Existence of solutions for a differential inclusion problem with singular coefficients involving the p(x)-Laplacian

  • Guowei Dai1Email author,

    Affiliated with

    • Ruyun Ma1 and

      Affiliated with

      • Qiaozhen Ma1

        Affiliated with

        Boundary Value Problems20122012:11

        DOI: 10.1186/1687-2770-2012-11

        Received: 5 November 2011

        Accepted: 9 February 2012

        Published: 9 February 2012

        Abstract

        Using the non-smooth critical point theory we investigate the existence and multiplicity of solutions for a differential inclusion problem with singular coefficients involving the p(x)-Laplacian.

        Mathematics Subject Classification 2000: 35D05; 35J20; 35J60; 35J70.

        Keywords

        p(x)-Laplacian differential inclusion singularity

        1 Introduction

        In this article, we study the existence and multiplicity of solutions for the differential inclusion problem with singular coefficients involving the p(x)-Laplacian of the form
        - div ( | u | p ( x ) - 2 u ) λ a 1 ( x ) G 1 ( x , u ) + μ a 2 ( x ) G 2 ( x , u ) in Ω , u = 0 on Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equ1_HTML.gif
        (1.1)

        where the following conditions are satisfied:

        (P) Ω is a bounded open domain in ℝ N , N ≥ 2, p C ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq1_HTML.gif, 1 < p- := infΩp(x) ≤ p+ := supΩp(x) < +∞, λ, μ ∈ ℝ.

        (A) For i = 1, 2, a i L r i ( x ) ( Ω ) , a i ( x ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq2_HTML.gif for x ∈ Ω, G i (x, u) is measurable with respect to x (for every u ∈ ℝ) and locally Lipschitz with respect to u (for a.e. x ∈ Ω), ∂G i : Ω × ℝ → ℝ is the Clarke sub-differential of G i and | ξ i | c 1 + c 2 | t | q i ( x ) - 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq3_HTML.gif for x ∈ Ω, t ∈ ℝ and ξ i ∈ ∂G i , where c i is a positive constant, r i , q i C Ω ̄ , r i - > 1 , q i - > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq4_HTML.gif, r i (x) > q i (x) for all x ∈ Ω, and
        q i ( x ) < r i ( x ) - q i ( x ) r i ( x ) p * ( x ) , x Ω ¯ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equ2_HTML.gif
        (1.2)
        here
        p * ( x ) = N p ( x ) N - p ( x ) if  p ( x ) < N , if  p ( x ) N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equ3_HTML.gif
        (1.3)

        ( A 1 ) q 1 + < p - . ( A 2 ) q 2 - > p + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq5_HTML.gif

        A typical example of (1.1) is the following problem involving subcritical Sobolev-Hardy exponents of the form
        - div ( | u | p ( x ) - 2 u ) λ 1 | x | s 1 ( x ) G 1 ( x , u ) + μ 1 | x | s 2 ( x ) G 2 ( x , u ) in Ω , u = 0 on Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equ4_HTML.gif
        (1.4)

        and in this case the assumption corresponding to (A) is the following

        ( A ) * 0 Ω ̄ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq6_HTML.gif, for i = 1, 2, ∂G i : Ω × ℝ → ℝ is the Clarke sub-differential of G i and | ξ i | c 1 + c 2 | t | q i ( x ) - 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq7_HTML.gif for x ∈ Ω, t ∈ ℝ and ξ i ∈ ∂G i , where c i is a positive constant, s i , q i C Ω ̄ , 0 s i - s i + < N , q i - > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq8_HTML.gif, and
        q i ( x ) < N - s i ( x ) q i ( x ) N p * ( x ) , x Ω ̄ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equ5_HTML.gif
        (1.5)

        The operator -div(|∇u| p(x)-2 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 electro-rheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electro-magnetic field [1, 2]. 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 baro-tropic gas through a porous medium [3, 4]. Another field of application of equations with variable exponent growth conditions is image processing [5]. The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to [611] for an overview of and references on this subject, and to [1221] for the study of the p(x)-Laplacian equations and the corresponding variational problems.

        Since many free boundary problems and obstacle problems may be reduced to partial differential equations with discontinuous nonlinearities, the existence of multiple solutions for Dirichlet boundary value problems with discontinuous nonlinearities has been widely investigated in recent years. Chang [22] extended the variational methods to a class of non-differentiable functionals, and directly applied the variational methods for non-differentiable functionals to prove some existence theorems for PDE with discontinuous nonlinearities. Later Kourogenis and Papageorgiou [23] obtained some nonsmooth critical point theories and applied these to nonlinear elliptic equations at resonance, involving the p-Laplacian with discontinuous nonlinearities. In the celebrated work [24, 25], Ricceri elaborated a Ricceri-type variational principle and a three critical points theorem for the Gâteaux differentiable functional, respectively. Later, Marano and Motreanu [26, 27] extended Ricceri's results to a large class of non-differentiable functionals and gave some applications to differential inclusion problems involving the p-Laplacian with discontinuous nonlinearities.

        In [21], by means of the critical point theory, Fan obtain the existence and multiplicity of solutions for (1.1) under the condition of G i ( x , ) C 1 ( ) and  g i = G i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq9_HTML.gif satisfying the Carathéodory condition for i = 1, 2, x ∈ Ω. The aim of the present article is to generalize the main results of [21] to the case of the functional of problem (1.1) is nonsmooth.

        This article is organized as follows: In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces and the generalized gradient of the locally Lipschitz function; In Section 3, we give the variational principle which is needed in the sequel; In Section 4, using the critical point theory, we prove the existence and multiplicity results for problem (1.1).

        2 Preliminaries

        2.1 Variable exponent Sobolev spaces

        Let Ω be a bounded open subset of ℝ N , denote L + ( Ω ) = { p L ( Ω ) : ess inf Ω p ( x ) 1 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq10_HTML.gif.

        For p L + ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq11_HTML.gif, denote
        p - = p - ( Ω ) = ess  inf x Ω p ( x ) , p + = p + ( Ω ) = ess  sup x Ω p ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equa_HTML.gif

        On the basic properties of the space W1,p(x)(Ω) we refer to [7, 2830]. Here we display some facts which will be used later.

        Denote by S(Ω) the set of all measurable real functions defined on Ω. Two functions in S(Ω) are considered as the same element of S(Ω) when they are equal almost everywhere. For p L + ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq12_HTML.gif, define the spaces L p(x) (Ω) and W1,p(x)(Ω) by
        L p ( x ) Ω = u S ( Ω ) : Ω | u ( x ) | p ( x ) d x < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equb_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%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equc_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%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equd_HTML.gif
        with the norm
        | | 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%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Eque_HTML.gif

        Denote by W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq13_HTML.gif the closure of C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq14_HTML.gif in W1,p(x)(Ω) . Hereafter, we always assume that p - > 1.

        Proposition 2.1. [7, 31] The spaces L p(x) (Ω) , W1,p(x)(Ω) and W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq15_HTML.gifare separable and reflexive Banach spaces.

        Proposition 2.2. [7, 31] The conjugate space of L p(x) (Ω) is L p 0 ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq16_HTML.gif, where 1 p ( x ) + 1 p 0 ( x ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq17_HTML.gif. For any uL p(x) (Ω) andv L p 0 ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq18_HTML.gif, Ω | u v | d x 2 | u | p ( x ) | v | p 0 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq19_HTML.gif.

        Proposition 2.3. [7, 31] In W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq20_HTML.gifthe Poincaré inequality holds, that is, there exists a positive constant c such that
        | u | L p ( x ) ( Ω ) c | u | L p ( x ) ( Ω ) u W 0 1 , p x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equf_HTML.gif

        So | u | L p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq21_HTML.gifis an equivalent norm in W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq22_HTML.gif.

        Proposition 2.4. [7, 28, 29, 31] Assume that the boundary of Ω possesses the cone property and p C ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq23_HTML.gif. If q C ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq24_HTML.gifand 1 q ( x ) < p * ( x ) f o r x Ω ̄ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq25_HTML.gif, then there is a compact embedding W1,p(x)(Ω) → L q(x) (Ω).

        Let us now consider the weighted variable exponent Lebesgue space.

        Let aS(Ω) and a(x) > 0 for x ∈ Ω. Define
        L a ( x ) p ( x ) Ω = u S ( Ω ) : Ω a ( x ) u ( x ) p ( x ) d x < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equg_HTML.gif
        with the norm
        u L a ( x ) p ( x ) Ω = u ( p ( x ) , a ( x ) ) = inf λ > 0 : Ω a ( x ) u ( x ) λ p ( x ) d x 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equh_HTML.gif

        then L a ( x ) p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq26_HTML.gif is a Banach space. The following proposition follows easily from the definition of | u | L a ( x ) p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq27_HTML.gif.

        Proposition 2.5. (see [7, 31]) Set ρ(u) =Ωa(x)|u(x)| p(x) dx. For u , u k L a ( x ) p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq28_HTML.gif, wehave

        (1) F o r u 0 , | u | ( p ( x ) , a ( x ) ) = λ ρ u λ = 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq29_HTML.gif

        (2) | u | ( p ( x ) , a ( x ) ) < 1 ( = 1 ; > 1 ) ρ ( u ) < 1 ( = 1 ; > 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq30_HTML.gif

        (3) I f | u | ( p ( x ) , a ( x ) ) > 1 , t h e n | u | ( p ( x ) , a ( x ) ) p - ρ u | u | ( p ( x ) , a ( x ) ) p + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq31_HTML.gif

        (4) I f | u | ( p ( x ) , a ( x ) ) < 1 , t h e n | u | ( p ( x ) , a ( x ) p + ρ ( u ) | u | ( p ( x ) , a ( x ) ) p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq32_HTML.gif

        (5) lim k | u k | ( p ( x ) , a ( x ) ) = 0 lim k ρ ( u k ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq33_HTML.gif

        (6) | u k | ( p ( x ) , a ( x ) ) ρ ( u k ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq34_HTML.gif

        Proposition 2.6. (see [21]) Assume that the boundary of Ω possesses the cone property and p C ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq35_HTML.gif. Suppose that aL r ( x )(Ω), a(x) > 0 for x ∈ Ω, r C ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq36_HTML.gifand r- > 1. If q C ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq37_HTML.gifand
        1 q ( x ) < r ( x ) - 1 r ( x ) p * ( x ) : = p a ( x ) * ( x ) , x Ω ̄ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equ6_HTML.gif
        (2.1)

        then there is a compact embedding W 1 , p ( x ) ( Ω ) L a ( x ) q ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq38_HTML.gif.

        The following proposition plays an important role in the present article.

        Proposition 2.7. Assume that the boundary of Ω possesses the cone property and p C ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq39_HTML.gif. Suppose that aL r ( x )(Ω), a(x) > 0 for x ∈ Ω, r C ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq40_HTML.gifand r(x) > q(x) for all x ∈ Ω. If q C ( Ω ̄ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq41_HTML.gifand
        1 q ( x ) < r ( x ) - q ( x ) r ( x ) p * ( x ) , x Ω ̄ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equ7_HTML.gif
        (2.2)

        then there is a compact embedding W 1 , p ( x ) ( Ω ) L ( a ( x ) ) q ( x ) q ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq42_HTML.gif.

        Proof. Set r 1 ( x ) = r ( x ) q ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq43_HTML.gif, then r 1 - > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq44_HTML.gif and ( a ( x ) ) q ( x ) L r 1 ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq45_HTML.gif. Moreover, from (2.2) we can get
        1 q ( x ) < r 1 ( x ) - 1 r 1 ( x ) p * ( x ) , x Ω ̄ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equi_HTML.gif

        Using Proposition 2.6, we see that the embedding W 1 , p ( x ) ( Ω ) L ( a ( x ) ) q ( x ) q ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq46_HTML.gif is compact.

        2.2 Generalized gradient of the locally Lipschitz function

        Let (X, || · ||) be a real Banach space and X* be its topological dual. A function f : X → ℝ is called locally Lipschitz if each point uX possesses a neighborhood Ω u such that |f(u1) - f(u2)| ≤ L||u1 - u2|| for all u1, u2 ∈ Ω u , for a constant L > 0 depending on Ω u . The generalized directional derivative of f at the point uX in the direction vX is
        f 0 ( u , v ) = lim sup w u , t 0 1 t ( f ( w + t v ) - f ( w ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equj_HTML.gif
        The generalized gradient of f at uX is defined by
        f ( u ) = { u * X * : u * , φ f 0 ( u ; φ ) for all φ X } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equk_HTML.gif

        which is a non-empty, convex and w*-compact subset of X, where 〈·,·〉 is the duality pairing between X* and X. We say that uX is a critical point of f if 0 ∈ ∂f(u). For further details, we refer the reader to Chang [22].

        We list some fundamental properties of the generalized directional derivative and gradient that will be used throughout the article.

        Proposition 2.8. (see [22, 32]) (1) Let j : X → ℝ be a continuously differentiable function. Thenj(u) = {j'(u)}, j0(u; z) coincides withj' (u), z X and (f + j)0(u, z) = f 0 (u; z) + 〈j' (u), z X for all u, zX.
        1. (2)
          The set-valued mapping u → ∂f(u) is upper semi-continuous in the sense that for each u0X, ε > 0, vX, there is a δ > 0, such that for each w ∈ ∂f (u) with ||w - u0|| < δ, there is w0 ∈ ∂f (u0)
          w - w 0 , v < ε . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equl_HTML.gif
           
        2. (3)
          (Lebourg's mean value theorem) Let u and v be two points in X. Then there exists a point w in the open segment joining u and v and x w * f ( w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq47_HTML.gif such that
          f ( u ) - f ( v ) = x w * , u - v X . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equm_HTML.gif
           
        3. (4)
          The function
          m ( u ) = min w f ( u ) w X * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equn_HTML.gif
           

        exists, and is lower semi continuous; i.e., lim inf u u 0 m ( u ) m ( u 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq48_HTML.gif.

        In the following we need the nonsmooth version of Palais-Smale condition.

        Definition 2.1. We say that φ satisfies the (PS) c -condition if any sequence {u n } ⊂ X such that φ(u n ) → c and m(u n ) → 0, as n → +∞, has a strongly convergent subsequence, where m(u n ) = inf{||u*|| X* : u* ∈ ∂φ (u n )}.

        In what follows we write the (PS) c -condition as simply the PS-condition if it holds for every level c ∈ ℝ for the Palais-Smale condition at level c.

        3 Variational principle

        In this section we assume that Ω and p(x) satisfy the assumption (P). For simplicity we write X = W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq49_HTML.gif and ||u|| = |∇u| p ( x ) for uX. Denote by u n ⇀ u and u n u the weak convergence and strong convergence of sequence {u n } in X, respectively, denote by c and c i the generic positive constants, B ρ = {uX : ||u|| < ρ}, S ρ = {uX : ||u|| = ρ}.

        Set
        F ( x , t ) = λ a 1 ( x ) G 1 ( x , t ) + μ a 2 ( x ) G 2 ( x , t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equ8_HTML.gif
        (3.1)

        where a i and G i (i = 1, 2) are as in (A).

        Define the integral functional
        φ ( u ) = Ω 1 p ( x ) | u | p ( x ) d x - Ω F ( x , u ) d x , u X . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equ9_HTML.gif
        (3.2)
        We write
        J ( u ) = Ω 1 p ( x ) | u | p ( x ) d x , Ψ ( u ) = Ω F ( x , u ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equo_HTML.gif

        then it is easy to see that JC1(X, ℝ) and φ = J - Ψ.

        Below we give several propositions that will be used later.

        Proposition 3.1. (see [19]) The functional J : X → ℝ is convex. The mapping J' : XX* is a strictly monotone, bounded homeomorphism, and is of (S+) type, namely
        u n u a n d lim ¯ n J ( u n ) ( u n - u ) 0 i m p l i e s u n u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equp_HTML.gif

        Proposition 3.2. Ψ is weakly-strongly continuous, i.e., u n u implies Ψ(u n ) → Ψ(u).

        Proof. Define ϒ1 = ∫ΩG1(x, u) dx and ϒ2 = ∫ΩG2(x, u) dx. In order to prove Ψ is weakly-strongly continuous, we only need to prove ϒ1 and ϒ2 are weakly-strongly continuous. Since the proofs of ϒ1 and ϒ2 are identical, we will just prove ϒ1.

        We assume u n u in X. Then by Proposition 2.8.3, we have
        ϒ 1 ( u n ) - ϒ 1 ( u ) = Ω ( G 1 ( x , u n ) - G 1 ( x , u ) ) d x = Ω ξ n ( x ) ( u n - u ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equq_HTML.gif
        where ξ n ∈ ∂G1(,τ n (x)) for some τ n (x) in the open segment joining u and u n . From Chang [22] we know that ξ n L q 1 0 ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq50_HTML.gif. So using Proposition 2.5, we have
        ϒ 1 ( u n ) - ϒ 1 ( u ) 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equr_HTML.gif

        Proposition 3.3. Assume (A) holds and F satisfies the following condition:

        (B) F ( x , u ) θ λ a 1 ( x ) ξ 1 , u + θ μ a 2 ( x ) ξ 2 , u + b ( x ) + i = 1 m d i ( x ) | u | k i ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq51_HTML.giffora.e.x ∈ Ω, alluX and ξ1 ∈ ∂G1, ξ2 ∈ ∂G2, where θ is a constant, θ < 1 p + , b L 1 ( Ω ) , d i L h i ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq52_HTML.gif, h i , k i C ( Ω ̄ ) , k i ( x ) < h i ( x ) - 1 h i ( x ) p * ( x ) f o r x Ω ¯ , k i + < p - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq53_HTML.gif.

        Then φ satisfies the nonsmooth (PS) condition on X.

        Proof. Let {u n } be a nonsmooth (PS) sequence, then by (B) we have
        c + 1 + u n φ ( u n ) - θ ω , u n = Ω 1 p ( x ) - θ u n p ( x ) d x - Ω ( F ( x , u n ) - θ λ a 1 ( x ) ξ 1 , u n - θ μ a 2 ( x ) ξ 2 , u n ) d x 1 p + - θ | | u n | | p - - c 1 - Ω b ( x ) + i = 1 m d i ( x ) | u n | k i ( x ) d x 1 p + - θ | | u n | | p - - c 2 - i = 1 m | u n | ( k i ( x ) , d i ( x ) ) k i + 1 p + - θ | | u n | | p - - c 2 - c 3 i = 1 m | | u n | | k i + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equs_HTML.gif

        and consequently {u n } is bounded.

        Thus by passing to a subsequence if necessary, we may assume that u n u in X as n → ∞. We have
        J ( u n ) , u n - u - Ω λ ξ 1 n ( x ) a 1 ( x ) ( u n - u ) - Ω μ ξ 2 n ( x ) a 2 ( x ) ( u n - u ) d x ε n | | u n - u | | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equt_HTML.gif
        with ε n ↓ 0, where ξ in (x) ∈ ∂G i (x, u n ) for a.e. x ∈ Ω, i = 1, 2. From Chang [22] or Theorem 1.3.10 of [33], we know that ξ i n ( x ) L q 1 0 ( x ) , i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq54_HTML.gif. Since X is embedded compactly in L ( a i ( x ) ) q i ( x ) q i ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq55_HTML.gif, we have that u n u as n → ∞ in L ( a i ( x ) ) q i ( x ) q i ( x ) ( Ω ) , i = 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq56_HTML.gif. So using Proposition 2.2, we have
        Ω ξ i n ( x ) a i ( x ) ( u n - u ) d x 0 as n , i = 1 , 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equu_HTML.gif
        Therefore we obtain lim sup n J ( u n ) , u n - u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq57_HTML.gif. But we know that J' is a mapping of type (S+). Thus we have
        u n u in X . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equv_HTML.gif

        Remark 3.1. Note that our condition (1.2) is stronger than (1.2) of [21]. Because Ψ' is weakly-strongly continuous in [21], to verify that φ satisfies (PS) condition on X, it is enough to verify that any (PS) sequence is bounded. However, in this paper we do not know whether ξ(u) is weakly-strongly continuous, where ξ(u) ∈ ⇀Ψ. Therefore, it will be very useful to consider this problem.

        Below we denote
        F 1 ( x , t ) = λ a 1 ( x ) G 1 ( x , t ) , F 2 ( x , t ) = μ a 2 ( x ) G 2 ( x , t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equw_HTML.gif

        We shall use the following conditions.

        (B1) ∃ c0> 0 such that G2(x, t) ≥ - c0 for x ∈ Ω and t ∈ ℝ.

        (B2) θ 0 , 1 p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq58_HTML.gif and M > 0 such that 0 < G2(x, u) ≤ θu, ξ2〉 for x∈ Ω, uX and |u|M, ξ2 ∈ ⇀G2.

        Corollary 3.1. Assume (P), (A) and (A1) hold. Then φ satisfies nonsmooth (PS) condition on X provided either one of the following conditions is satisfied.

        (1). λ ∈ ℝ and μ = 0.

        (2). λ ∈ ℝ, μ = 0 and (B1) holds.

        (3). λ ∈ ℝ, μ ∈ ℝ and (B2) holds.

        Proof. In case (1) or (2), we have, for x ∈ Ω and t ∈ ℝ,
        F ( x , t ) F 1 ( x , t ) + | μ | c 0 a 2 ( x ) ( c 1 a 1 ( x ) + | μ | c 0 a 2 ( x ) ) + c 2 a 1 ( x ) | t | q 1 ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equx_HTML.gif

        which shows that the condition (B) with θ = 0 is satisfied.

        In case (3), noting that (B2) and (A) imply (B1), by the conclusion (1) and (2) we know φ satisfies (PS) condition if μ ≤ 0. Below assume μ > 0. The conditions (B2) and (A) imply that, for x ∈ Ω and uX,
        G 2 ( x , u ) θ u , ξ 2 + c 3 , and F 2 ( x , u ) θ μ a 2 ( x ) u , ξ 2  +  c 3 μ a 2 ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equy_HTML.gif
        so we have
        F ( x , u ) - θ λ a 1 ( x ) ξ 1 , u - θ μ a 2 ( x ) ξ 2 , u = ( F 1 ( x , u ) - θ λ a 1 ( x ) ξ 1 , u ) + ( F 2 ( x , u ) - θ μ a 2 ( x ) ξ 2 , u ) c 1 a 1 ( x ) + c 2 a 1 ( x ) | u | q 1 ( x ) + c 3 μ a 2 ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equz_HTML.gif

        which shows (B) holds. The proof is complete. ■

        As X is a separable and reflexive Banach space, there exist (see [[34], Section 17]) e n n = 1 X and f n n = 1 X * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq59_HTML.gif such that
        f n e m = δ n , m = 1 if n = m 0 if n m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equaa_HTML.gif
        X = span ¯ { e n : n = 1 , 2 , , } , X * = span ¯ W * { f n : n = 1 , 2 , , } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equab_HTML.gif
        For k = 1, 2, . . . , denote
        X k = span{ e k } , Y k = j = 1 k X j , Z k = j = k X j ¯ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equ10_HTML.gif
        (3.3)
        Proposition 3.5. [35] Assume that Ψ : X → ℝ is weakly-strongly continuous and Ψ (0) = 0. Let γ > 0 be given. Set
        β k = β k ( γ ) = sup u Z k , u γ | Ψ ( u ) | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equac_HTML.gif

        Then β k → 0 as k → ∞.

        Proposition 3.6. (Nonsmooth Mountain pass theorem, see [23, 33]) If X is a reflexive Banach space, φ : X → ℝ is a locally Lipschitz function which satisfies the nonsmooth (PS) c -condition, and for some r > 0 and e1X with ||e1|| > r, max{φ(0), φ(e1)} ≤·inf{φ(u) : ||u|| = r}. Then φ has a nontrivial critical uX such that the critical value c = φ(u) is characterized by the following minimax principle
        c = inf γ Γ max t [ 0 , 1 ] φ ( γ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equad_HTML.gif

        where Γ = {γC([0, 1], X) : γ(0) = 0, γ(1) = e1}.

        Proposition 3.7. (Nonsmooth Fountain theorem, see [36]) Assume (F1) X is a Banach space, φ : X → ℝ be an invariant locally Lipschitz functional, the subspaces X k , Y k and Z k are defined by (3.3).

        If, for every k ∈ ℕ, there exist ρ k > r k > 0 such that

        (F 2 ) a k : = inf u Z k u = r k φ ( u ) , k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq60_HTML.gif

        (F 3 ) b k : = max u Y k u = ρ k φ ( u ) 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq61_HTML.gif

        (F4) φ satisfies the nonsmooth (PS) c condition for every c > 0, then φ has an unbounded sequence of critical values.

        Proposition 3.8. (Nonsmooth dual Fountain theorem, see [37]) Assume (F1) is satisfied and there is a k0> 0 such that, for each kk0, there exists ρ k > γ k > 0 such that

        (D 1 ) a k : = inf u Z k u = ρ k φ ( u ) 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq62_HTML.gif

        (D 2 ) b k : = max u Y k u = r k φ ( u ) < 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq63_HTML.gif

        (D 3 ) d k : = inf u % Z k | | u | | ρ k φ ( u ) 0 , k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq64_HTML.gif

        (D4) φ satisfies the nonsmooth ( PS ) c * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq65_HTML.gifcondition for every c [ d k 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq66_HTML.gif, then φ has a sequence of negative critical values converging to 0.

        Remark 3.2. We say φ that satisfies the nonsmooth ( PS ) c * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq67_HTML.gif condition at level c ∈ ℝ (with respect to (Y n )) if any sequence {u n } ⊂ X such that
        n j , u n j Y n j , φ ( u n j ) c , m | Y n j ( u n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equae_HTML.gif

        contains a subsequence converging to a critical point of φ.

        4 Existence and multiplicity of solutions

        In this section, using the critical point theory, we give the existence and multiplicity results for problem (1.1). We shall use the following assumptions:

        ( O 1 ) δ 1 > 0 , c 3 > 0 and q 3 C ( Ω ̄ ) with q 3 ( x ) < p a 1 ( x ) * ( x ) for x Ω ̄ and q 3 + < p - , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq68_HTML.gif such that
        G 1 ( x , t ) c 3 t q 3 ( x ) , x Ω , t 0 , δ 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equaf_HTML.gif
        ( O 2 ) δ 2 > 0 , c 4 > 0 and q 4 C ( Ω ̄ ) with q 4 ( x ) < p a 2 ( x ) * ( x ) for x Ω ̄ and q 4 - > p + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq69_HTML.gif such that
        | G 2 ( x , t ) | c 4 | t | q 4 ( x ) , x Ω , | t | δ 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equag_HTML.gif
        1. (S)

          For i = 1, 2, G i (x, -t) = G i (x, t), ∀x ∈ Ω, ∀t ∈ ℝ.

           

        Remark 4.1.

        (1) It follows from (A), (A2) and (O2) that
        G 2 ( x , t ) c 4 | t | q 4 ( x ) + c 5 | t | q 2 ( x ) , x Ω , t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equah_HTML.gif
        (2) It follows from (A) and (B2) that (see [33, p. 298])
        G ( x , t ) c 6 | t | 1 / θ - c 7 , x Ω , t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equai_HTML.gif

        The following is the main result of this article.

        Theorem 4.1. Assume (P), (A), (A1) hold.

        (1) If (B1) holds, then for every λ ∈ ℝ and μ ≤ 0, problem (1.1) has a solution which is a minimizer of the corresponding functional φ.

        (2) If (B1), (A2), (O1), (O2) hold, then for every λ > 0 and μ ≤ 0, problem (1.1) has a nontrivial solution v1such that v1is a minimizer of φ and φ(v1) < 0.

        (3) If (A2), (B2), (O2) hold, then for every μ > 0, there exists λ0(μ) > 0 such that when |λ| ≤ λ0(μ), problem (1.1) has a nontrivial solution u1such that φ(u1) > 0.

        (4) If (A2), (B2), (O1), (O2) holds, then for every μ > 0, there exists λ0(μ) > 0 such that when 0 < λλ0(μ), problem (1.1) has two nontrivial solutions u1and v1such that φ(u1) > 0 and φ(v1) < 0.

        (5) If (A2), (B2), (O1), (O2) and (S) holds, then for every μ > 0 and λ ∈ ℝ, problem (1.1) has a sequence of solutionsu k } such that φu k ) → ∞ as k → ∞.

        (6) If (A2), (B2), (O1), (O2) and (S) holds, then for every λ > 0 and μ ∈ ℝ, problem (1.1) has a sequence of solutionsv k } such that φv k ) < 0 and φv k ) → 0 as k → ∞.

        Proof. We will use c, c' and c i as a generic positive constant. By Corollary 3.1, under the assumptions of Theorem 4.1, φ satisfies nonsmooth (PS) condition. We write
        Ψ 1 ( u ) = λ Ω a 1 ( x ) G 1 ( x , u ) d x , Ψ 2 ( u )  =  μ Ω a 2 ( x ) G 2 ( x , u ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equaj_HTML.gif
        then Ψ = Ψ1 + Ψ2, φ(u) = J(u) - Ψ (u) = J(u) - Ψ1(u) - Ψ2(u). Firstly, we use Ψ i ^ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq70_HTML.gif to denote its extension to L q i ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq71_HTML.gif, where i = 1, 2. From (A) and Theorem 1.3.10 of [33] (or Chang [22]), we see that Ψ i ^ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq72_HTML.gif(u) is locally Lipschitz on L q i ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq73_HTML.gif and Ψ i ^ ( u ) { ξ i ( x ) L q i 0 ( Ω ) : ξ i ( u ) G i ( x , u ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq74_HTML.gif for a.e. x ∈ Ω and i = 1, 2. In view of Proposition 2.4 and Theorem 2.2 of [22], we have that Ψ i = Ψ i ^ | X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq75_HTML.gif is also locally Lipschitz, and ∂Ψ1(u) ⊆ λΩa1(x) ∂G1(x, u) dx, ∂Ψ2(u) ⊆ μΩa2(x) ∂G1(x, u) dx, (see [38]), where Ψ i ^ | X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq76_HTML.gif stands for the restriction of Ψ i ^ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq77_HTML.gif to X for i = 1, 2. Therefore, φ is a locally Lipschitz functional on X.
        1. (1)
          Let λ ∈ ℝ and μ ≤ 0. By (A),
          | Ψ 1 ( u ) | c 1 Ω a 1 ( x ) | u | q 1 ( x ) d x + c 2 c 1 ( | u | ( q 1 ( x ) , a 1 ( x ) ) q 1 + + c 3 c 4 | | u | | q 1 + + c 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equak_HTML.gif
           
        By (B1), Ψ2(u) ≤ - μc 0 Ωa2(x) dx = c5. Hence φ ( u ) 1 p + | | u | | p - - c 4 | | u | | q 1 + - c 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq78_HTML.gif. By (A1), q 1 + < p - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq79_HTML.gif, so φ is coercive, that is, φ(u) → ∞ as ||u|| → ∞. Thus φ has a minimize which is a solution of (1.1).
        1. (2)
          Let λ > 0, μ ≤ 0 and the assumptions of (2) hold. By the above conclusion (1), φ has a minimize v1. Take v 0 C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq80_HTML.gif such that 0 ≤ v0(x) ≤ min{δ1, δ2}, Ω a 1 ( x ) v 0 ( x ) q 3 ( x ) d x = d 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq81_HTML.gif and Ω a 2 ( x ) v 0 ( x ) q 4 ( x ) d x = d 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq82_HTML.gif. By (O1) and (O2) we have, for t ∈ (0, 1) small enough,
          φ ( t v 0 ) = Ω 1 p ( x ) | t v 0 | p ( x ) d x - λ Ω a 1 ( x ) G 1 ( x , t v 0 ( x ) ) d x - μ Ω a 2 ( x ) G 2 ( x , t v 0 ( x ) ) d x t p - Ω 1 p ( x ) | v 0 | p ( x ) d x - λ Ω a 1 ( x ) c 3 ( t v 0 ( x ) ) q 3 ( x ) d x - μ Ω a 2 ( x ) c 4 ( t v 0 ( x ) ) q 4 ( x ) d x t p - Ω 1 p ( x ) | v 0 | p ( x ) d x - t q 3 + λ c 3 d 1 - t q 4 - μ c 4 d 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equal_HTML.gif
           
        Since q 3 + < p - < q 4 - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq83_HTML.gif, we can find t0 ∈ (0, 1) such that φ(t0v0) < 0, and this shows φ(v1) = inf u X φ(u) < 0. So v1 ≠ 0 because φ(0) = 0. The conclusion (2) is proved.
        1. (3)
          Let μ > 0 and the assumptions of (3) hold. By Remark 4.1.(1), for sufficiently small ||u||
          Ψ 2 ( u ) μ Ω a 2 ( x ) c 4 | u | q 4 ( x ) + c 5 | u | q 2 ( x ) d x μ c 4 | u | ( q 4 ( x ) , a 2 ( x ) ) q 4 - + μ c 5 | u | ( q 2 ( x ) , a 2 ( x ) ) q 2 - μ c 8 | | u | | q 4 - + | | u | | q 2 - . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equam_HTML.gif
           
        Since p + < q 2 - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq84_HTML.gif and p + < q 4 - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq85_HTML.gif, there exists γ > 0 and α > 0 such that J(u) - Ψ2(u) ≥ α for uS γ . We can find λ0(μ) > 0 such that when |λ| ≤ λ0(μ), Ψ1(u) ≤ α/2 for uS γ . So when |λ| ≤ λ0(μ), φ(u) ≥ α/2 > 0 for uS γ . By Remark 4.1.(2), noting that 1 θ > p + > q 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq86_HTML.gif, we can find a u0X such that ||u0|| > γ and φ(u0) < 0. By Proposition 3.6 problem (1.1) has a nontrivial solution u1 such that φ(u1) > 0.
        1. (4)

          Let μ > 0 and the assumptions of (4) hold. By the conclusion (3), we know that, there exists λ0(μ) > 0 such that when 0 < λλ0(μ), problem (1.1) has a nontrivial solution u1 such that φ(u1) > 0. Let γ and α be as in the proof of (3), that is, φ(u) ≥ α/2 > 0 for uS γ . By (O1), (O2) and the proof of (2), there exists wX such that ||w|| < γ and φ(w) < 0. It is clear that there is v1B γ , a minimizer of φ on B γ . Thus v1 is a nontrivial solution of (1.1) and φ(v1) < 0.

           
        2. (5)
          Let μ > 0, λ ∈ ℝ and the assumptions of (5) hold. By (S), we can use the nonsmooth version Fountain theorem with the antipodal action of ℤ2 to prove (5) (see Proposition 3.7). Denote
          Ψ ( u ) = Ω F ( x , u ) d x = λ Ω a 1 ( x ) G 1 ( x , u ) d x + μ Ω a 2 ( x ) G 2 ( x , u ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equan_HTML.gif
           
        Let β k (γ) be as in Proposition 3.5. By Proposition 3.5, for each positive integer n, there exists a positive integer k0(n) such that β k (n) ≤ 1 for all kk0(n). We may assume k0(n) < k0(n + 1) for each n. We define {γ k : k = 1, 2, . . . , } by
        γ k = n if  k 0 ( n ) k < k 0 ( n + 1 ) 1 if 1 k < k 0 ( 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equao_HTML.gif
        Note that γ k → ∞ as k → ∞. Then for uZ k with ||u|| = γ k we have
        φ ( u ) = Ω 1 p ( x ) | u | p ( x ) d x - Ψ ( u ) 1 p + ( γ k ) p - - 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equap_HTML.gif
        and consequently
        inf u Z k , u = γ k φ u as k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equaq_HTML.gif

        i.e., the condition (F2) of Proposition 3.7 is satisfied.

        By (A), (A1), (B2) and Remark 4.1.(2), we have
        φ ( u ) 1 p - | | u | | p + + c 1 | λ | ( | u | ( q 1 ( x ) , a 1 ( x ) ) ) q 1 + - c 6 μ | u | ( 1 / θ , a 2 ( x ) ) 1 / θ + c 9 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equar_HTML.gif
        Noting that 1 θ > p + > q 1 + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq87_HTML.gif and all norms on a finite dimensional vector space are equivalent each other, we can see that, for each Y k , φ(u) → - ∞ as uY k and ||u|| → ∞. Thus for each k there exists ρ k > γ k such that φ(u) < 0 for uY k S ρk , so the condition (F3) of Proposition 3.7 is satisfied. As was noted earlier, φ satisfies nonsmooth (PS) condition. By Proposition 3.7 the conclusion (5) is true.
        1. (6)
          Let λ > 0, μ ∈ ℝ and the assumptions of (5) hold. Let us verify the conditions of the Nonsmooth dual Fountain theorem (see Proposition 3.8). By (S), φ is invariant on the antipodal action of ℤ2. For Ψ(u) = ∫ΩF(x, u)dx = Ψ1(u)+ Ψ2(u) let β k (1) be as in Proposition 3.5, that is
          β k ( 1 ) = sup u Z k , | | u | | 1 | Ψ ( u ) | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equas_HTML.gif
           
        By Proposition 3.5, there exists a positive integer k0 such that β k ( 1 ) 1 2 p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq88_HTML.gif for all kk0. Setting ρ k = 1, then for kk0 and uZ k S1, we have
        φ ( u ) 1 p + - 1 2 p + = 1 2 p + > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equat_HTML.gif

        which shows that the condition (D1) of Proposition 3.8 is satisfied.

        Since X = W 0 1 , p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq89_HTML.gif is the closure of C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq90_HTML.gif in W 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq91_HTML.gif, we may choose {Y k : k = 1, 2, . . . , }, a sequence of finite dimensional vector subspaces of X defined by (3.5), such that Y k C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq92_HTML.gif for all k. For each Y k , because all norms on Y k are equivalent each other, there is ε ∈ (0, 1) such that for every u Y k B ε , | u | min { δ 1 , δ 2 } , | u | ( q 3 ( x ) , a 1 ( x ) ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq93_HTML.gif and | u | ( q 4 ( x ) , a 2 ( x ) ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq94_HTML.gif By (O1) and (O2), for uY k B ε we have
        φ ( u ) 1 p - | | u | | p - - λ c 3 Ω a 1 ( x ) | u | q 3 ( x ) d x + | μ | c 4 Ω a 2 ( x ) | u | q 4 ( x ) d x 1 p - | | u | | p - - λ c 3 | u | ( q 3 ( x ) , a 1 ( x ) ) q 3 + + | μ | c 4 | u | ( q 4 ( x ) , a 2 ( x ) ) q 4 - . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equau_HTML.gif
        Because q 3 + < p - < q 4 - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq95_HTML.gif there exists γ k ∈ (0, ε) such that
        b k : = max u Y k , | | u | | = γ k φ u < 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equav_HTML.gif

        thus the condition (D2) of Proposition 3.8 is satisfied.

        Because Y k Z k ≠ ∅ and γ k < ρ k , we have
        d k : = inf u Z k , | | u | | ρ k φ u b k : = max u Y k , | | u | | = r k φ u < 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equaw_HTML.gif

        On the other hand, for any uZ k with ||u|| ≤ 1 = ρ k , we have φ(u) = J(u) - Ψ(u) ≥ -Ψ(u) ≥ k (1). Noting that β k → 0 as k → ∞, we obtain d k → 0, i.e., (D3) of Proposition 3.8 is satisfied.

        Finally let us prove that φ satisfies nonsmooth ( PS ) c * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq96_HTML.gif condition for every cR. Suppose { u n j } X , n j , u n j Y n j , φ u n j c and m | Y n j u n j 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq97_HTML.gif. Similar to the process of verifying the (PS) condition in the proof of Proposition 3.3, we can get u n j u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq98_HTML.gif in X. Let us prove 0 ∈ ∂φ(u) below. Notice that
        0 m ( u ) = m ( u ) - m ( u n j ) + m ( u n j ) = m ( u ) - m ( u n j ) + m | Y n j ( u n j ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equax_HTML.gif
        Using Proposition 2.8.4, Going to limit in the right side of above equation, we have
        m ( u ) 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equay_HTML.gif

        so m(u) ≡ 0, i.e., 0 ∈ ∂φ(u), this shows that φ satisfies the nonsmooth ( PS ) c * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_IEq99_HTML.gif condition for every c ∈ ℝ. So all conditions of Proposition 3.8 are satisfied and the conclusion (6) follows from Proposition 3.8. The proof of Theorem 4.1 is complete.    ■

        Remark 4.2

        Theorem 4.1 includes several important special cases. In particular, in the case of the problem (1.4), i.e., in the case that
        a 1 ( x ) = 1 | x | s 1 ( x ) , a 2 ( x )  =  1 | x | s 2 ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-11/MediaObjects/13661_2011_Article_116_Equaz_HTML.gif

        all conditions of Theorem 4.1 are satisfied provided (P), (A*), (A1), and (A2) hold.

        Declarations

        Acknowledgements

        The authors are very grateful to the anonymous referees for their valuable suggestions. Research supported by the NSFC (Nos. 11061030, 10971087), 1107RJZA223 and the Fundamental Research Funds for the Gansu Universities.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Northwest Normal University

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