A note on the existence of solutions for a class of quasilinear elliptic equations: an Orlicz-Sobolev space setting

  • Yang Yang1Email author and

    Affiliated with

    • Jihui Zhang2

      Affiliated with

      Boundary Value Problems20122012:136

      DOI: 10.1186/1687-2770-2012-136

      Received: 13 April 2012

      Accepted: 8 October 2012

      Published: 22 November 2012

      Abstract

      In this note, we study the existence and multiplicity of solutions for the quasilinear elliptic problem as follows:

      { div ( a ( | u | ) u ) = f ( x , u ) , in  Ω ; u = 0 , on  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equa_HTML.gif

      where Ω R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq1_HTML.gif is a bounded domain with a smooth boundary. The existence and multiplicity of solutions are obtained by a version of the symmetric mountain pass theorem.

      Keywords

      Orlicz-Sobolev spaces symmetric mountain pass theorem quasilinear elliptic equations

      1 Introduction

      In this note, we discuss the existence and multiplicity of solutions of the following boundary value problem:
      { div ( a ( | u | ) u ) = f ( x , u ) , in  Ω ; u = 0 , on  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equ1_HTML.gif
      (1.1)
      where Ω R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq1_HTML.gif is a bounded domain with a smooth boundary Ω. The function a is such that p : R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq2_HTML.gif defined by
      p ( t ) = { a ( | t | ) t , t 0 ; 0 , t = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equb_HTML.gif
      is an increasing homeomorphism from R onto itself and the continuous function f ( x , t ) C ( Ω ¯ × R , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq3_HTML.gif satisfies f ( x , 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq4_HTML.gif, x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq5_HTML.gif. Especially, when a ( t ) = | t | p 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq6_HTML.gif, the problem (1.1) is the well-known p-Laplacian equation. There is a large number of papers on the existence of solutions for the p-Laplacian equation. But the problem (1.1) possesses more complicated nonlinearities. For example, it is inhomogeneous and has an important physical background, e.g.,
      1. (a)

        nonlinear elasticity: P ( t ) = ( 1 + t 2 ) γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq7_HTML.gif, γ > 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq8_HTML.gif;

         
      2. (b)

        plasticity: P ( t ) = t α ( log ( 1 + t ) ) β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq9_HTML.gif, α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq10_HTML.gif, β > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq11_HTML.gif;

         
      3. (c)

        generalized Newtonian fluids: P ( t ) = 0 t s 1 α ( sinh 1 s ) β d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq12_HTML.gif, 0 α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq13_HTML.gif, β > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq11_HTML.gif.

         

      So, in the discussions, some special techniques are needed, and the problem (1.1) has been studied in an Orlicz-Sobolev space and received considerable attention in recent years; see, for instance, the papers [19]. In paper [9], Fang and Tan discussed the problem (1.1) under the conditions that f ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq14_HTML.gif was odd in t. They got the first result that when h + < p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq15_HTML.gif, and f ( x , t ) C t q 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq16_HTML.gif for 0 < t < δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq17_HTML.gif, q < p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq18_HTML.gif, the problem (1.1) had a sequence of solutions by genus theory. The second result is that when f ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq14_HTML.gif satisfies 0 < α F ( x , t ) t f ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq19_HTML.gif, x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq20_HTML.gif, t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq21_HTML.gif, α > p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq22_HTML.gif and f ( x , t ) = o ( p ( | t | ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq23_HTML.gif as | t | 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq24_HTML.gif, the problem (1.1) has infinitely many pairs of solutions which correspond to the positive critical values by the symmetric mountain pass theorem.

      Motivated by their results, in this note, we discuss the problem (1.1) when f ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq14_HTML.gif is still odd in t but it satisfies weaker conditions than [9]; and furthermore, we need not know the behaviors of f ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq14_HTML.gif near the zero. If h + > p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq25_HTML.gif, we can get multiplicity of solutions by a version of the symmetric mountain pass theorem.

      The paper is organized as follows. In Section 2, we present some preliminary knowledge on the Orlicz-Sobolev spaces and give the main result. In Section 3, we make the proof.

      2 Preliminaries

      Obviously, the problem (1.1) allows a nonhomogeneous function p in the differential operator defining the problem (1.1). To deal with this situation, we introduce an Orlicz-Sobolev space setting for the problem (1.1) as follows.

      Let
      P ( t ) = 0 t p ( s ) d s , P ˜ ( t ) = 0 t p 1 ( s ) d s , t R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equc_HTML.gif

      then P and P ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq26_HTML.gif are complementary N-functions (see [10]), which define the Orlicz spaces L P : = L P ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq27_HTML.gif and L P ˜ : = L P ˜ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq28_HTML.gif respectively.

      Throughout this paper, we assume the following condition on P:
      ( p ) 1 < p : = inf t > 0 t p ( t ) P ( t ) p + : = sup t > 0 t p ( t ) P ( t ) < + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equd_HTML.gif
      Under the condition (p), the Orlicz space L P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq29_HTML.gif coincides with the set (equivalence classes) of measurable functions u : Ω R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq30_HTML.gif such that
      Ω P ( | u | ) d x < + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Eque_HTML.gif
      and is equipped with the (Luxemburg) norm, i.e.,
      | u | P : = inf { k > 0 : Ω P ( | u | k ) d x < 1 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equf_HTML.gif
      We will denote by W 1 , P ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq31_HTML.gif the corresponding Orlicz-Sobolev space with the norm
      u W 1 , P ( Ω ) : = | u | P + u P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equg_HTML.gif
      and define W 0 1 , P ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq32_HTML.gif as the closure of C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq33_HTML.gif in W 1 , P ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq31_HTML.gif. In this note, we will use the following equivalent norm on W 0 1 , P ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq32_HTML.gif:
      u : = inf { k > 0 : Ω P ( | u | k ) d x < 1 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equh_HTML.gif
      Now, we introduce the Orlicz-Sobolev conjugate P http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq34_HTML.gif of P, which is given by
      P 1 ( t ) : = 0 t p 1 ( τ ) τ N + 1 N d τ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equi_HTML.gif
      where we suppose that
      lim t 0 t 1 p 1 ( τ ) τ N + 1 N d τ < + , lim t 1 t p 1 ( τ ) τ N + 1 N d τ = + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equj_HTML.gif

      Let p : = inf t > 0 t P ( t ) P ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq35_HTML.gif, p + : = sup t > 0 t P ( t ) P ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq36_HTML.gif. Throughout this paper, we assume that p + < p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq37_HTML.gif. Now, we will make the following assumptions on f ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq14_HTML.gif.

      ( f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq38_HTML.gif) There exists an odd increasing homeomorphism h from R to R, and nonnegative constants c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq39_HTML.gif, c 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq40_HTML.gif such that
      | f ( x , t ) | c 1 + c 2 h ( | t | ) , t R , x Ω ¯ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equk_HTML.gif
      and lim t + H ( t ) P ( k t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq41_HTML.gif, k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq42_HTML.gif, where
      H ( t ) : = 0 t h ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equl_HTML.gif
      Let
      H ˜ ( t ) : = 0 t h 1 ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equm_HTML.gif

      then we can obtain complementary N-functions which define corresponding Orlicz spaces L H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq43_HTML.gif and L H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq44_HTML.gif.

      Similar to the condition (p), we also assume the following condition on H:
      ( h ) 1 < h : = inf t > 0 t h ( t ) H ( t ) h + : = sup t > 0 t h ( t ) H ( t ) < + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equn_HTML.gif

      In order to prove our results, we now state some useful lemmas.

      Lemma 2.1 [10]

      Under the condition (p), the spaces L P ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq45_HTML.gif, W 0 1 , P ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq32_HTML.gif and W 1 , P ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq31_HTML.gif are separable and reflexive Banach spaces.

      Lemma 2.2 [10]

      Under the condition ( f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq38_HTML.gif), the embedding W 0 1 , P ( Ω ) L H ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq46_HTML.gif is compact.

      Lemma 2.3 [2]

      Let ρ ( u ) = Ω P ( u ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq47_HTML.gif, we have
      1. (1)

        if | u | P < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq48_HTML.gif, then | u | P p + ρ ( u ) | u | P p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq49_HTML.gif;

         
      2. (2)

        if | u | P > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq50_HTML.gif, then | u | P p ρ ( u ) | u | P p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq51_HTML.gif;

         
      3. (3)

        if 0 < t < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq52_HTML.gif, then t p + P ( u ) P ( t u ) t p P ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq53_HTML.gif;

         
      4. (4)

        if t > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq54_HTML.gif, then t p P ( u ) P ( t u ) t p + P ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq55_HTML.gif.

         

      Lemma 2.4 [1113]

      Let E = V + X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq56_HTML.gif, where E is a real Banach space and V is finite dimensional. Suppose I C 1 ( E , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq57_HTML.gif is an even functional satisfying I ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq58_HTML.gif and

      ( I 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq59_HTML.gif) there is a constant ρ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq60_HTML.gif such that I | B ρ X 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq61_HTML.gif;

      ( I 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq62_HTML.gif) there is a subspace W of E with dim V < dim W < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq63_HTML.gif and there is M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq64_HTML.gif such that max u W I ( u ) < M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq65_HTML.gif;

      ( I 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq66_HTML.gif) considering M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq64_HTML.gif given by ( I 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq62_HTML.gif), I satisfies (PS) c for 0 c M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq67_HTML.gif.

      Then I possesses at least dim W dim V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq68_HTML.gif pairs of nontrivial critical points.

      Using the version of the symmetric mountain pass theorem mentioned above, we can state our result as follows.

      Theorem 2.1 Assume that f ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq14_HTML.gif is odd in t, satisfies ( f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq38_HTML.gif) with p < h + p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq69_HTML.gif and the following assumptions:

      ( f 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq70_HTML.gif) there exist η > p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq71_HTML.gif and 1 < σ < p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq72_HTML.gif, and a 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq73_HTML.gif, a 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq74_HTML.gif, such that 1 η f ( x , t ) t F ( x , t ) a 1 a 2 | t | σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq75_HTML.gif for every t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq76_HTML.gif, a.e. in Ω.

      ( f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq77_HTML.gif) there is Ω 0 Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq78_HTML.gif with | Ω 0 | > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq79_HTML.gif such that lim inf | t | F ( x , t ) / | t | p + = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq80_HTML.gif uniformly a.e. in Ω 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq81_HTML.gif.

      Then for any given k N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq82_HTML.gif, the problem (1.1) possesses at least k pairs of nontrivial solutions.

      3 Main results and proofs

      In this section, we assume that N 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq83_HTML.gif and E = W 0 1 , P ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq84_HTML.gif, u E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq85_HTML.gif is called a weak solution of the problem (1.1) if
      Ω a ( | u | ) u ϕ d x = Ω f ( x , u ) ϕ d x , ϕ E . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equo_HTML.gif
      Set
      I ( u ) = Ω P ( | u | ) d x Ω F ( x , u ) d x , u E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equp_HTML.gif

      and we know that the critical points of I are just the weak solutions of the problem (1.1).

      For E is a separable and reflexive Banach space, then there exist (see [9]) { e n } n = 1 E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq86_HTML.gif and { e n } n = 1 E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq87_HTML.gif such that
      e n ( e m ) = δ n , m = { 1 , if  n = m ; 0 , if  n m . and e n ( v ) = α n for  v = i = 1 α i e i E . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equq_HTML.gif
      Now, we set V j = { u W 0 1 , P ( Ω ) : e i ( u ) = 0 , i > j } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq88_HTML.gif, X j = { u W 0 1 , P ( Ω ) : e i ( u ) = 0 , i j } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq89_HTML.gif, so
      W 0 1 , P ( Ω ) = V j X j . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equ2_HTML.gif
      (3.1)

      Lemma 3.1 Given δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq90_HTML.gif, there is j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq91_HTML.gif such that for all u X j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq92_HTML.gif, | u | H δ u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq93_HTML.gif.

      Proof We prove the lemma by contradiction. Suppose that there exist δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq90_HTML.gif and u j X j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq94_HTML.gif for every j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq91_HTML.gif such that | u j | H δ u j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq95_HTML.gif. Taking v j = u j | u j | H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq96_HTML.gif, we have | v j | H = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq97_HTML.gif for every j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq91_HTML.gif and v j 1 δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq98_HTML.gif. Hence, { v j } W 0 1 , P ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq99_HTML.gif is a bounded sequence, and we may suppose, without loss of generality, that v j v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq100_HTML.gif in W 0 1 , P ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq32_HTML.gif. Furthermore, e n ( v ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq101_HTML.gif for every n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq102_HTML.gif since e n ( v j ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq103_HTML.gif for all j n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq104_HTML.gif. This shows that v = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq105_HTML.gif. On the other hand, by the compactness of embedding W 0 1 , P ( Ω ) L H ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq46_HTML.gif, we conclude that | v | H = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq106_HTML.gif. This proves the lemma. □

      Lemma 3.2 Suppose f satisfies ( f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq38_HTML.gif), then there exist j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq91_HTML.gif and ρ , α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq107_HTML.gif such that
      I | B ρ X j α . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equr_HTML.gif
      Proof Now suppose that u > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq108_HTML.gif. From ( f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq38_HTML.gif), we know that
      I ( u ) = Ω P ( | u | ) d x Ω F ( x , u ) d x u p C 1 | u | H h + C 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equs_HTML.gif
      Consequently, considering δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq90_HTML.gif to be chosen posteriorly by Lemma 3.1, we have for all u X j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq92_HTML.gif and j sufficiently large,
      I ( u ) u p ( 1 C 1 δ h + u h + p ) C 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equt_HTML.gif

      Now, taking u = ρ ( δ ) = ( 1 2 C δ h + ) 1 h + p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq109_HTML.gif and noting that ρ ( δ ) + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq110_HTML.gif, if δ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq111_HTML.gif, we can choose δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq90_HTML.gif such that 1 2 ρ p > C 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq112_HTML.gif, ρ > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq113_HTML.gif, and I ( u ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq114_HTML.gif for every u X j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq92_HTML.gif, u = ρ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq115_HTML.gif, the proof is complete. □

      Lemma 3.3 Suppose f satisfies ( f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq77_HTML.gif). Then given m N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq116_HTML.gif, there exist a subspace W of W 0 1 , P ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq32_HTML.gif and a constant M m > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq117_HTML.gif such that dim W = m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq118_HTML.gif and max u W I ( u ) < M m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq119_HTML.gif.

      Proof Let x 0 Ω 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq120_HTML.gif and r 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq121_HTML.gif be such that B ( x 0 , r 0 ) ¯ Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq122_HTML.gif, and 0 < | B ( x 0 , r 0 ) ¯ Ω 0 | < | Ω 0 | 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq123_HTML.gif. First, we take v 1 C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq124_HTML.gif with supp ( v 1 ) = B ( x 0 , r 0 ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq125_HTML.gif. Considering Ω 1 = Ω 0 [ B ( x 0 , r 0 ) ¯ Ω 0 ] Ω ˆ 0 = Ω B ( x 0 , r 0 ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq126_HTML.gif, we have | Ω 1 | | Ω 0 | 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq127_HTML.gif. Let x 1 Ω 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq128_HTML.gif and r 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq129_HTML.gif be such that B ( x 1 , r 1 ) ¯ Ω ˆ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq130_HTML.gif, and 0 < | B ( x 1 , r 1 ) ¯ Ω 1 | < | Ω 1 | 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq131_HTML.gif. Next, we take v 2 C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq132_HTML.gif with supp ( v 2 ) = B ( x 1 , r 1 ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq133_HTML.gif. After a finite number of steps, we get v 1 , v 2 , , v m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq134_HTML.gif such that supp ( v i ) supp ( v j ) = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq135_HTML.gif, i j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq136_HTML.gif, and | supp ( v j ) Ω 0 | > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq137_HTML.gif for all i , j { 1 , 2 , , m } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq138_HTML.gif. Let W = span { v 1 , v 2 , , v m } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq139_HTML.gif, by construction, dim W = m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq118_HTML.gif, and Ω | v | p + d x > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq140_HTML.gif for every v W { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq141_HTML.gif.

      Since max u W { 0 } I ( u ) = max t > 0 , v W B 1 ( 0 ) ( Ω P ( t | v | ) d x Ω F ( x , t v ) d x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq142_HTML.gif, if t > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq54_HTML.gif, then I ( t v ) t p + Ω F ( x , t v ) d x = t p + ( 1 1 t p + Ω F ( x , t v ) d x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq143_HTML.gif. Now, it suffices to verify that
      lim t 1 t p + Ω F ( x , t v ) d x > 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equu_HTML.gif
      From the condition ( f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq77_HTML.gif), given L > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq144_HTML.gif, there is C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq145_HTML.gif such that for every s R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq146_HTML.gif, a.e. x in Ω 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq81_HTML.gif,
      F ( x , s ) L | s | p + C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equv_HTML.gif
      Consequently, for v B 1 ( 0 ) W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq147_HTML.gif and t > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq54_HTML.gif,
      Ω F ( x , t v ) d x L t p + Ω 0 | v | p + d x C t h + Ω Ω 0 H ( v ) d x C 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equw_HTML.gif
      and
      lim t Ω f ( x , t v ) d x t p + L Ω 0 | v | p + d x C Ω Ω 0 H ( v ) d x L r C R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equx_HTML.gif

      where r = min { Ω 0 | v | p + d x , v B 1 ( 0 ) W } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq148_HTML.gif and R = max { Ω Ω 0 H ( v ) d x , v B 1 ( 0 ) W } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq149_HTML.gif. Observing that W is finite dimensional and we have R < + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq150_HTML.gif, r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq151_HTML.gif, the inequality is obtained by taking L > 1 r ( 1 + C R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq152_HTML.gif; the proof is complete. □

      Lemma 3.4 Suppose f satisfies ( f 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq70_HTML.gif), then I satisfies the (PS) condition.

      Proof We suppose that u n > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq153_HTML.gif,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_Equy_HTML.gif

      Noting that 1 < σ < p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq72_HTML.gif, η > p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq71_HTML.gif, { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq154_HTML.gif is bounded. By [9], Lemma 3.1, we know that I satisfies the (PS) condition. □

      Proof of Theorem 2.1 First, we recall that W 0 1 , P ( Ω ) = V j X j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq155_HTML.gif, where V j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq156_HTML.gif and X j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq157_HTML.gif are defined in (3.1). Invoking Lemma 3.2, we find j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq91_HTML.gif, and I satisfies I 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq59_HTML.gif with X = X j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq158_HTML.gif. Now, by Lemma 3.3, there is a subspace W of W 0 1 , P ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq32_HTML.gif with dim W = k + j = k + dim V j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq159_HTML.gif and such that I satisfies ( I 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq62_HTML.gif). Since I ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq58_HTML.gif and I is even, we may apply Lemma 2.4 to conclude that I possesses at least k pairs of nontrivial critical points. The proof is complete. □

      Declarations

      Acknowledgements

      Project supported by Natural Science Foundation of China, Tian Yuan Special Foundation (No. 11226116), Natural Science Foundation of Jiangsu Province of China for Young Scholar (No. BK201209), the China Scholarship Council, the Fundamental Research Funds for the Central Universities (No. JUSRP11118) and Foundation for young teachers of Jiangnan University (No. 2008LQN008).

      Authors’ Affiliations

      (1)
      School of Science, Jiangnan University
      (2)
      School of Mathematics Science, Nanjing Normal University

      References

      1. Clément PH, García-Huidobro M, Manásevich R, Schmitt K: Mountain pass type solutions for quasilinear elliptic equations. Calc. Var. Partial Differ. Equ. 2000, 11: 33–62. 10.1007/s005260050002View Article
      2. Fukagai N, Ito M, Narukawa MK:Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-136/MediaObjects/13661_2012_Article_236_IEq160_HTML.gif. Funkc. Ekvacioj 2006, 49: 235–267. 10.1619/fesi.49.235MathSciNetView Article
      3. Fukagai N, Narukawa K: On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems. Ann. Mat. Pura Appl. 2007, 186: 539–564. 10.1007/s10231-006-0018-xMathSciNetView Article
      4. García-Huidobro M, Le V, Manásevich R, Schmitt K: On the principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting. Nonlinear Differ. Equ. Appl. 1999, 6: 207–225. 10.1007/s000300050073View Article
      5. Tan, Z, Fang, F: Orlicz-Sobolev versus Hölder local minimizer and multiplicity results for quasilinear elliptic equations. Preprint
      6. Mihǎilescu M, Rădulescu V: Nonhomogeneous Neumann problems in Orlicz-Sobolev spaces. C. R. Math. 2008, 346: 401–406. 10.1016/j.crma.2008.02.020View Article
      7. Bonanno G, Bisci GM, Rǎdulescu VD: Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz-Sobolev spaces. Nonlinear Anal. 2012, 75: 4441–4456. 10.1016/j.na.2011.12.016MathSciNetView Article
      8. Černý R: Generalized n -Laplacian: quasilinear nonhomogenous problem with critical growth. Nonlinear Anal. 2011, 74: 3419–3439. 10.1016/j.na.2011.03.002MathSciNetView Article
      9. Fang F, Tan Z: Existence and multiplicity of solutions for a class of quasilinear elliptic equations: an Orlicz-Sobolev space setting. J. Math. Anal. Appl. 2012, 389: 420–428. 10.1016/j.jmaa.2011.11.078MathSciNetView Article
      10. Adams RA, Fournier JJF: Sobolev Spaces. 2nd edition. Academic Press, Amsterdam; 2003.
      11. Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973, 14: 349–381. 10.1016/0022-1236(73)90051-7MathSciNetView Article
      12. Bartolo P, Benci V, Fortunato D: Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal. TMA 1983, 7: 981–1012. 10.1016/0362-546X(83)90115-3MathSciNetView Article
      13. Silva, EAB: Critical point theorems and applications to differential equations. PhD thesis, University of Wisconsin-Madison (1988)

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