Existence and multiplicity of solutions for nonlocal p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq1_HTML.gif-Laplacian problems in R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq2_HTML.gif

  • Erlin Guo1Email author and

    Affiliated with

    • Peihao Zhao1

      Affiliated with

      Boundary Value Problems20122012:79

      DOI: 10.1186/1687-2770-2012-79

      Received: 13 March 2012

      Accepted: 9 July 2012

      Published: 26 July 2012

      Abstract

      In this paper, we study the nonlocal p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq1_HTML.gif-Laplacian problem of the following form

      { M ( R N 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x ) ( div ( | u | p ( x ) 2 u ) + | u | p ( x ) 2 u ) = f ( x , u ) in R N , u W 1 , p ( ) ( R N ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equa_HTML.gif

      By using the method of weight function and the theory of the variable exponent Sobolev space, under appropriate assumptions on f and M, we obtain some results on the existence and multiplicity of solutions of this problem. Moreover, we get much better results with f in a special form.

      MSC:35B38, 35D05, 35J20.

      Keywords

      critical points p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq1_HTML.gif-Laplacian nonlocal problem variable exponent Sobolev spaces

      1 Introduction

      In this paper, we consider the following problem:
      ( P ) { M ( R N 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x ) ( div ( | u | p ( x ) 2 u ) + | u | p ( x ) 2 u ) = f ( x , u ) in R N , u W 1 , p ( ) ( R N ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equb_HTML.gif

      where p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq1_HTML.gif is a function defined on R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq2_HTML.gif, M ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq3_HTML.gif is a continuous function, f : Ω × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq4_HTML.gif satisfies the Caratheodory condition.

      The operator p ( x ) u = div ( | u | p ( x ) 2 u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq5_HTML.gif is called p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq1_HTML.gif-Laplacian, which becomes p-Laplacian when p ( x ) p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq6_HTML.gif (a constant). The p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq1_HTML.gif-Laplacian possesses more complicated nonlinearities than p-Laplacian; for example, p-Laplacian is ( p 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq7_HTML.gif-homogeneous, that is, p ( λ u ) = λ p 1 p ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq8_HTML.gif for every λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq9_HTML.gif; but the p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq1_HTML.gif-Laplacian operator, when p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq1_HTML.gif is not a constant, is not homogeneous. These problems with variable exponent are interesting in applications and raise many difficult mathematical problems. Some of the models leading to these problems of this type are the models of motion of electrorheological fluids, the mathematical models of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium. We refer the reader to [17] for the study of p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq1_HTML.gif-Laplacian equations and the corresponding variational problems.

      Kirchhoff has investigated the equation
      ρ 2 u t 2 ( P 0 h + E 2 L 0 L | u x | 2 d x ) 2 u x 2 = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equc_HTML.gif
      which is called the Kirchhoff equation. This equation is an extension of the classical d’Alembert’s wave equation by considering the effect of the changes in the length of the string during vibrations. A distinguishing feature of the Kirchhoff equation is that the equation contains a nonlocal coefficient ( P 0 h + E 2 L 0 L | u x | 2 d x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq10_HTML.gif which depends on the average 1 2 L 0 L | u x | 2 d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq11_HTML.gif of the kinetic energy 1 2 | u x | 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq12_HTML.gif on [ 0 , L ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq13_HTML.gif. Various equations of Kirchhoff type have been studied by many authors, especially after the work of Lions [8], where a functional analysis framework for the problem was proposed; see, e.g., [924] for some interesting results and further references. And now the study of a nonlocal elliptic problem has already been extended to the case involving the p-Laplacian; see, e.g., [25, 26]. Corrêa and Figueiredo in [16] present several sufficient conditions for the existence of positive solutions to a class of nonlocal boundary value problems of the p-Kirchhoff type equation. Recently, the Kirchhoff type equation involving the p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq1_HTML.gif-Laplacian of the form
      u t t M ( Ω 1 p ( x ) | u | p ( x ) d x ) p ( x ) u + Q ( t , x , u , u t ) + f ( x , u ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equd_HTML.gif

      has been investigated by Autuori, Pucci and Salvatori [27]. In [28] Fan studied p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq1_HTML.gif-Kirchhoff type equations with Dirichlet boundary value problems. Many papers are about these problems in bounded domains. According to the information I have, for Kirchhoff-type problems in R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq2_HTML.gif, the results are seldom, in [29] Jin and Wu obtained three existence results of infinitely many radial solutions for Kirchhoff-type problems in R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq2_HTML.gif, and in [30] Ji established the existence of infinitely many radially symmetric solutions of Kirchhoff-type p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq14_HTML.gif-Laplacian equations in R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq15_HTML.gif. The main difficulty here arises from the lack of compactness. Jin [29] and Ji [30] investigated these problems in radial symmetric spaces. In this paper, to deal with problem (P), we overcome the difficulty caused by the absence of compactness through the method of weight function. We establish conditions ensuring the existence and multiplicity of solutions for the problem.

      This paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces. In Section 3, we obtain the solutions with negative energy by the coercivity of functionals, and in Section 4, we obtain the solutions with positive energy by the Mountain Pass Theorem. Finally in Section 5, we obtain the infinity of solutions by the Fountain Theorem and the Dual Fountain Theorem when f satisfies a special form.

      2 Preliminaries

      In order to discuss problem (P), we need some theories on space W 1 , p ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq16_HTML.gif which we call variable exponent Sobolev space. Firstly, we state some basic properties of space W 1 , p ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq16_HTML.gif which will be used later (for details, see [6, 31, 32]).

      Let Ω be an open domain of R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq2_HTML.gif, denote by S ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq17_HTML.gif the set of all measurable real functions defined on Ω, elements in S ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq17_HTML.gif which are equal to each other and almost everywhere are considered as one element, and denote
      C + ( Ω ¯ ) = { p | p C ( Ω ¯ ) , p ( x ) > 1 , x Ω ¯ } , p + = sup x Ω ¯ p ( x ) , p = inf x Ω ¯ p ( x ) , p C ( Ω ¯ ) , L p ( ) ( Ω ) = { u | u is a measurable real-valued function on Ω , Ω | u | p ( x ) d x < } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Eque_HTML.gif
      we can introduce the norm on L p ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq18_HTML.gif by
      | u | p ( x ) = inf { λ > 0 : Ω | u ( x ) λ | p ( x ) d x 1 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equf_HTML.gif

      and ( L p ( ) ( Ω ) , | | p ( ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq19_HTML.gif becomes a Banach space. We call it a variable exponent Lebesgue space.

      The space W 1 , p ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq16_HTML.gif is defined by
      W 1 , p ( ) ( Ω ) = { u L p ( ) ( Ω ) | | u | L p ( ) ( Ω ) } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equg_HTML.gif
      and it can be equipped with the norm
      u = | u | p ( x ) + | u | p ( x ) , u W 1 , p ( ) ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equh_HTML.gif

      where | u | p ( x ) = u p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq20_HTML.gif; and we denote by W 0 1 , p ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq21_HTML.gif the closure of C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq22_HTML.gif in W 1 , p ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq16_HTML.gif, p = N p ( x ) N p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq23_HTML.gif, p = ( N 1 ) p ( x ) N p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq24_HTML.gif, when p ( x ) < N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq25_HTML.gif, and p = p = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq26_HTML.gif, when p ( x ) > N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq27_HTML.gif.

      Proposition 2.1 (see [6] and [31])

      1. (1)
        If p C + ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq28_HTML.gif, the space ( L p ( ) ( Ω ) , | | p ( ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq19_HTML.gifis a separable, uniform convex Banach space, and its dual space is L q ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq29_HTML.gif, where 1 / q ( x ) + 1 / p ( x ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq30_HTML.gif. For any u L p ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq31_HTML.gifand v L q ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq32_HTML.gif, we have
        Ω | u v | d x ( 1 p + 1 q ) | u | p ( x ) | v | q ( x ) 2 | u | p ( x ) | v | q ( x ) ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equi_HTML.gif
         
      2. (2)
        If 1 p ( x ) + 1 q ( x ) + 1 r ( x ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq33_HTML.gif, then for any u L p ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq31_HTML.gif, v L q ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq34_HTML.gif, and w L r ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq35_HTML.gif,
        Ω | u v w | d x ( 1 p + 1 q + 1 r ) | u | p ( x ) | v | q ( x ) | w | r ( x ) 3 | u | p ( x ) | v | q ( x ) | w | r ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equj_HTML.gif
         

      Proposition 2.2 (see [6])

      If f : Ω × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq36_HTML.gif is a Caratheodory function and satisfies
      | f ( x , s ) | a ( x ) + b | s | p 1 ( x ) p 2 ( x ) , for any x Ω , s R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equk_HTML.gif

      where p 1 , p 2 C + ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq37_HTML.gif, a L p 2 ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq38_HTML.gif, a ( x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq39_HTML.gifand b 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq40_HTML.gifis a constant, then the superposition operator from L p 1 ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq41_HTML.gifto L p 2 ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq42_HTML.gifdefined by ( N f ( u ) ) ( x ) = f ( x , u ( x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq43_HTML.gifis a continuous and bounded operator.

      Proposition 2.3 (see [6])

      If we denote
      ρ ( u ) = Ω | u | p ( x ) d x , u L p ( ) ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equl_HTML.gif
      then for u , u n L p ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq44_HTML.gif
      1. (1)

        | u ( x ) | p ( x ) < 1 ( = 1 ; > 1 ) ρ ( u ) < 1 ( = 1 ; > 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq45_HTML.gif;

         
      2. (2)

        | u ( x ) | p ( x ) > 1 | u | p ( x ) p ρ ( u ) | u | p ( x ) p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq46_HTML.gif; | u ( x ) | p ( x ) < 1 | u | p ( x ) p ρ ( u ) | u | p ( x ) p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq47_HTML.gif;

         
      3. (3)

        | u n ( x ) | p ( x ) 0 ρ ( u n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq48_HTML.gifas n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq49_HTML.gif; | u n ( x ) | p ( x ) ρ ( u n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq50_HTML.gifas n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq49_HTML.gif.

         

      Proposition 2.4 (see [6])

      If u , u n L p ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq44_HTML.gif, n = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq51_HTML.gif, then the following statements are equivalent to each other
      1. (1)

        lim k | u k u | p ( x ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq52_HTML.gif;

         
      2. (2)

        lim k ρ ( u k u ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq53_HTML.gif;

         
      3. (3)

        u k u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq54_HTML.gifin measure in Ω and lim k ρ ( u k ) = ρ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq55_HTML.gif.

         
      Proposition 2.5 (see [6])
      1. (1)

        If p C + ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq56_HTML.gif, then W 0 1 , p ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq21_HTML.gifand W 1 , p ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq16_HTML.gifare separable reflexive Banach spaces.

         

      Proposition 2.6 If p : Ω R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq57_HTML.gifis Lipschitz continuous and p + < N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq58_HTML.gif, then for q C + ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq59_HTML.gifwith p ( x ) q ( x ) p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq60_HTML.gif, there is a continuous embedding W 1 , p ( ) ( Ω ) L q ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq61_HTML.gif.

      For any measurable functions α, β, use the symbol α β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq62_HTML.gif to denote
      ess inf x Ω ¯ ( β ( x ) α ( x ) ) > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equm_HTML.gif

      Proposition 2.7 Let Ω be a bounded domain in R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq2_HTML.gif, p C + ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq28_HTML.gif, p + < N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq58_HTML.gif. Then for any q L + ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq63_HTML.gifwith q p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq64_HTML.gif, there is a compact embedding W 1 , p ( ) ( Ω ) L q ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq61_HTML.gif.

      Proposition 2.8 (Poincare inequality)

      There is a constant C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq65_HTML.gif, such that
      | u | p ( x ) C | u | p ( x ) u W 0 1 , p ( ) ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equn_HTML.gif

      So, | u | p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq66_HTML.gifis a norm equivalent to the norm u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq67_HTML.gifin the space W 0 1 , p ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq21_HTML.gif.

      3 Solutions with negative energy

      In the following sections, we consider problem (P), the nonlocal p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq1_HTML.gif-Laplacian problem with variational form, where M is a real function satisfying the following condition: (M1) = M : ( 0 , + ) ( 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq68_HTML.gif is continuous and bounded.. And we assume that N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq69_HTML.gif, p : R n R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq70_HTML.gif is Lipschitz continuous, 1 < p p + < N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq71_HTML.gif, f : R n × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq72_HTML.gif satisfies Caratheodory conditions.

      For simplicity, we write X = W 1 , p ( ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq73_HTML.gif. Denote by C a general positive constant (the exact value may change from line to line).

      Let t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq74_HTML.gif, u X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq75_HTML.gif, define
      M ˆ ( t ) = 0 t M ( s ) d s , I 1 ( u ) = R N 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x , J ( u ) = M ˆ ( I 1 ( u ) ) = M ˆ ( R N 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x ) , Φ ( u ) = R N F ( x , u ) d x , E ( u ) = J ( u ) Φ ( u ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equo_HTML.gif

      where F ( x , u ) = 0 u f ( x , t ) d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq76_HTML.gif.

      Before giving our main results, we first give several lemmas that will be used later.

      Lemma 3.1 (see [2] and [28])

      Let (M1) hold. Then the following statements hold:
      1. (1)

        M ˆ C 0 ( [ 0 , ) ) C 1 ( ( 0 , ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq77_HTML.gif, M ˆ ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq78_HTML.gif, M ˆ ( t ) = M ( t ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq79_HTML.giffor t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq80_HTML.gif.

         
      2. (2)
        J C 0 ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq81_HTML.gif, J ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq82_HTML.gif, J C 1 ( X { 0 } ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq83_HTML.gif, and
        J ( u ) v = M ( R N 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x ) R N ( | u | p ( x ) 2 u v + | u | p ( x ) 2 u v ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equp_HTML.gif
         

      for u , v X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq84_HTML.gif.

      Lemma 3.2 (see [2])

      Suppose
      | f ( x , t ) | i = 1 m b i ( x ) | t | q i ( x ) 1 , ( x , t ) R N × R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equq_HTML.gif
      where b i ( x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq85_HTML.gif, b i ( x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq86_HTML.gif, b i L r i ( R N ) L ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq87_HTML.gif, r i , q i L + ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq88_HTML.gif, q i p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq89_HTML.gif, and there are s i L + ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq90_HTML.gifsuch that
      p ( x ) s i ( x ) p ( x ) , 1 r i ( x ) + q i ( x ) s i ( x ) = 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equr_HTML.gif

      Then Φ C 1 ( X , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq91_HTML.gifand Φ, Φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq92_HTML.gifare weakly-strongly continuous, i.e., u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq93_HTML.gifimplies Φ ( u n ) Φ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq94_HTML.gifand Φ ( u n ) Φ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq95_HTML.gif.

      Lemma 3.3
      1. (1)

        The functional J : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq96_HTML.gifis sequentially weakly lower semi-continuous, Φ : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq97_HTML.gifis sequentially weakly continuous, and thus E is sequentially weakly lower semi-continuous.

         
      2. (2)
        For any open set D X { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq98_HTML.gifwith D ¯ X { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq99_HTML.gif, the mappings J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq100_HTML.gifand E : D ¯ X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq101_HTML.gifare bounded, and are of type ( S + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq102_HTML.gif, namely,
        u n u and lim n ¯ J ( u n ) ( u n u ) 0 , implies u n u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equs_HTML.gif
         

      Proof Since the function M ˆ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq103_HTML.gif is increasing and the functional I 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq104_HTML.gif is sequentially weakly lower semi-continuous, we conclude that the functional J : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq96_HTML.gif is sequentially weakly lower semi-continuous. From Lemma 3.2, we know that Φ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq105_HTML.gifand Φ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq106_HTML.gif are sequentially weakly-strongly continuous. Now let D ¯ X { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq99_HTML.gif. It is clear that the mapping J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq100_HTML.gif and E : D ¯ X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq107_HTML.gif are bounded. To prove that J : D ¯ X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq108_HTML.gif is of type ( S + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq102_HTML.gif, assuming that { u n } D ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq109_HTML.gif, u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq93_HTML.gif in X and lim sup n J ( u n ) ( u n u ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq110_HTML.gif, then there exist positive constants c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq111_HTML.gif and c 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq112_HTML.gif such that c 1 M ( R N 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x ) c 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq113_HTML.gif. Noting that J ( u n ) = M ( R N 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x ) L p ( ) ( u n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq114_HTML.gif. It follows from lim sup n J ( u n ) ( u n u ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq115_HTML.gif that lim sup n L p ( ) ( u n ) ( u n u ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq116_HTML.gif, where L p ( ) ( u ) v = R N ( | u | p ( x ) 2 u v + | u | p ( x ) 2 u v ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq117_HTML.gif. Since L p ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq118_HTML.gif is of type ( S + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq102_HTML.gif. Moreover, since Φ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq106_HTML.gif is sequentially weakly-strongly continuous, the mapping E : D ¯ X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq107_HTML.gif is of type ( S + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq102_HTML.gif. □

      Definition 3.1 Let c R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq119_HTML.gif. A C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq120_HTML.gif-functional E : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq121_HTML.gif satisfies ( P . S ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq122_HTML.gif condition if and only if every sequence { u j } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq123_HTML.gif in X such that lim j E ( u j ) = c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq124_HTML.gif, and lim j E ( u j ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq125_HTML.gif in X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq126_HTML.gif has a convergent subsequence.

      Lemma 3.4 (see [28])

      Suppose f satisfies the hypotheses in Lemma 3.2, and let (M1) hold. Then, for any c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq127_HTML.gif, every bounded ( P . S ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq122_HTML.gifsequence for E, i.e., a bounded sequence { u n } X { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq128_HTML.gifsuch that E ( u n ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq129_HTML.gifand E ( u n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq130_HTML.gif, has a strongly convergent subsequence.

      As X is a separable and reflexive Banach space, there exist { e n } n = 1 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq131_HTML.gif and { f n } n = 1 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq132_HTML.gif such that
      f n ( e m ) = δ n , m = { 1 if n = m , 0 if n m , X = span ¯ { e n : n = 1 , 2 , } , X = span ¯ W { f n : n = 1 , 2 , } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equt_HTML.gif
      For k = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq133_HTML.gif , 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%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equu_HTML.gif

      Lemma 3.5 (see [2])

      Assume that Φ : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq134_HTML.gifis weakly-strongly continuous and Φ ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq135_HTML.gif, γ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq136_HTML.gifis a given positive number. Set
      β k = sup u k Z k , u γ | Φ ( u ) | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equv_HTML.gif

      then β k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq137_HTML.gifas k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq138_HTML.gif.

      Theorem 3.1 Suppose f satisfies the hypotheses in Lemma 3.2, let (M1) hold and the following conditions hold: (M2) = There are positive constants α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq139_HTML.gif, M and C such that M ˆ ( t ) C t α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq140_HTML.giffor t M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq141_HTML.gif.; (H1) = q + < α 1 p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq142_HTML.gif..Then the functional E is coercive and attains its infimum in X at some u 0 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq143_HTML.gif. Therefore, u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq144_HTML.gifis a solution of (P) if E is differentiable at u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq144_HTML.gif, and in particular, if u 0 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq145_HTML.gif.

      Proof We have concluded that E is weakly lower semi-continuous. Let us prove that E is coercive on X, i.e., E ( u ) + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq146_HTML.gif as u + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq147_HTML.gif. For simplicity, we assume that m = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq148_HTML.gif and denote b 1 = b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq149_HTML.gif, q 1 = q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq150_HTML.gif, s 1 = s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq151_HTML.gif, r 1 = r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq152_HTML.gif. We have that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equw_HTML.gif
      When u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq67_HTML.gif is large enough, we have
      E ( u ) = J ( u ) Φ ( u ) C 2 u α 1 p C 4 u q + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equx_HTML.gif

      and hence E is coercive. Since E is sequentially weakly lower semi-continuous and X is reflexive, E attains its infimum in X at some u 0 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq143_HTML.gif. In the case where E is differentiable at u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq144_HTML.gif, u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq144_HTML.gif is a solution of (P). □

      Theorem 3.2 Suppose f satisfies the hypotheses in Lemma 3.2. Let (M1), (M2), (H1) and the following conditions hold: (M3) = There is a positive constant α 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq153_HTML.gifsuch that lim sup t 0 + M ˆ ( t ) t α 2 < + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq154_HTML.gif.; (f1) = There exists a positive constant δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq155_HTML.gif,
      f ( x , t ) b 0 ( x ) t q 0 ( x ) 1 for x R N and 0 < t δ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equy_HTML.gif

      where b 0 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq156_HTML.gif, b 0 ( x ) C ( R N , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq157_HTML.gif, b 0 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq158_HTML.gif, q 0 ( x ) L + ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq159_HTML.gif, q 0 + < p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq160_HTML.gif.; (H2) = q 0 + < α 2 p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq161_HTML.gif..Then (P) has at least one nontrivial solution which is a global minimizer of the energy functional E.

      Proof From Theorem 3.1 we know that E has a global minimizer u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq144_HTML.gif. It is clear that F ( x , 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq162_HTML.gif and consequently E ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq163_HTML.gif. As b 0 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq156_HTML.gif and b 0 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq158_HTML.gif, we can find a bounded open set Ω R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq164_HTML.gif such that b 0 ( x ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq165_HTML.gif for x Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq166_HTML.gif. The space W 0 k , p ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq167_HTML.gif is a subspace of X. Take w C 0 ( Ω ) { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq168_HTML.gif. Then, by (f1), (M3) and (H2), for sufficiently small λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq9_HTML.gif, we have that
      E ( λ w ) = M ˆ ( R N λ p ( x ) p ( x ) ( | w | p ( x ) + | w | p ( x ) ) d x ) R N F ( x , λ w ) d x C 1 ( R N λ p ( x ) p ( x ) ( | w | p ( x ) + | w | p ( x ) ) d x ) α 2 Ω F ( x , λ w ) d x C 4 λ α 2 p C 5 λ q 0 + < 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equz_HTML.gif

      Hence E ( u 0 ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq169_HTML.gif which shows u 0 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq145_HTML.gif. □

      Theorem 3.3 Let the hypotheses of Theorem 3.2 hold, and f satisfy the following condition: (f2) = f ( x , t ) = f ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq170_HTML.giffor x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq171_HTML.gifand t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq172_HTML.gif..Then (P) has a sequence of solutions { ± u k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq173_HTML.gifsuch that E ( ± u k ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq174_HTML.gif, and E ( ± u k ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq175_HTML.gifas k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq138_HTML.gif.

      Proof Denote by γ ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq176_HTML.gif the genus of A. Denote
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equaa_HTML.gif

      we have < c 1 c 2 c k c k + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq177_HTML.gif.

      From the condition on b 0 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq178_HTML.gif, there exists a bounded open set Ω R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq179_HTML.gif such that b 0 ( x ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq165_HTML.gif for x Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq166_HTML.gif. The space W 0 k , p ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq180_HTML.gif is a subspace of X. For any k, we can choose a k-dimensional linear subspace E k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq181_HTML.gif of W 0 k , p ( ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq180_HTML.gif such that E k C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq182_HTML.gif. As the norms on E k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq181_HTML.gif are equivalent to each other, there exists ρ k ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq183_HTML.gif such that u E k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq184_HTML.gif with u ρ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq185_HTML.gif implies | u | L δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq186_HTML.gif. S ρ k ( k ) = { u E k : u = ρ k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq187_HTML.gif is compact, and then there exists a constant d k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq188_HTML.gif such that
      Ω b 0 ( x ) q 0 ( x ) | u | q 0 ( x ) d x d k , u S ρ k ( k ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equab_HTML.gif
      For u S ρ k ( k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq189_HTML.gif and t ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq190_HTML.gif, we have
      E ( t u ) t α 2 p p ρ k p Ω b 0 ( x ) q 0 ( x ) t q 0 ( x ) | u | q 0 ( x ) d x t α 2 p p ρ k p t q 0 + d k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equac_HTML.gif

      As q 0 + < α 2 p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq161_HTML.gif, we can find t k ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq191_HTML.gif and ε k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq192_HTML.gif such that E ( t k u ) ε k < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq193_HTML.gif, u S ρ k ( k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq194_HTML.gif, which implies E ( u ) ε k < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq195_HTML.gif, u S t k ρ k ( k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq196_HTML.gif. Since γ ( S t k ρ k ( k ) ) = k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq197_HTML.gif, we get the conclusion c k ε k < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq198_HTML.gif.

      By the genus theory, each c k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq199_HTML.gif is a critical value of E, hence there is a sequence of solutions { ± u k : k = 1 , 2 , , } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq200_HTML.gif of problem (P) such that E ( ± u k ) = c k < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq201_HTML.gif.

      At last, we will prove c k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq202_HTML.gif as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq138_HTML.gif. By the coercive of E, there exists a constant γ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq136_HTML.gif such that E ( u ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq203_HTML.gif when u γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq204_HTML.gif. For any A Σ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq205_HTML.gif, let Y k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq206_HTML.gif and Z k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq207_HTML.gif be the subspace of X as mentioned above. According to the properties of genus, we know that A Z k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq208_HTML.gif. Set
      β k = sup u Z k , u γ | Φ ( u ) | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equad_HTML.gif

      we know β k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq137_HTML.gif as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq138_HTML.gif. When u Z k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq209_HTML.gif and u γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq210_HTML.gif, we have E ( u ) β k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq211_HTML.gif, and then c k β k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq212_HTML.gif, which concludes c k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq202_HTML.gif as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq138_HTML.gif. □

      Theorem 3.4 Let the hypotheses of Lemma 3.2, (f1), (M1), (M2), (M3), (H1), (H2) and the following condition hold, (f+) = f ( x , t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq213_HTML.giffor x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq171_HTML.gifand t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq74_HTML.gif..Then (P) has at least one nontrivial nonnegative solution with negative energy.

      Proof Define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equae_HTML.gif

      Then, like in the proof of Theorem 3.2, using truncation functions above, similarly to the proof of Theorem 3.4 in [28], we can prove that E ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq214_HTML.gif has a nontrivial global minimizer u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq144_HTML.gif and u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq144_HTML.gif is a nontrivial nonnegative solution of (P). □

      4 Solution with positive energy

      In this section we will find the Mountain Pass type critical points of the energy functional E associated with problem (P).

      Lemma 4.1 Let (f1), (M1) and the following conditions hold: (M2) = α 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq215_HTML.gif, M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq216_HTML.gif, and C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq65_HTML.gifsuch that
      M ˆ ( t ) C t α 1 for t M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equaf_HTML.gif

      with α 1 p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq217_HTML.gifhold.; (M4) = λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq218_HTML.gif, M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq216_HTML.gifsuch that

      λ M ˆ ( t ) M ( t ) t for t M . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equag_HTML.gif
      ; (f3) = μ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq219_HTML.gif, M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq216_HTML.gifsuch that
      0 μ F ( x , t ) f ( x , t ) t , for | t | M and x R N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equah_HTML.gif
      ; (H3) = λ p + < μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq220_HTML.gif..Then E satisfies condition ( P . S ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq122_HTML.giffor any c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq127_HTML.gif.
      Proof By (M4), for u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq67_HTML.gif large enough, we have
      λ p + J ( u ) p + M ( R N 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x ) R N 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x M ( R N 1 p ( x ) ( | u | p ( x ) + | u | p ( x ) ) d x ) R N ( | u | p ( x ) + | u | p ( x ) ) d x = J ( u ) u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equai_HTML.gif
      By (f3) we conclude that there exists C 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq221_HTML.gif such that
      C 1 μ R N F ( x , u ) d x R N f ( x , u ) u d x + C 1 , u X , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equaj_HTML.gif
      and thus, given any ε ( 0 , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq222_HTML.gif, there exists M ε M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq223_HTML.gif such that
      ( μ ε ) R N F ( x , u ) d x R N f ( x , u ) u d x , if R N F ( x , u ) d x M ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equak_HTML.gif
      we claim that there exists C ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq224_HTML.gif such that
      Φ ( u ) u ( μ ε ) Φ ( u ) C ε for u X , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equal_HTML.gif

      the notation of this conclusion can be seen in [28].

      Now let { u n } X { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq128_HTML.gif, E ( u n ) c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq225_HTML.gif and E ( u n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq130_HTML.gif. By (H3), there exists ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq226_HTML.gif small enough such that λ p + < ( μ ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq227_HTML.gif. Then, since { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq228_HTML.gif is a ( P . S ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq122_HTML.gif sequence, for sufficiently large n, we have
      ( μ ε ) c + 1 + u ( μ ε ) E ( u n ) E ( u n ) u n ( ( μ ε ) λ p + ) J ( u n ) + ( λ p + J ( u n ) J ( u n ) u n ) + ( Φ ( u n ) u n ( μ ε ) Φ ( u n ) ) C 2 u n α 1 p C 3 C ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equam_HTML.gif

      we conclude that { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq229_HTML.gif is bounded, since α 1 p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq217_HTML.gif. By Lemma 3.4, E satisfies condition ( P . S ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq122_HTML.gif for c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq127_HTML.gif. □

      Lemma 4.2 Under the hypotheses of Lemma 4.1, for any w X { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq230_HTML.gif, E ( s w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq231_HTML.gifas s + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq232_HTML.gif.

      Proof Let w X { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq230_HTML.gif be given. From (M4) for sufficiently large t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq80_HTML.gif we have
      M ˆ ( t ) C 1 t λ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equan_HTML.gif
      and then it follows that
      J ( s w ) d 1 s λ p + for s large enough , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equao_HTML.gif
      where d 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq233_HTML.gif is a positive constant depending on w. From (f4) for | t | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq234_HTML.gif large enough we have
      F ( x , t ) C 2 | t | μ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equap_HTML.gif
      which implies that
      Φ ( s w ) = R N F ( x , s w ) d x d 2 s μ for s large enough , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equaq_HTML.gif
      where d 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq235_HTML.gif is a positive constant depending on w. Hence for s large enough, we have
      E ( s w ) d 1 s λ p + d 2 s μ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equar_HTML.gif

      and then E ( s w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq231_HTML.gif as s + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq236_HTML.gif. □

      Lemma 4.3 Under the hypotheses of Lemma 3.2, (M1) holds and the following conditions hold: (M5) = There is a positive constant α 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq237_HTML.gifsuch that lim sup t 0 + M ˆ ( t ) t α 3 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq238_HTML.gif.; (f4) = There exists r 1 ( x ) C 0 ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq239_HTML.gifsuch that 1 < r 1 ( x ) < p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq240_HTML.giffor x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq171_HTML.gifand
      lim inf t 0 | F ( x , t ) | | t | r 1 ( x ) < + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equas_HTML.gif

      uniformly in x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq171_HTML.gif.; (H4) = α 3 p + < r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq241_HTML.gif..

      Then there exist positive constants ρ and δ such that E ( u ) δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq242_HTML.giffor u = ρ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq243_HTML.gif.

      Proof It follows from (M5) that
      J ( u ) C 1 u α 3 p + for u small enough . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equat_HTML.gif
      It follows from the hypotheses of Lemma 3.2 and (f4) that
      | Φ ( u ) | C 2 u r 1 for u small enough . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equau_HTML.gif

      Thus by (H4), we obtain the assertion of Lemma 4.3. □

      By the famous Mountain Pass lemma, from Lemmas 4.1-4.3, we have the following:

      Theorem 4.1 Let all hypotheses of Lemmas 4.1-4.3 hold. Then (P) has a nontrivial solution with positive energy.

      5 The case of concave-convex nonlinearity

      In this section, we will obtain much better results with f in a special form. We have the following theorem:

      Theorem 5.1 Let f ( x , u ) = a ( x ) | u | α ( x ) 2 u + b ( x ) | u | q ( x ) 2 u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq244_HTML.gif, where

      α , q L + ( R N ) , 1 < α α + < p p + < q , q p , a ( x ) > 0 , a L ( R N ) L r 1 ( ) ( R N ) , 1 r 1 ( x ) + α ( x ) s 1 ( x ) = 1 , b ( x ) > 0 , b L ( R N ) L r 2 ( ) ( R N ) , 1 r 2 ( x ) + α ( x ) s 2 ( x ) = 1 , p ( x ) s 1 ( x ) p ( x ) , p ( x ) s 2 ( x ) p ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equav_HTML.gif
      Then we have
      1. (1)

        If (M1), (M2), (M4), (H3) hold and we also assume that α + < α 1 p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq245_HTML.gifand λ p + < q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq246_HTML.gif, then problem (P) has solutions { ± u k } k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq247_HTML.gifsuch that E ( ± u k ) + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq248_HTML.gifas k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq138_HTML.gif.

         
      2. (2)

        If (M1), (M4), (M5), (H3) hold and we also assume that α < α 3 p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq249_HTML.gifand α + < λ p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq250_HTML.gif, then problem (P) has solutions { ± v k } k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq251_HTML.gifsuch that E ( ± v k ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq252_HTML.gif, E ( ± v k ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq253_HTML.gifas k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq138_HTML.gif.

         

      We will use the following ‘Fountain Theorem’ and the ‘Dual Fountain Theorem’ to prove Theorem 5.1.

      Proposition 5.1 (Fountain Theorem, see [11])

      Assume(A1) = X is a Banach space, E C 1 ( X , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq254_HTML.gifis an even functional, the subspaces X k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq255_HTML.gif, Y k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq206_HTML.gifand Z k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq207_HTML.gifare defined by (3.2)..

      If for each k = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq133_HTML.gif, there exists ρ k > r k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq256_HTML.gifsuch that(A2) = inf u Z k , u = r k E ( u ) + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq257_HTML.gifas k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq138_HTML.gif.; (A3) = max u Y k , u = ρ k E ( u ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq258_HTML.gif.; (A4) = E satisfies the ( P S ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq259_HTML.gifcondition for every c > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq260_HTML.gif. Then E has a sequence of critical values tending to +∞..

      Proposition 5.2 (Dual Fountain Theorem, see [11])

      Assume (A1) is satisfied and there is a k 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq261_HTML.gifso as to for each k k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq262_HTML.gif, there exists ρ k > r k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq256_HTML.gifsuch that(B1) = inf u Z k , u = ρ k E ( u ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq263_HTML.gif.; (B2) = b k : = max u Y k , u = r k E ( u ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq264_HTML.gif.; (B3) = d k : = inf u Z k , u ρ k E ( u ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq265_HTML.gifas k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq138_HTML.gif.; (B4) = E satisfies ( P S ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq266_HTML.gifcondition for every c [ d k 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq267_HTML.gif. Then E has a sequence of negative critical values converging to 0..

      Definition 5.1 We say that E satisfies the ( P S ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq266_HTML.gif condition (with respect to ( Y k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq268_HTML.gif), if any sequence { u n j } X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq269_HTML.gif such that n j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq270_HTML.gif, u n j Y n j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq271_HTML.gif, E ( u n j ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq272_HTML.gif and ( E Y n j ) ( u n j ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq273_HTML.gif, contains a subsequence converging to a critical point of E.

      Proof of Theorem 5.1 Firstly, we verify the ( P S ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq266_HTML.gif condition for every c R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq119_HTML.gif. Suppose { u n j } X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq274_HTML.gif, n j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq270_HTML.gif, E ( u n j ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq272_HTML.gif and ( E Y n j ) ( u n j ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq275_HTML.gif. It is easy to obtain that f ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq276_HTML.gif satisfies condition ( f 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq277_HTML.gif), when it has this special form. So similar to the method in Lemma 4.1, we have that
      ( μ ε ) c + 1 + u n j ( μ ε ) E ( u n ) E ( u n j ) u n j ( ( μ ε ) λ p + ) J ( u n j ) + ( λ p + J ( u n j ) J ( u n j ) u n j ) + ( Φ ( u n j ) u n j ( μ ε ) Φ ( u n j ) ) C 2 u n j α 1 p C 3 C ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equaw_HTML.gif
      hence, we can get that { u n j } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq278_HTML.gif is bounded. Going if necessary to a subspace, we can assume that u n j u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq279_HTML.gif in X. As X = n j Y n j ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq280_HTML.gif, we can choose v n j Y n j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq281_HTML.gif such that v n j u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq282_HTML.gif. Hence
      lim n j E ( u n j ) ( u n j u ) = lim n j E ( u n j ) ( u n j v n j ) + lim n j E ( u n j ) ( v n j u ) = lim n j ( E Y n j ) ( u n j ) ( u n j v n j ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equax_HTML.gif

      As E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq283_HTML.gif is of ( S + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq284_HTML.gif type, we can conclude u n j u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq285_HTML.gif; furthermore, we have E ( u n j ) E ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq286_HTML.gif.

      It only remains to prove E ( u ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq287_HTML.gif. For any w k Y k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq288_HTML.gif and n j k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq289_HTML.gif we have
      E ( u ) w k = ( E ( u ) E ( u n j ) ) w k + E ( u n j ) w k = ( E ( u ) E ( u n j ) ) w k + ( E Y n j ) ( u n j ) w k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equay_HTML.gif
      Going to the limit on the right side of the above equation reaches
      E ( u ) w k = 0 , w k Y k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equaz_HTML.gif
      so E ( u ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq287_HTML.gif, this shows that E satisfies the ( P S ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq266_HTML.gif condition for every c R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq119_HTML.gif. Obviously, E also satisfies the ( P S ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq259_HTML.gif condition for every c R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq119_HTML.gif.
      1. (1)
        We will prove that if k is large enough, then there exist ρ k > r k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq256_HTML.gif such that (A2) and (A3) are satisfied. (A2) For k = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq133_HTML.gif , denote
        θ k = sup v Z k , v 1 R N a ( x ) α ( x ) | v | α ( x ) d x , β k = sup v Z k , v 1 R N b ( x ) q ( x ) | v | q ( x ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equba_HTML.gif
         
      then θ k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq290_HTML.gif, β k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq291_HTML.gif, and θ k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq292_HTML.gif, β k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq137_HTML.gif, as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq138_HTML.gif. When u Z k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq209_HTML.gif, u M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq293_HTML.gif,
      E ( u ) 1 p + u α 1 p u α + θ k u q + β k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equbb_HTML.gif
      For sufficiently large k, we have θ k < 1 2 p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq294_HTML.gif. As α + < α 1 p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq245_HTML.gif, we get
      E ( u ) 1 2 p + u α 1 p u q + β k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equbc_HTML.gif
      Choose r k = ( p 2 p + q + β k ) 1 q + α 1 p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq295_HTML.gif, we have
      E ( u ) ( p 2 p + q + ) q + q + α 1 p q + p p ( 1 β k ) p q α 1 p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equbd_HTML.gif

      Since β k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq137_HTML.gif, we have inf u Z k , u = r k E ( u ) + as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq296_HTML.gif. (A2) is satisfied.

      (A3) For k = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq133_HTML.gif , denote
      e k = inf v Y k , v = 1 R N b ( x ) q ( x ) | v | q ( x ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Eqube_HTML.gif
      Then e k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq297_HTML.gif. For any v Y k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq298_HTML.gif, with v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq299_HTML.gif and t large enough, since dim Y k < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq300_HTML.gif, all norms are equivalent in Y k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq206_HTML.gif, we have
      E ( t v ) 1 p t λ p + e k t q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equbf_HTML.gif
      As q > λ p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq301_HTML.gif, there exists ρ k > r k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq302_HTML.gif such that t = ρ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq303_HTML.gif concludes E ( t v ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq304_HTML.gif and then
      max u Y k , u = ρ k E ( u ) 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equbg_HTML.gif

      so (A2) is satisfied.

      Conclusion (1) is reached by the Fountain Theorem.
      1. (2)

        We use the Dual Fountain Theorem to prove conclusion (2), and now it remains for us to prove that there exist ρ k > r k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq256_HTML.gif such that if k is large enough (B1), (B2) and (B3) are satisfied.

         
      (B1) Let θ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq305_HTML.gif and β k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq306_HTML.gif be defined as above, when v Z k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq307_HTML.gif, v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq299_HTML.gif and t small enough we have
      E ( t v ) 1 p + t α 3 p + t α θ k t q β k 1 p + t α 3 p + t α θ k t p + β k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equbh_HTML.gif
      For sufficiently large k we have β k < 1 2 p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq308_HTML.gif, thus
      E ( t v ) 1 2 p + t α 3 p + t α θ k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equbi_HTML.gif
      Choose ρ k = ( 2 p + β k ) 1 α 3 p + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq309_HTML.gif, then for sufficiently large k, ρ k < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq310_HTML.gif. When t = ρ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq303_HTML.gif, v Z k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq307_HTML.gif with v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq299_HTML.gif, we have E ( t v ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq311_HTML.gif, which implies
      inf u Z k , u = ρ k E ( u ) 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equbj_HTML.gif

      Hence (B1) is satisfied.

      (B2) For k = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq133_HTML.gif , denote
      δ k = inf v Y k , v = 1 R N a ( x ) α ( x ) | v | α ( x ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equbk_HTML.gif
      then δ k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq312_HTML.gif. For v Y k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq298_HTML.gif, v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq299_HTML.gif and t small enough, we have
      E ( t v ) 1 p t λ p δ k t α + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equbl_HTML.gif
      since dim Y k < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq300_HTML.gif and α + < λ p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq250_HTML.gif, we get
      b k : = max u Y k , u = r k E ( u ) < 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equbm_HTML.gif

      with r k ( 0 , ρ k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq313_HTML.gif small enough. Hence (B2) is satisfied.

      (B3) From the proof above and Y k Z k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq314_HTML.gif, we have
      d k : = inf u Z k , u ρ k E ( u ) b k : = max u Y k , u = r k E ( u ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equbn_HTML.gif
      For v Z k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq307_HTML.gif, v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq299_HTML.gif and u = t v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq315_HTML.gif small enough, we have
      E ( u ) = E ( t v ) 1 2 p + t α 3 p + t α θ k t α θ k ρ k α θ k θ k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_Equbo_HTML.gif

      hence d k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-79/MediaObjects/13661_2012_Article_180_IEq316_HTML.gif. Hence (B3) is satisfied.

      Conclusion (2) is reached by the Dual Fountain Theorem. □

      Declarations

      Acknowledgements

      The authors thank the two referees for their careful reading and helpful comments on the study. Research was supported by the National Natural Science Foundation of China (10971088), (10971087) and the Fundamental Research Funds for the Central Universities (lzujbky-2012-180).

      Authors’ Affiliations

      (1)
      School of Mathematics and Statistics, Lanzhou University

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