Existence of solutions for semilinear elliptic equations on R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq1_HTML.gif

  • Ruichang Pei1, 2Email author and

    Affiliated with

    • Jihui Zhang1

      Affiliated with

      Boundary Value Problems20132013:163

      DOI: 10.1186/1687-2770-2013-163

      Received: 28 March 2013

      Accepted: 8 June 2013

      Published: 9 July 2013

      Abstract

      In this paper, the existence of at least one nontrivial solution for a class of semilinear elliptic equations on R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq1_HTML.gif is established by using the linking methods.

      Keywords

      Schrödinger equation subcritical exponent local linking

      1 Introduction

      In this paper we consider the question of the existence of solutions for a class of semilinear equations of the form
      ( P λ ) u + λ u = g ( x , u ) , x R N , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equa_HTML.gif
      where λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq2_HTML.gif is a parameter and the nonlinearity g C ( R N × R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq3_HTML.gif is asymptotically linear, i.e.,
      lim | t | g ( x , t ) t = V ( x ) , lim | x | V ( x ) = v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equ1_HTML.gif
      (1.1)
      for some V ( x ) C ( R N , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq4_HTML.gif and v R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq5_HTML.gif. In case this equation is considered in a bounded domain Ω R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq6_HTML.gif (with, say, the Dirichlet boundary condition), there is a large amount of literature on existence and multiplicity results, with the case of resonance being of particular interest (see [13]). We recall that the problem is said to be at resonance if λ σ ( S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq7_HTML.gif, where σ ( S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq8_HTML.gif denotes the spectrum of S, the ‘asymptotic linearization’ of the problem. In other words, S : D ( S ) L 2 ( Ω ) L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq9_HTML.gif is the operator given by
      S u ( x ) = u ( x ) V ( x ) u ( x ) , D ( S ) = H 0 1 ( Ω ) H 2 ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equ2_HTML.gif
      (1.2)

      On the other hand, a systematic study of such asymptotically linear problems set in unbounded domains or the whole space R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq10_HTML.gif is more recent and presents a number of mathematical difficulties (see [4, 5]). As an example, we note that in the case of problem ( P λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq11_HTML.gif), the asymptotic linearization operator S (now defined on D ( S ) = H 2 ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq12_HTML.gif) has a much more complicated spectrum (including an essential part [ v , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq13_HTML.gif), which in turn makes the study of this problem more challenging. In [4], motivated by the paper [5], Tehrani and Costa studied the existence of positive solutions to ( P λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq11_HTML.gif) by using the mountain pass theorem if g ( x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq14_HTML.gif satisfies some strong asymptotically linear conditions. Comparing with previous paper [4], in [6], Tehrani obtained the existence of a (possibly sign-changing) solution for problem ( P λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq11_HTML.gif) under essentially condition (1.1) only. In fact, he proved the following.

      Theorem 1.0 [6]

      Let g 0 ( x , s ) : = g ( x , s ) V ( x ) s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq15_HTML.gif and assume that
      1. (G)
        for every ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq16_HTML.gif, there exists 0 b ϵ ( x ) L 2 ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq17_HTML.gif such that
        | g 0 ( x , s ) | b ϵ ( x ) + ϵ | s | a.e. x R N , s R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equb_HTML.gif
         

      If Λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq18_HTML.gif or max { 0 , v } < λ < Λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq19_HTML.gif and λ σ p ( S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq20_HTML.gif, then ( P λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq11_HTML.gif) has a solution in H 1 ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq21_HTML.gif.

      Now, one naturally asks: Are there nontrivial solutions for problem ( P λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq11_HTML.gif) if λ σ ( S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq7_HTML.gif in the above theorem? Obviously, this case is resonance. But, this problem is not easy because we face the difficulties of verifying that the energy functional satisfies the (PS) condition if we still follow the idea of [6]. Here, there is still an interesting problem: Are there nontrivial solutions for problem ( P λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq11_HTML.gif) if λ σ ( S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq7_HTML.gif and g 0 ( x , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq22_HTML.gif (in Theorem 1.0) is more generalized superlinear? We will answer the above problems affirmatively by using Li and Willem’s local linking methods (see [7]).

      Next, we recall a few basic facts in the theory of Schrödinger operators which are relevant to our discussion (see [6]).
      1. 1.

        Since lim | x | V ( x ) = v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq23_HTML.gif, one has σ ess ( S ) = [ v , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq24_HTML.gif.

         
      2. 2.
        The bottom of the spectrum σ ( S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq8_HTML.gif of the operator S is given by
        Λ = λ 0 = inf 0 u H 2 ( R N ) | u | 2 V ( x ) u 2 u 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equc_HTML.gif
         
      Therefore we clearly have Λ v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq25_HTML.gif. If Λ < v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq26_HTML.gif, then by using the concentration compactness principle of Lions, one shows that Λ is the principle eigenvalue of S with a positive eigenfunction Φ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq27_HTML.gif:
      S Φ 0 = λ 0 Φ 0 , Φ 0 H 2 ( R N ) , Φ 0 > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equd_HTML.gif
      1. 3.
        The spectrum of S in ( , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq28_HTML.gif, namely σ ( S ) ( , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq29_HTML.gif, is at most a countable set, which we denote by
        Λ = λ 0 < λ 1 < λ 2 < , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Eque_HTML.gif
         

      where each λ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq30_HTML.gif is an isolated eigenvalue of S of the finite multiplicity. Let E λ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq31_HTML.gifdenote the eigenspace of S corresponding to the eigenvalue λ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq32_HTML.gif.

      Now, we state our main results. In this paper, we always assume that lim | x | V ( x ) = v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq33_HTML.gif and v < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq34_HTML.gif. The conditions imposed on g 0 ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq35_HTML.gif (see Theorem 1.0) are as follows:

      (H1) g 0 C ( R N × R , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq36_HTML.gif, and there are constants C 1 , C 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq37_HTML.gif such that
      | g 0 ( x , t ) | C 1 + C 2 | t | s 1 , x R N , t R , s ( 2 , p ) ( N 3 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equf_HTML.gif

      where p = 2 N N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq38_HTML.gif;

      (H2) g 0 ( x , t ) = ( | t | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq39_HTML.gif, | t | 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq40_HTML.gif, uniformly on R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq10_HTML.gif;

      (H3) lim | t | g 0 ( x , t ) t = + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq41_HTML.gif uniformly on R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq10_HTML.gif;

      (H4) There is a constant θ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq42_HTML.gif such that for all ( x , t ) R N × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq43_HTML.gif and s [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq44_HTML.gif,
      θ ( g 0 ( x , t ) t 2 G 0 ( x , t ) ) ( s g 0 ( x , s t ) t 2 G 0 ( x , s t ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equg_HTML.gif

      where G 0 ( x , t ) = 0 t g 0 ( x , s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq45_HTML.gif;

      (H5) For some δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq46_HTML.gif, either
      G 0 ( x , t ) 0 for  | t | δ , x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equh_HTML.gif
      or
      G 0 ( x , t ) 0 for  | t | δ , x R N ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equi_HTML.gif

      (H6) lim | x | sup | t | r g 0 ( x , t ) | t | = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq47_HTML.gif for every r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq48_HTML.gif.

      Theorem 1.1 Assume that conditions (H1)-(H4) hold. Ifλ is an eigenvalue of S ( λ < v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq49_HTML.gif, assume also that (H5) and (H6) hold. Then the problem ( P λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq11_HTML.gif) has at least one nontrivial solution.

      Remark 1.1 It follows from the condition (H3) that our nonlinearity g 0 ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq35_HTML.gif does not satisfy the classical condition of Ambrosetti and Rabinowitz:

      (AR) There is μ > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq50_HTML.gif such that 0 < μ G 0 ( x , u ) u g 0 ( x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq51_HTML.gif for all x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq52_HTML.gif and u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq53_HTML.gif.

      In recent years, there have been some papers devoted to replacing (AR) with more natural conditions (see [810]). But our methods are different from the references therein.

      We also consider asymptotically quadratic functions. We assume that:

      (H7) For every ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq16_HTML.gif, there exists 0 b ϵ ( x ) L 2 ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq54_HTML.gif such that
      | g 0 ( x , s ) | b ϵ ( x ) + ϵ | s | a.e.  x R N , s R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equj_HTML.gif

      and λ k < λ < λ k + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq55_HTML.gif.

      Theorem 1.2 Assume that conditions (H2), (H6), (H7) and one of the following conditions hold:

      (A1) λ j < 0 < λ j + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq56_HTML.gif, j k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq57_HTML.gif;

      (A2) λ j = 0 < λ j + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq58_HTML.gif, j k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq57_HTML.gif for some δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq46_HTML.gif,
      G 0 ( x , u ) 0 for | u | > δ , x R N ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equk_HTML.gif
      (A3) λ j < 0 = λ j + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq59_HTML.gif, j k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq57_HTML.gif for some δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq46_HTML.gif,
      G 0 ( x , u ) 0 for | u | δ , x R N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equl_HTML.gif

      Then problem ( P λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq11_HTML.gif) has at least one nontrivial solution.

      2 Preliminaries

      Let X be a Banach space with a direct sum decomposition
      X = X 1 X 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equm_HTML.gif
      Consider two sequences of subspaces
      X 0 1 X 1 1 X 1 , X 0 2 X 1 2 X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equn_HTML.gif
      such that
      X j = n N X n j , j = 1 , 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equo_HTML.gif
      For every multi-index α = ( α 1 , α 2 ) N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq60_HTML.gif, let X α = X α 1 X α 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq61_HTML.gif. We know that
      α β α 1 β 1 , α 2 β 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equp_HTML.gif

      A sequence ( α n ) N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq62_HTML.gif is admissible if, for every α N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq63_HTML.gif, there is m N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq64_HTML.gif such that n m α n α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq65_HTML.gif. For every I : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq66_HTML.gif, we denote by I α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq67_HTML.gif the function I restricted X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq68_HTML.gif.

      Definition 2.1 Let I be locally Lipschitz on X and c R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq69_HTML.gif. The functional I satisfies the ( C ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq70_HTML.gif condition if every sequence ( u α n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq71_HTML.gif such that ( α n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq72_HTML.gif is admissible and
      u α n X α n , I ( u α n ) c , ( 1 + u α n ) I ( u α n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equq_HTML.gif

      contains a subsequence which converges to a critical point of I.

      Definition 2.2 Let I be locally Lipschitz on X and c R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq69_HTML.gif. The functional I satisfies the ( C ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq73_HTML.gif condition if every sequence ( u α n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq71_HTML.gif such that ( α n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq72_HTML.gif is admissible and
      u α n X α n , sup n I ( u α n ) c , ( 1 + u α n ) I ( u α n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equr_HTML.gif

      contains a subsequence which converges to a critical point of I.

      Remark 2.1 1. The ( C ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq73_HTML.gif condition implies the ( C ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq70_HTML.gif condition for every c R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq69_HTML.gif.
      1. 2.

        When the ( C ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq70_HTML.gif sequence is bounded, then the sequence is a ( PS ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq74_HTML.gif sequence (see [11]).

         
      2. 3.
        Without loss of generality, we assume that the norm in X satisfies
        u 1 + u 2 2 = u 1 2 + u 2 2 , u j X j , j = 1 , 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equs_HTML.gif
         
      Definition 2.3 Let X be a Banach space with a direct sum decomposition
      X = X 1 X 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equt_HTML.gif
      The function I C 1 ( X , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq75_HTML.gif has a local linking at 0, with respect to ( X 1 , X 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq76_HTML.gif if, for some r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq48_HTML.gif,
      I ( u ) 0 , u X 1 , u r , I ( u ) 0 , u X 2 , u r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equu_HTML.gif

      Lemma 2.1 (see [7])

      Suppose that I C 1 ( X , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq75_HTML.gif satisfies the following assumptions:

      (B1) I has a local linking at 0 and X 1 { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq77_HTML.gif;

      (B2) I satisfies ( PS ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq78_HTML.gif;

      (B3) I maps bounded sets into bounded sets;

      (B4) For every m N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq79_HTML.gif, I ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq80_HTML.gif, u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq81_HTML.gif, u X = X m 1 X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq82_HTML.gif. Then I has at least two critical points.

      Remark 2.2 Assume that I satisfies the ( C ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq70_HTML.gif condition. Then this theorem still holds.

      Let X be a real Hilbert space and let I C 1 ( X , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq75_HTML.gif. The gradient of I has the form
      I ( u ) = A u + B ( u ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equv_HTML.gif

      where A is a bounded self-adjoint operator, 0 is not the essential spectrum of A, and B is a nonlinear compact mapping.

      We assume that there exist an orthogonal decomposition,
      X = X 1 + X 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equw_HTML.gif
      and two sequences of finite-dimensional subspaces,
      X 0 1 X 1 1 X 1 1 X 1 , X 0 2 X 1 2 X 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equx_HTML.gif
      such that
      X j = n N X n j ¯ , j = 1 , 2 , A X n j X n j , j = 1 , 2 , n N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equy_HTML.gif
      For every multi-index α = ( α 1 , α 2 ) N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq60_HTML.gif, we denote by X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq68_HTML.gif the space
      X α 1 X α 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equz_HTML.gif

      by p α : X X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq83_HTML.gif the orthogonal projector onto X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq68_HTML.gif, and by M ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq84_HTML.gif the Morse index of a self-adjoint operator L.

      Lemma 2.2 (see [7])

      I satisfies the following assumptions:
      1. (i)

        I has a local linking at 0 with respect to ( X 1 , X 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq76_HTML.gif;

         
      2. (ii)
        There exists a compact self-adjoint operator B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq85_HTML.gif such that
        B ( u ) = B ( u ) + ( u ) , u ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equaa_HTML.gif
         
      3. (iii)

        A + B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq86_HTML.gif is invertible;

         
      4. (vi)
        For infinitely many multiple-indices α : = ( n , n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq87_HTML.gif,
        M ( ( A + P α B ) | X α ) dim X n 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equab_HTML.gif
         

      Then I has at least two critical points.

      3 The proof of main results

      Proof of Theorem 1.1 (1) We shall apply Lemma 2.1 to the functional
      I ( u ) = 1 2 ( | u | 2 V ( x ) | u | 2 ) + 1 2 λ u 2 Ω G 0 ( x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equac_HTML.gif
      defined on X = H 1 ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq88_HTML.gif. We consider only the case λ σ ( S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq7_HTML.gif, and
      G 0 ( x , u ) 0 for  | u | δ , x R N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equ3_HTML.gif
      (3.1)

      Then other case is similar and simple.

      Let X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq89_HTML.gif be a finite dimensional space spanned by the eigenfunctions corresponding to negative eigenvalues of S + λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq90_HTML.gif and let X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq91_HTML.gif be its orthogonal complement in X. Choose a Hilbertian basis e n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq92_HTML.gif ( n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq93_HTML.gif) for X and define
      X n 1 = span ( e 0 , e 1 , , e n ) , n N ; X n 2 = X 2 , n N ; X 1 = n N X n 1 ¯ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equad_HTML.gif
      By the condition (H1) and Sobolev inequalities, it is easy to see that the functional I belongs to C 1 ( X , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq94_HTML.gif and maps bounded sets to bounded sets.
      1. (2)

        We claim that I has a local linking at 0 with respect to ( X 1 , X 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq76_HTML.gif. Decompose X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq91_HTML.gif into V + W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq95_HTML.gif when V = E λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq96_HTML.gif, W = ( X 2 + V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq97_HTML.gif. Also, set u = v + w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq98_HTML.gif, u X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq99_HTML.gif, v V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq100_HTML.gif, w W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq101_HTML.gif.

         
      For the convenience of our proof, we state some facts for the norm of the whole space X. It is well known that there is an equivalent norm http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq102_HTML.gif on X = H 1 ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq88_HTML.gif such that
      ( | u | 2 V ( x ) | u | 2 ) = u 2 , u X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equae_HTML.gif
      and
      ( | u | 2 V ( x ) | u | 2 ) = u 2 , u W . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equaf_HTML.gif
      By the equivalence of norm in the finite-dimensional space, there exists C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq103_HTML.gif such that
      v C v , v V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equ4_HTML.gif
      (3.2)
      It follows from (H1) and (H2) that for any ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq16_HTML.gif, there exists C ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq104_HTML.gif such that
      | G 0 ( x , u ) | ϵ u 2 + C ϵ | u | s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equ5_HTML.gif
      (3.3)
      Hence, we obtain
      I ( u ) m u 2 + c u s + 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equag_HTML.gif
      where m > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq105_HTML.gif, c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq106_HTML.gif is a constant and hence, for r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq48_HTML.gif small enough,
      I ( u ) 0 , u X 2 , u X r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equah_HTML.gif
      Let u = v + w X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq107_HTML.gif be such that u X r 1 = δ 2 C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq108_HTML.gif and let
      A 1 = { x R N : | w ( x ) | δ 2 } , A 2 = R N A 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equai_HTML.gif
      From (3.2), we have
      | v ( x ) | v C v δ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equaj_HTML.gif
      for all u r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq109_HTML.gif and x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq110_HTML.gif. On the one hand, one has | u ( x ) | | v ( x ) | + | w ( x ) | v + δ 2 δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq111_HTML.gif for all x A 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq112_HTML.gif. Hence, from (H5), we obtain
      A 1 G 0 ( x , u ) d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equak_HTML.gif
      On the other hand, we have
      | u ( x ) | | v ( x ) | + | w ( x ) | δ 2 + | w ( x ) | 2 | w ( x ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equal_HTML.gif
      for all x A 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq113_HTML.gif. It follows from (3.3) that
      G 0 ( x , u ) ϵ u 2 + C ϵ | u | s + 1 4 ϵ w 2 + 2 s + 1 C ϵ | w | s + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equam_HTML.gif
      for all x A 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq113_HTML.gif and all u X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq114_HTML.gif with u r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq109_HTML.gif, which implies that
      G 0 ( x , u ) 4 ϵ A 2 w 2 d x + A 2 2 s + 1 C ϵ | w | s + 1 d x 4 ( C 3 ) 2 ϵ w 2 + ( 2 C 3 ) λ + 1 C ϵ w s + 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equan_HTML.gif
      where C 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq115_HTML.gif is a constant. Hence, there exist positive constants C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq116_HTML.gif, C 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq117_HTML.gif and C 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq118_HTML.gif such that
      I ( u ) = 1 2 w 2 A 2 G 0 ( x , u ) d x A 1 G 0 ( x , u ) d x C w 2 4 ( C 3 ) 2 ϵ w 2 ( 2 C 3 ) λ + 1 C ϵ w s + 1 A 1 G ( x , u ) d x C 4 w 2 C 5 w s + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equao_HTML.gif
      for all u X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq99_HTML.gif with u r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq109_HTML.gif, which implies that
      I ( u ) 0 , u X 1  with  u r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equap_HTML.gif
      for 0 < r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq119_HTML.gif small enough.
      1. (3)
        We claim that I satisfies ( C ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq120_HTML.gif. Consider a sequence ( u α n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq71_HTML.gif such that ( u α n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq71_HTML.gif is admissible and
        u α n X α n , I ( u α n ) c , ( 1 + u α n ) I ( u α n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equ6_HTML.gif
        (3.4)
         
      and
      lim n ( 1 2 g 0 ( x , u α n ) u α n G 0 ( x , u α n ) ) = c . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equ7_HTML.gif
      (3.5)
      Let w α n = u α n 1 u α n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq121_HTML.gif. Up to a subsequence, we have
      w α n w in  X , w α n w in  L loc 2 , w α n ( x ) w ( x ) a.e.  x R N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equaq_HTML.gif
      If w = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq122_HTML.gif, we choose a sequence { t n } [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq123_HTML.gif such that
      I ( t n u α n ) = max t [ 0 , 1 ] I ( t u α n ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equar_HTML.gif
      For any m > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq105_HTML.gif, let v α n = 2 m w α n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq124_HTML.gif. Now, we claim that
      lim n G 0 ( x , v α n ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equas_HTML.gif
      Let ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq16_HTML.gif; for r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq125_HTML.gif, then,
      | v α n | r G 0 ( x , v α n ) d x C 6 r p 2 | v α n | r | v α n | 2 d x C 7 r p 2 | v α n | 2 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equat_HTML.gif
      Since p < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq126_HTML.gif, we may fix r large enough such that
      | | v α n | r G 0 ( x , v α n ) d x | ϵ 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equau_HTML.gif
      for all n. Moreover, by (H6), there exists R > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq127_HTML.gif such that
      | | v α n | r G 0 ( x , v α n ) d x | | v α n | 2 2 sup | t | r , | x | R | G 0 ( x , t ) | t 2 ϵ 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equav_HTML.gif
      for all n. Finally, since v α n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq128_HTML.gif in L s ( B R ( 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq129_HTML.gif for s [ 2 , 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq130_HTML.gif, we can use (H1) again to derive
      | | v α n | r | x | R G 0 ( x , v α n ) d x | ϵ 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equaw_HTML.gif

      for n large enough. Combining the above three formulas, our claim holds.

      So, for n large enough, 2 m u α n 1 ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq131_HTML.gif, we have
      I ( t n u α n ) I ( v α n ) m ϵ m 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equ8_HTML.gif
      (3.6)

      where ϵ is a small enough constant.

      That is, I ( t n u α n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq132_HTML.gif. Now, I ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq133_HTML.gif, I ( u α n ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq134_HTML.gif, we know that t n [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq135_HTML.gif and
      ( | ( t n u α n ) | 2 V ( x ) t n 2 | u α n | 2 ) + λ t n 2 | u α n | 2 g 0 ( x , t n u α n ) t n u α n = t n d d t | t = t n I ( t u α n ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equ9_HTML.gif
      (3.7)
      Therefore, using (H4), we have
      1 2 g 0 ( x , u α n ) u α n G 0 ( x , u α n ) 1 θ ( 1 2 g 0 ( x , t n u α n ) t n u α n G 0 ( x , t n u α n ) ) + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equax_HTML.gif

      This contradicts (3.5).

      If w 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq136_HTML.gif, then the set = { x R N : w ( x ) 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq137_HTML.gif has a positive Lebesgue measure. For x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq138_HTML.gif, we have | u α n ( x ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq139_HTML.gif. Hence, by (H3), we have
      g 0 ( x , u α n ( x ) ) u α n ( x ) | u α n ( x ) | 2 | w α n ( x ) | 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equ10_HTML.gif
      (3.8)
      From (3.4), we obtain
      1 ( 1 ) ( w 0 + w = 0 ) g 0 ( x , u α n ( x ) ) u α n ( x ) | u α n ( x ) | 2 | w α n ( x ) | 2 d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equ11_HTML.gif
      (3.9)

      By (3.8), the right-hand side of (3.9) + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq140_HTML.gif. This is a contradiction.

      In any case, we obtain a contradiction. Therefore, { u α n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq141_HTML.gif is bounded.

      Next, we denote { u α n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq141_HTML.gif as { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq142_HTML.gif and prove { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq142_HTML.gif contains a convergent subsequence.

      In fact, we know that { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq142_HTML.gif is bounded in X. Passing to a subsequence, we may assume that u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq143_HTML.gif in X. In order to establish strong convergence, it suffices to show that
      u n u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equay_HTML.gif

      By the condition (H6) and I ( u n ) , u n u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq144_HTML.gif, we can similarly conclude it according to the above proof of our claim.

      Finally, we claim that for every m N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq79_HTML.gif,
      I ( u ) as  u , u X m 1 X 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equaz_HTML.gif
      By (H2) and (H3), there exist large enough M and some positive constant T such that
      G 0 ( x , t ) M t 2 , x R N , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equba_HTML.gif
      So, for any u X m 1 X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_IEq145_HTML.gif, we have
      I ( t u ) = 1 2 t 2 ( | u | 2 V ( x ) | u | 2 ) + t 2 2 λ u 2 G 0 ( x , t u ) 1 2 t 2 ( | u | 2 V ( x ) | u | 2 ) + t 2 2 λ u 2 M t 2 u 2 as  t + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-163/MediaObjects/13661_2013_Article_419_Equbb_HTML.gif

      Hence, our claim holds. □

      Proof of Theorem 1.2 We omit the proof which depends on Lemma 2.2 and is similar to the preceding one since our result is a variant of Ding Yanheng’s Theorem 1.2 (see [12]). □

      Declarations

      Acknowledgements

      The authors would like to thank the referees for valuable comments and suggestions in improving this paper. This work was supported by the National NSF (Grant No. 10671156) of China.

      Authors’ Affiliations

      (1)
      Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University
      (2)
      School of Mathematics and Statistics, Tianshui Normal University

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      © Pei and Zhang; licensee Springer. 2013

      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.