Existence result for semilinear elliptic systems involving critical exponents

  • S Khademloo1Email author,

    Affiliated with

    • M Farzinejad1 and

      Affiliated with

      • O Khazaee kohpar1

        Affiliated with

        Boundary Value Problems20122012:119

        DOI: 10.1186/1687-2770-2012-119

        Received: 14 July 2012

        Accepted: 4 October 2012

        Published: 24 October 2012

        Abstract

        In this paper we deal with the existence of a positive solution for a class of semilinear systems of multi-singular elliptic equations which involve Sobolev critical exponents. In fact, by the analytic techniques and variational methods, we prove that there exists at least one positive solution for the system.

        MSC: 35J60, 35B33.

        Keywords

        semilinear elliptic system nontrivial solution critical exponent variational method

        1 Introduction

        We consider the following elliptic system:
        { L u = σ α 2 u | u | α 2 | ν | β + η u | u | 2 2 + a 1 u + a 2 ν , Ω , L ν = σ β 2 ν | ν | β 2 | u | α + λ ν | ν | 2 2 + a 2 u + a 3 ν , Ω , u = ν = 0 , Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ1_HTML.gif
        (1.1)

        where Ω R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq1_HTML.gif ( N 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq2_HTML.gif) is a smooth bounded domain such that ξ i Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq3_HTML.gif, i = 1 , 2 , , k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq4_HTML.gif, k 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq5_HTML.gif, are different points, 0 μ i < μ ¯ : = ( N 2 2 ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq6_HTML.gif, L : = Δ i = 1 k μ i | x ξ i | 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq7_HTML.gif, η , λ , σ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq8_HTML.gif, a 1 , a 2 , a 3 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq9_HTML.gif, 1 < α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq10_HTML.gif, β < 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq11_HTML.gif, α + β = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq12_HTML.gif.

        We work in the product space H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq13_HTML.gif, where the space H : = H 0 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq14_HTML.gif is the completion of C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq15_HTML.gif with respect to the norm ( Ω | | 2 d x ) 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq16_HTML.gif.

        In resent years many publications [13] concerning semilinear elliptic equations involving singular points and the critical Sobolev exponent have appeared. Particularly in the last decade or so, many authors used the variational method and analytic techniques to study the existence of positive solutions of systems of the form of (1.1) or its variations; see, for example, [48].

        Before stating the main result, we clarify some terminology. Since our method is variational in nature, we need to define the energy functional of (1.1) on H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq13_HTML.gif
        J ( u , ν ) = 1 2 Ω ( | u | 2 + | ν | 2 i = 1 k μ i ( u 2 + ν 2 ) | x ξ i | 2 ) d x 1 2 Ω ( a 1 u 2 + 2 a 2 u ν + a 3 ν 2 ) d x σ σ 2 Ω | u | α | u | β d x η 2 Ω | u | 2 d x λ 2 Ω | ν | 2 d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equa_HTML.gif
        Then J ( u , ν ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq17_HTML.gif belongs to C 1 ( H × H , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq18_HTML.gif. A pair of functions ( u 0 , ν 0 ) H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq19_HTML.gif is said to be a solution of (1.1) if ( u 0 , ν 0 ) ( 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq20_HTML.gif, and for all ( φ , ϕ ) H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq21_HTML.gif, we have
        Ω ( u 0 φ + ν 0 ϕ i = 1 k μ i ( u 0 φ + ν 0 ϕ ) | x ξ i | 2 ( a 1 u 0 φ + a 2 φ ν 0 + a 2 ϕ u 0 + a 3 ν 0 ϕ ) σ ( α φ u 0 | u 0 | α 2 | ν 0 | β + β ϕ ν 0 | ν 0 | β 2 | u 0 | α ) η | u 0 | 2 2 u 0 φ λ | ν 0 | 2 2 ν 0 ϕ ) d x = J ( u 0 , ν 0 ) , ( φ , ϕ ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equb_HTML.gif
        Standard elliptic arguments show that
        u , ν C 2 ( Ω { ξ 1 , , ξ k } ) C 1 ( Ω ¯ { ξ 1 , , ξ k } ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equc_HTML.gif

        The following assumptions are needed:

        ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif) η + λ + σ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq23_HTML.gif, 0 μ 1 μ 2 μ k < μ ¯ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq24_HTML.gif and i = 1 k μ i < μ ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq25_HTML.gif, α + β > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq26_HTML.gif, α + β = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq12_HTML.gif, a 1 , a 2 , a 3 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq27_HTML.gif, a 1 a 3 a 2 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq28_HTML.gif,

        ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq29_HTML.gif) 0 < λ 1 λ 2 < Λ 1 ( μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq30_HTML.gif, where Λ 1 ( μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq31_HTML.gif is the first eigenvalue of L, λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq32_HTML.gif, λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq33_HTML.gif are the eigenvalues of the matrix A = ( a 1 a 2 a 2 a 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq34_HTML.gif.

        The quadratic from Q ( u , ν ) : = ( u , ν ) A ( u , ν ) T = a 1 u 2 + 2 a 2 u ν + a 3 ν 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq35_HTML.gif is positively defined and satisfies
        λ 1 ( u 2 + ν 2 ) a 1 u 2 + 2 a 2 u ν + a 3 ν 2 λ 2 ( u 2 + ν 2 ) u , ν H . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ2_HTML.gif
        (1.2)

        Our main results are as follows.

        Theorem 1.1 Suppose ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif) holds. Then for any solution ( u , v ) H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq36_HTML.gif of problem (1.1), there exists a positive constant C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq37_HTML.gif such that
        max { | u ( x ) | , | ν ( x ) | } C 1 | x ξ i | ( μ ¯ μ ¯ μ i ) , x B ρ 1 ( ξ i ) { ξ i } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equd_HTML.gif

        where ρ 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq38_HTML.gif and B ρ 1 ( ξ i ) Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq39_HTML.gif.

        Theorem 1.2 Suppose ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif) holds. Then for any positive solution ( u , v ) H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq36_HTML.gif of problem (1.1), there exists a positive constant C 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq40_HTML.gif such that B ρ 2 ( ξ i ) Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq41_HTML.gif and
        min { u ( x ) , ν ( x ) } C 2 | x ξ i | ( μ ¯ μ ¯ μ i ) , x B ρ 2 ( ξ i ) { ξ i } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Eque_HTML.gif

        where ρ 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq42_HTML.gif.

        Theorem 1.3 Suppose ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif), ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq29_HTML.gif) hold. Then the problem (1.1) has a positive solution.

        2 Preliminaries

        On H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq43_HTML.gif, we use the norm
        ( u , ν ) H × H = ( Ω ( | u | 2 + | ν | 2 ) d x ) 1 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equf_HTML.gif
        Using the Young inequality, the following best constant is well defined:
        S μ i : = inf u D 1 , 2 ( R N ) { 0 } R N ( | u | 2 μ i u 2 | x ξ i | 2 ) d x ( R N | u | 2 d x ) 2 / 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ3_HTML.gif
        (2.1)

        where D 1 , 2 ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq44_HTML.gif is the completion of C 0 ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq45_HTML.gif with respect to the norm ( R N | | 2 d x ) 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq46_HTML.gif.

        We infer that S μ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq47_HTML.gif is attained in R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq48_HTML.gif by the functions
        V μ i , ε ξ i ( x ) = ε 2 N 2 U μ i ( x ξ i ε ) , ε > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equg_HTML.gif
        where
        U μ i ( x ξ i ) = ( 2 N ( μ ¯ μ i ) μ ¯ ) μ ¯ / 2 | x ξ i | ( μ ¯ μ ¯ μ i ) ( 1 + | x ξ i | 2 μ ¯ μ i μ ¯ ) μ ¯ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equh_HTML.gif
        For all η , λ , σ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq8_HTML.gif, η + λ + σ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq23_HTML.gif, α + β > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq26_HTML.gif, α + β = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq12_HTML.gif, by the Young and Hardy-Sobolev inequalities, the following constant is well defined on D : = ( D 1 , 2 ( R N ) { 0 } ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq49_HTML.gif:
        S η , λ , σ ( μ i ) : = inf ( u , ν ) D R N ( | u | 2 + | ν | 2 μ i u 2 + ν 2 | x ξ i | 2 ) d x ( R N ( η | u | 2 + λ | ν | 2 + σ | u | α | ν | β ) d x ) 2 / 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ4_HTML.gif
        (2.2)
        Set
        u μ , ε ξ ( x ) = ψ ( x ) V μ , ε ξ ( x ) = ε 2 N 2 ψ ( x ) U μ ( x ξ ε ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equi_HTML.gif
        where ξ Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq50_HTML.gif, 0 μ < μ ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq51_HTML.gif, ψ C 0 ( B ρ ( ξ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq52_HTML.gif satisfies 0 ψ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq53_HTML.gif and ψ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq54_HTML.gif, x B ρ 2 ( ξ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq55_HTML.gif, for all ρ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq56_HTML.gif small. Then for any 0 < μ < μ ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq57_HTML.gif, by [9] we have the following estimates:
        Ω ( | u μ , ε ξ | 2 μ ( u μ , ε ξ ) 2 | x ξ | 2 ) d x = S μ N 2 + o ( ε 2 μ ¯ μ ) , Ω | u μ , ε ξ | 2 d x = S μ N 2 + o ( ε 2 N N 2 μ ¯ μ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equj_HTML.gif
        and for any a R N { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq58_HTML.gif,
        Ω | u μ , ε 0 | 2 | x + ξ | 2 d x = { ε 2 | ξ | 2 R N U μ 2 ( x ) d x + o ( ε 2 ) if  μ < μ ¯ 1 , C μ 2 ω N | ξ | 2 ε 2 | log ε | + o ( ε 2 ) if  μ = μ ¯ 1 , Ω | u μ , ε ξ | 2 d x = { o 1 ( ε 2 ) if  0 μ < μ ¯ 1 , o 1 ( ε 2 | log ε | ) if  μ = μ ¯ 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equk_HTML.gif

        where C μ = ( 4 N ( μ ¯ μ ) N 2 ) N 2 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq59_HTML.gif, ω N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq60_HTML.gif is the volume of the unit ball in R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq48_HTML.gif.

        3 Asymptotic behavior of solutions

        Proof of Theorem 1.1 Suppose ( u 0 , ν 0 ) H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq61_HTML.gif is a nontrivial solution to problem (1.1). For all 0 μ i μ ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq62_HTML.gif define
        u ( x ) = | x ξ i | γ u 0 ( x ) and ν ( x ) = | x ξ i | γ ν 0 ( x ) , where  γ = ( μ ¯ μ ¯ μ i ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equl_HTML.gif
        It is not difficult to verify that u , ν H 0 1 ( Ω , | x ξ i | 2 γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq63_HTML.gif and satisfy
        { div ( | x ξ i | 2 γ u ) = σ α 2 | x ξ i | 2 γ u | u | α 2 | ν | β + η | x ξ i | 2 γ u | u | 2 2 + a 1 | x ξ i | 2 γ u + a 2 | x ξ i | 2 γ ν + j = 1 j i k μ j | x ξ j | 2 | x ξ i | 2 γ u , div ( | x ξ i | 2 γ ν ) = σ β 2 | x ξ i | 2 γ ν | ν | β 2 | u | α + λ | x ξ i | 2 γ ν | ν | 2 2 + a 2 | x ξ i | 2 γ u + a 3 | x ξ i | 2 γ ν + j = 1 j i k μ j | x ξ j | 2 | x ξ i | 2 γ ν . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ5_HTML.gif
        (3.1)
        Let R > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq64_HTML.gif small enough such that B R ( ξ i ) Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq65_HTML.gif and ξ i B R ( ξ i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq66_HTML.gif for j i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq67_HTML.gif. Also, let φ i C 0 ( B R ( ξ i ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq68_HTML.gif be a cut-off function. Set
        u n : = min { | u | , n } ; ν n : = min { | ν | , n } ; ϕ 1 i : = φ i 2 u u n 2 ( s 1 ) , ϕ 2 i : = φ i 2 ν ν n 2 ( s 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equm_HTML.gif
        where s , n > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq69_HTML.gif. Multiplying the first equation of (3.1) by ϕ 1 i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq70_HTML.gif and the second one by ϕ 2 i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq71_HTML.gif respectively and integrating, we have
        Ω | x ξ i | 2 γ u ϕ 1 i = σ α 2 Ω | x ξ i | 2 γ u | u | α 2 | ν | β ϕ 1 i + η Ω | x ξ i | 2 γ × u | u | 2 2 ϕ 1 i + a 1 Ω | x ξ i | 2 γ u ϕ 1 i + a 2 Ω | x ξ i | 2 γ × ν ϕ 1 i + j = 1 j i k μ j | x ξ j | 2 | x ξ i | 2 γ u ϕ 1 i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equn_HTML.gif

        Note that ϕ 1 i = 2 φ i u u n 2 ( s 1 ) φ i + φ i 2 u n 2 ( s 1 ) u + 2 ( s 1 ) φ i 2 u n 2 ( s 1 ) u n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq72_HTML.gif.

        Then
        Ω | x ξ i | 2 γ u ϕ 1 i = 2 Ω | x ξ i | 2 γ φ i u u n 2 ( s 1 ) φ i u + Ω | x ξ i | 2 γ φ i 2 × u n 2 ( s 1 ) | u | 2 + 2 ( s 1 ) Ω | x ξ i | 2 γ φ i 2 u n 2 ( s 1 ) | u n | 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equo_HTML.gif
        By the Cauchy inequality and the Young inequality, we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ6_HTML.gif
        (3.2)

        The same result holds for Ω | x ξ i | 2 γ ϕ 2 i ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq73_HTML.gif.

        By letting ψ 1 ( x ) = φ i u u n ( s 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq74_HTML.gif, ψ 2 ( x ) = φ i ν ν n ( s 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq75_HTML.gif, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ7_HTML.gif
        (3.3)
        Using Caffarelli-Kohn-Nirenberg inequality [10], we infer that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ8_HTML.gif
        (3.4)
        Define
        ω ( x ) : = max { u ( x ) , ν ( x ) } , ω n ( x ) : = min { ω ( x ) , n } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equp_HTML.gif
        Then ω n ( x ) : = max { u n ( x ) , ν n ( x ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq76_HTML.gif. Now, from the Hölder inequality, we deduce that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ9_HTML.gif
        (3.5)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ10_HTML.gif
        (3.6)
        In the sequel, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ11_HTML.gif
        (3.7)
        By the choice of φ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq77_HTML.gif, we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ12_HTML.gif
        (3.8)
        So, from (3.4) to (3.8) it follows that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ13_HTML.gif
        (3.9)
        Take s = 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq78_HTML.gif and φ i ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq79_HTML.gif to be a constant near the zero. Letting n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq80_HTML.gif, we infer that ω L 2 2 2 ( B R ( ξ i ) , | x ξ i | 2 γ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq81_HTML.gif and so
        u , ν L 2 2 2 ( B R ( ξ i ) , | x ξ i | 2 γ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ14_HTML.gif
        (3.10)

        Suppose r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq82_HTML.gif is sufficiently small such that r + l < R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq83_HTML.gif and φ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq77_HTML.gif is a cut-off function with the properties | φ i | < 1 l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq84_HTML.gif and φ i ( x ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq85_HTML.gif in B r ( ξ i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq86_HTML.gif.

        Set t : = 2 2 2 ( 2 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq87_HTML.gif, δ : = 2 γ t 2 γ ( t 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq88_HTML.gif.

        Then we have the following results:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ15_HTML.gif
        (3.11)
        where we used the Hölder inequality. From (3.9) in combination with (3.11), it follows that
        ( B r + l ( ξ i ) | x ξ i | 2 γ | ω | 2 s ) 1 2 s C 1 2 s l 1 2 s ( B r + l ( ξ i ) | x ξ i | 2 γ | ω | p ¯ 0 s ) 1 p ¯ 0 s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ16_HTML.gif
        (3.12)

        where p ¯ 0 = 2 t t 1 < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq89_HTML.gif.

        Denote s = χ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq90_HTML.gif, χ = p p ¯ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq91_HTML.gif and l = ρ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq92_HTML.gif, j 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq93_HTML.gif, where χ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq94_HTML.gif, 2 γ < N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq95_HTML.gif and p ¯ 0 χ j = p χ j 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq96_HTML.gif. Using (3.12) recursively, we get
        ( B r ( ξ i ) | ω | 2 χ j ) 1 2 χ j r γ χ j ( B r ( ξ i ) | ω | 2 χ j | x ξ i | 2 γ ) 1 2 χ j r γ χ j C i = 1 j 1 2 χ j ρ i = 1 j i χ i ( B r + ρ ( ξ i ) | x ξ i | 2 γ | ω | 2 ) 1 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equq_HTML.gif
        we have χ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq97_HTML.gif as j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq98_HTML.gif. Note that the infinite sums on the right-hand side converge, then we obtain that ω L ( B r ( ξ i ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq99_HTML.gif, particularly, we have u , ν L ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq100_HTML.gif. Thus,
        u 0 ( x ) = | x ξ i | γ u ( x ) M 1 | x ξ i | γ for  x B r ( ξ i ) { ξ i } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equr_HTML.gif
        where M 1 = max { u L ( B r ( ξ i ) ) , 1 i k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq101_HTML.gif.
        ν 0 ( x ) = | x ξ i | γ ν ( x ) M 2 | x ξ i | γ for  x B r ( ξ i ) { ξ i } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equs_HTML.gif

        where M 1 = max { ν L ( B r ( ξ i ) ) , 1 i k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq102_HTML.gif. The proof is complete. □

        Proof of Theorem 1.2 Suppose ( u 0 , ν 0 ) H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq61_HTML.gif is a positive solution to problem (1.1). For all 0 μ i μ ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq62_HTML.gif, set
        u ( x ) = | x ξ i | γ u 0 ( x ) and ν ( x ) = | x ξ i | γ ν 0 ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equt_HTML.gif
        Then
        { div ( | x ξ i | 2 γ u ) = σ α 2 | x ξ i | 2 γ u α 1 ν β + η | x ξ i | 2 γ u 2 1 + a 1 | x ξ i | 2 γ u + a 2 | x ξ i | 2 γ ν + j = 1 j i k μ j | x ξ j | 2 | x ξ i | 2 γ u , div ( | x ξ i | 2 γ ν ) = σ β 2 | x ξ i | 2 γ ν β 1 u α + λ | x ξ i | 2 γ ν 2 1 + a 2 | x ξ i | 2 γ u + a 3 | x ξ i | 2 γ ν + j = 1 j i k μ j | x ξ j | 2 | x ξ i | 2 γ ν . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ17_HTML.gif
        (3.13)
        Choose 0 < ρ 0 < ρ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq103_HTML.gif and define n ( t ) = min | x ξ i | = t u ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq104_HTML.gif for ρ 0 t ρ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq105_HTML.gif. Let
        n ( ρ 0 ) = A ρ 0 2 μ ¯ μ i + B , n ( ρ ) = A ρ 2 μ ¯ μ i + B , where A = n ( ρ ) n ( ρ 0 ) ρ 2 μ ¯ μ i ρ 0 2 μ ¯ μ i , B = n ( ρ 0 ) ρ 2 μ ¯ μ i n ( ρ ) ρ 0 2 μ ¯ μ i ρ 2 μ ¯ μ i ρ 0 2 μ ¯ μ i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equu_HTML.gif
        It is easy to verify that
        div ( | x ξ i | 2 ( μ ¯ μ ¯ μ i ) ( A | x ξ i | 2 μ ¯ μ i + B ) ) = 0 x Ω { ξ i } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ18_HTML.gif
        (3.14)
        Combining (3.13) with (3.14), we get
        div ( | x ξ i | 2 ( μ ¯ μ ¯ μ i ) ( u A | x ξ i | 2 μ ¯ μ i + B ) ) 0 , x B ρ ( ξ i ) B ρ 0 ( ξ i ) , u ( x ) ( A | x ξ i | 2 μ ¯ μ i + B ) 0 , x ( B ρ ( ξ i ) B ρ 0 ( ξ i ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equv_HTML.gif
        Therefore, by the maximum principle in B ρ ( ξ i ) B ρ 0 ( ξ i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq106_HTML.gif, we obtain
        u ( x ) ( A | x ξ i | 2 μ ¯ μ i + B ) 0 , x B ρ ( ξ i ) B ρ 0 ( ξ i ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equw_HTML.gif
        Thus, for all x B ρ ( ξ i ) B ρ 0 ( ξ i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq107_HTML.gif,
        u ( x ) A | x ξ i | 2 μ ¯ μ i + B = | x ξ i | 2 μ ¯ μ i ρ 2 μ ¯ μ i ρ 0 2 μ ¯ μ i ρ 2 μ ¯ μ i n ( ρ 0 ) + ρ 0 2 μ ¯ μ i | x ξ i | 2 μ ¯ μ i ρ 0 2 μ ¯ μ i ρ 2 μ ¯ μ i n ( ρ ) | x ξ i | 2 μ ¯ μ i ρ 0 2 μ ¯ μ i | x ξ i | 2 μ ¯ μ i ρ 0 2 μ ¯ μ i ρ 2 μ ¯ μ i | x ξ i | 2 μ ¯ μ i n ( ρ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equx_HTML.gif

        Taking ρ 0 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq108_HTML.gif, we conclude u ( x ) n ( ρ ) = min | x ξ i | = ρ u ( x ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq109_HTML.gif for all x B ρ ( ξ i ) { ξ i } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq110_HTML.gif.

        Similar result also holds for ν ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq111_HTML.gif. Therefore, we have
        u 0 ( x ) = | x ξ i | γ u ( x ) | x ξ i | γ min | x ξ i | = ρ u ( x ) = | x ξ i | γ C i | x ξ i | γ min i = 1 , 2 , , k C i = | x ξ i | γ N 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equy_HTML.gif
        For any x B ρ ( ξ i ) { ξ i } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq110_HTML.gif,
        ν 0 ( x ) = | x ξ i | γ ν ( x ) | x ξ i | γ min | x ξ i | = ρ ν ( x ) = | x ξ i | γ C i ´ | x ξ i | γ min i = 1 , 2 , , k C i ´ = | x ξ i | γ N 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equz_HTML.gif

        For any x B ρ ( ξ i ) { ξ i } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq110_HTML.gif. This proves the theorem. □

        4 Local ( P S ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq112_HTML.gif-condition and the existence of positive solutions

        We first establish a compactness result.

        Lemma 4.1 Suppose that ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif) holds. Then J satisfies the ( P S ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq112_HTML.gif-condition for all
        c < c : = 1 N min { S η , λ , σ N 2 ( μ 1 ) , , S η , λ , σ N 2 ( μ k ) , ( S 0 ) N 2 } = 1 N S η , λ , σ N 2 ( μ k ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equaa_HTML.gif

        Proof Suppose that { ( u n , ν n ) } H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq113_HTML.gif satisfies J ( u n , ν n ) c < c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq114_HTML.gif and J ( u n , ν n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq115_HTML.gif. The standard argument shows that { ( u n , ν n ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq116_HTML.gif is bounded in H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq13_HTML.gif.

        For some ( u , ν ) H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq117_HTML.gif, we have
        ( u n , ν n ) ( u , ν ) weakly in  H × H , ( u n , ν n ) ( u , ν ) weakly in  L 2 ( Ω , | x ξ i | 2 ) × L 2 ( Ω , | x ξ i | 2 ) , ( u n , ν n ) ( u , ν ) weakly in  L 2 ( Ω ) × L 2 ( Ω ) , ( u n , ν n ) ( u , ν ) strongly in  L q 1 ( Ω ) × L q 2 ( Ω ) , q 1 , q 2 [ 1 , 2 ) , ( u n , ν n ) ( u , ν ) a.e. in  Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equab_HTML.gif
        Therefore, ( u , ν ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq118_HTML.gif is a solution to (1.1). Then by the concentration-compactness principle [1113] and up to a subsequence, there exist an at most countable set J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq119_HTML.gif, a set of different points { x j } j J Ω ξ i i = 1 k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq120_HTML.gif, nonnegative real numbers τ ˜ x j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq121_HTML.gif, ν ˜ x j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq122_HTML.gif, j J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq123_HTML.gif, and τ ˜ ξ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq124_HTML.gif, ν ˜ ξ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq125_HTML.gif, γ ˜ ξ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq126_HTML.gif ( 1 i k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq127_HTML.gif) such that the following convergence holds in the sense of measures:
        | u n | 2 + | ν n | 2 d τ ˜ | u | 2 + | ν | 2 + j J τ ˜ x j δ x j + i = 1 k τ ˜ ξ i δ ξ i , u n 2 + ν n 2 | x ξ i | 2 d γ ˜ = u 2 + ν 2 | x ξ i | 2 + γ ˜ ξ i δ ξ i , η | u n | 2 + λ | ν n | 2 + σ | u n | α | ν n | β d ν ˜ = η | u | 2 + λ | ν | 2 + σ | u | α | ν | β + j J ν ˜ x j δ x j + i = 1 k ν ˜ ξ i δ ξ i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equac_HTML.gif
        By the Sobolev inequalities [10], we have
        S μ i ν ˜ ξ i 2 2 τ ˜ ξ i μ i γ ˜ ξ i , 1 i k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ19_HTML.gif
        (4.1)

        We claim that J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq119_HTML.gif is finite, and for any j J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq123_HTML.gif, ν ˜ x j = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq128_HTML.gif or ν ˜ x j S 0 N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq129_HTML.gif.

        In fact, let ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq130_HTML.gif be small enough for any 1 i k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq127_HTML.gif, ξ i B ε ( x j ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq131_HTML.gif and B ε ( x i ) B ε ( x j ) = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq132_HTML.gif for i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq133_HTML.gif, i , j J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq134_HTML.gif. Let ϕ ε j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq135_HTML.gif be a smooth cut-off function centered at x j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq136_HTML.gif such that 0 ϕ ε j 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq137_HTML.gif, ϕ ε j = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq138_HTML.gif for | x x j | ε 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq139_HTML.gif, ϕ ε j = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq140_HTML.gif for | x x j | ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq141_HTML.gif and | ϕ ε j | 4 ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq142_HTML.gif. Then
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equad_HTML.gif
        Then we have
        0 = lim ε 0 lim n J λ ( u n , ν n ) , ( u n ϕ ε j , ν n ϕ ε j ) τ ˜ x j ν ˜ x j . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equae_HTML.gif

        By the Sobolev inequality, S 0 ν ˜ x j 2 2 τ ˜ x j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq143_HTML.gif; and then we deduce that ν ˜ x j = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq128_HTML.gif or ν ˜ x j S 0 N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq129_HTML.gif, which implies that J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq119_HTML.gif is finite.

        Now, we consider the possibility of concentration at points ξ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq144_HTML.gif ( 1 i k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq127_HTML.gif), for ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq130_HTML.gif small enough that x j B ε ( ξ i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq145_HTML.gif for all j J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq123_HTML.gif and B ε ( ξ i ) B ε ( ξ j ) = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq146_HTML.gif for i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq133_HTML.gif and 1 i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq147_HTML.gif, j k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq148_HTML.gif. Let φ ε i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq149_HTML.gif be a smooth cut-off function centered at ξ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq144_HTML.gif such that 0 φ ε i 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq150_HTML.gif, φ ε i = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq151_HTML.gif for | x ξ i | ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq152_HTML.gif and | φ ε i | 4 ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq153_HTML.gif. Then
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equaf_HTML.gif
        Thus, we have
        0 = lim ε 0 lim n J λ ( u n , ν n ) , ( u n φ ε i , ν n φ ε i ) τ ˜ ξ i μ i γ ˜ ξ i ν ˜ ξ i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ20_HTML.gif
        (4.2)
        From (4.1) and (4.2) we derive that S μ i ν ˜ ξ i 2 2 ν ˜ ξ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq154_HTML.gif, 1 i k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq127_HTML.gif, and then either ν ˜ ξ i = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq155_HTML.gif or ν ˜ ξ i S μ i N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq156_HTML.gif. On the other hand, from the above arguments, we conclude that
        c = lim n ( J ( u n , ν n ) 1 2 J ( u n , ν n ) , ( u n , ν n ) ) = 1 N lim n Ω ( η | u n | 2 + λ | ν n | 2 + σ | u n | α | ν n | β ) d x = 1 N ( Ω ( η | u | 2 + λ | ν | 2 + σ | u | α | ν | β ) d x + j J ν ˜ x j + i = 1 k ν ˜ ξ i ) = 1 N ( j J ν ˜ x j + i = 1 k ν ˜ ξ i ) + J ( u , ν ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equag_HTML.gif
        If ν ˜ ξ i = ν ˜ x j = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq157_HTML.gif for all i { 1 , , k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq158_HTML.gif and j J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq123_HTML.gif, then c = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq159_HTML.gif, which contradicts the assumption that c > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq160_HTML.gif. On the other hand, if there exists an i { 1 , , k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq158_HTML.gif such that ν ˜ ξ i 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq161_HTML.gif or there exists a j J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq123_HTML.gif with ν ˜ x j 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq162_HTML.gif, then we infer that
        c 1 N min { ( S η , λ , σ ( 0 ) ) N / 2 , ( S η , λ , σ ( μ 1 ) ) N / 2 , , ( S η , λ , σ ( μ k ) ) N / 2 } = 1 N ( S η , λ , σ ( μ k ) ) N / 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equah_HTML.gif

        which contradicts our assumptions. Hence, ( u n , ν n ) ( u , ν ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq163_HTML.gif, as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq164_HTML.gif in H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq43_HTML.gif. □

        First, under the assumptions ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif), ( H 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq29_HTML.gif), we have the following notations:
        f η , λ , σ ( τ ) = ( 1 + τ 2 ) S μ k ( η + σ τ β + λ τ 2 ) 2 2 , τ > 0 ; f η , λ , σ ( τ min ) : = min τ > 0 f η , λ , σ ( τ ) > 0 , σ > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equai_HTML.gif
        where τ min > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq165_HTML.gif is a minimal point of f η , λ , σ ( τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq166_HTML.gif, and therefore a root of the equation
        α σ τ β σ β τ β 2 2 λ τ 2 2 + 2 η = 0 , τ > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equaj_HTML.gif
        Lemma 4.2 Suppose that ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif) holds. Then we have
        1. (i)

          S η , λ , σ ( μ ) = f η , λ , σ ( τ min ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq167_HTML.gif

           
        2. (ii)

          S η , λ , σ ( μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq168_HTML.gif has the minimizers ( V μ , ε ξ ( x ) , τ min V μ , ε ξ ( x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq169_HTML.gif, ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq170_HTML.gif, where V μ , ε ξ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq171_HTML.gif are the extremal functions of S η , λ , σ ( μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq172_HTML.gif defined as in (2.2).

           

        Proof The argument is similar to that of [6]. □

        Lemma 4.3 Under the assumptions of ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif), we have
        sup J ( t u ε , μ k , t ( τ min u ε , μ k ) ) < c = 1 N ( S η , λ , σ ( μ k ) ) N / 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equak_HTML.gif
        Proof Suppose ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif) holds. Define the function
        g ( t ) : = J ( t u ε , μ k , t ( τ min u ε , μ k ) ) , t 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equal_HTML.gif
        Note that lim t + g ( t ) = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq173_HTML.gif and g ( t ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq174_HTML.gif as t is close to 0. Thus, sup t 0 g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq175_HTML.gif is attained at some finite t ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq176_HTML.gif with g ( t ε ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq177_HTML.gif. Furthermore, c < t ε < c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq178_HTML.gif, where c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq179_HTML.gif and c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq180_HTML.gif are the positive constants independent of ε. By using (1.2), we have
        g ( t ) t 2 2 ( 1 + τ min 2 ) ( Ω ( | u ε , μ k | 2 μ k u ε , μ k 2 | x ξ i | 2 λ 1 u ε , μ k 2 ) d x ) t 2 2 ( σ τ min β + η + λ τ min 2 ) Ω | u ε , μ k | 2 d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equam_HTML.gif
        Note that
        max ( t 2 2 B 1 t 2 2 B 2 ) = 1 N ( B 1 B 2 2 / 2 ) N / 2 , B 1 > 0 , B 2 > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ21_HTML.gif
        (4.3)

        and 0 μ μ ¯ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq181_HTML.gif and so 2 < 2 μ ¯ μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq182_HTML.gif.

        From (4.3), Lemma 4.2 and Lemma 4.3, it follows that
        g ( t ε ) 1 N ( ( 1 + τ min 2 ) Ω ( | u ε , μ k | 2 μ k u ε , μ k 2 | x ξ k | 2 λ 1 u ε , μ k 2 ) d x ( ( σ τ min β + η + λ τ min 2 ) Ω | u ε , μ k | 2 d x ) 2 / 2 ) N / 2 1 N ( f ( τ min ) S ( μ k ) × ( S ( μ k ) ) N / 2 + O ( ε 2 μ ¯ μ ) C ε 2 ( S ( μ k ) ) ( N 2 ) / 2 + O ( ε 2 μ ¯ μ ) ) 1 N ( f ( τ min ) ) N / 2 + O ( ε 2 μ ¯ μ ) C ε 2 < 1 N ( S η , λ , σ ( μ k ) ) N / 2 = c , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equan_HTML.gif
        so g ( t ε ) < c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq183_HTML.gif. Hence, g ( t ε ) < c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq183_HTML.gif, t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq184_HTML.gif and
        sup t 0 g ( t ) = sup t 0 J ( t u ε , μ k , t ( τ min u ε , μ k ) ) < c , if  μ < μ ¯ 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equ22_HTML.gif
        (4.4)

         □

        Proof of Theorem 1.3 Set c : = inf h Γ max t [ 0 , 1 ] J ( h ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq185_HTML.gif, where
        Γ = { h C ( [ 0 , 1 ] , H × H ) | h ( 0 ) = ( 0 , 0 ) , J ( h ( 1 ) ) < 0 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equao_HTML.gif
        Suppose that ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq22_HTML.gif) holds. For all ( u , ν ) H × H { ( 0 , 0 ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq186_HTML.gif, from the Young and Hardy-Sobolev inequalities, it follows that
        J ( u , ν ) C ( u 2 + ν 2 ) C ( u 2 + ν 2 ) C ( u , ν ) 2 C ( u , ν ) 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equap_HTML.gif
        and there exists a constant ρ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq56_HTML.gif small such that
        b : = inf ( u , ν ) = ρ J ( u , ν ) > 0 = J ( 0 , 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equaq_HTML.gif

        Since J ( t u , t ν ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq187_HTML.gif as t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq188_HTML.gif, there exists t 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq189_HTML.gif such that ( t 0 u , t 0 ν ) > ρ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq190_HTML.gif and J ( t 0 u , t 0 ν ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq191_HTML.gif. By the mountain-pass theorem [14], there exists a sequence { ( u n , ν n ) } H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq192_HTML.gif such that J ( u n , ν n ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq193_HTML.gif and J ( u n , ν n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq194_HTML.gif, as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq164_HTML.gif.

        From Lemma 4.2 it follows that
        0 < c sup t [ 0 , 1 ] J ( t t 0 u ε , μ k , t t 0 τ min u ε , μ k ) sup t 0 J ( t u ε , μ k , t τ min u ε , μ k ) < c . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_Equar_HTML.gif

        By Lemma 4.1 there exists a subsequence of { ( u n , ν n ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq116_HTML.gif, still denoted by { ( u n , ν n ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq116_HTML.gif, such that ( u n , ν n ) ( u , ν ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq195_HTML.gif strongly in H × H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq43_HTML.gif. Thus, we get a critical point ( u , ν ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq118_HTML.gif of J satisfying (1.1), and c is a critical value. Set u + = max { u , 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq196_HTML.gif.

        Replacing respectively u, ν with u + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq197_HTML.gif and ν + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq198_HTML.gif in terms of the right-hand side of (1.1) and repeating the above process, we can get a nonnegative nontrivial solution ( u , ν ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq118_HTML.gif of (1.1). If u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq199_HTML.gif, we get ν 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq200_HTML.gif by (1.1) and the assumption a 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq201_HTML.gif. Similarly, if ν 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq200_HTML.gif, we also have u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq199_HTML.gif. There, u , ν 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq202_HTML.gif. From the maximum principle, it follows that u , ν > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-119/MediaObjects/13661_2012_Article_243_IEq203_HTML.gif in Ω. □

        Declarations

        Authors’ Affiliations

        (1)
        Department of Basic Sciences, Babol University of Technology

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        © Khademloo et al.; licensee Springer 2012

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