Existence of positive solutions for variable exponent elliptic systems

  • Samira Ala1Email author,

    Affiliated with

    • Ghasem Alizadeh Afrouzi2,

      Affiliated with

      • Qihu Zhang3 and

        Affiliated with

        • Asadollah Niknam4

          Affiliated with

          Boundary Value Problems20122012:37

          DOI: 10.1186/1687-2770-2012-37

          Received: 30 October 2011

          Accepted: 3 April 2012

          Published: 3 April 2012

          Abstract

          We consider the system of differential equations

          - Δ p ( x ) u = λ p ( x ) [ g ( x ) a ( u ) + f ( v ) ] in  Ω , - Δ p ( x ) v = λ p ( x ) [ g ( x ) b ( v ) + h ( u ) ] in  Ω , u = v = 0 on  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equa_HTML.gif

          where Ω ⊂ ℝ N is a bounded domain with C 2 boundary ∂Ω, 1 < p(x) ∈C1 ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq1_HTML.gif is a function. Δ p ( x ) u  = div  ( | u | p ( x ) - 2 u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq2_HTML.gif is called p(x)-Laplacian. We discuss the existence of positive solution via sub-super solutions without assuming sign conditions on f(0), h(0).

          MSC: 35J60; 35B30; 35B40.

          Keywords

          positive solutions p(x)-Laplacian problems sub-supersolution

          1. Introduction

          The study of diferential equations and variational problems with variable exponent has been a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc., (see[13]). Many results have been obtained on this kind of problems, for example [1, 38]. In [7], Fan gives the regularity of weak solutions for differential equations with variable exponent. On the existence of solutions for elliptic systems with variable exponent, we refer to [8, 9]. In this article, we mainly consider the existence of positive weak solutions for the system
          ( P ) - Δ p ( x ) u = λ p ( x ) [ g ( x ) a ( u ) + f ( v ) ] in  Ω , - Δ p ( x ) v = λ p ( x ) [ g ( x ) b ( v ) + h ( u ) ] in  Ω , u = v = 0 on  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equb_HTML.gif
          where Ω ⊂ ℝ N is a bounded domain with C2 boundary Ω, 1 < p(x) ∈ C1 ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq1_HTML.gif is a function. The operator Δ p ( x ) u  = div  ( | u | p ( x ) - 2 u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq2_HTML.gif is called p(x)-Laplacian. Especially, if p(x) ≡ p (a constant), (P) is the well-known p-Laplacian system. There are many articles on the existence of solutions for p-Laplacian elliptic systems, for example [5, 10]. Owing to the nonhomogeneity of p(x)-Laplacian problems are more complicated than those of p-Laplacian, many results and methods for p-Laplacian are invalid for p(x)-Laplacian; for example, if Ω is bounded, then the Rayleigh quotient
          λ p ( x ) = inf u W 0 1 , p ( x ) ( Ω ) \ { 0 } Ω 1 p ( x ) | u | p ( x ) d x Ω 1 p ( x ) | u | p ( x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equc_HTML.gif

          is zero in general, and only under some special conditions λ p (x)> 0 (see [11]), and maybe the first eigenvalue and the first eigenfunction of p(x)-Laplacian do not exist, but the fact that the first eigenvalue λ p > 0 and the existence of the first eigenfunction are very important in the study of p-Laplacian problems. There are more difficulties in discussing the existence of solutions of variable exponent problems.

          Hai and Shivaji [10], consider the existence of positive weak solutions for the following p-Laplacian problems
          ( I ) - Δ p u = λ f ( v ) in  Ω , - Δ p v = λ g ( u ) in  Ω , u = v = 0 on  Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equd_HTML.gif
          the first eigenfunction is used to construct the subsolution of p-Laplacian problems success-fully. On the condition that λ is large enough and
          lim u + f M ( g ( u ) ) 1 ( p - 1 ) u p - 1 = 0 , for every M > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Eque_HTML.gif

          the authors give the existence of positive solutions for problem (I).

          Chen [5], considers the existence and nonexistence of positive weak solution to the following quasilinear elliptic system:
          ( II ) - Δ p u = λ f ( u , v ) = λ u α v γ in  Ω , - Δ q v = λ g ( u , v ) = λ u δ v β in  Ω , u = v = 0 on  Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equf_HTML.gif

          the first eigenfunction is used to construct the subsolution of problem(II), the main results are as following

          (i) If α, β ≥ 0, γ, δ > 0, θ = (p - 1 - α)(q - 1 - β) - γδ > 0, then problem (II) 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 (II) has no nontrivial nonnegative weak solution.

          On the p(x)-Laplacian problems, maybe the first eigenvalue and the first eigenfunction of p(x)-Laplacian do not exist. Even if the first eigenfunction of p(x)-Laplacian exist, because of the nonhomogeneity of p(x)-Laplacian, the first eigenfunction cannot be used to construct the subsolution of p(x)-Laplacian problems. Zhang [12] investigated the existence of positive solutions of the system
          - Δ p ( x ) u = λ p ( x ) f ( v ) in  Ω , - Δ p ( x ) v = λ p ( x ) g ( u ) in  Ω , u = v = 0 on  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equg_HTML.gif
          In this article, we consider the existence of positive solutions of the system
          - Δ p ( x ) u = λ p ( x ) F ( x , u , v ) in  Ω , - Δ p ( x ) v = λ p ( x ) G ( x , u , v ) in  Ω , u = v = 0 on  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equh_HTML.gif

          where p(x) ∈ C1 ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq1_HTML.gif is a function, F(x, u, v) = [g(x)a(u) + f(v)], G(x, u, v) = [g(x)b(v) +h(u)], λ is a positive parameter and Ω ⊂ ℝ N is a bounded domain.

          To study p(x)-Laplacian problems, we need some theory on the spaces L p (x)(Ω), W1,p(x)(Ω) and properties of p(x)-Laplacian which we will use later (see [6, 13]). If Ω ⊂ ℝ N is an open domain, write
          C + ( Ω ) = { h : h C ( Ω ) , h ( x ) > 1 f o r x Ω } , h + = sup  x Ω h ( x ) , h - = inf x Ω h ( x ) , for any  h C ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equi_HTML.gif

          Throughout the article, we will assume that:

          (H1) Ω ⊂ ℝ N is an open bounded domain with C2 boundary Ω.

          (H2) p(x) ∈ C1 ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq1_HTML.gif and 1 < p- ≤ p+.

          (H3) a, bC1([0, )) are nonnegative, nondecreasing functions such that
          lim u + a ( u ) u p - - 1 = 0 , lim u + b ( u ) u p - - 1 = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equj_HTML.gif
          (H4) f, h : [0, +) → R are C1, monotone functions, lim u →+∞ f(u) = +, lim u →+∞ h(u) = +, and
          lim u + f M ( h ( u ) ) 1 ( p - - 1 ) u p - - 1 = 0 , M > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equk_HTML.gif

          (H5) g : [0, +) (0, +) is a continuous function such that L 1 = min x Ω ̄ g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq3_HTML.gif, and L 2 = max x Ω ̄ g ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq4_HTML.gif

          Denote
          L p ( x ) ( Ω ) = u | u is a measurable real - valued function , Ω | u ( x ) | p ( x ) d x < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equl_HTML.gif
          We introduce the norm on L p (x)(Ω) by
          | u | p ( x ) = inf λ > 0 : Ω u ( x ) λ p ( x ) d x 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equm_HTML.gif

          and (L p (x)(Ω), |.| p (x)) becomes a Banach space, we call it generalized Lebesgue space. The space (L p (x)(Ω), |.| p (x)) is a separable, reflexive, and uniform convex Banach space (see [[6], Theorems 1.10 and 1.14]).

          The space W1,p(x)(Ω) is defined by W1,p(x)(Ω) = {uL p (x): |u|L p (x)}, and it is equipped with the norm
          u = | u | p ( x ) + | u | p ( x ) , u W 1 , p ( x ) ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equn_HTML.gif
          We denote by W01,p(x)(Ω) is the closure of C 0 Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq5_HTML.gif in W1,p(x)(Ω). W1,p(x)(Ω) and W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq6_HTML.gif are separable, reflexive, and uniform convex Banach space (see [[6], Theorem 2.1] We define
          ( L ( u ) , v ) = Ω | u | p ( x ) - 2 u v d x , v , u W 0 1 , p ( x ) ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equo_HTML.gif

          then L : W 0 1 , p ( x ) ( Ω ) ( W 0 1 , p ( x ) ( Ω ) ) * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq7_HTML.gif is a continuous, bounded, and strictly monotone operator, and it is a homeomorphism (see [[14], Theorem 3.1]).

          If u , v W 0 1 , p ( x ) ( Ω ) , ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq8_HTML.gif is called a weak solution of (P) if it satisfies
          Ω | u | p ( x ) - 2 u q d x = Ω λ p ( x ) F ( x , u , v ) q d x , q W 0 1 , p ( x ) ( Ω ) , Ω | v | p ( x ) - 2 v q d x = Ω λ p ( x ) G ( x , u , v ) q d x , q W 0 1 , p ( x ) ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equp_HTML.gif
          Define A : W 1 , p ( x ) ( Ω ) ( W 0 1 , p ( x ) ( Ω ) ) * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq9_HTML.gifas
          < A u , φ > = Ω ( | u | p ( x ) - 2 u φ + l ( x , u ) φ ) d x , u W 1 , p ( x ) ( Ω ) , φ W 0 1 , p ( x ) ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equq_HTML.gif

          where l(x, u) is continuous on Ω ̄ × http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq10_HTML.gif, and l(x, .) is increasing. It is easy to check that A is a continuous bounded mapping. Copying the proof of [15], we have the following lemma.

          Lemma 1.1. (Comparison Principle). Let u, vW1,p(x)(Ω) satisfying Au - Av ≥ 0 in ( W 0 1 , p ( x ) ( Ω ) ) * , φ ( x ) = min { u ( x ) - v ( x ) , 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq11_HTML.gif. If φ ( x ) W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq12_HTML.gif (i.e., u ≥ v on ∂ Ω ), then u ≥ v a.e. in Ω.

          Here and hereafter, we will use the notation d(x, Ω) to denote the distance of x ∈ Ω to the boundary of Ω.

          Denote d(x) = d(x, Ω) and Ω ϵ = { x Ω | d ( x , Ω ) < ϵ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq13_HTML.gif. Since Ω is C2 regularly, then there exists a constant δ ∈ (0, 1) such that d ( x ) C 2 ( Ω 3 δ ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq14_HTML.gif, and |d(x)| ≡ 1.

          Denote
          v 1 ( x ) = γ d ( x ) , d ( x ) < δ , γ δ + δ d ( x ) γ 2 δ - t δ 2 p - - 1 ( L 1 + 1 ) 2 p - - 1 d t , δ d ( x ) < 2 δ , γ δ + δ 2 δ γ 2 δ - t δ 2 p - - 1 ( L 1 + 1 ) 2 p - - 1 d t , 2 δ d ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equr_HTML.gif
          Obviously, 0 v 1 ( x ) C 1 ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq15_HTML.gif. Considering
          - Δ p ( x ) w ( x ) = η in  Ω , w = 0 on  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ1_HTML.gif
          (1)

          we have the following result

          Lemma 1.2. (see [16]). If positive parameter η is large enough and w is the unique solution of (1), then we have

          (i) For any θ∈ (0, 1) there exists a positive constant C1 such that
          C 1 η 1 p + - 1 + θ max  x Ω ̄ w ( x ) ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equs_HTML.gif
          (ii) There exists a positive constant C 2 such that
          max  x Ω ̄ w ( x ) C 2 η 1 p - - 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equt_HTML.gif

          2. Existence results

          In the following, when there be no misunderstanding, we always use C i to denote positive constants.

          Theorem 2.1. On the conditions of (H1) - (H5), then (P) has a positive solution when λ is large enough.

          Proof. We shall establish Theorem 2.1 by constructing a positive subsolution (Φ1, Φ2) and supersolution (z1, z2) of (P), such that Φ1 ≤ z1 and Φ2 ≤ z2. That is (Φ1, Φ2) and (z1, z2) satisfies
          Ω | Φ 1 | p ( x ) - 2 Φ 1 q d x Ω λ p ( x ) g ( x ) a ( Φ 1 ) q d x + Ω λ p ( x ) f ( Φ 2 ) q d x , Ω | Φ 2 | p ( x ) - 2 Φ 2 q d x Ω λ p ( x ) g ( x ) b ( Φ 2 ) q d x + Ω λ p ( x ) h ( Φ 1 ) q d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equu_HTML.gif
          and
          Ω | z 1 | p ( x ) - 2 z 1 q d x Ω λ p ( x ) g ( x ) a ( z 1 ) q d x + Ω λ p ( x ) f ( z 2 ) q d x , Ω | z 2 | p ( x ) - 2 z 2 q d x Ω λ p ( x ) g ( x ) b ( z 2 ) q d x + Ω λ p ( x ) h ( z 1 ) q d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equv_HTML.gif

          for all q W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq16_HTML.gif with q ≥ 0. According to the sub-supersolution method for p(x)-Laplacian equations (see [16]), then (P) has a positive solution.

          Step 1. We construct a subsolution of (P).

          Let σ ∈ (0, δ) is small enough. Denote
          ϕ ( x ) = e k d ( x ) - 1 , d ( x ) < σ , e k σ - 1 + σ d ( x ) k e k σ 2 δ - t 2 δ - σ 2 p - - 1 d t , σ d ( x ) < 2 δ , e k σ - 1 + σ 2 δ k e k σ 2 δ - t 2 δ - σ 2 p - - 1 d t , 2 δ d ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equw_HTML.gif
          It is easy to see that ϕ C 1 ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq17_HTML.gif. Denote
          α = min inf  p ( x ) - 1 4 ( sup | p ( x ) | + 1 ) , 1 , ζ = min { a ( 0 ) L 1 + f ( 0 ) , b ( 0 ) L 1 + h ( 0 ) , - 1 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equx_HTML.gif
          By computation
          - Δ p ( x ) ϕ = - k ( k e k d ( x ) ) p ( x ) - 1 ( p ( x ) - 1 ) + ( d ( x ) + ln k k ) p d + d k , d ( x ) < σ , 1 2 δ - σ 2 ( p ( x ) - 1 ) p - - 1 - 2 δ - d 2 δ - σ ln  k e k σ 2 δ - d 2 δ - σ 2 p - - 1 p d + Δ d × ( k e k σ ) p ( x ) - 1 2 δ - d 2 δ - σ 2 ( p ( x ) - 1 ) p - - 1 - 1 ( L 1 + 1 ) , σ < d ( x ) < 2 δ , 0 , 2 δ < d ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equy_HTML.gif
          From (H3) and (H4), there exists a positive constant M > 1 such that
          f ( M - 1 ) 1 , h ( M - 1 ) 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equz_HTML.gif
          Let σ = 1 k ln M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq18_HTML.gif, then
          σ k = ln  M . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ2_HTML.gif
          (2)
          If k is sufficiently large, from (2), we have
          - Δ p ( x ) ϕ - k p ( x ) α , d ( x ) < σ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ3_HTML.gif
          (3)
          Let -λζ = , then
          k p ( x ) α - λ p ( x ) ζ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equaa_HTML.gif
          from (3), then we have
          - Δ p ( x ) ϕ λ p ( x ) ( a ( 0 ) L 1 + f ( 0 ) ) λ p ( x ) ( g ( x ) a ( ϕ ) + f ( ϕ ) ) , d ( x ) < σ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ4_HTML.gif
          (4)
          Since d ( x ) C 2 ( Ω 3 δ ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq19_HTML.gif, then there exists a positive constant C3 such that
          - Δ p ( x ) ϕ ( k e k σ ) p ( x ) - 1 2 δ - d 2 δ - σ 2 ( p ( x ) - 1 ) p - - 1 - 1 . 2 ( p ( x ) - 1 ) ( 2 δ - σ ) ( p - - 1 ) - 2 δ - d 2 δ - σ ln  k e k σ 2 δ - d 2 δ - σ 2 p - - 1 p d + Δ d C 3 ( k e k σ ) p ( x ) - 1 In  k , σ < d ( x ) < 2 δ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equab_HTML.gif
          If k is sufficiently large, let -λζ = , we have
          C 3 ( k e k σ ) p ( x ) - 1 ln  k = C 3 ( k M ) p ( x ) - 1 ln  k λ p ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equac_HTML.gif
          then
          - Δ p ( x ) ϕ λ p ( x ) ( L 1 + 1 ) , σ < d ( x ) < 2 δ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equad_HTML.gif
          Since ϕ (x) 0 and a, f are monotone, when λ is large enough, then we have
          - Δ p ( x ) ϕ λ p ( x ) ( g ( x ) a ( ϕ ) + f ( ϕ ) ) , σ < d ( x ) < 2 δ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ5_HTML.gif
          (5)
          Obviously
          - Δ p ( x ) ϕ = 0 λ p ( x ) ( L 1 + 1 ) λ p ( x ) ( g ( x ) a ( ϕ ) + f ( ϕ ) ) , 2 δ < d ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ6_HTML.gif
          (6)
          Combining (4), (5), and (6), we can conclude that
          - Δ p ( x ) ϕ λ p ( x ) ( g ( x ) a ( ϕ ) + f ( ϕ ) ) , a . e .  on  Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ7_HTML.gif
          (7)
          Similarly
          - Δ p ( x ) ϕ λ p ( x ) ( g ( x ) b ( ϕ ) + h ( ϕ ) ) , a . e . on  Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ8_HTML.gif
          (8)

          From (7) and (8), we can see that (ϕ1, ϕ2) = (ϕ, ϕ) is a subsolution of (P).

          Step 2. We construct a supersolution of (P).

          We consider
          - Δ p ( x ) z 1 = λ p + μ ( L 2 + 1 ) in  Ω , - Δ p ( x ) z 2 = λ p + ( L 2 + 1 ) h ( β ( λ p + ( L 2 + 1 ) μ ) ) in  Ω , z 1 = z 2 = 0 on  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equae_HTML.gif

          where β = β ( λ p + ( L 2 + 1 ) μ ) = max x Ω ̄ z 1 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq20_HTML.gif. We shall prove that (z1, z2) is a supersolution for (p).

          For q W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq21_HTML.gif with q ≥ 0, it is easy to see that
          Ω | z 2 | p ( x ) - 2 z 2 q d x = Ω λ p + ( L 2 + 1 ) h ( β ( λ p + ( L 2 + 1 ) μ ) ) q d x Ω λ p + L 2 h ( β ( λ p + ( L 2 + 1 ) μ ) ) q d x + Ω λ p + h ( z 1 ) q d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ9_HTML.gif
          (9)
          Since lim u + f M ( h ( u ) ) 1 ( p - - 1 ) u p - - 1 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq22_HTML.gif,when μ is sufficiently large, combining Lemma 1.2 and (H3), then we have
          h ( β ( λ p + ( L 2 + 1 ) μ ) ) b C 2 λ p + ( L 2 + 1 ) h ( β ( λ p + ( L 2 + 1 ) μ ) ) 1 p - - 1 b ( z 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ10_HTML.gif
          (10)
          Hence
          Ω | z 2 | p ( x ) - 2 z 2 q d x Ω λ p + g ( x ) b ( z 2 ) q d x + Ω λ p + h ( z 1 ) q d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ11_HTML.gif
          (11)
          Also
          Ω | z 1 | p ( x ) - 2 z 1 q d x = Ω λ p + ( L 2 + 1 ) μ q d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equaf_HTML.gif
          By (H3), (H4), when μ is sufficiently large, combining Lemma 1.2 and (H3), we have
          ( L 2 + 1 ) μ 1 λ p + 1 C 2 β ( λ p + ( L 2 + 1 ) μ ) p - - 1 L 2 a ( β ( λ p + ( L 2 + 1 ) μ ) ) + f C 2 λ p + ( L 2 + 1 ) h ( β ( λ p + ( L 2 + 1 ) μ ) ) 1 p - - 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equag_HTML.gif
          Then
          Ω | z 1 | p ( x ) - 2 z 1 q d x Ω λ p + g ( x ) a ( z 1 ) q d x + Ω λ p + f ( z 2 ) q d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ12_HTML.gif
          (12)

          According to (11) and (12), we can conclude that (z1, z2) is a supersolution for (P).

          It only remains to prove that ϕ1 ≤ z1 and ϕ2 ≤ z2.

          In the definition of v1(x), let γ = δ 2 ( max x Ω ̄ ϕ ( x ) + max x Ω ̄ | ϕ ( x ) | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq23_HTML.gif. We claim that
          ϕ ( x ) v 1 ( x ) , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ13_HTML.gif
          (13)
          From the definition of v1, it is easy to see that
          ϕ ( x ) 2 max  x Ω ̄ ϕ ( x ) v 1 ( x ) , when  d ( x ) = δ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equah_HTML.gif
          and
          ϕ ( x ) 2 max  x Ω ̄ ϕ ( x ) v 1 ( x ) , when  d ( x ) δ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equai_HTML.gif
          It only remains to prove that
          ϕ ( x ) v 1 ( x ) , when  d ( x ) < δ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equaj_HTML.gif
          Since v 1 - ϕ C 1 ( Ω δ ¯ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq24_HTML.gif then there exists a point x 0 Ω δ ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq25_HTML.gifsuch that
          v 1 ( x 0 ) - ϕ ( x 0 ) = min x 0 Ω δ ¯ [ v 1 ( x ) - ϕ ( x ) ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equak_HTML.gif
          If v1(x0) - ϕ(x0) < 0, it is easy to see that 0 < d(x0) < δ, and then
          v 1 ( x 0 ) - ϕ ( x 0 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equal_HTML.gif
          From the definition of v1, we have
          | v 1 ( x 0 ) | = γ = 2 δ max  x Ω ̄ ϕ ( x ) + max  x Ω ̄ | ϕ ( x ) | > | ϕ ( x 0 ) | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equam_HTML.gif

          It is a contradiction to ∇v1(x0) -ϕ(x0) = 0. Thus (13) is valid.

          Obviously, there exists a positive constant C3 such that
          γ C 3 λ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equan_HTML.gif
          Since d ( x ) C 2 ( Ω 3 δ ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq26_HTML.gif, according to the proof of Lemma 1.2, then there exists a positive constant C4 such that
          - Δ p ( x ) v 1 ( x ) C * γ p ( x ) - 1 + θ C 4 λ p ( x ) - 1 + θ , a . e . in  Ω , where  θ ( 0 , 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equao_HTML.gif
          When η λ p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq27_HTML.gif is large enough, we have
          - Δ p ( x ) v 1 ( x ) η . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equap_HTML.gif
          According to the comparison principle, we have
          v 1 ( x ) w ( x ) , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ14_HTML.gif
          (14)
          From (13) and (14), when η λ p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq28_HTML.gif and λ ≥ 1 is sufficiently large, we have
          ϕ ( x ) v 1 ( x ) w ( x ) , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ15_HTML.gif
          (15)
          According to the comparison principle, when μ is large enough, we have
          v 1 ( x ) w ( x ) z 1 ( x ) , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equaq_HTML.gif
          Combining the definition of v1(x) and (15), it is easy to see that
          ϕ 1 ( x ) = ϕ ( x ) v 1 ( x ) w ( x ) z 1 ( x ) , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equar_HTML.gif

          When μ ≥ 1 and λ is large enough, from Lemma 1.2, we can see that β ( λ p + ( L 2 + 1 ) μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq29_HTML.gif is large enough, then λ p + ( L 2 + 1 ) h ( β ( λ p + ( L 2 + 1 ) μ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq30_HTML.gif is large enough. Similarly, we have ϕ2 ≤ z2. This completes the proof. □

          3. Asymptotic behavior of positive solutions

          In this section, when parameter λ → +, we will discuss the asymptotic behavior of maximum of solutions about parameter λ, and the asymptotic behavior of solutions near boundary about parameter λ.

          Theorem 3.1. On the conditions of (H1)-(H5), if (u, v) is a solution of (P) which has been given in Theorem 2.1, then

          (i) There exist positive constants C1 and C2 such that
          C 1 λ max  x Ω ̄ u ( x ) C 2 λ p + ( L 2 + 1 ) μ 1 p - - 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ16_HTML.gif
          (16)
          C 1 λ max  x Ω ̄ v ( x ) C 2 λ p + ( L 2 + 1 ) h C 2 ( λ p + ( L 2 + 1 ) μ ) 1 p - - 1 1 p - - 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ17_HTML.gif
          (17)
          (ii) for any θ ∈ (0, 1), there exist positive constants C3 and C4 such that
          C 3 λ d ( x ) u ( x ) C 4 ( λ p + ( L 2 + 1 ) μ ) 1 / ( p - - 1 ) ( d ( x ) ) θ , a s d ( x ) 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ18_HTML.gif
          (18)
          C 3 λ d ( x ) v ( x ) C 4 λ p + ( L 2 + 1 ) h C 2 ( λ p + ( L 2 + 1 ) μ ) 1 p - - 1 1 p - - 1 ( d ( x ) ) θ , a s d ( x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equ19_HTML.gif
          (19)

          where μ satisfies (10).

          Proof. (i) Obviously, when 2δ ≤ d(x), we have
          u ( x ) , v ( x ) ϕ ( x ) = e k σ - 1 + σ 2 δ k e k σ 2 δ - t 2 δ - σ 2 p - - 1 d t - λ ζ α σ 2 δ M 2 δ - t 2 δ - σ 2 p - - 1 d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equas_HTML.gif
          then there exists a positive constant C1 such that
          C 1 λ max  x Ω ̄ u ( x ) and C 1 λ max  x Ω ̄ v ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equat_HTML.gif
          It is easy to see
          u ( x ) z 1 ( x ) max  x Ω ̄ z 1 ( x ) C 2 ( λ p + ( L 2 + 1 ) μ ) 1 p - - 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equau_HTML.gif
          then
          max  x Ω ̄ u ( x ) C 2 λ p + ( L 2 + 1 ) μ 1 p - - 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equav_HTML.gif
          Similarly
          max  x Ω ̄ v ( x ) C 2 λ p + ( L 2 + 1 ) h C 2 λ p + ( L 2 + 1 ) μ 1 p - - 1 1 p - - 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equaw_HTML.gif

          Thus (16) and (17) are valid.

          (ii) Denote
          v 3 ( x ) = α ( d ( x ) ) θ , d ( x ) ρ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equax_HTML.gif

          where θ ∈ (0, 1) is a positive constant, ρ ∈ (0, δ) is small enough.

          Obviously, v3(x) ∈ C1 ρ ), By computation
          - Δ p ( x ) v 3 ( x ) = - ( α θ ) p ( x ) - 1 ( θ - 1 ) ( p ( x ) - 1 ) ( d ( x ) ) ( θ - 1 ) ( p ( x ) - 1 ) - 1 ( 1 + Π ( x ) ) , d ( x ) < ρ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equay_HTML.gif
          where
          Π x = d p d In α θ θ - 1 p x - 1 + d p d In d p x - 1 + d Δ d θ - 1 p x - 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equaz_HTML.gif
          Let α = 1 ρ C 2 λ p + L 2 + 1 μ 1 / p - - 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq31_HTML.gif where ρ > 0 is small enough, it is easy to see that
          α p x - 1 λ p + μ L 2 + 1  and  Π x 1 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equba_HTML.gif
          where ρ > 0 is small enough, then we have
          - Δ p x v 3 x λ p + μ L 2 + 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equbb_HTML.gif
          Obviously v3(x) ≥ z1(x) on Ω ρ . According to the comparison principle, we have v3(x) ≥ z1 (x) on Ω ρ. Thus
          u x C 4 λ p + L 2 + 1 μ 1 / p - - 1 d x θ , a s d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equbc_HTML.gif
          Let α = 1 ρ C 2 λ p + L 2 + 1 h C 2 λ p + L 2 + 1 μ 1 p - - 1 1 p - - 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq32_HTML.gif when ρ > 0 is small enough, it is easy to see that
          α p x - 1 λ p + L 2 + 1 h C 2 λ p + L 2 + 1 μ 1 p - - 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equbd_HTML.gif
          Similarly, when ρ > 0 is small enough, we have
          v x C 4 λ p + L 2 + 1 h C 2 λ p + L 2 + 1 μ 1 p - - 1 1 p - - 1 d x θ a s d x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Eqube_HTML.gif
          Obviously, when d(x) < σ, we have
          u x , v x ϕ x = e k d x - 1 C 3 λ d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_Equbf_HTML.gif

          Thus (18) and (19) are valid. This completes the proof. □

          Declarations

          Acknowledgements

          The authors would like to appreciate the referees for their helpful comments and suggestions. The third author partly supported by the National Science Foundation of China (10701066 & 10971087).

          Authors’ Affiliations

          (1)
          Department of Mathematics, Sciences and Research, Islamic Azad University (IAU)
          (2)
          Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran
          (3)
          Department of Mathematics and Information Science, Zhengzhou University of Light Industry
          (4)
          Department of Mathematics, Ferdowsi University of Mashhad

          References

          1. Chen Y, Levine S, Rao M: Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math 2006, 66(4):1383-1406.MathSciNetView Article
          2. Ruzicka M: Electrorheological fluids: Modeling and mathematical theory. In Lecture Notes in Math. Volume 1784. Springer-Verlag, Berlin; 2000.
          3. Zhikov VV: Averaging of functionals of the calculus of variations and elasticity theory. Math USSR Izv 1987, 29: 33-36.View Article
          4. Acerbi E, Mingione G: Regularity results for a class of functionals with nonstandard growth. Arch Rat Mech Anal 2001, 156: 121-140.MathSciNetView Article
          5. Chen M: On positive weak solutions for a class of quasilinear elliptic systems. Nonlinear Anal 2005, 62: 751-756.MathSciNetView Article
          6. Fan XL, Zhao D: On the spaces L p (x)(Ω) and W 1, p(x)(Ω). J Math Anal Appl 2001, 263: 424-446.MathSciNetView Article
          7. Fan XL: Global C1, αregularity for variable exponent elliptic equations in divergence form. J Di Equ 2007, 235: 397-417.View Article
          8. El Hamidi A: Existence results to elliptic systems with nonstandard growth conditions. J Math Anal Appl 2004, 300: 30-42.MathSciNetView Article
          9. Zhang QH: Existence of positive solutions for a class of p ( x )-Laplacian systems. J Math Anal Appl 2007, 333: 591-603.MathSciNetView Article
          10. Hai DD, Shivaji R: An existence result on positive solutions of p -Laplacian systems. Nonlinear Anal 2004, 56: 1007-1010.MathSciNetView Article
          11. Fan XL, Zhang QH, Zhao D: Eigenvalues of p ( x )-Laplacian Dirichlet problem. J Math Anal Appl 2005, 302: 306-317.MathSciNetView Article
          12. Zhang QH: Existence and asymptotic behavior of positive solutions for variable exponent elliptic systems. Nonlinear Anal 2009, 70: 305-316.MathSciNetView Article
          13. Samko SG: Densness of C 0 R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-37/MediaObjects/13661_2011_Article_131_IEq33_HTML.gifin the generalized Sobolev spaces Wm,p(x) ( R N ). Dokl Ross Akad Nauk 1999, 369(4):451-454.MathSciNet
          14. Fan XL, Zhang QH: Existence of solutions for p ( x )-Laplacian Dirichlet problem. Nonlinear Anal 2003, 52: 1843-1852.MathSciNetView Article
          15. Zhang QH: A strong maximum principle for differential equations with nonstandard p ( x )-growth con-ditions. J Math Anal Appl 2005, 312(1):24-32.MathSciNetView Article
          16. Fan XL: On the sub-supersolution method for p ( x )-Laplacian equations. J Math Anal Appl 2007, 330: 665-682.MathSciNetView Article

          Copyright

          © Ala et al; licensee Springer. 2012

          This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.