Existence and multiplicity of positive solutions for a class of p ( x )-Kirchhoff type equations

  • Ruyun Ma1,

    Affiliated with

    • Guowei Dai1Email author and

      Affiliated with

      • Chenghua Gao1

        Affiliated with

        Boundary Value Problems20122012:16

        DOI: 10.1186/1687-2770-2012-16

        Received: 24 September 2011

        Accepted: 13 February 2012

        Published: 13 February 2012

        Abstract

        In this article, we study the existence and multiplicity of positive solutions for the Neumann boundary value problems involving the p(x)-Kirchhoff of the form

        - M Ω 1 p ( x ) ( u p ( x ) + λ u p ( x ) ) d x ( div ( u p ( x ) - 2 u ) - λ u p ( x ) - 2 u ) = f ( x , u ) in Ω , u v = 0 on Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equa_HTML.gif

        Using the sub-supersolution method and the variational method, under appropriate assumptions on f and M, we prove that there exists λ* > 0 such that the problem has at least two positive solutions if λ > λ*, at least one positive solution if λ = λ* and no positive solution if λ < λ*. To prove these results we establish a special strong comparison principle for the Neumann problem.

        2000 Mathematical Subject Classification: 35D05; 35D10; 35J60.

        Keywords

        p(x)-Kirchhoff positive solution sub-supersolution method comparison principle

        1 Introduction

        In this article we study the following problem
        - M ( t ) ( div ( u p ( x ) - 2 u ) - λ u p ( x ) - 2 u ) = f ( x , u ) in Ω , u v = 0 on Ω , ( P λ f ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equb_HTML.gif

        where Ω is a bounded domain of ℝ N with smooth boundary Ω and N ≥ 1, u v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq1_HTML.gif is the outer unit normal derivative, λ ∈ ℝ is a parameter, p = p ( x ) C 1 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq2_HTML.gif with 1 < p-: = infΩ p(x) ≤ p+ := supΩ p(x) < +∞, f C ( Ω ¯ × , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq3_HTML.gif, M(t) is a function with Ω 1 p ( x ) ( u p ( x ) + λ u p ( x ) ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq4_HTML.gif and satisfies the following condition:

        (M0) M(t): [0, +∞) → (m0, +∞) is a continuous and increasing function with m0 > 0.

        The operator -div(|∇u|p(x)-2u) := -Δp(x)u is said to be the p(x)-Laplacian, and becomes p-Laplacian when p(x) ≡ p (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than the p-Laplacian; 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 [13]. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium [4, 5]. Another field of application of equations with variable exponent growth conditions is image processing [6]. The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to [711] for an overview of and references on this subject, and to [1216] for the study of the variable exponent equations and the corresponding variational problems.

        The problem P λ 1 f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq5_HTML.gif is a generalization of the stationary problem of a model introduced by Kirchhoff [17]. More precisely, Kirchhoff proposed a model given by the equation
        ρ 2 u t 2 - ρ 0 h + E 2 L 0 L u x 2 d x 2 u x 2 = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ1_HTML.gif
        (1.2)
        where ρ, ρ0, h, E, L are constants, which extends the classical D'Alembert's wave equation, by considering the effect of the changing in the length of the string during the vibration. A distinguishing feature of Equation (1.2) is that the equation contains a nonlocal coefficient ρ 0 h + E 2 L 0 L u x 2 d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq6_HTML.gif which depends on the average 1 L 0 L u x 2 d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq7_HTML.gif, and hence the equation is no longer a pointwise identity. The equation
        - ( a + b Ω u 2 d x ) Δ u = f ( x , u ) in Ω , u = 0 on Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ2_HTML.gif
        (1.3)

        is related to the stationary analogue of the Equation (1.2). Equation (1.3) received much attention only after Lions [18] proposed an abstract framework to the problem. Some important and interesting results can be found, for example, in [1922]. Moreover, nonlocal boundary value problems like (1.3) can be used for modeling several physical and biological systems where u describes a process which depends on the average of itself, such as the population density [2326]. The study of Kirchhoff type equations has already been extended to the case involving the p-Laplacian (for details, see [2729]) and p(x)-Laplacian (see [3033]).

        Many authors have studied the Neumann problems involving the p-Laplacian, see e.g., [3436] and the references therein. In [34, 35] the authors have studied the problem P λ 1 f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq5_HTML.gif in the cases of p(x) ≡ p = 2, M(t) ≡ 1 and of p(x) ≡ p > 1, M(t) ≡ 1, respectively. In [36], Fan and Deng studied the Neumann problems with p(x)-Laplacian, with the nonlinear potential f(x, u) under appropriate assumptions. By using the sub-supersolution method and variation method, the authors get the multiplicity of positive solutions of P λ 1 f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq5_HTML.gif with M(t) ≡ 1. The aim of the present paper is to generalize the main results of [3436] to the p(x)-Kirchhoff case. For simplicity we shall restrict to the 0-Neumann boundary value problems, but the methods used in this article are also suitable for the inhomogeneous Neumann boundary value problems.

        In this article we use the following notations:
        F ( x , t ) = 0 t f ( x , s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equc_HTML.gif
        Λ = {λ ∈ ℝ: there exists at least a positive solution of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif},
        λ * = inf Λ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equd_HTML.gif

        The main results of this article are the following theorems. Throughout the article we always suppose that the condition (M0) holds.

        Theorem 1.1. Suppose that f satisfies the following conditions:
        f ( x , t ) 0 , f ( x , t ) 0 x Ω , t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ3_HTML.gif
        (1.4)
        and
        for each x Ω , f ( x , t ) is nondecreasing with respect to t 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ4_HTML.gif
        (1.5)

        Then Λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq9_HTML.gif, λ* ≥ 0 and*, +∞) ⊂ Λ. Moreover, for every λ > λ* problem P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif has a minimal positive solution uλ in [0,w1], where w1 is the unique solution of P λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq10_HTML.gif and u λ 1 < u λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq11_HTML.gif if λ* < λ2 < λ1.

        Theorem 1.2. Under the assumptions of Theorem 1.1, also suppose that there exist positive constants M, c1 and c2 such that
        f ( x , t ) c 1 + c 2 t q ( x ) - 1 , x Ω , t M , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ5_HTML.gif
        (1.6)
        where q C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq12_HTML.gif and 1 ≤ q(x) < p*(x) for x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq13_HTML.gif, μ ∈ (0,1) such that
        M ^ ( t ) ( 1 - μ ) M ( t ) t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ6_HTML.gif
        (1.7)
        where M ^ ( t ) = 0 t M ( τ ) d τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq14_HTML.gif and M1 > 0, θ > p + 1 - μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq15_HTML.gif such that
        0 < θ F ( x , t ) t f ( x , t ) , x Ω , t M 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ7_HTML.gif
        (1.8)

        Then for each λ ∈ (λ*, +∞), P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif has at least two positive solutions uλ and vλ, where uλ is a local minimizer of the energy functional and uλvλ.

        Theorem 1.3. (1) Suppose that f satisfies (1.4),
        f ( x , 0 ) f ( x , t ) for t > 0 and x Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ8_HTML.gif
        (1.9)
        and the following conditions:
        f ( x , t ) c 3 + c 4 t r ( x ) - 1 , x Ω , t M 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ9_HTML.gif
        (1.10)

        where M2, c3 and c4 are positive constants, r C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq16_HTML.gif and 1 ≤ r(x) < p(x) for x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq13_HTML.gif. Then λ* = 0.

        (2) If f satisfies (1.4)-(1.8), then λ* ∈ Λ.

        Example 1.1. Let M(t) = a + bt, where a and b are positive constants. It is clear that
        M ( t ) a > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Eque_HTML.gif
        Taking μ = 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq17_HTML.gif, we have
        M ^ ( t ) = 0 t M ( s ) d s = a t + 1 2 b t 2 1 2 ( a + b t ) t = ( 1 - μ ) M ( t ) t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equf_HTML.gif

        So the conditions (M0) and (1.7) are satisfied.

        The underlying idea for proving Theorems 1.1-1.3 is similar to the one of [36]. The special features of this class of problems considered in the present article are that they involve the nonlocal coefficient M(t). To prove Theorems 1.1-1.3, we use the results of [37] on the global C1,αregularity of the weak solutions for the p(x)-Laplacian equations. The main method used in this article is the sub-supersolution method for the Neumann problems involving the p(x)-Kirchhoff. A main difficulty for proving Theorem 1.1 is that a special strong comparison principle is required. It is well known that, when p ≠ 2, the strong comparison principles for the p-Laplacian equations are very complicated (see e.g. [3841]). In [13, 42, 43] the required strong comparison principles for the Dirichlet problems have be established, however, they cannot be applied to the Neumann problems. To prove Theorem 1.1, we establish a special strong comparison principle for the Neumann problem P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif (see Lemma 4.6 in Section 4), which is also valid for the inhomogeneous Neumann boundary value problems.

        In Section 2, we give some preliminary knowledge. In Section 3, we establish a general principle of sub-supersolution method for the problem P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif based on the regularity results. In Section 4, we give the proof of Theorems 1.1-1.3.

        2 Preliminaries

        In order to discuss problem P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif, we need some theories on W1,p(x)(Ω) which we call variable exponent Sobolev space. Firstly we state some basic properties of spaces W1,p(x)(Ω) which will be used later (for details, see [17]). 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.

        Write
        C + ( Ω ¯ ) = { h : h C ( Ω ¯ ) , h ( x ) > 1 for any x Ω ¯ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equg_HTML.gif
        and
        L p ( x ) ( Ω ) = u S ( Ω ) : Ω u ( x ) p ( x ) d x < + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equh_HTML.gif
        with the norm
        u L p ( x ) ( Ω ) = u p ( x ) = inf λ > 0 : Ω u ( x ) λ p ( x ) d x 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equi_HTML.gif
        and
        W 1 , p ( x ) ( Ω ) = { u L p ( x ) ( Ω ) : u L p ( x ) ( Ω ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equj_HTML.gif
        with the norm
        u = u W 1 , p ( x ) ( Ω ) = u L p ( x ) ( Ω ) + u L p ( x ) ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equk_HTML.gif

        Denote by W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq18_HTML.gif the closure of C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq19_HTML.gif in W1,p(x)(Ω). The spaces Lp(x)(Ω), W1,p(x)(Ω) and W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq18_HTML.gif are all separable Banach spaces. When p- > 1 these spaces are reflexive.

        Let λ > 0. Define for uW1,p(x)(Ω),
        u λ = inf σ > 0 : Ω u σ p ( x ) + λ u σ p ( x ) d x 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equl_HTML.gif

        Then ||u||λ is a norm on W1,p(x)(Ω) equivalent to u W 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq20_HTML.gif.

        By the definition of ||u||λ we have the following

        Proposition 2.1. [11, 14] Put ρ λ ( u ) = Ω u p ( x ) + λ u p ( x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq21_HTML.gif for λ > 0 and uW1,p(x)(Ω). We have:

        (1) u λ 1 u λ p - ρ λ ( u ) u λ p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq22_HTML.gif;

        (2) u λ 1 u λ p + ρ λ ( u ) u λ p - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq23_HTML.gif;

        (3) lim k + u k λ = 0 lim k + ρ λ ( u k ) = 0 ( a s k + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq24_HTML.gif;

        (4) lim k + u k λ = + lim k + ρ λ ( u k ) = + ( a s k + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq25_HTML.gif.

        Proposition 2.2. [14] If u, u k W1,p(x)(Ω), k = 1,2,..., then the following statements are equivalent each other:
        1. (i)

          lim k + u k - u λ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq26_HTML.gif;

           
        2. (ii)

          lim k + ρ λ ( u k - u ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq27_HTML.gif;

           
        3. (iii)

          u k u in measure in Ω and lim k + ρ λ ( u k ) = ρ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq28_HTML.gif.

           
        Proposition 2.3. [14] Let p C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq29_HTML.gif. If q C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq12_HTML.gif satisfies the condition
        1 q ( x ) < p * ( x ) , x Ω ¯ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ10_HTML.gif
        (2.1)

        then there is a compact embedding W1,p(x)(Ω) ↪ Lq(x)(Ω).

        Proposition 2.4. [14] The conjugate space of Lp(x)(Ω) is Lq(x)(Ω), where 1 q ( x ) + 1 p ( x ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq30_HTML.gif. For any uLp(x)(Ω) and vLq(x)(Ω), we have the following Hölder-type inequality
        Ω u v d x 1 p - + 1 q - u p ( x ) v q ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equm_HTML.gif
        Now, we discuss the properties of p(x)-Kirchhoff-Laplace operator
        Φ K ( u ) : = - M Ω 1 p ( x ) ( u p ( x ) + λ u p ( x ) ) d x ( div u p ( x ) - 2 u - λ u p ( x ) - 2 u ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equn_HTML.gif
        where λ > 0 is a parameter. Denotes
        Φ ( u ) : M ^ Ω 1 p ( x ) ( u p ( x ) + λ u p ( x ) ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ11_HTML.gif
        (2.2)
        For simplicity we write X = W1,p(x)(Ω), denote by u n u and u n u the weak convergence and strong convergence of sequence {u n } in X, respectively. It is obvious that the functional Φ is a Gâteaux differentiable whose Gâteaux derivative at the point uX is the functional Φ'(u) ∈ X*, given by
        Φ ( u ) , v = M Ω 1 p ( x ) u p ( x ) + λ u p ( x ) d x Ω ( u p ( x ) - 2 u v + λ u p ( x ) - 2 u v ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ12_HTML.gif
        (2.3)

        where 〈·, ·〉 is the duality pairing between X and X*. Therefore, the p(x)-Kirchhoff-Laplace operator is the derivative operator of Φ in the weak sense. We have the following properties about the derivative operator of Φ.

        Proposition 2.5. If (M0) holds, then
        1. (i)

          Φ': XX* is a continuous, bounded and strictly monotone operator;

           
        2. (ii)

          Φ' is a mapping of type (S+), i.e., if u n u in X and lim n + ¯ Φ ( u n ) - Φ ( u ) , u n - u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq31_HTML.gif, then u n u in X;

           
        3. (iii)

          Φ'(u): XX* is a homeomorphism;

           
        4. (iv)

          Φ is weakly lower semicontinuous.

           

        Proof. Applying the similar method to prove [15, Theorem 2.1], with obvious changes, we can obtain the conclusions of this proposition.

        3 Sub-supersolution principle

        In this section we give a general principle of sub-supersolution method for the problem P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif based on the regularity results and the comparison principle.

        Definition 3.1. uX is called a weak solution of the problem P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif if for all vX,
        M Ω 1 p ( x ) ( u p ( x ) + λ u p ( x ) ) d x Ω ( u p ( x ) - 2 u v + λ u p ( x ) - 2 u v ) d x = Ω f ( x , u ) v d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equo_HTML.gif

        In this article, we need the global regularity results for the weak solution of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif. Applying Theorems 4.1 and 4.4 of [44] and Theorem 1.3 of [37], we can easily get the following results involving of the regularity of weak solutions of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif.

        Proposition 3.1. (1) If f satisfies (1.6), then uL(Ω) for every weak solution u of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif.
        1. (2)
          Let uXL (Ω) be a solution of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif. If the function p is log-Hölder continuous on Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq32_HTML.gif, i.e., there is a positive constant H such that
          p ( x ) - p ( y ) H - log x - y f o r x , y Ω ¯ w i t h x - y 1 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ13_HTML.gif
          (3.2)
           
        then u C 0 , α ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq33_HTML.gif for some α ∈ (0,1).
        1. (3)

          If in (2), the condition (3.2) is replaced by that p is Hölder continuous on Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq32_HTML.gif, then u C 1 , α ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq34_HTML.gif for some α ∈ (0,1).

           

        For u, vS(Ω), we write uv if u(x) ≤ v(x) for a.e. x ∈ Ω. In view of (M0), applying Theorem 1.1 of [16], we have the following strong maximum principle.

        Proposition 3.2. Suppose that p ( x ) C + ( Ω ¯ ) C 1 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq35_HTML.gif, uX, u ≥ 0 and u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq36_HTML.gif in Ω. If
        - M ( t ) ( div ( u p ( x ) - 2 u ) - d ( x ) u p ( x ) - 2 u ) 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equp_HTML.gif

        where t = Ω 1 p ( x ) u p ( x ) + 1 p ( x ) d ( x ) u p ( x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq37_HTML.gif, M(t) ≥ m0 > 0, 0 ≤ d(x) ∈ L(Ω), q ( x ) C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq38_HTML.gif with p(x) ≤ q(x) ≤ p* (x), then u > 0 in Ω.

        Definition 3.2. uX is called a subsolution (resp. supersolution) of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif if for all vX with v ≥ 0, u ≤ 0 (resp. ≥) on Ω and
        M Ω 1 p ( x ) | u | p ( x ) + 1 p ( x ) λ | u | p ( x ) d x Ω u p ( x ) - 2 u v + λ u p ( x ) - 2 u v d x ( resp. ) Ω f ( x , u ) v d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equq_HTML.gif
        Theorem 3.1. Let λ > 0 and q C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq12_HTML.gif satisfies (2.1). Then for each h L q ( x ) q ( x ) - 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq39_HTML.gif, the problem
        - M Ω 1 p ( x ) u p ( x ) + 1 p ( x ) λ u p ( x ) d x div ( u p ( x ) - 2 u ) - λ u p ( x ) - 2 u = h ( x ) in Ω u v = 0 on Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ14_HTML.gif
        (3.3λ)

        has a unique solution uX.

        Proof. According to Propositions 2.3 and 2.4, ( f , v ) : = Ω f ( x ) v d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq40_HTML.gif (for any vX) defines a continuous linear functional on X. Since Φ' is a homeomorphism, P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif has a unique solution.

        Let q C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq12_HTML.gif satisfy (2.1). For h L q ( x ) q ( x ) - 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq39_HTML.gif, we denote by K(h) = Kλ(h) = u the unique solution of (3.3λ). K = Kλ is called the solution operator for (3.3λ). From the regularity results and the embedding theorems we can obtain the properties of the solution operator K as follows.

        Proposition 3.3. (1) The mapping K : L q ( x ) q ( x ) - 1 ( Ω ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq41_HTML.gif is continuous and bounded. Moreover, the mapping K : L q ( x ) q ( x ) - 1 ( Ω ) L q ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq42_HTML.gif is completely continuous since the embedding XLq(x)(Ω) is compact.
        1. (2)

          If p is log-Hölder continuous on Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq32_HTML.gif, then the mapping K : L ( Ω ) C 0 , α ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq43_HTML.gif is bounded, and hence the mapping K : L ( Ω ) C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq44_HTML.gif is completely continuous.

           
        2. (3)

          If p is Hölder continuous on Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq32_HTML.gif, then the mapping K : L ( Ω ) C 1 , α ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq45_HTML.gif is bounded, and hence the mapping K : L ( Ω ) C 1 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq46_HTML.gif is completely continuous.

           

        Using the similar proof to [36], we have

        Proposition 3.4. If h L q ( x ) q ( x ) - 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq39_HTML.gif and h ≥ 0, where q C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq12_HTML.gif satisfies (2.1), then K(h) ≥ 0. If pC1(Ω), hL(Ω) and h ≥ 0, then K(h) > 0 on Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq32_HTML.gif.

        Now we give a comparison principle as follows.

        Theorem 3.2. Let u, vX, φ W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq47_HTML.gif. If
        M I 0 ( u ) Ω u p ( x ) - 2 u φ + λ u p ( x ) - 2 u φ d x M I 0 ( v ) Ω v p ( x ) - 2 v φ + λ v p ( x ) - 2 v φ d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ15_HTML.gif
        (3.4)

        with φ ≥ 0 and uv on ∂ Ω, I 0 ( u ) : = Ω 1 p ( x ) u p ( x ) + 1 p ( x ) λ u p ( x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq48_HTML.gif, then uv in Ω.

        Proof. Taking φ = (u - v)+ as a test function in (3.4), we have
        Φ ( u ) - Φ ( v ) , φ = M Ω u p ( x ) + λ u p ( x ) p ( x ) d x Ω u p ( x ) - 2 u φ + λ u p ( x ) - 2 u φ d x - M Ω v p ( x ) + λ v p ( x ) p ( x ) d x Ω v p ( x ) - 2 v φ + λ v p ( x ) - 2 v φ d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equr_HTML.gif
        Using the similar proof to Theorem 2.1 of [15] with obvious changes, we can show that
        Φ ( u ) - Φ ( v ) , φ m 0 Ω 1 2 ( u p ( x ) - 2 - v p ( x ) - 2 ) ( u 2 - v 2 ) d x + m 0 λ Ω 1 2 ( u p ( x ) - 2 - v p ( x ) - 2 ) ( u 2 - v 2 ) d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equs_HTML.gif

        Therefore, we get 〈Φ'(u) - Φ'(v), φ〉 = 0. Proposition 2.5 implies that φ ≡ 0 or uv in Ω. It follows that uv in Ω.

        It follows from Theorem 3.2 that the solution operator K is increasing under the condition (M0), that is, K(u) ≤ K(v) if uv.

        In this article we will use the following sub-supersolution principle, the proof of which is based on the well known fixed point theorem for the increasing operator on the order interval (see e.g., [45]) and is similar to that given in [12] for Dirichlet problems involving the p(x)-Laplacian.

        Theorem 3.3. (A sub-supersolution principle) Suppose that u0, v0XL(Ω), u0 and v0 are a subsolution and a supersolution of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif respectively, and u0v0. If f satisfies the condition:
        f ( x , t ) is nondecreasing in t [ inf u 0 ( x ) , sup v 0 ( x ) ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ16_HTML.gif
        (3.5)

        then P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif has a minimal solution u * and a maximal solution v* in the order interval [u0,v0], i.e., u0u*v*v0 and if u is any solution of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif such that u0uv0, then u*uv*.

        The energy functional corresponding to P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif is
        J λ ( u ) = Φ ( u ) - Ω F ( x , u ) d x , u X . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ17_HTML.gif
        (3.6)

        The critical points of Jλ are just the solutions of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif. Many authors, for example, Chang [46], Brezis and Nirenberg [47] and Ambrosetti et al. [48], have combined the sub-supersolution method with the variational method and studied successfully the semilinear elliptic problems, where a key lemma is that a local minimizer of the associated energy functional in the C1-topology is also a local minimizer in the H1-topology. Such lemma have been extended to the case of the p-Laplacian equations (see [43, 49]) and also to the case of the p(x)-Laplacian equations (see [12, Theorem 3.1]). In [50], Fan extended the Brezis-Nirenberg type theorem to the case of the p(x)-Kirchhoff [50, Theorem 1.1]. The Theorem 1.1 of [50] concerns with the Dirichlet problems, but the method for proving the theorem is also valid for the Neumann problems. Thus we have the following

        Theorem 3.4. Let λ > 0 and (1.6) holds. If u C 1 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq49_HTML.gif is a local minimizer of Jλ in the C 1 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq50_HTML.gif-topology, then u is also a local minimizer of Jλ in the X-topology.

        4 Proof of theorems

        In this section we shall prove Theorems 1.1-1.3. Since only the positive solutions are considered, without loss of generality, we can assume that
        f ( x , t ) = f ( x , 0 ) for t < 0 and x Ω , ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equt_HTML.gif
        otherwise we may replace f(x,t) by f(+)(x,t), where
        f ( + ) ( x , t ) = f ( x , t ) if t 0 , f ( x , 0 ) if t < 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equu_HTML.gif

        The proof of Theorem 1.1 consists of the following several Lemmata 4.1-4.6.

        Lemma 4.1. Let (1.4) hold. Then λ > 0 if λ ∈ Λ.

        Proof. Let λ ∈ Λ and u be a positive solution of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif. Taking v ≡ 1 as a test function in Definition 3.1. (1) yields
        M Ω λ p ( x ) u p ( x ) d x λ Ω u p ( x ) - 1 d x = Ω f ( x , u ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ18_HTML.gif
        (4.1)

        which implies λ > 0 because the value of the right side in (4.1) is positive.

        Lemma 4.2. Let (1.4) and (1.5) hold. Then Λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq9_HTML.gif.

        Proof. By Theorem 3.1, Propositions 3.4 and 3.3. (3), the problem
        - M Ω 1 p ( x ) ( u p ( x ) + u p ( x ) ) d x div ( u p ( x ) - 2 u ) - u p ( x ) - 2 u = 0 in Ω u ν = 0 on Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ19_HTML.gif
        (4.2)
        has a unique positive solution w 1 C 1 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq51_HTML.gif and w1(x) ≥ ε > 0 for x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq13_HTML.gif. We can assume ε ≤ 1. Put d = sup { f ( x , w 1 ( x ) ) : x Ω ¯ } , M 3 = d m 0 ε p + - 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq52_HTML.gif and λ1 = 1 + M3. Then
        - M Ω 1 p ( x ) ( w 1 p ( x ) + λ 1 w 1 p ( x ) ) d x Δ p ( x ) w 1 - λ 1 w 1 p ( x ) - 1 = - M Ω 1 p ( x ) ( w 1 p ( x ) + λ 1 w 1 p ( x ) ) d x Δ p ( x ) w 1 - w 1 p ( x ) - 1 + M Ω 1 p ( x ) ( w 1 p ( x ) + λ 1 w 1 p ( x ) ) d x M 3 w 1 p ( x ) - 1 m 0 M 3 ε p + - 1 = d f ( x , w 1 ( x ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equv_HTML.gif

        This shows that w1 is a supersolution of the problem P λ 1 f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq5_HTML.gif. Obviously 0 is a subsolution of P λ 1 f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq5_HTML.gif. By Theorem 3.3, P λ 1 f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq5_HTML.gif has a solution u λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq53_HTML.gif such that 0 u λ 1 w 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq54_HTML.gif. By Proposition 3.4, u λ 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq55_HTML.gif on Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq32_HTML.gif. So λ1 ∈ Λ and Λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq9_HTML.gif.

        Lemma 4.3. Let (1.4) and (1.5) hold. If λ0 ∈ Λ, then λ ∈ Λ for all λ > λ0.

        Proof. Let λ0 ∈ Λ and λ > λ0. Let u λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq56_HTML.gif be a positive solution of P λ 0 f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq57_HTML.gif. Then, we have
        - Δ p ( x ) u λ 0 + λ u λ 0 p ( x ) - 1 - Δ p ( x ) u λ 0 + λ 0 u λ 0 p ( x ) - 1 = f ( x , u λ 0 ) M Ω 1 p ( x ) ( u λ 0 p ( x ) + λ 0 u λ 0 p ( x ) ) d x f ( x , u λ 0 ) M Ω 1 p ( x ) ( u λ 0 p ( x ) + λ u λ 0 p ( x ) ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equw_HTML.gif

        thanks to (M0). This shows that u λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq56_HTML.gif is a supersolution of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif. We know that 0 is a subsolution of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif By Theorem 3.3, P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif has a solution uλ such that 0 u λ u λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq58_HTML.gif. By Proposition 3.4, uλ > 0 on Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq32_HTML.gif. Thus λ ∈ Λ.

        Lemma 4.4. Let (1.4) and (1.5) hold. Then for every λ > λ*, there exists a minimal positive solution uλ of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif such that u λ 1 u λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq59_HTML.gif if λ* < λ2 < λ1.

        Proof. The proof is similar to [36, Lemma 3.4], we omit it here.

        Lemma 4.5. Let (1.4) and (1.5) hold. Let λ1, λ2 ∈ Λ and λ2 < λ < λ1. Suppose that u λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq53_HTML.gif and u λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq60_HTML.gif are the positive solutions of P λ 1 f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq5_HTML.gif and P λ 2 f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq61_HTML.gif respectively and u λ 1 u λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq59_HTML.gif. Then there exists a positive solution vλ of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif such that u λ 1 v λ u λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq62_HTML.gif and vλ is a global minimizer of the restriction of Jλ to the order interval u λ 1 , u λ 2 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq63_HTML.gif.

        Proof. Define f ̃ : Ω ¯ × http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq64_HTML.gif by
        f ̃ ( x , t ) = f ( x , u λ 1 ( x ) ) , if t < u λ 1 ( x ) f ( x , t ) , if u λ 1 ( x ) t u λ 2 ( x ) f ( x , u λ 2 ( x ) ) , if t > u λ 2 ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equx_HTML.gif
        Define F ̃ ( x , t ) = 0 t f ̃ ( x , s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq65_HTML.gif and for all uX,
        J ̃ λ ( u ) = M ^ Ω u p ( x ) + λ u p ( x ) p ( x ) d x - Ω F ̃ ( x , u ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equy_HTML.gif
        It is easy to see that the global minimum of J ̃ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq66_HTML.gif on X is achieved at some vλX. Thus vλ is a solution of the following problem
        - M Ω u p ( x ) + λ u p ( x ) p ( x ) d x div ( u p ( x ) - 2 u ) - λ u p ( x ) - 2 u = f ̃ ( x , u ) in Ω u ν = 0 on Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ20_HTML.gif
        (4.3)
        and v λ C 1 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq67_HTML.gif. Noting that
        f x , u λ 1 = f ̃ x , u λ 1 f ̃ ( x , v λ ) f ¯ ( x , u λ 2 ) = f ( x , u λ 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equz_HTML.gif

        and λ2 < λ < λ1, since K is increasing operator, we obtain that u λ 1 v λ u λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq62_HTML.gif. So f ̃ ( x , v λ ) = f ( x , v λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq68_HTML.gif, and vλ is a positive solution of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif. It is easy to see that there exists a constant c such that J λ ( u ) J ̃ λ ( u ) + c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq69_HTML.gif for u u λ 1 , u λ 2 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq70_HTML.gif. Hence vλ is a global minimizer of J λ | u λ 1 , u λ 2 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq71_HTML.gif.

        A key lemma of this paper is the following strong comparison principle.

        Lemma 4.6 (A strong comparison principle). Let (1.4) and (1.5) hold. Let λ1, λ2 ∈ Λ and λ2 < λ1. Suppose that u λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq53_HTML.gif and u λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq60_HTML.gif are the positive solutions of ( 1 . 1 λ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq72_HTML.gif and ( 1 . 1 λ 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq73_HTML.gif respectively. Then u λ 1 < u λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq11_HTML.gif on Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq32_HTML.gif.

        Proof. Since u λ 1 , u λ 2 C 1 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq74_HTML.gif and u λ 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq55_HTML.gif on Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq32_HTML.gif, in view of Lemma 4.4, there exist two positive constants b1 ≤ 1 and b2 such that
        b 1 u λ 1 u λ 2 b 2 on Ω ¯ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equaa_HTML.gif
        For ε 0 , b 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq75_HTML.gif, setting v ε = u λ 2 - ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq76_HTML.gif, then
        - M Ω v ε p ( x ) + λ 1 v ε p ( x ) p ( x ) d x ( div ( v ε p ( x ) - 2 v ε ) - λ 1 v ε p ( x ) - 1 ) = - M Ω u λ 2 p ( x ) + λ 1 v ε p ( x ) p ( x ) d x ( div ( u λ 2 p ( x ) - 2 u λ 2 ) - λ 2 v ε p ( x ) - 1 + ( λ 2 - λ 1 ) v ε p ( x ) - 1 ) = - M Ω u λ 2 p ( x ) + λ 1 v ε p ( x ) p ( x ) d x ( div ( u λ 2 p ( x ) - 2 u λ 2 ) - λ 2 u λ 2 p ( x ) - 1 ) + M Ω u λ 2 p ( x ) + λ 1 v ε p ( x ) p ( x ) d x ( λ 1 - λ 2 ) v ε p ( x ) - 1 - λ 2 u λ 2 p ( x ) - 1 - v ε p ( x ) - 1 f ( x , u λ 2 ) M Ω u λ 2 p ( x ) + λ 1 v ε p ( x ) p ( x ) d x M Ω u λ 2 p ( x ) + λ 2 u λ 2 p ( x ) p ( x ) d x + M Ω u λ 2 p ( x ) + λ 1 v ε p ( x ) p ( x ) d x ( λ 1 - λ 2 ) b 1 2 p + - 1 - λ 2 u λ 2 p ( x ) - 1 - v ε p ( x ) - 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equab_HTML.gif
        Taking an ε > 0 sufficiently small such that
        λ 2 u λ 2 p ( x ) - 1 - v ε p ( x ) - 1 < ( λ 1 - λ 2 ) b 1 2 p + - 1 for x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equac_HTML.gif
        and
        λ 1 Ω 1 p ( x ) v ε p ( x ) d x λ 2 Ω 1 p ( x ) u λ 2 p ( x ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equad_HTML.gif
        then
        - M Ω v ε p ( x ) + λ 1 v ε p ( x ) p ( x ) d x ( d i v ( v ε p ( x ) - 2 v ε ) - λ 1 v ε p ( x ) - 1 ) = g ( x ) f ( x , u λ 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equae_HTML.gif
        consequently, v ε is a solution of the problem
        - M Ω v ε p ( x ) + λ 1 v ε p ( x ) p ( x ) d x div( v ε p ( x ) - 2 v ε ) -  λ 1 v ε p ( x ) - 1 = g ( x ) in Ω u ν = 0 on Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equaf_HTML.gif

        where g ( x ) f ( x , u λ 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq77_HTML.gif. With other words, v ε = K λ 1 ( g ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq78_HTML.gif, where K λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq79_HTML.gif is the solution operator of ( 3 . 1 λ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq80_HTML.gif. Since u λ 1 = K λ 1 ( h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq81_HTML.gif, where h ( x ) = f ( x , u λ 1 ) f ( x , u λ 2 ) g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq82_HTML.gif, noting that K λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq79_HTML.gif is increasing, we have v ε u λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq83_HTML.gif, that is, u λ 2 - ε u λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq84_HTML.gif on Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq32_HTML.gif.

        The proof of Theorem 1.1 is complete. Let us now turn to the proof of Theorem 1.2.

        Proof of Theorem 1.2. Let (1.4)-(1.8) hold. Let λ > λ*. Take λ1, λ2 ∈ Λ such that λ2 < λ < λ1 and let u λ 1 u λ u λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq85_HTML.gif be as in Lemma 4.5.

        We claim that uλ is a local minimizer of Jλ in the X-topology.

        Indeed, Lemma 4.6 implies that u λ 1 < u λ < u λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq86_HTML.gif on Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq32_HTML.gif. It follows that there is a C0-neighborhood U of uλ such that U [ u λ 1 , u λ 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq87_HTML.gif, consequently uλ is a local minimizer of Jλ in the C0-topology, and of course, also in the C1-topology. By Theorem 3.4, uλ is also a local minimizer of Jλ in the X-topology.

        Define
        f ̃ λ ( x , t ) = f ( x , t ) , if t > u λ ( x ) , f ( x , u λ ( x ) ) , if t u λ ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equag_HTML.gif
        and F ̃ λ ( x , t ) = 0 t f ̃ λ ( x , s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq88_HTML.gif. Consider the problem
        - M Ω u p ( x ) + λ u p ( x ) p ( x ) d x div ( u p ( x ) - 2 u ) - λ u p ( x ) - 2 u = f ̃ λ ( x , u ) in Ω u ν = 0 on Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equah_HTML.gif

        and denote by J ̃ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq89_HTML.gif the energy functional corresponding to (4.4λ). By the definition of f ̃ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq90_HTML.gif, we have f ̃ λ ( x , u ( x ) ) f ( x , u λ ( x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq91_HTML.gif for every uX. Hence, for each solution u of (4.4λ), we have that uuλ, consequently f ̃ λ ( x , u ) = f ( x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq92_HTML.gif and u is also a solution of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif. It is easy to see that u λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq53_HTML.gif and u λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq60_HTML.gif are a subsolution and a supersolution of (4.4λ) respectively. By Theorems 3.3 and 1.2, there exists u λ * [ u λ 1 , u λ 2 ] C 1 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq93_HTML.gif such that u λ * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq94_HTML.gif is a solution of (4.4λ) and is a local minimizer of J ̃ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq89_HTML.gif in the C1-topology. As was noted above, we know that u λ * u λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq95_HTML.gif and u λ * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq94_HTML.gif is also a solution of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif. If u λ * u λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq96_HTML.gif, then the assertion of Theorem 1.2 already holds, hence we can assume that u λ * = u λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq97_HTML.gif. Now uλ is a local minimizer of J ̃ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq89_HTML.gif in the C1-topology, and so also in the X-topology. We can assume that uλ is a strictly local minimizer of J ̃ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq89_HTML.gif in the X-topology, otherwise we have obtained the assertion of Theorem 1.2. It is easy to verify that, under the assumptions of Theorem 1.3, J ̃ λ C 1 ( X , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq98_HTML.gif and J ̃ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq89_HTML.gif satisfies the (P.S.) condition (see e.g., [30]). It follows from the condition (1.7) and (1.8) that { J ̃ λ ( u ) : u X } = - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq99_HTML.gif (see e.g., [30]). Using the mountain pass lemma (see [51]), we know that (4.4λ) has a solution vλ such that vλuλ. vλ, as a solution of (4.4λ), must satisfy vλuλ, and vλ is also a solution of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif. The proof of Theorem 1.2 is complete.

        Proof of Theorem 1.3. (1) Let f satisfy (1.4), (1.9), and (1.10). For given any λ > 0, consider the energy functional Jλ defined by (3.3). By (1.10) and noting that r(x) < p(x) for x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq13_HTML.gif, there is a positive constant M4 such that
        F ( x , t ) λ m 0 2 p + t p ( x ) , x Ω , t M 4 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equ21_HTML.gif
        (4.5)
        For uX with ||u||λ ≥ 1, we have that
        J λ ( u ) m 0 p + Ω p ( x ) + λ u p ( x ) d x - λ m 0 2 p + Ω u p ( x ) d x - c 5 m 0 p + Ω u p ( x ) d x + λ m 0 2 p + Ω u p ( x ) d x - c 5 m 0 2 p + u λ p - - c 5 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equai_HTML.gif

        where c5 is a positive constant. This shows that Jλ(u) → +∞ as ||u||λ → +∞, that is, Jλ is coercive. In view of Proposition 2.5. (iv), the condition (1.10) also implies that Jλ is weakly sequentially lower semi-continuous. Thus Jλ has a global minimizer u0. Put v0(x) = |u0(x)| for x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq13_HTML.gif. It is easy to see that Jλ(v0) ≤ Jλ(u0), consequently, v0 is a global minimizer of Jλ and is a positive solution of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif. This shows that λ ∈ Λ for all λ > 0. Hence λ* = 0 and the statement (1) is proved.

        To prove Theorem 1.3. (2) we give the following lemma.

        Lemma 4.7. Let (1.4) and (1.5) hold. Then for each λ > λ*, P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif has a positive solution uλ such that Jλ(uλ) ≤ 0.

        Proof. Let λ > λ*. Take λ2 ∈ (λ*, λ) and let u λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq60_HTML.gif be a positive solution of P λ 2 f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq61_HTML.gif. then u λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq60_HTML.gif is a supersolution of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif. We know that 0 is a subsolution of P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif. Analogous to the proof of Lemma 4.5, we can prove that P λ f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq8_HTML.gif has a positive solution u λ [ 0 , u λ 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq100_HTML.gif such that J λ ( u λ ) = inf { J λ ( u ) : u [ 0 , u λ 2 ] } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq101_HTML.gif. So Jλ(uλ) ≤ Jλ(0) = 0.

        Proof of Theorem 1.3. (2). Let (1.4)-(1.8) hold. Let λ n > λ* and λ n → λ* as n → +∞. By Lemma 4.7, for each n, P λ n f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq102_HTML.gif has a positive solution u λ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq103_HTML.gif such that J λ n ( u λ n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq104_HTML.gif, that is
        M ^ Ω u λ n p ( x ) + λ n u λ n p ( x ) p ( x ) d x Ω F ( x , u λ n ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equaj_HTML.gif
        Since u λ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq103_HTML.gif is a solution of P λ n f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq102_HTML.gif, we have that
        M Ω u λ n p ( x ) + λ n u λ n p ( x ) p ( x ) d x Ω u λ n p ( x ) + λ n u λ n p ( x ) d x = Ω f ( x , u λ n ) u λ n d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equak_HTML.gif
        It follows from (1.8) that there exists a positive constant c6 such that
        Ω F ( x , u λ n ) d x c 6 + 1 θ Ω f ( x , u λ n ) u λ n d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equal_HTML.gif
        Thus, using condition (1.7), we have that
        1 - μ p + M Ω u λ n p ( x ) + λ n u λ n p ( x ) p ( x ) d x Ω ( u λ n p ( x ) + λ n u λ n p ( x ) ) d x c 6 + 1 θ Ω f ( x , u λ n ) u λ n d x c 6 + 1 θ M Ω u λ n p ( x ) + λ n u λ n p ( x ) p ( x ) d x Ω ( u λ n p ( x ) + λ n u λ n p ( x ) ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equam_HTML.gif
        and consequently,
        m 0 1 - μ p + - 1 θ u λ n λ n p - c 7 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equan_HTML.gif
        where the positive constant c7 is independent of n. This shows that { u λ n λ n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq105_HTML.gif is bounded. Noting that λ n → λ* > 0, we have that { u λ n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq106_HTML.gif is bounded. Without loss of generality, we can assume that u λ n u * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq107_HTML.gif in X and u λ n ( x ) u * ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq108_HTML.gif for a.e. x ∈ Ω. By (1.6) and the L(Ω)-regularity results of [44], the boundedness of { u λ n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq106_HTML.gif implies the boundedness of u λ n L ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq109_HTML.gif. By the C 1 , α ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq110_HTML.gif-regularity results of [37], the boundedness of u λ n L ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq109_HTML.gif implies the boundedness of u λ n C 1 , α ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq111_HTML.gif, where α ∈ (0, 1) is a constant. Thus we have u λ n u * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq107_HTML.gif in C 1 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq112_HTML.gif. For every vX, since u λ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq103_HTML.gif is a solution of P λ n f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq102_HTML.gif, we have that, for each n,
        M Ω u λ n p ( x ) + λ n u λ n p ( x ) p ( x ) d x Ω u λ n p ( x ) - 2 u λ n v + λ n u λ n p ( x ) - 2 u λ n v d x = Ω f ( x , u λ n ) v d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equao_HTML.gif
        Passing the limit of above equality as n → +∞, yields
        M Ω u * p ( x ) + λ * u * p ( x ) p ( x ) d x Ω u * p ( x ) - 2 u * v + λ * u * p ( x ) - 2 u * v d x = Ω f ( x , u * ) v d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_Equap_HTML.gif

        which shows that u* is a solution of P λ * f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq113_HTML.gif. Obviously u* ≥ 0 and u * 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq114_HTML.gif. Hence u* is a positive solution of P λ * f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-16/MediaObjects/13661_2011_Article_128_IEq113_HTML.gif and λ* ∈ Λ.

        Declarations

        Acknowledgements

        The authors are very grateful to the anonymous referees for their valuable suggestions. Research supported by the NSFC (Nos. 11061030 and 10971087), NWNU-LKQN-10-21.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Northwest Normal University

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