Existence of positive solutions for a class of

Rend. Circ. Mat. Palermo, II. Ser DOI 10.1007/s12215-017-0297-7

Existence of positive solutions for a class of ( p (x), q (x))-Laplacian systems Rafik Guefaifia1 · Salah Boulaaras2,3

Received: 5 October 2016 / Accepted: 13 January 2017 © Springer-Verlag Italia 2017

Abstract Using sub-super solutions method, we study the existence of weak positive solutions for a new class of the system of differential equations with respect to the symmetry conditions. Our results are natural extensions from the previous ones in Fan (J Math Anal Appl 330:665–682 2007), Zhang (Nonlinear Anal 70:305–316 2009). Keywords Differential equations · p(x) -Laplacian · Positive solutions · p(x)-growth conditions Mathematics Subject Classification 35J60 · 35B30 · 35B40

1 Introduction In this paper, we consider the system of differential equations ⎧ p(x) ⎪ ⎨ − p(x) u = λ [λ1 f (v) + μ1 h (u)] in −q(x) v = λq(x) [λ2 g (u) + μ2 τ (v)] in , ⎪ ⎩ u = v = 0 on ∂,

(1.1)

In memory of the father of second author (1910–1999) Mr. Mahmoud ben Mouha Boulaaras.

B

Salah Boulaaras [email protected]; [email protected] Rafik Guefaifia [email protected]

1

Department of Mathematics, Faculty of Exact Sciences, University Tebessa, 12002 Tebessa, Algeria

2

Department of Mathematics, College of Sciences and Arts, Al-Rass, Qassim University, Qassim, Kingdom of Saudi Arabia

3

Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Oran, Algeria

123

R. Guefaifia et al. N where domain with C 2 boundary ∂, and 1 < p (x) , q (x) ∈ ⊂ R is a bounded smooth 1 − C are functions with 1 < p := inf p (x) ≤ p + := sup p (x) < ∞, 1 < q − := inf q (x) ≤ q + := sup q (x) and p(x) is a p(x)-Laplacian defined as p(x) u = div |∇u| p(x)−2 ∇u

and λ, λ1 , λ2 , μ1 , μ2 are positive parameters, while f, g, h, τ are monotone functions in [0, +∞[ such that lim

u→+∞

f (u) = +∞, lim g (u) = +∞, lim h (u) = +∞, lim τ (u) = +∞ u→+∞

u→+∞

u→+∞

and satisfying some natural growth condition at u = ∞. The study of differential equations and variational problems with nonstandard p(x)-growth conditions is a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc. (see [6,18]). Many existence results have been obtained on this kind of problems, see for example [1,4–6,8,9,11,12,14] and in [17], a new class of anisotropic quasilinear elliptic equations with a power like variable reaction term has been investigated. In the last years in [3,7,11,11,16], the regularity and existence of solutions for differential equations with nonstandard p(x)-growth conditions has been studied, and p-Laplacian elliptic systems with p (x) = q (x) = p (a constant) has been archived. We point out that the extension from p-Laplace operator to p(x)-Laplace operator is not trivial, since the p(x)-Laplacian has a more complicated structure then the p-Laplace operator, such as it is nonhomogeneous. Moreover, many results and methods for p-Laplacians are not valid for the p(x)-Laplacian; for example, if is bounded, then the Rayleigh quotient

1 p(x) dx p(x) |∇u| λ p(x) = inf (1.2)

1 p(x) 1, p(x) dx u∈W0 (){0} p(x) |u| is zero in general, and only under some special conditions λ p(x) is positive (see [14]). Maybe the first eigenvalue and the first eigenfunction of the p (x) −Laplacian do not exist, but the fact that the first eigenvalue λ p is positive and the existence of the first eigenfunction are very important in the study of p-Laplacian problem. There are more difficulties in discussing the existence of solutions of variable exponent problems. In [16], the authors considered the existence of positive weak solutions for the following p-Laplacian problem ⎧ − p u = λ f (v) in ⎪ ⎪ ⎪ ⎪ ⎨ − p v = λg (u) in (1.3) ⎪ ⎪ ⎪ ⎪ ⎩ u = v = 0 on ∂, where the first eigenfunction has been used to construct the subsolution of p−Laplacian problem. Under the condition that 1 f M (g (u)) p−1 = 0 for all M > 0, (1.4) lim u→+∞ u p−1 the authors gave the existence of positive solutions for problem (1.3) provided that λ is large enough.

123

Existence of positive solutions for a class of …

In [7], the existence and nonexistence of positive weak solutions to the following quasilinear elliptic system ⎧ ⎪ − p u = λu α v γ in ⎪ ⎪ ⎪ ⎨ (1.5) −q v = λu δ v β in ⎪ ⎪ ⎪ ⎪ ⎩ u = v = 0 on ∂, has been considered where the first eigenfunction has been used to construct the subsolution of problem (1.5) and he obtained the following results: (i) If α, β ≥ 0, γ , δ > 0, θ = ( p − 1 − α) (q − 1 − β) − γ δ > 0, then the problem (1.5) has a positive weak solution for each λ > 0. (ii) If θ = 0 and pγ = q ( p − 1 − α), then there exists λ0 > 0 such that for 0 < λ < λ0 , then problem (1.5) has no nontrivial nonnegative weak solution. For further generalizations of system (1.5) we refer to [5] and [15]. As already discussed before, on the p (x)-Laplacian problems, maybe the first eigenvalue and the first eigenfunction of the p (x)-Laplacian do not exist. Even if the first eigenfunction of the p (x)-Laplacian exists. Because of the nonhomogeneous of the p (x)-Laplacian, the first eigenfunction cannot be used to construct the subsolutions of p (x)-Laplacian problems. Moreover, in [2], [20,23], the authors studied the existence of solutions for the problem 1.3, where some symmetry conditions are imposed. Then in [22] investigated the existence of positive solutions of the system ⎧ − p(x) u = λ p(x) f (v) in ⎪ ⎪ ⎪ ⎪ ⎨ (1.6) − p(x) u = λ p(x) g (u) in ⎪ ⎪ ⎪ ⎪ ⎩ u = v = 0 on ∂, without any symmetry conditions. Motivated by the ideas introduced in [22], where the authors proved the existence of a positive solution when λ is large enough and satisfies the condition 1.4 and they did not assume any symmetric condition, and did not assume any sign condition on f (0) and g(0). Also the authors proved the existence of positive solutions with multiparameter, in this paper we extend this given system of differential equations, where we establish the existence of a positive solution for a new class of this system with respect to the symmetry conditions by constructing a positive subsolution and supersolution and p, q ∈ C 1 are function, λ, λ1 , λ2 , μ1 ,and μ2 are positive parameters and ⊂ R N is a bounded domain and we did not assume any sign condition on f (0), g(0), h(0), τ (0).

2 Preliminary results 1, p(x)

In order to discuss problem (1.1), we need some theories on W0 () which we call 1, p(x) variable exponent Sobolev space. Firstly we state some basic properties of spaces W0 () which will be used later (for details, see [9]). Let us define ⎧ ⎫ ⎨ ⎬ |u (x)| p(x) d x < ∞ . L p(x) () = u : u is a measurable real-valued function such that ⎩ ⎭

123

R. Guefaifia et al.

We introduce the norm on L p(x) () by ⎧ ⎨ |u (x)| p(x) = inf λ > 0 : ⎩

⎫ ⎬ u (x) p(x) dx ≤ 1 λ ⎭

W 1, p(x) () = u ∈ L p(x) () ; |∇u| ∈ L p(x) () ,

and with the norm

u = |u| p(x) + |∇u| p(x) , ∀u ∈ W 1, p(x) (). 1, p(x)

We denote by W0

() the closure of C0∞ () in W 1, p(x) (). 1, p(x)

Proposition 1 (ref. [16]). The spaces L p(x) () , W 1, p(x) () and W0 and reflexive Banach spaces.

() are separable

Throughout the paper, we will assume that: (H 1) p, q ∈ C 1 and 1 < p − ≤ p + , 1 < q − ≤ q + (H 2) f, g, h, τ : [0, +∞[→ R are C 1 , monotone functions such that lim

u→+∞

f (u) = +∞, lim g (u) = +∞, lim h (u) = +∞, lim τ (u) = +∞; u→+∞

f M(g(u)) q

(H 3) limu→+∞ (H 4) limu→+∞ We define

1 − −1

u→+∞

= 0, for all M > 0;

−

u p −1 h(u) = 0, − u p −1

u→+∞

and limu→+∞

τ (u) − u q −1

= 0.

1, p(x)

|∇u| p(x)−2 ∇u∇vd x, ∀u, v ∈ W0

L (u) , v =

() .

∗ 1, p(x) 1, p(x) Then L : W0 () → W0 () is a continuous, bounded and strictly monotone operator, and it is a homeomorphism (see [[13], Theorem 3.1]).

1, p(x)

Define A : W0

A (u) , ϕ =

1, p(x)

() → W0

()

∗

as

1, p(x) |∇u| p(x)−2 ∇u∇ϕ + h (x, u) ϕ d x, for all u, ϕ ∈ W0 () ,

where h (x, u) is continuous on × R and h (x, .) is increasing. It is easy to check that A is a continuous bounded mapping. Copying the proof of [21], we have the following lemma. 1, p(x)

Lemma 1 [Comparison principle] Let u, v ∈ W0 () satisfy Au − Av ≥ 0 in ∗ 1, p(x) 1, p(x) W0 () , ϕ (x) = min {u (x) − v (x) , 0}. If ϕ (x) ∈ W0 () (i.e u ≥ v on ∂), then u ≥ v a.e. in . 1, p(x) 1,q(x) Definition 1 Let (u, v) ∈ W0 () × W0 () , (u, v) is said a weak solution of (1.1) if it satisfies

123


⎧

p(x)−2 ⎪ ∇u.∇ϕd x = λ p(x) [λ1 f (v) + μ1 h (u)] ϕd x, ⎨ |∇u|

q(x)−2 ⎪ ∇v.∇ϕd x = λq(x) [λ2 g (u) + μ2 τ (v)] ψd x, ⎩ |∇v|

1, p(x)

for all (ϕ, ψ) ∈ W0

1,q(x)

() × W0

() with (ϕ, ψ) ≥ 0.

Here and hereafter, we will use the notation d (x, ∂) to denote the distance of x ∈ to . Denote d (x) = d (x, ∂) and ∂ε = {x ∈ : d (x, ∂) < ε} . Since ∂ is C 2 regularly, there exists a constant δ ∈ (0, 1) such that d (x) ∈ C 2 ∂3δ and |∇d (x)| = 1. Denote ⎧ γ d (x) , d (x) < δ ⎪ ⎪ ⎪ ⎪ ⎪ 2 d(x) ⎪

2δ−t −2 ⎨ p− −1 p −1 (λ + μ ) γ δ + γ dt, δ ≤ d (x) < 2δ, 1 1 δ v1 (x) = δ ⎪ ⎪ 2 ⎪

2δ 2δ−t −2 ⎪ p− −1 ⎪ ⎪ dt, 2δ ≤ d (x) ⎩ γ δ + γ δ p −1 (λ1 + μ1 ) δ

and

⎧ γ d (x) , d (x) < δ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ d(x)

2δ−t −2 ⎨ q − −1 p −1 (λ + μ ) γ δ + γ dt, δ ≤ d (x) < 2δ, 2 2 δ v2 (x) = δ ⎪ ⎪ ⎪ 2 ⎪

2δ −2 ⎪ q − −1 ⎪ p −1 (λ + μ ) ⎪ γ δ + γ 2δ−t dt, 2δ ≤ d (x) . 2 2 ⎩ δ δ

Obviously, 0 ≤ v1 (x) , v2 (x) ∈ C 1 . Considering ⎧ ⎨ − p(x) ω (x) = η in ⎩

(2.1) ω = on ∂,

we have the following result Lemma 2 (Lemma 2.1 in [12]). If positive parameter η is large enough and ω is the unique solution of (2.1), then we have (i) For any θ ∈ (0, 1) there exists a positive constant C1 such that 1

C1 η p+ −1+θ ≤ max ω (x) x∈

(ii) There exists a positive constant C2 such that 1

max ω (x) ≤ C2 η p− −1 . x∈

3 Main result In the following, when there is no misunderstanding, we always use Ci to denote positive constants.

123

R. Guefaifia et al.

Theorem 1 Assume that the conditions (H 1) − (H 4) are satisfied. Then problem (1.1) has a positive solution when λ is large enough. Proof We shall establish Theorem 1 by constructing a positive subsolution (φ1 , φ2 ) and supersolution (z 1 , z 2 ) of (1.1) such that φ1 ≤ z 1 and φ2 ≤ z 2 , that is, (φ1 , φ2 ) and (z 1 , z 2 ) satisfies ⎧

p(x)−2 ⎪ ∇φ1 .∇ϕd x ≤ λ p(x) [λ1 f (φ2 ) + μ1 h (φ1 )] ϕd x, ⎨ |∇φ1 |

q(x)−2 ⎪ ∇φ2 .∇ψd x ≤ λ p(x) [λ2 g (φ1 ) + μ2 τ (φ2 )] ψd x, ⎩ |∇2 |

⎧

p(x)−2 ⎪ ∇z 1 .∇ϕd x ≥ λ p(x) [λ1 f (z 2 ) + μ1 h (z 1 )] ϕd x, ⎨ |∇z 1 |

q(x)−2 ⎪ ∇z 2 .∇ψd x ≥ λq(x) [λ2 g (z 1 ) + μ2 τ (z 2 )] ψd x, ⎩ |∇z 2 |

and

1, p(x) 1,q(x) for all (ϕ, ψ) ∈ W0 () × W0 () with (ϕ, ψ) ≥ 0. According to the sub-super solution method for p (x) −Laplacian equations (see [12]), problem (1.1) has a positive solution. Step 1. We will construct a subsolution of (1.1). Let σ ∈ (0, δ) is small enough. Denote ⎧ ⎪ ekd(x) − 1, d (x) < σ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ 2 d(x) ⎨ kσ

− kσ 2δ−t p− −1 (λ + μ ) p −1 dt, σ ≤ d (x) < 2δ, − 1 + ke e 1 1 φ1 (x) = 2δ−σ σ ⎪ ⎪ ⎪ 2 2 ⎪ 2δ ⎪ ⎪ ekσ − 1 + kekσ 2δ−t p− −1 (λ + μ ) p− −1 dt, 2δ ≤ d (x) ⎪ ⎩ 1 1 2δ−σ σ

and ⎧ ⎪ ekd(x) − 1, d (x) < σ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ 2 d(x) ⎨

− kσ − 1 + kσ 2δ−t p− −1 (λ + μ ) q −1 dt, σ ≤ d (x) < 2δ, ke e 2 2 φ2 (x) = 2δ−σ σ ⎪ ⎪ ⎪ 2 2 ⎪

2δ − ⎪ ⎪ kσ − 1 + kekσ 2δ−t p− −1 (λ + μ ) q −1 dt, 2δ ≤ d (x) . ⎪ e ⎩ 2 2 2δ−σ σ

It is easy to see that φ1 , φ2 ∈ C 1 . Denote inf p (x) − 1 inf q (x) − 1 α = min , ,1 4 (sup |∇ p (x)| + 1) 4 (sup |∇q (x)| + 1) and ζ = min {λ1 f (0) + μ1 h (0) , λ2 g (0) + μ2 τ (0) , −1} .

123


By some simple computations we can obtain ⎧ p(x)−1 ⎪ −k ekd(x) ( p (x) − 1) + d (x) + lnkk ∇ p∇d + d ⎪ k , d (x) < σ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2δ−d p−2−1 ⎪ 1 2( p(x)−1) 2δ−d kσ ⎨ ln ke − 2δ−σ ∇ p∇d + d , 2δ−σ 2δ−σ p − −1 − p(x) φ1 = ⎪ 2( p(x)−1) ⎪ ⎪ ⎪ × K ekσ p(x)−1 2δ−d p− −1 −1 (λ + μ ) , σ ≤ d (x) < 2δ, ⎪ 1 1 ⎪ 2δ−σ ⎪ ⎪ ⎪ ⎩ 0, 2δ ≤ d (x) and

−q(x) φ2 =

⎧ q(x)−1 ⎪ −k ekd(x) , d (x) < σ (q (x) − 1) + d (x) + lnkk ∇q∇d + d ⎪ k ⎪ ⎪ ⎪ ⎪ ⎪ −2 ⎪ ⎪ 1 2(q(x)−1) ⎨ ln kekσ 2δ−d q −1 ∇q∇d + d − 2δ−d − 2δ−σ

q −1

2δ−σ

2δ−σ

⎪ ⎪ −1 q(x)−1 2δ−d 2(q(x)−1) ⎪ q − −1 ⎪ ⎪ × K ekσ (λ2 + μ2 ) , ⎪ 2δ−σ ⎪ ⎪ ⎪ ⎩ 0, 2δ ≤ d (x).

σ ≤ d (x) < 2δ,

From (H 3) there exists a positive constant M > 1 such that f (M − 1) ≥ 1, g (M − 1) ≥ 1, h (M − 1) ≥ 1, τ (M − 1) ≥ 1. Let σ =

1 k

ln M, then σ k = ln M

(3.1)

If k is sufficiently large, from (3.1), we have − p(x) φ1 ≤ −k p(x) α, d (x) < σ. Let λζ = kα, then

(3.2)

k p(x) α ≥ −λ p(x) ζ.

From (3.2), we have ⎧ ⎨ − p(x) φ1 ≤ λ p(x) ζ ≤ λ p(x) (λ1 f (0) + μ1 h (0)) (3.3)

⎩

≤ λ p(x) (λ1 f (φ2 ) + μ1 h (φ1 )) , d (x) < σ. Since d (x) ∈ C 2 ∂3δ , there exists a positive constant C3 such that

− p(x) φ1 ≤ K e

kσ

p(x)−1 2δ − d 2δ − σ

2 ( p (x) − 1) −1 p− − 1 (λ1 + μ1 )

×

1 2 ( p (x) − 1) 2δ − d − 2δ − σ p− − 1 2δ − σ 2δ − d p−2−1 kσ ln ke ∇ p∇d + d 2δ − σ

p(x)−1 ≤ C3 K ekσ (λ1 + μ1 ) ln k, σ ≤ d (x) < 2δ.

123

R. Guefaifia et al.

If k is sufficiently large, let λζ = kα, then we have p(x)−1 C3 kekσ (λ1 + μ1 ) ln k = C3 (k M) p(x)−1 (λ1 + μ1 ) ln k ≤ λ p(x) (λ1 + μ1 ), then − p(x) φ1 ≤ λ p(x) (λ1 + μ1 ) , σ ≤ d (x) < 2δ.

(3.4)

Since φ1 (x) , φ2 (x) and f, h are monotone, when λ is large enough we have − p(x) φ1 ≤ λ p(x) (λ1 f (φ2 ) + μ1 h (φ1 )) , σ ≤ d (x) < 2δ and − p(x) φ1 = 0 ≤ λ p(x) (λ1 + μ1 ) ≤ λ p(x) (λ1 f (φ2 ) + μ1 h (φ1 )), 2δ ≤ d (x).

(3.5)

Combining (3.3), (3.5) and (3.6), we can deduce that − p(x) φ1 ≤ λ p(x) (λ1 f (φ2 ) + μ1 h (φ1 )), a.e. on .

(3.6)

− q(x) φ2 ≤ λq(x) (λ2 g (φ1 ) + μ2 τ (φ2 )), a.e. on .

(3.7)

Similarly From (3.6) and (3.7), we can see that (φ1 , φ2 ) is a subsolution of problem (1.1). Step 2. We will construct a supersolution of problem (1.1), we consider ⎧ − p(x) z 1 = λ p+ (λ1 + μ1 ) μ in , ⎪ ⎪ ⎪ ⎪ ⎨ −q(x) z 2 = λq+ (λ2 + μ2 ) g β λ p+ (λ1 + μ1 ) μ in , ⎪ ⎪ ⎪ ⎪ ⎩ z 1 = z 2 = 0 on ∂, p+ where β = β λ (λ1 + μ1 ) μ = maxx∈ z 1 (x). We shall prove that (z 1 , z 2 ) is a supersolution of problem (1.1). 1,q(x) For ψ ∈ W0 () with ψ ≥ 0, it is easy to see that

⎧

q(x)−2 ∇z 2 .∇ψd x = λq+ (λ2 + μ2 ) g β λ p+ (λ1 + μ1 ) μ ψd x ⎪ ⎪ |∇z 2 | ⎨

⎪ ⎪ ⎩

≥

λq+ λ2 g (z 1 ) ψd x +

λq+ μ2 g β λ p+ (λ1 + μ1 ) μ ψd x.

By (H 4), for μ a large enough, using Lemma 2, we have g β λ p+ (λ1 + μ1 ) μ ≥ τ C2 λq+ (λ2 + μ2 ) g β λ p+ (λ1 + μ1 ) μ ≥ τ (z 2 ).

(3.8)

Hence

|∇z 2 |q(x)−2 ∇z 2 .∇ψd x ≥

λ

q+

λ2 g (z 1 ) ψd x +

λq+ μ2 τ (z 2 ) ψd x.

1, p(x)

Also, for ϕ ∈ W0

123

1 q − −1

() with ϕ ≥ 0, it is easy to see that p(x)−2 |∇z 1 | ∇z 1 .∇ϕd x = λ p+ (λ1 + μ1 ) μϕd x.

(3.9)


By (H 3), (H 4) and Lemma 2, when μ is sufficiently large, we have

− p −1 1 p+ μ β λ + μ (λ ) 1 1 λ p+ C2 ≥ μ1 h β λ p+ (λ1 + μ1 ) μ + λ1 f C2 λq+ (λ2 + μ2 ) g β λ p+ (λ1 + μ1 ) μ

(λ1 + μ1 ) μ ≥

Then

1

.

|∇z 1 | p(x)−2 ∇z 1 .∇ϕd x ≥

1 q − −1

λ p+ λ1 f (z 2 ) ϕd x +

λ p+ μ1 h (z 1 ) ϕd x.

(3.10)

According to (3.9) and (3.10), we can conclude that (z 1 , z 2 ) is a supersolution of problem (1.1). It only remains to prove that φ1 ≤ z 1 and φ2 ≤ z 2 . In the definition of v1 (x), let 2 γ = max φ1 (x) + max |∇φ1 | (x) . δ We claim that φ1 (x) ≤ v1 (x) , ∀x ∈ .

(3.11)

From the definition of v1 , it is easy to see that φ1 (x) ≤ 2 max φ1 (x) ≤ v1 (x) , when d (x) = δ,

φ1 (x) ≤ 2 max φ1 (x) ≤ v1 (x) , when d (x) ≥ δ

and Since v1 − φ1 ∈ C

1

φ1 (x) ≤ v1 (x) , when d (x) < δ.

∂δ , there exists a point x0 ∈ ∂δ such that v1 (x0 ) − φ1 (x0 ) = min (v1 (x0 ) − φ1 (x0 )) . x0 ∈∂δ

If v1 (x0 ) − φ1 (x0 ) < 0, it is easy to see that 0 < d (x) < δ and then ∇v1 (x0 ) − ∇φ1 (x0 ) = 0. From the definition of v1 , we have 2 |∇v1 (x0 )| = γ = max φ1 (x0 ) + max |∇φ1 | (x0 ) > |∇φ1 | (x0 ) . δ It is a contradiction to ∇v1 (x0 ) − ∇φ1 (x0 ) = 0. Thus (3.11) is valid. Obviously, there exists constant C3 such that γ ≤ C3 λ. a positive Since d (x) ∈ C 2 ∂3δ , according to the proof of Lemma 2, there exists a positive constant C4 such that

123

R. Guefaifia et al.

− p(x) v1 (x) ≤ C∗ γ p(x)−1+θ ≤ C4 λ p(x)−1+θ a.e in , where θ ∈ (0, 1). +

When η ≥ λ p is large enough, we have − p(x) v1 (x) ≤ η. According to the comparison principle, we have v1 (x) ≤ ω (x), for all x ∈ . From (3.11) and (3.12), when η ≥ λ

p+

(3.12)

and λ ≥ 1 is sufficiently large, we have

φ1 (x) ≤ v1 (x) ≤ ω (x), for all x ∈ .

(3.13)

According to the comparison principle, when μ is large enough, we have v1 (x) ≤ ω (x) ≤ z 1 (x), for all x ∈ . Combining the definition of v1 (x) and (3.13), it is easy to see that φ1 (x) ≤ v1 (x) ≤ ω (x) ≤ z 1 (x), for all x ∈ .

When μ ≥ 1 and λ is large enough, from Lemma 2, we can see that β λ p+ (λ1 + μ1 ) μ is large enough, then λq+ (λ2 + μ2 ) g β λ p+ (λ1 + μ1 ) μ is a large enough. Similarly, we have φ2 ≤ z 2 . This completes the proof.

Acknowledgements The second author gratefully acknowledges Qassim University in Kingdom of Saudi Arabia, and this presented work is in memory of his father (1910–1999) Mr. Mahmoud ben Mouha Boulaaras. All authors of this paper would like to thank the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions, which helped the authors improve the paper.

References 1. Acerbi, E., Mingione, G.: Regularity results for a class of functional with nonstandard growth. Arch. Ration. Mech. Anal. 156, 121–140 (2001) 2. Afrouzi, G.A., Ghorbani, H.: Positive solutions for a class of p(x)−Laplacian problems. Glasgow Math. J. 51, 571–578 (2009) 3. Ali, J., Shivaji, R.: Positive solutions for a class of p− Laplacian systems with multiple parameter. J. Math. Anal. Appl. 335, 1013–1019 (2007) 4. Azizieh, C., Clement, P., Mitidieri, E.: Existence and a priori estimates for positive solutions of p-Laplace systems. J. Differ. Equ. 184, 422–442 (2002) 5. Bidaut-Veron, M.F., Pohozaev, S.: Nonexistence results and estimates for some nonlinear elliptic problems. J. Anal. Math. 84, 1–49 (2001) 6. Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functional in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006) 7. Chen, M.: On positive weak solutions for a class of quasilinear elliptic systems. Nonlinear Anal. 62, 751–756 (2005) 8. Coscia, A., Mingione, G.: Hölder continuity of the gradient of p(x)−harmonic mappings. C. R. Acad. Sci. Ser. I-Math. 328, 363–368 (1999) 9. El Hamidi, A.: Existence results to elliptic systems with nonstandard growth conditions. J. Math. Anal. Appl. 300, 30–42 (2004) 10. Fan, X.L., Zhao, D.: On the spaces L p(x) () and W m, p(x) (). J. Math. Anal. Appl. 263, 424–446 (2001) 11. Fan, X.L.: Global C 1,α regularity for variable exponent elliptic equations in divergence form. J. Differ. Equ. 235, 397–417 (2007) 12. Fan, X.L.: On the subsupersolution method for p(x)−Laplacian equations. J. Math. Anal. Appl. 330, 665–682 (2007) 13. Fan, X.L., Zhang, Q.H.: Existence of solutions for p(x)− Laplacian Dirichlet problem. Nonlinear Anal. 52, 1843–1852 (2003)

123

Existence of positive solutions for a class of … 14. Fan, X.L., Zhang, Q.H., Zhao, D.: Eigenvalues of p(x)− Laplacian Dirichlet problem. J. Math. Anal. Appl. 302, 306–317 (2005) 15. Filippucci, R.: Quasilinear elliptic systems in R N with multipower forcing terms depending on the gradient. J. Differ. Equ. 255, 1839–1866 (2013) 16. Hai, D.D., Shivaji, R.: An existence result on positive solutions of p−Laplacian systems. Nonlinear Anal. 56, 1007–1010 (2004) 17. Mihailescu, M., Pucci, P., Radulescu, V.: Eigenvalue problems for anisotropic quasi-linear elliptic equations with variable exponent. J. Math. Anal. Appl. 340, 687–698 (2008) 18. Ruzicka, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Berlin (2002) 19. Samko, S.G.: Densness of C0∞ (N ) in the generalized Sobolev spaces W m, p(x) (N ). Dokl. Ross. Akad. Nauk. 369(4), 451–454 (1999) 20. Zhang, Q.H.: Existence of positive solutions for a class of p(x)−Laplacian systems. J. Math. Anal. Appl. 333, 591–603 (2007) 21. Zhang, Q.H.: A strong maximum principle for differential equations with nonstandard p(x)−growth conditions. J. Math. Anal. Appl. 312, 24–32 (2005) 22. Zhang, Q.H.: Existence and asymptotic behavior of positive solutions for variable exponent elliptic systems. Nonlinear Anal. 70, 305–316 (2009) 23. Zhang, Q.H.: Existence of positive solutions for elliptic systems with nonstandard p(x)−growth conditions via sub-supersolution method. Nonlinear Anal. 67, 1055–1067 (2007)

123