Existence of multiple solutions for the Brezis-Nirenberg-type problem with singular coefficients

  • Yang Yang1Email author,

    Affiliated with

    • Jihui Zhang2 and

      Affiliated with

      • Xiusong Gu3

        Affiliated with

        Boundary Value Problems20122012:137

        DOI: 10.1186/1687-2770-2012-137

        Received: 4 September 2012

        Accepted: 19 September 2012

        Published: 26 November 2012

        Abstract

        By energy estimates and by establishing a local (PS) condition, we obtain the multiplicity of solutions to a class of Brezis-Nirenberg-type problem with singular coefficients via minimax methods and the Krasnoselskii genus theory.

        Keywords

        Brezis-Nirenberg-type problem minimax method

        1 Introduction and main results

        This paper is concerned with multiple solutions for the semilinear Brezis-Nirenberg-type problem with singular coefficients
        { div ( D u | x | 2 a ) = λ | u | 2 2 u | x | 2 b + β | u | q 2 u | x | α , x Ω ; u = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ1_HTML.gif
        (1)

        where Ω R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq1_HTML.gif is a bounded smooth domain, and 0 Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq2_HTML.gif, < a < n 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq3_HTML.gif, a b < a + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq4_HTML.gif, 2 = 2 n n 2 d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq5_HTML.gif, d = a + 1 b ( 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq6_HTML.gif, 1 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq7_HTML.gif, α < ( 1 + a ) q + n ( 1 q 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq8_HTML.gif. β > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq9_HTML.gif, λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq10_HTML.gif are two real parameters.

        The starting point of the variational approach to the problem is the Caffarelli-Kohn-Nirenberg inequality (see [1]): There is a constant C a , b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq11_HTML.gif such that
        ( R n | x | 2 b | u | 2 d x ) 2 / 2 C a , b R n | x | 2 a | D u | 2 d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ2_HTML.gif
        (2)
        for all u C 0 ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq12_HTML.gif, where
        < a < n 2 2 , a b < a + 1 , 2 = 2 n n 2 d , d = a + 1 b . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equa_HTML.gif
        Let D a 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq13_HTML.gif be the completion of C 0 ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq14_HTML.gif with respect to the weighted norm http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq15_HTML.gif defined by
        u = ( Ω | x | 2 a | D u | 2 d x ) 1 / 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equb_HTML.gif
        From the boundedness of Ω and the standard approximation arguments, it is easy to see that (2) holds for any u D a 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq16_HTML.gif in the sense:
        ( Ω | x | α | u | r d x ) 2 / r C Ω | x | 2 a | D u | 2 d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ3_HTML.gif
        (3)
        for 1 r 2 = 2 n n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq17_HTML.gif, α r ( 1 + a ) + n ( 1 r 1 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq18_HTML.gif, that is, the embedding D a 1 , 2 ( Ω ) L r ( Ω , | x | α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq19_HTML.gif is continuous, where L r ( Ω , | x | α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq20_HTML.gif is the weighted L r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq21_HTML.gif space with the norm
        u r , α : = u L r ( Ω , | x | α ) = ( Ω | x | α | u | r d x ) 1 / r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equc_HTML.gif
        On D a 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq13_HTML.gif, we can define the energy functional
        J ( u ) = 1 2 Ω | x | 2 a | D u | 2 d x λ 2 Ω | x | 2 b | u | 2 d x β q Ω | x | α | u | q d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ4_HTML.gif
        (4)

        From (4), J is well defined in D a 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq13_HTML.gif, and J C 1 ( D a 1 , 2 ( Ω ) , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq22_HTML.gif. Furthermore, the critical points of J are weak solutions of problem (1).

        Breiz-Nirenberg-type problems have been generalized to many situations such as
        { div ( D u | x | 2 a ) μ u | x | 2 ( a + 1 ) = | u | 2 2 u | x | 2 b + λ u | x | 2 ( a + 1 ) c , x Ω , u = 0 , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ5_HTML.gif
        (5)

        Xuan et al.[2] derived the explicit formula for the extremal functions of the best embedding constant by applying the Bliss lemma [3]. They got a nontrivial solution for problem (5) including the resonant and nonresonant cases by variational methods. He and Zou [4] studied problem (5) and obtained the multiplicity of solutions with the aid of a pseudo-index theory. In [5], problem (5) has been extended to the p-Laplace case by Xuan.

        The purpose of this paper is to study the multiplicity of solutions for the Breiz-Nirenberg-type problem (1) with the aid of a minimax method. We obtain multiple nontrivial solutions of (1) by proving the local (PS) condition and energy estimates.

        Our main results are the following.

        Theorem 1.1 Suppose 1 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq23_HTML.gif, then
        1. (i)

          β > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq24_HTML.gif, λ 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq25_HTML.gif such that if 0 < λ < λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq26_HTML.gif, problem (1) has a sequence of solutions { u m } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq27_HTML.gif with J ( u m ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq28_HTML.gif and J ( u m ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq29_HTML.gif as m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq30_HTML.gif.

           
        2. (ii)

          λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq31_HTML.gif, β 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq32_HTML.gif such that if 0 < β < β 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq33_HTML.gif, problem (1) has a sequence of solutions { u m } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq27_HTML.gif with J ( u m ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq28_HTML.gif and J ( u m ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq29_HTML.gif as m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq30_HTML.gif.

           

        2 Preliminary results

        Lemma 2.1[5]

        Suppose that Ω R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq1_HTML.gifis an open bounded domain with C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq34_HTML.gifboundary and 0 Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq35_HTML.gif, < a < ( n 2 ) / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq36_HTML.gif. The embedding D a 1 , 2 ( Ω ) L r ( Ω , | x | α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq37_HTML.gifis compact if 1 r < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq38_HTML.gif, α < ( 1 + a ) r + n ( 1 r 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq39_HTML.gif.

        Lemma 2.2 (Concentration compactness principle [5])

        Let < a < ( n 2 ) / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq36_HTML.gif, a b a + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq40_HTML.gif, 2 = 2 n / ( n 2 d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq41_HTML.gif, d = 1 + a b [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq42_HTML.gif, and M ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq43_HTML.gifbe the space of bounded measures on R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq44_HTML.gif. Suppose that { u m } D a 1 , 2 ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq45_HTML.gifis a sequence such that
        u m u in  D a 1 , 2 ( R n ) , μ m : = | | x | a D u m | 2 d x μ in  M ( R n ) , ν m : = | | x | b u m | 2 d x ν in  M ( R n ) , u m u a.e. on  R n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equd_HTML.gif
        Then there are the following statements:
        1. (1)
          There exists some at most countable set I, a family { x ( i ) : i I } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq46_HTML.gif of distinct points in R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq44_HTML.gif, and a family { ν ( i ) : i I } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq47_HTML.gif of positive numbers such that
          ν = | | x | b u | 2 d x + i I ν ( i ) δ x ( i ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ6_HTML.gif
          (6)
           
        where δ x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq48_HTML.gifis the Dirac-mass of mass 1 concentrated at x R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq49_HTML.gif.
        1. (2)
          The following inequality holds
          μ | | x | a D u | 2 d x + i I μ ( i ) δ x ( i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ7_HTML.gif
          (7)
           
        for some family { μ ( i ) > 0 : i I } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq50_HTML.gif satisfying
        S ( ν ( i ) ) 2 / 2 μ ( i ) for all  i I , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ8_HTML.gif
        (8)
        where S : = inf u D a 1 , 2 ( R n ) { 0 } E a , b ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq51_HTML.gifto be the best embedding constants, and
        E a , b ( u ) = R n | x | 2 a | D u | 2 d x ( R n | x | 2 b | u | 2 d x ) 2 / 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Eque_HTML.gif

        In particular, i I ( ν ( i ) ) 2 / 2 < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq52_HTML.gif.

        Lemma 2.3 Assume { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq53_HTML.gifis a (PS) c sequence with c < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq54_HTML.gif, 1 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq23_HTML.gif, then
        1. (1)

          λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq31_HTML.gif, there exists β 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq55_HTML.gif such that for any 0 < β < β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq56_HTML.gif, { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq53_HTML.gif has a convergent subsequence in D a 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq57_HTML.gif.

           
        2. (2)

          β > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq24_HTML.gif, there exists λ 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq58_HTML.gif such that for any 0 < λ < λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq59_HTML.gif, { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq53_HTML.gif has a convergent subsequence in D a 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq57_HTML.gif.

           

        Proof (1) The boundedness of (PS) c sequence.

        For { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq53_HTML.gif is a (PS) c sequence, then
        J ( u n ) = 1 2 Ω | x | 2 a | D u n | 2 d x λ 2 Ω | x | 2 b | u n | 2 d x β q Ω | x | α | u n | q d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ9_HTML.gif
        (9)
        J ( u n ) , u n = Ω | x | 2 a | D u n | 2 d x λ Ω | x | 2 b | u n | 2 d x β Ω | x | α | u n | q d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ10_HTML.gif
        (10)
        So, we get
        o ( 1 ) ( 1 + u n ) + | c | J ( u n ) 1 2 J ( u n ) , u n = d n u n 2 ( 1 q 1 2 ) β Ω | u n | q | x | α d x d n u n 2 ( 1 q 1 2 ) β C α u n q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equf_HTML.gif
        We have the boundedness of { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq53_HTML.gif for 1 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq23_HTML.gif, then there exists a subsequence, we still denote it by { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq53_HTML.gif, such that
        u n u in  D a 1 , 2 ( Ω ) , u n u in  L 2 ( Ω , | x | 2 b ) , u n u in  L r ( Ω , | x | α ) , 1 r < 2 n n 2 , α < ( 1 + a ) r + n ( 1 r 2 ) , u n u a.e. on  Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equg_HTML.gif
        From the concentration compactness principle, there exist nonnegative measures μ, ν and a countable family { x i } Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq60_HTML.gif such that
        | x | 2 b | u n | 2 d x ν = | x | 2 b | u | 2 d x + i I ν ( i ) δ x ( i ) , | x | 2 a | D u n | 2 d x μ | x | 2 a | D u | 2 d x + S i I ( ν ( i ) ) 2 / 2 δ x ( i ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equh_HTML.gif
        1. (2)

          Up to a subsequence, u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq61_HTML.gif in L 2 ( Ω , | x | 2 b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq62_HTML.gif.

           
        Since { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq53_HTML.gif is bounded in D a 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq13_HTML.gif, we may suppose, without loss of generality, that there exists T ( L 2 ( Ω , | x | 2 a ) ) n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq63_HTML.gif such that
        D u n T in  ( L 2 ( Ω , | x | 2 a ) ) n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equi_HTML.gif
        On the other hand, | u n | 2 2 u n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq64_HTML.gif is also bounded in L 2 ( Ω , | x | 2 b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq65_HTML.gif and
        | u n | 2 2 u n | u | 2 2 u in  L 2 ( Ω , | x | 2 b ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equj_HTML.gif
        Note that
        o ( 1 ) φ = J ( u n ) , φ = Ω | x | 2 a D u n D φ d x λ Ω | x | 2 b | u n | 2 2 u n φ d x β Ω | x | α | u n | q 2 u n φ d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ11_HTML.gif
        (11)
        taking n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq66_HTML.gif in (11), we have
        Ω | x | 2 a T D φ d x = λ Ω | x | 2 b | u | 2 2 u φ d x + β Ω | x | α | u | q 2 u φ d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ12_HTML.gif
        (12)
        for any φ D a 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq67_HTML.gif. Let φ = ψ u n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq68_HTML.gif in (12), where ψ C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq69_HTML.gif, then it follows that
        Ω | x | 2 a D u n u n D ψ d x + Ω | x | 2 a | D u n | 2 ψ d x = λ Ω | x | 2 b | u n | 2 ψ d x + β Ω | x | α | u n | q ψ d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ13_HTML.gif
        (13)
        Taking n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq66_HTML.gif in (13), we have
        Ω | x | 2 a u T D ψ d x + Ω ψ d μ = λ Ω ψ d ν + β Ω | x | α | u | q ψ d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ14_HTML.gif
        (14)
        Let φ = ψ u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq70_HTML.gif in (12), then it follows that
        Ω | x | 2 a T ψ u d x + Ω | x | 2 a T u d ψ = λ Ω | x | 2 b | u | 2 ψ d x + β Ω | x | α | u | q ψ d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ15_HTML.gif
        (15)
        Thus, it implies that
        Ω ψ d μ = λ i I ν i ψ ( x i ) + Ω | x | 2 a T D u ψ d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equ16_HTML.gif
        (16)
        which implies that
        S ( ν i ) 2 / 2 μ i = λ ν i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equk_HTML.gif

        Hence, ν i ( λ 1 S ) n / 2 d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq71_HTML.gif if ν i 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq72_HTML.gif.

        On the other hand,
        0 > c = lim n ( J ( u n ) 1 2 J ( u n ) , u n ) = lim n ( d n u n 2 β ( 1 q 1 2 ) Ω | x | α | u n | q d x ) d n u 2 β C u q , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equl_HTML.gif
        then u q C β q / ( 2 q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq73_HTML.gif, so that
        0 > c = lim n ( J ( u n ) 1 2 J ( u n ) , u n ) = lim n ( d n u n 2 β ( 1 q 1 2 ) Ω | x | α | u n | q d x ) d n μ i β C β q / ( 2 q ) d n S n 2 d ( λ 1 ) n 2 d 2 d C β 2 2 q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equm_HTML.gif

        However, if β > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq9_HTML.gif is given, we can choose λ 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq58_HTML.gif so small that for every 0 < λ < λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq59_HTML.gif, the last term on the right-hand side above is greater than 0, which is a contradiction. Similarly, if λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq10_HTML.gif is given, we can take β 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq55_HTML.gif so small that for every 0 < β < β 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq56_HTML.gif, the last term on the right-hand side above is greater than 0. Then ν i = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq74_HTML.gif for each i.

        Up to now, we have shown that
        lim n Ω | x | 2 b | u n | 2 d x = Ω | x | 2 b | u | 2 d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equn_HTML.gif
        So, by the Breiz-Lieb lemma,
        o ( 1 ) u n = u n 2 λ Ω | x | 2 b | u n | 2 d x β Ω | x | α | u n | q d x = u n u 2 u 2 λ Ω | x | 2 b | u | 2 d x β Ω | x | α | u | q d x = u n u 2 + o ( 1 ) u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equo_HTML.gif

        since J ( u ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq75_HTML.gif. Thus, we prove that { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq53_HTML.gif strongly converges to u in D a 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq13_HTML.gif. □

        3 Existence of infinitely many solutions

        In this section, we use the minimax procedure to prove the existence of infinitely many solutions. Let Σ be the class of subsets of D a 1 , 2 ( Ω ) { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq76_HTML.gif, which are closed and symmetric with respect to the origin. For A Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq77_HTML.gif, we define the genus γ ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq78_HTML.gif by
        γ ( A ) = min { k N : ϕ C ( A , R k { 0 } ) , ϕ ( x ) = ϕ ( x ) } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equp_HTML.gif
        Assume that 1 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq23_HTML.gif, then we obtain
        J ( u ) = 1 2 Ω | x | 2 a | D u | 2 d x λ 2 Ω | x | 2 b | u | 2 d x λ q Ω | x | α | u | q d x 1 2 u 2 C b λ 2 u 2 β C α q u q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equq_HTML.gif
        Define
        h ( t ) = 1 2 t 2 λ C 1 t 2 β C 2 t q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equr_HTML.gif
        Then, given β > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq9_HTML.gif, there exists λ 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq79_HTML.gif so small that for every 0 < λ < λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq80_HTML.gif, there exists 0 < T 0 < T 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq81_HTML.gif such that h ( t ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq82_HTML.gif for 0 < t < T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq83_HTML.gif, h ( t ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq84_HTML.gif for T 0 < t < T 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq85_HTML.gif, h ( t ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq82_HTML.gif for t > T 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq86_HTML.gif. Similarly, given λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq10_HTML.gif, we can choose β 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq87_HTML.gif with the property that T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq88_HTML.gif, T 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq89_HTML.gif as above exist for each 0 < β < β 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq90_HTML.gif. Clearly, h ( T 0 ) = h ( T 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq91_HTML.gif. Following the same idea as in [68], we consider the truncated functional
        J ˜ ( u ) = 1 2 Ω | x | 2 a | D u | 2 d x λ 2 ψ ( u ) Ω | x | 2 b | u | 2 d x λ q Ω | x | α | u | q d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equs_HTML.gif

        where ψ ( u ) = τ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq92_HTML.gif, and τ : R + [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq93_HTML.gif is a nonincreasing C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq94_HTML.gif function such that τ ( t ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq95_HTML.gif if t T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq96_HTML.gif and τ ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq97_HTML.gif if t T 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq98_HTML.gif. The main properties of J ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq99_HTML.gif are the following.

        Lemma 3.1
        1. (1)

          J ˜ C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq100_HTML.gif and J ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq99_HTML.gif is bounded below.

           
        2. (2)

          If J ˜ ( u ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq101_HTML.gif, then u T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq102_HTML.gif and J ˜ ( u ) = J ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq103_HTML.gif.

           
        3. (3)

          For any λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq10_HTML.gif, there exists β 0 = min { β 1 , β 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq104_HTML.gif such that if 0 < β < β 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq33_HTML.gif and c < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq54_HTML.gif, then J ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq99_HTML.gif satisfies (PS) c condition.

           
        4. (4)

          for any β > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq9_HTML.gif, there exists λ 0 = min { λ 1 , λ 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq105_HTML.gif such that if 0 < λ < λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq26_HTML.gif and c < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq54_HTML.gif, then J ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq99_HTML.gif satisfies (PS) c condition.

           

        Proof (1) and (2) are immediate. To prove (3) and (4), observe that all (PS) c sequences for J ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq99_HTML.gif with c < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq54_HTML.gif must be bounded. Similar to the proof of Lemma 2.3, there exists a convergent subsequence. □

        Lemma 3.2 Given m N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq106_HTML.gif, there is ε m < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq107_HTML.gifsuch that
        γ ( { u D a 1 , 2 ( Ω ) : J ˜ ( u ) ε m } ) m . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equt_HTML.gif
        Proof Fix m and let H m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq108_HTML.gif be an m-dimensional subspace of D a 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq13_HTML.gif. Take u H m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq109_HTML.gif, u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq110_HTML.gif, write u = r m v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq111_HTML.gif with v H m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq112_HTML.gif, v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq113_HTML.gif and r m = u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq114_HTML.gif. Thus, for 0 < r m < T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq115_HTML.gif, since all the norms are equivalent, we have
        J ˜ ( u ) = J ( u ) = 1 2 Ω | x | 2 a | D u | 2 d x λ 2 Ω | x | 2 b | u | 2 d x λ q Ω | x | α | u | q d x 1 2 u 2 λ C 1 2 u 2 λ C 2 q u q = 1 2 r m 2 λ C 1 2 r m 2 λ C 2 q r m q : = ε m . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equu_HTML.gif
        Therefore, we can choose r m ( 0 , T 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq116_HTML.gif so small that J ˜ ( u ) ε m < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq117_HTML.gif. Let S r m = { u D a 1 , 2 ( Ω ) : u = r m } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq118_HTML.gif, then S r m H m J ˜ ε m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq119_HTML.gif. Hence, γ ( J ˜ ε m ) γ ( S r m H m ) = m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq120_HTML.gif. Denote Γ m = { A Σ : γ ( A ) m } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq121_HTML.gif and let
        c m = inf A Γ m sup u A J ˜ ( u ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equv_HTML.gif

        Then < c m ε m < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq122_HTML.gif because J ˜ ε m Γ m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq123_HTML.gif and J ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq99_HTML.gif is bounded from below. □

        Lemma 3.3 Let λ, β be as in (3) or (4) of Lemma  3.1. Then all c m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq124_HTML.gifare critical values of J ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq99_HTML.gifas c m 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq125_HTML.gif.

        Proof It is clear that c m c m + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq126_HTML.gif, c m < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq127_HTML.gif. Hence, c m c ¯ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq128_HTML.gif. Moreover, since all c m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq124_HTML.gif are critical values of J ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq99_HTML.gif, we claim that c ¯ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq129_HTML.gif. If c ¯ < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq130_HTML.gif, because K c ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq131_HTML.gif is compact and K c ¯ Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq132_HTML.gif, it follows that γ ( K c ¯ ) = N 0 < + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq133_HTML.gif and there exists δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq134_HTML.gif such that γ ( K c ¯ ) = γ ( N δ ( K c ¯ ) ) = N 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq135_HTML.gif. By the deformation lemma there exist ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq136_HTML.gif ( c ¯ + ε < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq137_HTML.gif) and an odd homeomorphism η such that
        η ( J ˜ c ¯ + ε N δ ( K c ¯ ) ) J ˜ c ¯ ε . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equw_HTML.gif
        Since c m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq124_HTML.gif is increasing and converges to c ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq138_HTML.gif, there exists m N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq106_HTML.gif such that c m > c ¯ ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq139_HTML.gif and c m + N 0 c ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq140_HTML.gif and there exists A Γ m + N 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq141_HTML.gif such that sup u A J ˜ ( u ) < c ¯ + ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq142_HTML.gif. By the properties of γ, we have
        γ ( A N δ ( K c ¯ ) ¯ ) γ ( A ) γ ( N δ ( K c ¯ ) ) m , γ ( A N δ ( K c ¯ ) ¯ ) m . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equx_HTML.gif
        Therefore,
        η ( A N δ ( K c ¯ ) ¯ ) Γ m . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equy_HTML.gif
        Consequently,
        sup u η ( A N δ ( K c ¯ ) ¯ ) J ˜ ( u ) c m > c ¯ ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_Equz_HTML.gif

        a contradiction, hence c m 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-137/MediaObjects/13661_2012_Article_222_IEq125_HTML.gif. □

        With Lemma 3.1 to Lemma 3.3, we have proved Theorem 1.1.

        Declarations

        Acknowledgements

        Project is supported by National Natural Science Foundation of China, Tian Yuan Special Foundation (No. 11226116), the China Scholarship Council, Natural Science Foundation of Jiangsu Province of China for Young Scholar (No. BK2012109), the Fundamental Research Funds for the Central Universities (No. JUSRP11118, JUSRP211A22) and Foundation for young teachers of Jiangnan University (No. 2008LQN008).

        Authors’ Affiliations

        (1)
        School of Science, Jiangnan University
        (2)
        School of Mathematical Science, Nanjing Normal University
        (3)
        Institute of Science, PLA University of Science and Technology

        References

        1. Caffarelli I, Kohn R, Nirenberg L: First order interpolation inequalities with weights. Compos. Math. 1984, 53: 259–275.MathSciNet
        2. Xuan B, Su S, Yan Y: Existence results for Brezis-Nirenberg problems with Hardy potential and singular coefficients. Nonlinear Anal. 2007, 67: 2091–2106. 10.1016/j.na.2006.09.018MathSciNetView Article
        3. Bliss G: An integral inequality. J. Lond. Math. Soc. 1930, 5: 40–46. 10.1112/jlms/s1-5.1.40MathSciNetView Article
        4. He XM, Zou WM: Multiple solutions for the Brezis-Nirenberg problem with a Hardy potential and singular coefficients. Comput. Math. Appl. 2008, 56: 1025–1031. 10.1016/j.camwa.2008.01.029MathSciNetView Article
        5. Xuan B: The solvability of quasilinear Brezis-Nirenberg-type problems with singular weights. Nonlinear Anal. 2005, 62: 703–725. 10.1016/j.na.2005.03.095MathSciNetView Article
        6. Bernis F, Garcia-Azorero J, Peral I: Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth-order. Adv. Differ. Equ. 1996, 1: 219–240.MathSciNet
        7. Wang YJ, Shen YT: Multiple and sign-changing solutions for a class of semilinear biharmonic equation. J. Differ. Equ. 2009, 246: 3109–3125. 10.1016/j.jde.2009.02.016View Article
        8. Wang YJ, Yin YM, Yao YT: Multiple solutions for quasilinear Schrödinger equations involving critical exponent. Appl. Math. Comput. 2010, 216: 849–856. 10.1016/j.amc.2010.01.091MathSciNetView Article

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        © Yang et al.; licensee Springer 2012

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