Hn of S ◦Ln that is a solution of (2.1).

Existence of solutions of sublinear

Notice that we can extend un and vn by zero outside ofBn, which providesthat (un, vn) ∈ H1(RN ) ×H1(RN ). �

Lemma 2.2. The functions un and vn are nonnegative,

(i.e. un(x) ≥ 0, and vn(x) ≥ 0 a.e. in RN ,∀ n ≥ 1).

Proof. For t ∈ R, we denote by t+ = max(t, 0) and t− = min(t, 0).Taking into account that (un, vn) is a solution of (2.1) and putting

(u−n exp (−un), 0) as test function, we get∫

{x∈Bn, un≤0}|∇un|2 exp (−un)dx+

∫Bn

|u−n |2 exp (−un)dx

+∫

Bn

(1

1 + 1n |∇un|2 − 1

)u−

n |∇un|2 exp (−un)dx

=∫

Bn

(a(x)|vn|p1 + 1

n |vn|p + f

)u−

n exp (−un)dx ≤∫

Bn

fu−n exp (−un)dx.

Having in mind that∫Bn

(1

1 + 1n |∇un|2 − 1

)u−

n |∇un|2 exp (−un)dx ≥ 0,

we deduce that∫Bn

|u−n |2 exp (−un)dx ≤

∫Bn

fu−n exp (−un)dx ≤ 0.

It leads to

u−n = 0 a.e. in R

N , ∀ n ≥ 1.

By the same argument used above, we prove that

vn ≥ 0 a.e. in RN , ∀ n ≥ 1.

Now, we are going to prove that un and vn are uniformly bounded in H1(RN )∩L∞(RN ). �

Lemma 2.3. The sequences un and vn are uniformly bounded in H1(RN ) ∩L∞(RN ).

Proof. Taking into account that (un, vn) is a solution of (2.1), putting (un, 0)as test function and having in mind that un ≥ 0 a.e. in Bn, we get∫

RN

(|∇un|2 + |un|2)dx ≤∫

RN

a(x)vpnundx+

∫RN

fundx

≤ ‖un‖2 ‖vn‖p2 ‖a‖r + ‖f‖2 ‖un‖2

≤ ‖un‖H1(RN ) ‖vn‖pH1(RN ) ‖a‖r + ‖f‖2 ‖un‖H1(RN )

≤ ‖(un, vn)‖p+1 ‖a‖r + ‖f‖2 ‖(un, vn)‖ (2.2)

where r is defined as in (H2). Also, taking (0, vn) as test function, we get∫RN

(|∇vn|2 + |vn|2)dx ≤ ‖(un, vn)‖q+1 ‖b‖s + ‖g‖2 ‖(un, vn)‖. (2.3)

M. Benrhouma NoDEA

where s is defined as in (H2). Combining (2.2) and (2.3), we obtain that

‖(un, vn)‖2 ≤ ‖(un, vn)‖p+1 ‖a‖r + ‖(un, vn)‖q+1 ‖b‖s + (‖f‖2 + ‖g‖2) ‖(un, vn)‖Since p+ 1, q + 1 < 2, then (un, vn) is uniformly bounded in H, which yieldsthat (un) and (vn) are uniformly bounded in H1(RN ). �

To prove the a priori estimate in L∞(Bn). Choose M > 0, taking intoaccount that (un, vn) is a solution of (2.1) and putting ((un + vn −M)+,(un + vn −M)+) as test function.∫

Dn

(|∇un|2 + 2∇un∇vn + |∇vn|2)dx+∫

Bn

(un + vn)(un + vn −M)+dx

+∫

Bn

(un + vn −M)+( |∇un|2

1 + 1n |∇un|2 +

|∇vn|21 + 1

n |∇vn|2)dx

=∫

Bn

(a(x)

vpn

1+ 1nv

pn

+b(x)uq

n

1+ 1nu

qn

)(un + vn −M)+dx

+∫

Bn

(f + g)(un + vn −M)+dx,

where Dn = {x ∈ Bn, un + vn −M ≥ 0}. By the previous equality we get∫Bn

(un + vn)(un + vn −M)+dx

≤∫

Bn

(a(x)vpn + b(x)uq

n + f + g)(un + vn −M)+dx

≤∫

Bn

(12(un + vn) + c1(a(x))

11−p

)(un + vn −M)+

+∫

Bn

(c2(b(x))

11−q + f + g

)(un + vn −M)+,

which implies that,

12

∫Bn

(un+vn)(un+vn −M)+dx

≤∫

Bn

(c1(a(x))1

1−p +c2(b(x))1

1−q + f+g)(un + vn −M)+dx

and12

∫Bn

|(un+vn −M)+|2dx

≤∫

Bn

(c1(a(x))1

1−p +c2(b(x))1

1−q +f+g− 12M)(un + vn −M)+dx.

Since a, b, f and g are in L∞(RN ), then for M large enough we get∫Bn

|(un + vn −M)+|2dx ≤ 0.

Existence of solutions of sublinear

So, (un + vn −M)+ = 0 a.e. in Bn. It leads, by Lemma 2.2, to

un ≤ M and vn ≤ M a.e. in RN , ∀ n ≥ 1.

3. Main result

In this section we study the strongly convergence of the approximated solution(un, vn).

Since un and vn are uniformly bounded inH1(RN ), then there exist u andv in H1(RN ) such that up to a subsequence un ⇀ u and vn ⇀ v in H1(RN ),un(x) → u(x) and vn(x) → v(x) a.e. in R

N . By the almost convergence of un

and vn, we get that u and v are nonnegative functions.We need the following lemma to prove that un and vn converge to u and

v respectively in H1(Ω), for every bounded domain Ω in RN .

Lemma 3.1. Assume (H1) − (H3) hold, then for every ψ ∈ C∞0 (RN ) such that

ψ ≥ 0, we have

limn→∞

∫RN

|∇(un − u)|2ψdx = 0 and limn→∞

∫RN

|∇(vn − v)|2ψdx = 0

Proof. In the literature, to the author’s knowledge, few results are known onthe strong convergence of the gradients of solutions to elliptic equations. Wecan, for example, quote [21,26,27,33,34]. In these works the proofs are basedon the following functions

Tk(s) ={s if |s| ≤ k,k s

|s| if |s| ≥ k, Gk(s) = s− Tk(s), ϕγ(s)=s exp(γs2), s∈R

where k > 0 and γ is a suitable constant. In the present work we should prove,in two steps, that

limn→∞

∫RN

|∇(un − u)|2ψdx = 0, ∀ ψ ∈ C∞0 (RN )

�Step 1 limn→∞

∫RN |∇(un − u)−|2ψdx = 0, ∀ ψ ∈ C∞

0 (RN ), ψ ≥ 0.

Proof. Let ψ ∈ C∞0 (RN ) such that ψ ≥ 0. There exists n0 ∈ N such that

supp(ψ) ⊂ Bn for any n ≥ n0. Choosing n ≥ n0, taking into account that(un, vn) is a solution of (2.1) and putting ((un − u)−ψ e−un , 0) as test function.We get∫

RN

∇un∇(un − u)−ψ e−undx+∫

RN

(un − u)−∇un∇ψ e−undx

−∫

RN

(un − u)−ψ|∇un|2 e−undx+∫

RN

un(un − u)−ψ e−undx

+∫

RN

|∇un|21 + 1

n |∇un|2 (un − u)−ψ e−undx

=∫

RN

(a(x)

vpn

1 + 1nv

pn

+ f

)(un − u)−ψ e−undx. (3.1)

M. Benrhouma NoDEA

Adding and subtracting∫

RN ∇u∇(un − u)−ψ e−undx to (3.1) and having inmind that∫

RN

(1

1 + 1n |∇un|2 − 1

)(un − u)−ψ|∇un|2 e−undx ≥ 0,

we derive that∫RN

|∇(un−u)−|2ψ e−undx+∫

RN

∇u∇(un − u)−ψ e−undx

+∫

RN

(un − u)−∇un∇ψ e−undx+∫

RN

un(un − u)−ψ e−undx

≤∫

RN

(a(x)

vpn

1 + 1nv

pn

+ f

)(un − u)−ψ e−undx (3.2)

On the other hand, we have∫RN

∇u∇(un − u)−ψ e−undx

=∫

RN

∇u∇ ((un − u)− e−un

)ψdx+

∫RN

∇u∇un(un − u)− e−unψdx(3.3)

By the weak convergence of ((un − u)− e−un) to 0 in H1(RN ), we obtain∫RN

∇u∇ ((un − u)− e−un

)ψdx → 0. (3.4)

Using Lemma 2.3, Holder inequality and by the Lebesgue dominated conver-gence theorem, we get∣∣∣∣∫

RN

∇u∇un(un − u)− e−unψdx

∣∣∣∣ ≤ C

(∫RN

|∇u|2|(un − u)−|2ψ2dx

) 12

→ 0.

(3.5)

Combining (3.3)–(3.5), we infer that∫RN

∇u∇(un − u)−ψ e−undx → 0. (3.6)

Likewise, we have∣∣∣∣∫

RN

(un − u)−∇un∇ψ e−undx

∣∣∣∣ ≤ C

(∫RN

|(un − u)−|2|∇ψ|2dx) 1

2

→ 0,

∣∣∣∣∫

RN

un(un − u)−ψ e−undx

∣∣∣∣ ≤ C1

(∫RN

|(un − u)−|2ψ2dx

) 12

→ 0.

(3.7)

and ∣∣∣∣∫

RN

(a(x)

vpn

1 + 1nv

pn

+ f

)(un − u)−ψ e−undx

∣∣∣∣≤ (|a|∞Mp + |f |∞)

∫RN

|(un − u)−|ψdx → 0, (3.8)

where M is defined in the proof of Lemma 2.3. �

Existence of solutions of sublinear

Moreover we have

e−M

∫RN

|∇(un − u)−|2ψdx ≤∫

RN

|∇(un − u)−|2ψ e−undx, (3.9)

Combining (3.2), (3.6)–(3.9), we deduce that

limn→∞

∫RN

|∇(un − u)−|2ψdx = 0, ∀ ψ ∈ C∞0 (RN ), ψ ≥ 0

Step 2 limn→∞∫

RN |∇(un − u)+|2ψdx = 0, ∀ ψ ∈ C∞0 (RN ), ψ ≥ 0.

Proof. Let ψ ∈ C∞0 (RN ) such that ψ ≥ 0. There exists n1 ∈ N such that

supp(ψ) ⊂ Bn for any n ≥ n1. Choosing n ≥ n1, taking into account that(un, vn) is a solution of (2.1) and putting ((un − u)+ψ, 0) as test function. Weget∫

RN

∇un∇(un − u)+ψdx+∫

RN

(un − u)+∇un∇ψdx+∫

RN

un(un − u)+ψdx

+∫

RN

|∇un|21+ 1

n |∇un|2 (un − u)+ψdx=∫

RN

(a(x)

vpn

1+ 1nv

pn

+ f

)(un − u)+ψdx.

(3.10)

Adding and subtracting∫

RN ∇u∇(un − u)+ψdx to (3.10),we derive that∫

RN

|∇(un − u)+|2dx+∫

RN

∇u∇(un − u)+ψdx+∫

RN

(un − u)+∇un∇ψdx

+∫

RN

un(un − u)+ψdx ≤∫

RN

(a(x)

vpn

1 + 1nv

pn

+ f

)(un − u)+ψdx. (3.11)

By the weak convergence of (un − u)+ to 0 in H1(RN ), we get∫RN

∇u∇(un − u)+ψdx → 0. (3.12)

By the Lebesgue dominated convergence theorem, we obtain∫RN

un(un − u)+ψdx → 0 and∫

RN

(a(x)

vpn

1 + 1nv

pn

+ f

)(un − u)+ψdx → 0.

(3.13)

Also, we have∣∣∣∣∫

RN

(un − u)+∇un∇ψdx∣∣∣∣ ≤ C2

(∫RN

|(un − u)+|2|∇ψ|2dx) 1

2

→ 0. (3.14)

Combining (3.11)–(3.14), we infer that

limn→∞

∫RN

|∇(un − u)+|2ψdx = 0, ∀ ψ ∈ C∞0 (RN ), ψ ≥ 0.

Follow the same technique used above and in two steps, we obtain

limn→∞

∫RN

|∇(vn − v)|2ψdx = 0, ∀ ψ ∈ C∞0 (RN ), ψ ≥ 0

�

M. Benrhouma NoDEA

Proof of Theorem 1.1 completed: Let (ϕ,ψ) ∈ C∞0 (RN )×C∞

0 (RN ), thereexists n2 ∈ N such that

Supp(ϕ) ⊂ Bn and Supp(ψ) ⊂ Bn, ∀ n ≥ n2.

Choosing n ≥ n2, taking into account that (un, vn) is a solution of (2.1) andputting (ϕ,ψ) as test function, we obain∫

RN

(∇un∇ϕ+ unϕ+ ∇vn∇ψ + vnψ +

ϕ|∇un|21 + 1

n |∇un|2 +ψ|∇vn|2

1 + 1n |∇vn|2|

)dx

=∫

RN

((a(x)

vpn

1 + 1nv

pn

+ f

)ϕ+

(b(x)

uqn

1 + 1nu

qn

+ g

)ψ

)dx. (3.15)

By the weak convergence of un and vn to u and v respectively in H1(RN ), weget ∫

RN

(∇un∇ϕ+ unϕ+ ∇vn∇ψ + vnψ) dx

→∫

RN

(∇u∇ϕ+ uϕ+ ∇v∇ψ + vψ) dx. (3.16)

By the Lebesgue dominated convergence theorem, we deduce that∫RN

a(x)vpnϕ

1 + 1nv

pndx →

∫RN

a(x)vpϕdx. (3.17)

and ∫RN

b(x)uqnψ

1 + 1nu

qndx →

∫RN

b(x)uqψdx. (3.18)

Using Lemma 3.1 and extracting a subsequence there exist F ∈ L2(supp(ϕ))and G ∈ L2(supp(ψ)) such that{∇un(x) → ∇u(x) and ∇vn(x) → ∇v(x) a.e.

|∇un| ≤ |F | and |∇vn| ≤ |G| .

Then, by virtue of Lebesgue’s dominated convergence theorem, we get∫RN

ϕ|∇un|21 + 1

n |∇un|2 dx →∫

RN

|∇u|2ϕdx (3.19)

and ∫RN

ψ|∇vn|21 + 1

n |∇vn|2|dx →∫

RN

|∇v|2ψdx. (3.20)

Passing to the limit in (3.15) and combining (3.16)–(3.20), we get∫RN

(∇u∇ϕ+ uϕ+ ∇v∇ψ + vψ + |∇u|2ϕ+ |∇v|2ψ)dx

−∫

RN

((a(x)vp + f)ϕ+ (b(x)uq + g)ψ) dx = 0 ∀(ϕ,ψ) ∈ C∞c (RN ) × C∞

c (RN ).

Besides, u and v are nonnegative functions and f and g are nontrivial functions,so (u, v) is a weak nonnegative and nontrivial solution of the system (1.1).

Existence of solutions of sublinear

References

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Mohamed BenrhoumaMathematics DepartmentSciences Faculty of MonastirUniversity of Monastir5019 MonastirTunisiae-mail: brhouma06@yahoo.fr

Received: 22 November 2013.

Revised: 23 January 2014.

Accepted: 28 January 2014.