Existence of solutions for quasilinear elliptic equations with superlinear nonlinearities

  • Jia Gao1Email author,

    Affiliated with

    • Huang Lina1 and

      Affiliated with

      • Zhang Xiaojuan1

        Affiliated with

        Boundary Value Problems20122012:90

        DOI: 10.1186/1687-2770-2012-90

        Received: 23 March 2012

        Accepted: 6 August 2012

        Published: 10 August 2012

        Abstract

        Working in a weighted Sobolev space, a new result involving superlinear nonlinearities for a quasilinear elliptic boundary value problem in a domain in R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq1_HTML.gif is established. The proofs rely on the Galerkin method, Brouwer’s theorem and a new weighted compact Sobolev-type embedding theorem due to V.L. Shapiro.

        MSC:35J25, 35J62, 65L60.

        Keywords

        weighted Sobolev space superlinear quasilinear elliptic equation

        1 Introduction

        Consider the following quasilinear elliptic problem:
        { M u = [ λ 1 u + f ( x , u ) ] ρ G , x Ω , u H p , q , ρ 1 ( Ω , Γ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ1_HTML.gif
        (1.1)
        where Ω is an open (possibly unbounded) set in R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq1_HTML.gif ( N 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq2_HTML.gif), λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq3_HTML.gif is the first eigenvalue of L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq4_HTML.gif ((2.3) below), and M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq5_HTML.gif is a singular quasilinear elliptic operator defined by
        M u = i , j = 1 N D i [ p i 1 2 ( x ) p j 1 2 ( x ) σ i 1 2 ( u ) σ j 1 2 ( u ) b i j ( x ) D j u ] + b 0 σ 0 q u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ2_HTML.gif
        (1.2)

        The nonlinear part f ( x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq6_HTML.gif in Eq. (1.1) satisfies certain superlinear conditions.

        There have been many results for quasilinear elliptic equations under the conditions of which the nonlinearities satisfy sublinear or linear growth in a weighted Sobolev space. One can refer to [16].

        However, there seem to be relatively few papers that consider the quasilinear elliptic equations with superlinearity, because the compactly embedding theorem cannot be obtained easily.

        The aim of this paper is to obtain an existence result for problem (1.1). Our methods combine the Galerkin-type techniques, Brouwer’s fixed-point theorem, and a new compactly embedding theorem established by V.L. Shapiro in [7].

        This paper is organized as follows. In Section 2, we introduce some necessary assumptions and main results. In Section 3, four fundamental lemmas are established. In Section 4, the proofs of the main results are given.

        2 Assumptions and main results

        In this section, we introduce some assumptions and give the main results in this paper.

        Let Γ Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq7_HTML.gif be a fixed closed set (it may be the empty set) and ρ ( x ) , p i ( x ) , q ( x ) C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq8_HTML.gif be weight functions. q ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq9_HTML.gif is nonnegative (maybe identically zero). Denote p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq10_HTML.gif as the vector function ( p 1 ( x ) , p 2 ( x ) , , p N ( x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq11_HTML.gif.

        Consider the following pre-Hilbert spaces
        C ρ 0 ( Ω ) = { u C 0 ( Ω ) | Ω | u | 2 ρ < } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equa_HTML.gif
        with inner product u , v ρ = Ω u v ρ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq12_HTML.gif, u , v C ρ 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq13_HTML.gif, and
        C p , q , ρ 1 ( Ω , Γ ) = { u C 0 ( Ω ¯ ) C 2 ( Ω ) | u ( x ) = 0 , x Γ ; Ω [ i = 1 N | D i u | 2 p i + u 2 ( q + ρ ) ] < } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equb_HTML.gif
        with the inner product
        u , v p , q , ρ = Ω ( i = 1 N p i D i u D i v + ( q + ρ ) u v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ3_HTML.gif
        (2.1)
        u , v C p , q , ρ 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq14_HTML.gif where D i u = u x i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq15_HTML.gif, i = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq16_HTML.gif. Let L ρ 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq17_HTML.gif be the Hilbert space obtained through the completion of C ρ 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq18_HTML.gif by using the method of Cauchy sequences with respect to the norm u ρ = u , u ρ 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq19_HTML.gif, and H p , q , ρ 1 ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq20_HTML.gif be the completion of the space C p , q , ρ 1 ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq21_HTML.gif with the norm u p , q , ρ = u , u p , q , ρ 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq22_HTML.gif. Similarly, we may have L p i 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq23_HTML.gif ( i = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq16_HTML.gif) and L q 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq24_HTML.gif. Consequently, (2.1) may lead to
        u , v p , q , ρ = i = 1 N D i u , D i v p i + u , v ρ + u , v q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ4_HTML.gif
        (2.2)
        Definition 2.1 For the quasilinear differential operator M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq5_HTML.gif, the two-form is
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equc_HTML.gif
        For the linear differential operator,
        L u = i , j = 1 N D i [ p i 1 2 p j 1 2 a i j ( x ) D j u ] + a 0 q u , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ5_HTML.gif
        (2.3)
        the two-form is
        L ( u , v ) = Ω i , j = 1 N p i 1 2 p j 1 2 a i j ( x ) D j u D i v + a 0 u , v q , u , v H p , q , ρ 1 ( Ω , Γ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equd_HTML.gif

        Definition 2.2 ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq25_HTML.gif is a simple- V L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq26_HTML.gif region if the following conditions ( V L 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq27_HTML.gif)-( V L 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq28_HTML.gif) hold:

        ( V L 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq27_HTML.gif) There exists a complete orthonormal system { φ n } n = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq29_HTML.gif in L ρ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq30_HTML.gif. Also, φ n H p , q , ρ 1 ( Ω , Γ ) C 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq31_HTML.gif, ∀n;

        ( V L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq32_HTML.gif) There exists a sequence of eigenvalues { λ n } n = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq33_HTML.gif, corresponding to the orthonormal sequence { φ n } n = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq29_HTML.gif, and satisfying 0 < λ 1 < λ 2 λ 3 λ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq34_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq35_HTML.gif, such that L ( φ n , v ) = λ n φ n , v ρ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq36_HTML.gif, v H p , q , ρ 1 ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq37_HTML.gif;

        ( V L 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq38_HTML.gif) Ω = Ω 1 × × Ω N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq39_HTML.gif, where Ω i R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq40_HTML.gif is an open set for i = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq16_HTML.gif;

        ( V L 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq41_HTML.gif) For each p i ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq42_HTML.gif and ρ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq43_HTML.gif in ( V L 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq27_HTML.gif)-( V L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq32_HTML.gif), associated with each Ω i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq44_HTML.gif there are positive functions p i ( s ) , ρ i ( s ) C 0 ( Ω i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq45_HTML.gif satisfying Ω i [ p i ( s ) + ρ i ( s ) ] d s < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq46_HTML.gif, and ρ ( x ) = ρ 1 ( x 1 ) ρ N ( x N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq47_HTML.gif, p i ( x ) = ρ 1 ( x 1 ) ρ i 1 ( x i 1 ) p i ( x i ) ρ i + 1 ( x i + 1 ) ρ N ( x N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq48_HTML.gif, for i = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq16_HTML.gif;

        ( V L 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq28_HTML.gif) For each Ω i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq44_HTML.gif, p i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq49_HTML.gif, ρ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq50_HTML.gif ( i = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq51_HTML.gif), there exists h i C 0 ( Ω i ) L ρ i θ ( Ω i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq52_HTML.gif for 2 < θ < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq53_HTML.gif with the property
        | Φ ( t ) | h i ( t ) Φ p i , ρ i , Φ C 1 ( Ω i ) , t Ω i , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Eque_HTML.gif

        where Φ p i , ρ i 2 = Ω i [ p i ( t ) | d Φ ( t ) d t | 2 + ρ i ( t ) Φ 2 ( t ) ] d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq54_HTML.gif.

        There are many examples to illustrate the Simple- V L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq26_HTML.gif region. One can refer to [7] and [8].

        Remark 2.1 From ( V L 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq38_HTML.gif) and ( V L 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq41_HTML.gif), it is easy to see that ρ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq43_HTML.gif, p i ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq42_HTML.gif are positive and
        Ω ρ ( x ) d x < , Ω p i ( x ) d x < , i = 1 , , N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ6_HTML.gif
        (2.4)
        Definition 2.3 M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq5_HTML.gif is near-related to L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq4_HTML.gif if the following condition holds:
        lim u p , q , ρ M ( u , v ) L ( u , v ) u p , q , ρ = 0 , uniformly for  v p , q , ρ 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equf_HTML.gif
        We make the following assumptions concerning the operators M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq55_HTML.gif and L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq4_HTML.gif: a i j ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq56_HTML.gif ( i , j = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq57_HTML.gif) and a 0 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq58_HTML.gif satisfy (so do b i j ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq59_HTML.gif and b 0 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq60_HTML.gif):
        { ( 1 ) a 0 ( x ) , a i j ( x ) C 0 ( Ω ) L ( Ω ) , i , j = 1 , , N ; ( 2 ) a i j ( x ) = a j i ( x ) , x Ω , i , j = 1 , , N ; ( 3 ) a 0 ( x ) β 0 > 0 ( b 0 ( x ) β 1 > 0 ) , x Ω ; ( 4 ) i , j = 1 N a i j ( x ) ξ i ξ j c 0 | ξ | 2 ( i , j = 1 N b i j ( x ) ξ i ξ j c 1 | ξ | 2 ) , ( 4 ) x Ω , ξ R N  where  c 0 > 0 ( c 1 > 0 ) , | ξ | 2 = i = 1 N ξ i 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ7_HTML.gif
        (2.5)

        It is assumed throughout the paper that σ i ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq61_HTML.gif ( i = 0 , 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq62_HTML.gif) meets:

        ( σ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq63_HTML.gif) σ i ( u ) : H p , q , ρ 1 ( Ω , Γ ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq64_HTML.gif is weakly sequentially continuous;

        ( σ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq65_HTML.gif) η 0 , η 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq66_HTML.gif, s.t. η 0 σ i ( u ) η 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq67_HTML.gif, u H p , q , ρ 1 ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq68_HTML.gif.

        f ( x , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq69_HTML.gif meets the following conditions:

        ( f 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq70_HTML.gif) f ( x , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq69_HTML.gif satisfies the Caratheodory conditions;

        ( f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq71_HTML.gif) (superlinear growth condition) There exists θ with 2 < θ < 2 N N 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq72_HTML.gif, such that
        | f ( x , s ) | h 0 ( x ) + K | s | θ 1 , s R ,  a.e.  x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equg_HTML.gif

        where h 0 ( x ) L ρ θ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq73_HTML.gif. K is a nonnegative constant and θ = θ θ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq74_HTML.gif.

        ( f 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq75_HTML.gif) There exists a nonnegative function h ˜ 0 ( x ) L ρ θ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq76_HTML.gif and a constant β > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq77_HTML.gif, such that
        s f ( x , s ) β | s | 2 + h ˜ 0 ( x ) | s | , s R ,  a.e.  x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equh_HTML.gif

        Remark 2.2 Observing that for N = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq78_HTML.gif, f ( x , s ) = g ( x ) s | s | 5 3 β s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq79_HTML.gif, where g ( x ) C 0 ( Ω ) L ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq80_HTML.gif is a positive function, and meets both ( f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq71_HTML.gif) and ( f 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq75_HTML.gif).

        Now we state our main results in this paper.

        Theorem 2.1 Assume that ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq25_HTML.gif is a simple- V L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq26_HTML.gif region, the operator M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq5_HTML.gif satisfies ( σ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq63_HTML.gif)-( σ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq65_HTML.gif), and is near-related to the operator L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq4_HTML.gif, (2.5) holds for both M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq5_HTML.gif and L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq4_HTML.gif, f meets ( f 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq70_HTML.gif)-( f 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq75_HTML.gif), and G [ H p , q , ρ 1 ( Ω , Γ ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq81_HTML.gif (the dual of H p , q , ρ 1 ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq82_HTML.gif). Then the problem (1.1) has at least one nontrivial weak solution, that is, there exists a u H p , q , ρ 1 ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq83_HTML.gif such that
        M ( u , v ) = λ 1 u , v ρ + Ω f ( x , u ) v ρ G ( v ) v H p , q , ρ 1 ( Ω , Γ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equi_HTML.gif

        To derive out Theorem 2.1, we first discuss the problem in S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq84_HTML.gif, which is the subspace of H p , q , ρ 1 ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq20_HTML.gif spanned by φ 1 , , φ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq85_HTML.gif. Then by virtue of the Galerkin method, the results will be extended to H p , q , ρ 1 ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq86_HTML.gif.

        3 Fundamental lemmas

        In this section, we introduce and establish four fundamental lemmas. Lemmas 3.1 and 3.2 give two useful embedding theorems. Lemma 3.3 constructs some approximation solutions in S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq84_HTML.gif. Lemma 3.4 studies the properties of the approximation solutions.

        Lemma 3.1 ([7])

        Assume that L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq4_HTML.gif is given by (2.3) and ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq87_HTML.gif is a simple- V L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq26_HTML.gif region. For N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq88_HTML.gif, then H p , q , ρ 1 ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq89_HTML.gif is compactly imbedded in L ρ θ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq90_HTML.gif forθ ( 2 < θ < 2 N N 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq72_HTML.gif); for N = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq91_HTML.gif, then H p , q , ρ 1 ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq20_HTML.gif is compactly imbedded in L ρ θ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq92_HTML.gif forθ ( 2 < θ < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq53_HTML.gif).

        Lemma 3.2 ([7])

        Assume that L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq4_HTML.gif is given by (2.3) and ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq87_HTML.gif is a simple- V L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq26_HTML.gif region. Then H p , q , ρ 1 ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq20_HTML.gif is compactly imbedded in L ρ 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq17_HTML.gif.

        Lemma 3.3 Let all the assumptions in Theorem 2.1 hold. Then for n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq93_HTML.gif, there exists a u n S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq94_HTML.gif such that
        M ( u n , v ) = ( λ 1 1 n ) u n , v ρ + Ω f ( x , u n ) v ρ G ( v ) , v S n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ8_HTML.gif
        (3.1)
        Proof For fixed n ( n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq93_HTML.gif) and α = ( α 1 , , α n ) R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq95_HTML.gif, set u = k = 1 n α k φ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq96_HTML.gif. From simple- V L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq26_HTML.gif conditions, we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ9_HTML.gif
        (3.2)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ10_HTML.gif
        (3.3)
        From (3) and (4) of (2.5), we have
        L ( u , u ) + u ρ 2 c 0 Ω i = 1 N p i | D i u | 2 + β 0 u q 2 + u ρ 2 l u p , q , ρ 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ11_HTML.gif
        (3.4)
        where l = min { c 0 , β 0 , 1 } > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq97_HTML.gif. Combining (3.3) with (3.4), we get
        u ρ 2 u p , q , ρ 2 λ n + 1 l u ρ 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ12_HTML.gif
        (3.5)
        For m 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq98_HTML.gif, a positive integer, we put
        f m ( x , s ) = { f ( x , m ) , m s ; f ( x , s ) , m s m ; f ( x , m ) , s m . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ13_HTML.gif
        (3.6)
        Note from ( f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq71_HTML.gif) that | f m ( x , s ) | h 0 ( x ) + K | m | θ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq99_HTML.gif for s R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq100_HTML.gif, a.e. x Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq101_HTML.gif. Also, from h 0 ( x ) L ρ θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq102_HTML.gif, the Hölder inequality, Minkowski inequality, and (2.4), for v S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq103_HTML.gif, we get
        Ω | f m ( x , u ) v ρ | f m ( x , u ) L ρ θ v L ρ θ T m v L ρ θ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ14_HTML.gif
        (3.7)

        where T m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq104_HTML.gif is a positive constant depending on m.

        The remaining proof is separated into two parts. The first part is to prove the claim (3.8) for f m ( x , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq105_HTML.gif. The second part is to get the conclusion by leaving m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq106_HTML.gif based on (3.8).

        Part 1. Fix m ( m 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq98_HTML.gif). To show there exists u n , m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq107_HTML.gif such that
        M ( u n , m , v ) = ( λ 1 1 n ) u n , m , v ρ + Ω f m ( x , u n , m ) v ρ G ( v ) , v S n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ15_HTML.gif
        (3.8)
        we set
        F k ( α ) = M ( u , φ k ) ( λ 1 1 n ) u , φ k ρ Ω f m ( x , u ) φ k ρ + G ( φ k ) , k = 1 , , n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equj_HTML.gif
        It is clear that k = 1 n F k ( α ) α k = I ( α ) + I I ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq108_HTML.gif, where
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ16_HTML.gif
        (3.9)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ17_HTML.gif
        (3.10)
        For (3.9), observing the fact that the operator M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq5_HTML.gif is near-related to L , G [ H p , q , ρ 1 ( Ω , Γ ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq109_HTML.gif, (3.2), (3.5), (3.7), and Lemma 3.1, we conclude that
        lim | α | M ( u , u ) L ( u , u ) | α | 2 = 0 , lim | α | | Ω f m ( x , u ) u ρ | | α | 2 = 0 , lim | α | | G ( u ) | | α | 2 = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equk_HTML.gif
        and
        lim | α | | I ( α ) | | α | 2 = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equl_HTML.gif
        For (3.10), by (3.2), we obtain
        I I ( α ) = k = 1 n ( λ k λ 1 + 1 n ) α k 2 1 n | α | 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ18_HTML.gif
        (3.11)

        Consequently, k = 1 n F k ( α ) α k | α | 2 2 n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq110_HTML.gif where | α | s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq111_HTML.gif (here s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq112_HTML.gif is a large enough constant). By virtue of the generalized Brouwer’s theorem [9], there exists γ n , m = ( γ n , m ( 1 ) , , γ n , m ( n ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq113_HTML.gif, such that F k ( γ n , m ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq114_HTML.gif, k = 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq115_HTML.gif. Taking u n , m = k = 1 n γ n , m ( k ) φ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq116_HTML.gif, then (3.8) holds.

        Part 2. We claim that { u n , m ρ } m = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq117_HTML.gif (n fixed) is uniformly bounded according to m.

        Arguing by contradiction, without loss of generality, suppose that
        lim m u n , m ρ = . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ19_HTML.gif
        (3.12)
        Taking v = u n , m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq118_HTML.gif in (3.8),
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ20_HTML.gif
        (3.13)
        holds, that is,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ21_HTML.gif
        (3.14)

        where u ˆ n , m ( k ) = φ k , u n , m ρ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq119_HTML.gif.

        On the other hand, using ( f 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq75_HTML.gif), for s m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq120_HTML.gif, we have
        s f m ( x , s ) = s m m f ( x , m ) h ˜ 0 ( x ) | s | , a.e.  x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ22_HTML.gif
        (3.15)
        Similarly, we can also obtain the same conclusion where m s m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq121_HTML.gif or s m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq122_HTML.gif. As a result,
        s f m ( x , s ) h ˜ 0 ( x ) | s | , s R ,  a.e.  x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ23_HTML.gif
        (3.16)
        (3.14) and (3.16) imply that
        1 n u n , m ρ 2 h ˜ 0 L ρ θ u n , m L ρ θ + | G ( u n , m ) | + L ( u n , m , u n , m ) M ( u n , m , u n , m ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ24_HTML.gif
        (3.17)
        Dividing both sides of (3.17) by u n , m ρ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq123_HTML.gif and leaving m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq106_HTML.gif, we obtain from the fact that M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq5_HTML.gif is near-related to L , G [ H p , q , ρ 1 ( Ω , Γ ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq124_HTML.gif, h ˜ 0 ( x ) L ρ θ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq125_HTML.gif, and (3.5) together with Lemma 3.1 that 1 n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq126_HTML.gif. However, n is a positive integer. So, we have arrived at a contradiction. (3.12) does not hold. Then
        K 1 > 0 , u n , m ρ K 1 , m 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ25_HTML.gif
        (3.18)
        (3.5) and (3.18) imply that there is a subsequence of { u n , m } m = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq127_HTML.gif (for ease of notation take the full sequence) and a u n S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq128_HTML.gif [10] such that
        { lim m u n , m u n p , q , ρ = 0 ; lim m u n , m ( x ) = u n ( x ) , a.e.  x Ω ; lim m D i u n , m ( x ) = D i u n ( x ) , a.e.  x Ω , i = 1 , , N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ26_HTML.gif
        (3.19)
        Therefore, from (3.19), we obtain
        lim m M ( u n , m , v ) = M ( u n , v ) , v S n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ27_HTML.gif
        (3.20)
        And recall Lemma 3.1 that
        lim m Ω | u n , m u n | θ ρ = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ28_HTML.gif
        (3.21)

        Moreover, there exists a W ( x ) L ρ θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq129_HTML.gif and a subsequence { u n , m j } j = 1 { u n , m } m = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq130_HTML.gif such that | u n , m j ( x ) | W ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq131_HTML.gif, a.e. x Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq101_HTML.gif for ∀j.

        Since by virtue of the Hölder inequality, ( f 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq70_HTML.gif), ( f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq71_HTML.gif) and the Lebesgue dominated convergence theorem, we get
        lim j Ω f m j ( x , u n , m j ) v ρ = Ω f ( x , u n ) v ρ , v S n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equm_HTML.gif

        Now replacing m by m j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq132_HTML.gif in (3.8) and taking the limit as j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq133_HTML.gif on both sides of the equation, we consequently obtain that (3.1) holds and Lemma 3.3 is completed. □

        Lemma 3.4 Let all the assumptions in Theorem 2.1 hold. Then the sequence { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq134_HTML.gif obtained in Lemma 3.3 is uniformly bounded in H p , q , ρ 1 ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq135_HTML.gif.

        Proof For { u n } n = 2 S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq136_HTML.gif in Lemma 3.3, set u n = k = 1 n u ˆ n ( k ) φ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq137_HTML.gif where u ˆ n ( k ) = φ k , u n ρ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq138_HTML.gif.

        We suppose that Lemma 3.4 is false. Without loss of generality, suppose that
        lim n u n p , q , ρ = . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ29_HTML.gif
        (3.22)
        To lead to a contradiction, taking v = u n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq139_HTML.gif in (3.1), then
        M ( u n , u n ) = ( λ 1 1 n ) u n , u n ρ + Ω f ( x , u n ) u n ρ G ( u n ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ30_HTML.gif
        (3.23)
        And we can get
        1 n u n ρ 2 Ω f ( x , u n ) u n ρ + | G ( u n ) | + L ( u n , u n ) M ( u n , u n ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ31_HTML.gif
        (3.24)
        In view of G [ H p , q , ρ 1 ( Ω , Γ ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq140_HTML.gif, ( f 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq75_HTML.gif), and (3.24),
        ( 1 n + β ) u n ρ 2 h ˜ 0 L ρ θ u n L ρ θ + | G ( u n ) | + L ( u n , u n ) M ( u n , u n ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ32_HTML.gif
        (3.25)
        Dividing both sides of (3.25) by u p , q , ρ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq141_HTML.gif and leaving n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq35_HTML.gif, from the fact that M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq5_HTML.gif is near-related to L , h ˜ 0 ( x ) L ρ θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq142_HTML.gif and Lemma 3.1, we get
        lim n u n ρ u n p , q , ρ = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ33_HTML.gif
        (3.26)
        Apply (3) and (4) of (2.5) in conjunction with ( σ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq65_HTML.gif) to show
        M ( u n , u n ) c 1 η 0 Ω i = 1 N p i | D i u n | 2 + β 1 η 0 u n q 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ34_HTML.gif
        (3.27)
        also, (1) and (2) of (2.5) to show
        L ( u n , u n ) K 2 Ω i = 1 N p i | D i u n | 2 + K 3 u n q 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ35_HTML.gif
        (3.28)
        where K 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq143_HTML.gif and K 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq144_HTML.gif are positive constants. Setting c 2 = min { c 1 η 0 , β 1 η 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq145_HTML.gif and c 3 = max { K 2 , K 3 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq146_HTML.gif, it is obvious c 2 , c 3 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq147_HTML.gif. By (3.27) and (3.28), we conclude that
        c 2 c 3 L ( u n , u n ) M ( u n , u n ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ36_HTML.gif
        (3.29)
        Using (3.29) and ( f 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq75_HTML.gif), from (3.25), we obtain
        c 2 c 3 L ( u n , u n ) + β u n ρ 2 λ 1 u n ρ 2 + h ˜ 0 L ρ θ u n L ρ θ + | G ( u n ) | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ37_HTML.gif
        (3.30)
        From (3.4), we get L ( u n , u n ) + u n ρ 2 l u n p , q , ρ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq148_HTML.gif. Set c 4 = min { c 2 c 3 , β } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq149_HTML.gif. It is easy to obtain
        c 4 l u n p , q , ρ 2 λ 1 u n ρ 2 + h ˜ 0 L ρ θ u n L ρ θ + | G ( u n ) | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ38_HTML.gif
        (3.31)
        Dividing both sides of (3.31) by u p , q , ρ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq150_HTML.gif and leaving n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq35_HTML.gif, by (3.26), we get
        c 4 l 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ39_HTML.gif
        (3.32)
        So, we have arrived at a contradiction. Thus, there holds
        u n p , q , ρ K 4 , n 2  and some  K 4 > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ40_HTML.gif
        (3.33)

        Lemma 3.4 is completed. □

        4 Proof of Theorem 2.1

        Proof Since H p , q , ρ 1 ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq20_HTML.gif is a separable Hilbert space, from Lemma 3.1 and Lemma 3.2, we conclude that there exist a subsequence of { u n } n = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq151_HTML.gif (which for ease of notation we take the full sequence) and a function u H p , q , ρ 1 ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq152_HTML.gif with the following properties [10]:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ41_HTML.gif
        (4.1)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ42_HTML.gif
        (4.2)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ43_HTML.gif
        (4.3)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ44_HTML.gif
        (4.4)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ45_HTML.gif
        (4.5)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ46_HTML.gif
        (4.6)
        Let v J S J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq153_HTML.gif where J 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq154_HTML.gif is a fixed but arbitrary positive integer. In fact, for n J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq155_HTML.gif, we have
        M ( u n , v J ) M ( u , v J ) = Ω i , j = 1 N { p i 1 2 p j 1 2 σ i 1 2 ( u n ) σ j 1 2 ( u n ) b i j ( x ) D j ( u n u ) D i v J + p i 1 2 p j 1 2 [ σ i 1 2 ( u n ) σ j 1 2 ( u n ) σ i 1 2 ( u ) σ j 1 2 ( u ) ] b i j ( x ) D j u D i v J } + b 0 [ σ 0 ( u n ) u n σ 0 ( u ) u ] , v J q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ47_HTML.gif
        (4.7)
        Observing (1) of (2.5), ( σ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq63_HTML.gif)-( σ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq65_HTML.gif), (4.3)-(4.5) and D i v J L p i 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq156_HTML.gif, we have
        lim n M ( u n , v J ) = M ( u , v J ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ48_HTML.gif
        (4.8)
        On the other hand, applying ( f 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq70_HTML.gif), ( f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq71_HTML.gif), (4.2), (4.3), the Hölder inequality, and the Lebesgue dominated convergence theorem, we obtain that
        lim n Ω f ( x , u n ) v J ρ = Ω f ( x , u ) v J ρ , a.e.  x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ49_HTML.gif
        (4.9)
        Also, by (3.1), we have
        M ( u n , v J ) = ( λ 1 1 n ) u n , v J ρ + Ω f ( x , u n ) v J ρ G ( v J ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ50_HTML.gif
        (4.10)
        For (4.10), leaving n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq35_HTML.gif, from (4.1), (4.8), and (4.9), we have
        M ( u , v J ) = λ 1 u , v J ρ + Ω f ( x , u ) v J ρ G ( v J ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ51_HTML.gif
        (4.11)
        Next, given v H p , q , ρ 1 ( Ω , Γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq157_HTML.gif, we define a projection P J : H p , q , ρ 1 ( Ω , Γ ) S J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq158_HTML.gif, that is,
        P J v = k = 1 J v ˆ ( k ) φ k S J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ52_HTML.gif
        (4.12)
        where v ˆ ( k ) = φ k , v ρ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq159_HTML.gif. It is easy to get lim j P J v v p , q , ρ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq160_HTML.gif. As a result, there hold
        { lim J M ( u , P J v ) = M ( u , v ) , lim J u , P J v ρ = u , v ρ , lim J Ω f ( x , u ) P J v ρ = Ω f ( x , u ) v ρ , lim J G ( P J v ) = G ( v ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equ53_HTML.gif
        (4.13)
        Replacing v J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq161_HTML.gif by P J v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq162_HTML.gif in (4.11), passing to the limit as J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_IEq163_HTML.gif on both sides, and using (4.13), we can obtain
        M ( u , v ) = λ 1 u , v ρ + Ω f ( x , u ) v ρ G ( v ) , v H p , q , ρ 1 ( Ω , Γ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-90/MediaObjects/13661_2012_Article_201_Equn_HTML.gif

        Hence, the proof of Theorem 2.1 is complete. □

        Declarations

        Acknowledgements

        The authors express their sincere thanks to the referees for their valuable suggestions. This work was supported financially by the National Natural Science Foundation of China (11171220).

        Authors’ Affiliations

        (1)
        College of Science, University of Shanghai for Science and Technology

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        Copyright

        © Gao et al.; licensee Springer 2012

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