Infinitely many periodic solutions for some second-order differential systems with p(t)-Laplacian

  • Liang Zhang1,

    Affiliated with

    • Xian Hua Tang1Email author and

      Affiliated with

      • Jing Chen1

        Affiliated with

        Boundary Value Problems20112011:33

        DOI: 10.1186/1687-2770-2011-33

        Received: 3 June 2011

        Accepted: 14 October 2011

        Published: 14 October 2011

        Abstract

        In this article, we investigate the existence of infinitely many periodic solutions for some nonautonomous second-order differential systems with p(t)-Laplacian. Some multiplicity results are obtained using critical point theory.

        2000 Mathematics Subject Classification: 34C37; 58E05; 70H05.

        Keywords

        p(t)-Laplacian Periodic solutions Critical point theory

        1. Introduction

        Consider the second-order differential system with p(t)-Laplacian
        { d d t ( | u ˙ ( t ) | p ( t ) 2 u ˙ ( t ) ) + | u ( t ) | p ( t ) 2 u ( t ) = F ( t , u ( t ) ) a . e . t [ 0 , T ] , u ( 0 ) u ( T ) = u ˙ ( 0 ) u ˙ ( T ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ1_HTML.gif
        (1.1)
        where T > 0, F: [0, T] × ℝ N → ℝ, and p(t) ∈ C([0, T], ℝ+) satisfies the following assumptions:
        1. (A)

          p(0) = p(T) and p - : = min 0 t T p ( t ) > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq1_HTML.gif, where q + > 1 which satisfies 1/p - + 1/q + = 1.

           

        Moreover, we suppose that F: [0, T] × ℝ N → ℝ satisfies the following assumptions:

        (A') F(t, x) is measurable in t for every x ∈ ℝ N and continuously differentiable in x for a.e. t ∈ [0, T], and there exist aC(ℝ+, ℝ+), bL1(0, T; ℝ+), such that
        | F ( t , x ) | a ( | x | ) b ( t ) , | F ( t , x ) | a ( | x | ) b ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equa_HTML.gif

        for all x ∈ ℝ N and a.e. t ∈ [0, T].

        The operator d d t ( | u ˙ ( t ) | p ( t ) - 2 u ˙ ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq2_HTML.gif is said to be p(t)-Laplacian, and becomes p-Laplacian when p(t) ≡ p (a constant). The p(t)-Laplacian possesses more complicated nonlinearity than p-Laplacian; for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth conditions has received considerable attention in recent years. These problems are interesting in applications and raise many mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electro-rheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electromagnetic field. Another field of application of equations with variable exponent growth conditions is image processing (see [1, 2]). The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to [312] for an overview on this subject.

        In 2003, Fan and Fan [13] studied the ordinary p(t)-Laplacian system and introduced a generalized Orlicz-Sobolev space W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif, which is different from the usual space W T 1 , p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq4_HTML.gif, then Wang and Yuan [14] obtained the existence and multiplicity of periodic solutions for ordinary p(t)-Laplacian system under the generalized Ambrosetti-Rabinowitz conditions. Fountain and Dual Fountain theorems were established by Bartsch and Willem [15, 16], and both theorems are effective tools for studying the existence of infinitely many large energy solutions and small energy solutions. When we impose some suitable conditions on the growth of the potential function at origin or at infinity, we get three multiplicity results of infinitely many periodic solutions for system (1.1) using the Fountain theorem, the Dual Fountain theorem, and the Symmetric Mountain Pass theorem.

        The rest of the article is divided as follows: Basic definitions and preliminary results are collected in Second 2. The main results and proofs are given in Section 3. The three examples are presented in Section 4 for illustrating our results.

        In this article, we denote by p + : = max 0 t T p ( t ) > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq5_HTML.gif throughout this article, and we use 〈·, ·〉 and |·| to denote the usual inner product and norm in ℝ N , respectively.

        2. Preliminaries

        In this section, we recall some known results in nonsmooth critical point theory, and the properties of space W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif are listed for the convenience of readers.

        Definition 2.1[14]. Let p(t) satisfies the condition (A), define
        L p ( t ) ( [ 0 , T ] , N ) = u L 1 ( [ 0 , T ] , N ) : 0 T | u | p ( t ) d t < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equb_HTML.gif
        with the norm
        | u | p ( t ) : = inf λ > 0 : 0 T u λ p ( t ) d t 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equc_HTML.gif
        For u L l o c 1 ( [ 0 , T ] , N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq6_HTML.gif, let u' denote the weak derivative of u, if u L loc 1 ( [ 0 , T ] , N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq7_HTML.gif and satisfies
        0 T u ϕ d t = - 0 T u ϕ d t , ϕ C 0 ( [ 0 , T ] , N ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equd_HTML.gif
        Define
        W 1 , p ( t ) ( [ 0 , T ] , N ) = { u L p ( t ) ( [ 0 , T ] , N ) : u L p ( t ) ( [ 0 , T ] , N ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Eque_HTML.gif

        with the norm u W 1 , p ( t ) : = | u | p ( t ) + | u | p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq8_HTML.gif.

        In this article, we will use the following equivalent norm on W1, p(t)([0, T], ℝ N ), i.e.,
        u : = inf λ > 0 : 0 T u λ p ( t ) + u ˙ λ p ( t ) d t 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equf_HTML.gif

        and some lemmas given in the following section have been proven under the norm of u W 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq9_HTML.gif, and it is obvious that they also hold under the norm ||u||.

        Remark 2.1. If p(t) = p, where p ∈ (1, ∞) is a constant, by the definition of |u|p(t), it is easy to get | u | p = ( 0 T | u ( t ) | p d t ) 1 p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq10_HTML.gif, which is the same with the usual norm in space L p .

        The space Lp(t)is a generalized Lebesgue space, and the space W1, p(t)is a generalized Sobolev space. Because most of the following lemmas have appeared in [13, 14, 17, 18], we omit their proofs.

        Lemma 2.1[13]. Lp(t)and W1, p(t)are both Banach spaces with the norms defined above, when p- > 1, they are reflexive.

        Lemma 2.2[14]. (i) The space Lp(t)is a separable, uniform convex Banach space, its conjugate space is Lq(t), for any uLp(t)and vLq(t), we have
        0 T u v d t 2 | u | p ( t ) | v | q ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equg_HTML.gif
        where 1 p ( t ) + 1 q ( t ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq11_HTML.gif.
        1. (ii)

          If p 1(t) and p 2(t) ∈ C([0, T], ℝ+) and p 1(t) ≤ p 2(t) for any t ∈ [0, T], then L p 2 ( t ) L p 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq12_HTML.gif, and the embedding is continuous.

           
        Lemma 2.3[14]. If we denote ρ ( u ) = 0 T | u ( t ) | p ( t ) d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq13_HTML.gif, ∀ uLp(t), then
        1. (i)

          |u|p(t)< 1 (= 1; > 1) ⇔ ρ(u) < 1 (= 1; > 1);

           
        2. (ii)

          | u | p ( t ) > 1 | u | p ( t ) p - ρ ( u ) | u | p ( t ) p + , | u | p ( t ) < 1 | u | p ( t ) p + ρ ( u ) | u | p ( t ) p - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq14_HTML.gif;

           
        3. (iii)

          |u|p(t)→ 0 ⇔ ρ(u) → 0; |u|p(t)→ ∞ ⇔ ρ(u) → ∞.

           
        4. (iv)

          For u ≠ 0, | u | p ( t ) = λ ρ ( u λ ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq15_HTML.gif.

           

        Similar to Lemma 2.3, we have

        Lemma 2.4. If we denote I ( u ) = 0 T ( | u ( t ) | p ( t ) + | u ˙ ( t ) | p ( t ) ) d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq16_HTML.gif, ∀ uW1,p(t), then
        1. (i)

          ||u|| < 1 (= 1; > 1) ⇔ I(u) < 1 (= 1; > 1);

           
        2. (ii)

          u > 1 u p - I ( u ) u p + , u < 1 u p + I ( u ) u p - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq17_HTML.gif;

           
        3. (iii)

          ||u|| → 0 ⇔ I(u) → 0; ||u|| → ∞ ⇔ I(u) → ∞.

           
        4. (iv)

          For u ≠ 0, u = λ I ( u λ ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq18_HTML.gif.

           
        Defnition 2.2[17].
        C T = C T ( , N ) : = { u C ( , N ) : u  is T  - periodic } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equh_HTML.gif

        with the norm u : = max t [ 0 , T ] | u ( t ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq19_HTML.gif.

        For a constant p ∈ (1, ∞), using another conception of weak derivative which is called T-weak derivative, Mawhin and Willem gave the definition of the space W T 1 , p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq4_HTML.gif by the following way.

        Definition 2.3[17]. Let uL1([0, T], ℝ N ) and vL1([0, T], ℝ N ), if
        0 T v ϕ d t = - 0 T u ϕ d t ϕ C T , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equi_HTML.gif

        then v is called a T-weak derivative of u and is denoted by u ˙ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq20_HTML.gif.

        Definition 2.4[17]. Define
        W T 1 , p ( [ 0 , T ] , N ) = { u L p ( [ 0 , T ] , N ) : u ˙ L p ( [ 0 , T ] , N ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equj_HTML.gif

        with the norm u W T 1 , p = ( | u | p p + | u ˙ | p p ) 1 p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq21_HTML.gif.

        Definition 2.5[13]. Define
        W T 1 , p ( t ) ( [ 0 , T ] , N ) = { u L p ( t ) ( [ 0 , T ] , N ) : u ˙ L p ( t ) ( [ 0 , T ] , N ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equk_HTML.gif

        and H T 1 , p ( t ) ( [ 0 , T ] , N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq22_HTML.gif to be the closure of C T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq23_HTML.gif in W1,p(t)([0, T], ℝ N ).

        Remark 2.2. From Definition 2.4, if u W T 1 , p ( t ) ( [ 0 , T ] , N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq24_HTML.gif, it is easy to conclude that u W T 1 , p - ( [ 0 , T ] , N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq25_HTML.gif.

        Lemma 2.5[13].
        1. (i)

          C T ( [ 0 , T ] , N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq26_HTML.gif is dense in W T 1 , p ( t ) ( [ 0 , T ] , N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq27_HTML.gif;

           
        2. (ii)

          W T 1 , p ( t ) ( [ 0 , T ] , N ) = H T 1 , p ( t ) ( [ 0 , T ] , N ) : = { u W 1 , p ( t ) ( [ 0 , T ] , N ) : u ( 0 ) = u ( T ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq28_HTML.gif;

           
        3. (iii)

          If u H T 1 , 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq29_HTML.gif, then the derivative u' is also the T-weak derivative u ˙ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq20_HTML.gif, i.e., u = u ˙ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq30_HTML.gif.

           
        Lemma 2.6[17]. Assume that u W T 1 , 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq31_HTML.gif, then
        1. (i)

          0 T u ˙ d t = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq32_HTML.gif,

           
        2. (ii)

          u has its continuous representation, which is still denoted by u ( t ) = 0 t u ˙ ( s ) d s + u ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq33_HTML.gif, u(0) = u(T),

           
        3. (iii)

          u ˙ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq20_HTML.gif is the classical derivative of u, if u ˙ C ( [ 0 , T ] , N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq34_HTML.gif.

           

        Since every closed linear subspace of a reflexive Banach space is also reflexive, we have

        Lemma 2.7[13]. H T 1 , p ( t ) ( [ 0 , T ] , N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq22_HTML.gif is a reflexive Banach space if p- > 1.

        Obviously, there are continuous embeddings L p ( t ) L p - , W 1 , p ( t ) W 1 , p - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq35_HTML.gif and H T 1 , p ( t ) H T 1 , p - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq36_HTML.gif. By the classical Sobolev embedding theorem, we obtain

        Lemma 2.8[13]. There is a continuous embedding
        W 1 , p ( t ) ( or H T 1 , p ( t ) ) C ( [ 0 , T ] , N ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equl_HTML.gif

        when p- > 1, the embedding is compact.

        Lemma 2.9[13]. Each of the following two norms is equivalent to the norm in W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif:
        1. (i)

          | u ˙ | p ( t ) + | u | q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq37_HTML.gif, 1 ≤ q ≤ ∞;

           
        2. (ii)

          | u ˙ | p ( t ) + | ū | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq38_HTML.gif, where ū = ( 1 T ) 0 T u ( t ) d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq39_HTML.gif.

           
        Lemma 2.10[13]. If u, u n Lp(t)(n = 1,2,...), then the following statements are equivalent to each other
        1. (i)

          lim n | u n - u | p ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq40_HTML.gif;

           
        2. (ii)

          lim n ρ ( u n - u ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq41_HTML.gif;

           
        3. (iii)

          u n u in measure in [0, T] and lim n ρ ( u n ) = ρ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq42_HTML.gif.

           
        Lemma 2.11[14]. The functional J defined by
        J ( u ) = 0 T 1 p ( t ) | u ˙ ( t ) | p ( t ) d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equm_HTML.gif
        is continuously differentiable on W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif and J' is given by
        J ( u ) , v = 0 T ( | u ˙ ( t ) | p ( t ) - 2 u ˙ ( t ) , v ˙ ( t ) ) d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ2_HTML.gif
        (2.1)
        and J' is a mapping of (S+), i.e., if u n u weakly in W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif and
        limsup n ( J ( u n ) - J ( u ) , u n - u ) 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equn_HTML.gif

        then u n has a convergent subsequence on W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif.

        Lemma 2.12[18]. Since W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif is a separable and reflexive Banach space, there exist { e n } n = 1 W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq43_HTML.gif and { f n } n = 1 ( W T 1 , p ( t ) ) * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq44_HTML.gif such that
        f n ( e m ) = δ n , m = { 1 , n = m , 0 , n m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equo_HTML.gif

        W T 1 , p ( t ) = span { e n : n = 1 , 2 , } ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq45_HTML.gif and ( W T 1 , p ( t ) ) * = span { f n : n = 1 , 2 , } ¯ W * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq46_HTML.gif.

        For k = 1, 2,..., denote
        X k = span { e k } , Y k = j = 1 k X j , Z k = j = k X j ¯ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ3_HTML.gif
        (2.2)
        Lemma 2.13[19]. Let X be a reflexive infinite Banach space, ϕC1(X, ℝ) is an even functional with the (C) condition and ϕ(0) = 0. If X = YV with dimY < ∞, and ϕ satisfies
        1. (i)

          there are constants σ, α > 0 such that φ | B σ V α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq47_HTML.gif,

           
        2. (ii)

          for any finite-dimensional subspace W of X, there exists positive constants R 2(W) such that ϕ(u) ≤ 0 for uW\B r (0), where B r (0) is an open ball in W of radius r centered at 0. Then ϕ possesses an unbounded sequence of critical values.

           

        Lemma 2.14[15]. Suppose

        (A1) ϕC1(X, ℝ) is an even functional, then the subspace X k , Y k , and Z k are defined by (2.2);

        If for every k ∈ ℕ, there exists ρ k > r k > 0 such that

        (A2) a k : = max u Y k , u = ρ k φ ( u ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq48_HTML.gif, where Y k : = j = 0 k X j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq49_HTML.gif;

        (A3) b k : = inf u Z k , u = r k φ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq50_HTML.gif, as k → ∞, where Z k : = j = k X j ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq51_HTML.gif;

        (A4) ϕ satisfies the (PS) c condition for every c > 0.

        Then ϕ has an unbounded sequence of critical values.

        Lemma 2.15[16]. Assume (A1) is satisfied, and there is a k0 > 0 so as to for each kk0, there exist ρ k > r k > 0 such that

        (A5) d k : = inf u Z k , u ρ k φ ( u ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq52_HTML.gif, as k → ∞;

        (A6) i k : = max u Y k , u = r k φ ( u ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq53_HTML.gif;

        (A7) inf u Z k , u = ρ k φ ( u ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq54_HTML.gif;

        (A8) ϕ satisfies the (PS) c * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq55_HTML.gif condition for every c ∈ [dk 0, 0).

        Then ϕ has a sequence of negative critical values converging to 0.

        Remark 2.3. ϕ satisfies the (PS) c * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq55_HTML.gif condition means that if any sequence { u n j } X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq56_HTML.gif such that n j → ∞, u n j Y n j , φ ( u n j ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq57_HTML.gif and ( φ | Y n j ) ( u n j ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq58_HTML.gif, then { u n j } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq59_HTML.gif contains a subsequence converging to critical point of ϕ. It is obvious that if ϕ satisfies the (PS) c * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq55_HTML.gif condition, then ϕ satisfies the (PS) c condition.

        3. Main results and proofs of the theorems

        Theorem 3.1. Let F(t, x) satisfies the condition (A'), and suppose the following conditions hold:

        (B1) there exist β > p + and r > 0 such that
        β F ( t , x ) ( F ( t , x ) , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equp_HTML.gif

        for a.e. t ∈ [0, T] and all |x| ≥ r in ℝ N ;

        (B2) there exist positive constants μ > p + and Q > 0 such that
        limsup | x | + F ( t , x ) | x | μ Q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equq_HTML.gif

        uniformly for a.e. t ∈ [0, T];

        (B3) there exists μ' > p+ and Q' > 0 such that
        liminf | x | + F ( t , x ) | x | μ Q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equr_HTML.gif

        uniformly for a.e. t ∈ [0, T];

        (B4) F(t, x) = F(t, -x) for t ∈ [0, T] and all x in ℝ N .

        Then system (1.1) has infinite solutions u k in W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif for every positive integer k such that ||u k || → +∞, as k → ∞.

        Remark 3.1. Suppose that F(t, ·) is continuously differentiable in x and p(t) ≡ p, then condition (B1) reduces to the well-known Ambrosetti-Rabinowitz condition (see [19]), which was introduced in the context of semi-linear elliptic problems. This condition implies that F(t, x) grows at a superquadratic rate as |x| → ∞. This kind of technical condition often appears as necessary to use variational methods when we solve super-linear differential equations such as elliptic problems, Dirac equations, Hamiltonian systems, wave equations, and Schrödinger equations.

        Theorem 3.2. Assume that F(t, x) satisfies (A'), (B1), (B3), and (B4) and the following assumption:

        (B5) 0 T F ( t , 0 ) d t = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq60_HTML.gif, and there exists r1 > p+ and M > 0 such that
        limsup | x | 0 | F ( t , x ) | | x | r 1 M . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equs_HTML.gif

        Then system (1.1) has infinite solutions u k in W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif for every positive integer k such that ||u k || → +∞, as k → ∞.

        Theorem 3.3. Assume that F(t, x) satisfies the following assumption:

        (B6) F(t, x):= a(t)|x| γ , where a(t) ∈ L (0, T; ℝ+) and 1 < γ < p- is a constant. Then system (1.1) has infinite solutions u k in W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif for every positive integer k.

        The proof of Theorem 3.1 is organized as follows: first, we show the functional ϕ defined by
        φ ( u ) = 0 T 1 p ( t ) | u ˙ ( t ) | p ( t ) d t + 0 T 1 p ( t ) | u ( t ) | p ( t ) d t - 0 T F ( t , u ( t ) ) d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equt_HTML.gif

        satisfies the (PS) condition, then we verify for ϕ the conditions in Lemma 2.14 item-by-item, then ϕ has an unbounded sequence of critical values.

        Proof of Theorem 3.1. Let { u n } W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq61_HTML.gif such that ϕ(u n ) is bounded and ϕ'(u n ) → 0 as n → ∞. First, we prove {u n } is a bounded sequence, otherwise, {u n } would be unbounded sequence, passing to a subsequence, still denoted by {u n }, such that ||u n || ≥ 1 and ||u n || → ∞. Note that
        φ ( u ) , v = 0 T ( | u ˙ ( t ) | p ( t ) - 2 u ˙ ( t ) , v ˙ ( t ) ) d t + 0 T ( | u ( t ) | p ( t ) - 2 u ( t ) - F ( t , u ( t ) ) , v ( t ) ) d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ4_HTML.gif
        (3.1)

        for all v W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq62_HTML.gif.

        It follows from (3.1) that
        0 T ( β p ( t ) 1 ) ( | u ˙ n ( t ) | p ( t ) + | u n ( t ) | p ( t ) ) d t = β φ ( u n ) φ ( u n ) , u n + 0 T [ β F ( t , u n ( t ) ) ( F ( t , u n ( t ) ) , u n ( t ) ) ] d t = β φ ( u n ) φ ( u n ) , u n + Ω 1 [ β F ( t , u n ( t ) ) ( F ( t , u n ( t ) ) , u n ( t ) ) ] d t + Ω 2 [ β F ( t , u n ( t ) ) ( F ( t , u n ( t ) ) , u n ( t ) ) ] d t β φ ( u n ) φ ( u n ) , u n + Ω 1 [ β F ( t , u n ( t ) ) ( F ( t , u n ( t ) ) , u n ( t ) ) ] d t β φ ( u n ) φ ( u n ) , u n + C 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ5_HTML.gif
        (3.2)

        where Ω1:= {t ∈ [0, T]; |u n (t)| ≤ r}, Ω2:= [0, T] \ Ω1 and C0 is a positive constant.

        However, from (3.2), we have
        β φ ( u n ) + C 0 β p + - 1 u n p - - φ ( u n ) u n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equu_HTML.gif

        Thus ||u n || is a bounded sequence in W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif.

        By Lemma 2.8, the sequence {u n } has a subsequence, also denoted by {u n }, such that
        u n u weakly in W T 1 , p ( t ) and u n u strongly in C ( [ 0 , T ] ; N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ6_HTML.gif
        (3.3)

        and ||u||C1||u|| by Lemma 2.8, where C1 is a positive constant.

        Therefore, we have
        φ ( u n ) - φ ( u ) , u n - u 0 as n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ7_HTML.gif
        (3.4)
        i.e.,
        φ ( u n ) φ ( u ) , u n u = 0 T ( F ( t , u n ( t ) ) F ( t , u ( t ) ) , u n ( t ) u ( t ) ) d t + 0 T ( | u n ( t ) | p ( t ) 2 u n ( t ) | u ( t ) | p ( t ) 2 u ( t ) , u n ( t ) u ( t ) ) d t + 0 T ( | u ˙ n ( t ) | p ( t ) 2 u ˙ n ( t ) | u ˙ ( t ) | p ( t ) 2 u ˙ ( t ) , u ˙ n ( t ) u ˙ ( t ) ) d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ8_HTML.gif
        (3.5)
        By (3.4) and (3.5), we get 〈J'(u) - J'(u n ), u - u n 〉 → 0, i.e.,
        0 T ( | u ˙ n ( t ) | p ( t ) - 2 u ˙ n ( t ) - | u ˙ ( t ) | p ( t ) - 2 u ˙ ( t ) , u ˙ n ( t ) - u ˙ ( t ) ) d t 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equv_HTML.gif

        so it follows Lemma 2.11 that {u n } admits a convergent subsequence.

        For any uY k , let
        u * : = ( 0 T | u ( t ) | μ d t ) 1 μ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ9_HTML.gif
        (3.6)
        and it is easy to verify that ||·||* defined by (3.6) is a norm of Y k . Since all the norms of a finite dimensional normed space are equivalent, so there exists positive constant C2 such that
        C 2 u u * for u Y k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ10_HTML.gif
        (3.7)
        In view of (B3), there exist two positive constants M1 and C3 such that
        F ( t , x ) M 1 | x | μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ11_HTML.gif
        (3.8)

        for a.e. t ∈ [0, T] and |x| ≥ C3.

        It follows (3.7) and (3.8) that
        φ ( u ) = 0 T 1 p ( t ) | u ˙ ( t ) | p ( t ) d t + 0 T 1 p ( t ) | u ( t ) | p ( t ) d t 0 T F ( t , u ( t ) ) d t 1 p ( u p + + 1 ) Ω 3 F ( t , u ( t ) ) d t Ω 4 F ( t , u ( t ) ) d t 1 p ( u p + + 1 ) M 1 Ω 3 | u ( t ) | μ d t Ω 4 F ( t , u ( t ) ) d t = 1 p ( u p + + 1 ) M 1 0 T | u ( t ) | μ d t + M 1 Ω 4 | u ( t ) | μ d t Ω 4 F ( t , u ( t ) ) d t 1 p ( u p + + 1 ) C 2 μ M 1 u μ + C 4 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equw_HTML.gif

        where Ω3:= {t ∈ [0, T]; |u(t)| ≥ C3}, Ω4:= [0, T] \ Ω3 and C4 is a positive constant.

        Since μ' > p+, there exist positive constants d k such that
        φ ( u ) 0 for all u Y k and u d k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ12_HTML.gif
        (3.9)
        For any uZ k , let
        u μ : = ( 0 T | u ( t ) | μ d t ) 1 μ and β k : = sup u Z k , u = 1 u μ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ13_HTML.gif
        (3.10)

        then we conclude β k → 0 as k → ∞.

        In fact, it is obvious that β k βk + 1> 0, so β k β ≥ 0 as k → ∞. For every k ∈ ℕ, there exists u k Z k such that
        u k = 1 and u k μ > β k 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ14_HTML.gif
        (3.11)

        As W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif is reflexive, {u k } has a weakly convergent subsequence, still denoted by {u k }, such that u k u. We claim u = 0.

        In fact, for any f m ∈ {f n : n = 1, 2...,}, we have f m (u k ) = 0, when k > m, so
        f m ( u k ) 0 , as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equx_HTML.gif

        for any f m ∈ {f n : n = 1, 2 ...,}, therefore u = 0.

        By Lemma 2.8, when u k ⇀ 0 in W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif, then u k → 0 strongly in C([0, T]; ℝ N ). So, we conclude β = 0 by (3.11).

        In view of (B2), there exist two positive constants M2 and C10 such that
        F ( t , x ) M 2 | x | μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ15_HTML.gif
        (3.12)

        uniformly for a.e. t ∈ [0, T] and |x| ≥ C5.

        When ||u|| ≥ 1, we conclude
        φ ( u ) = 0 T 1 p ( t ) | u ( t ) | p ( t ) d t + 0 T 1 p ( t ) | u ˙ ( t ) | p ( t ) d t - 0 T F ( t , u ( t ) ) d t (1) 1 p + 0 T ( | u ( t ) | p ( t ) + | u ˙ ( t ) | p ( t ) ) d t - Ω 5 F ( t , u ( t ) ) d t - Ω 6 F ( t , u ( t ) ) d t (2) 1 p + u p - - M 2 0 T | u ( t ) | μ d t + M 2 Ω 6 | u ( t ) | μ d t - Ω 6 F ( t , u ( t ) ) d t (3) 1 p + u p - - M 2 β k μ u μ - C 6 , (4) (5) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equy_HTML.gif

        where Ω5:= {t ∈ [0, T]; |u(t)| ≥ C5}, Ω6:= [0, T] \ Ω5 and C6 is a positive constant.

        Choosing r k = 1/β k , it is obvious that
        r k as k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equz_HTML.gif
        then
        b k : = inf u Z k , u = r k φ ( u ) as k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ16_HTML.gif
        (3.13)

        i.e., the condition (A3) in Lemma 2.14 is satisfied.

        In view of (3.9), let ρ k := max{d k , r k + 1}, then
        a k : = max u Y k , u = ρ k φ ( u ) 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equaa_HTML.gif

        and this shows the condition of (A2) in Lemma 2.14 is satisfied.

        We have proved the functional ϕ satisfies all the conditions of Lemma 2.14, then ϕ has an unbounded sequence of critical values c k = ϕ(u k ) by Lemma 2.14, we only need to show ||u k || → ∞ as k → ∞.

        In fact, since uk is a critical point of the functional ϕ, we have
        0 T | u ˙ k ( t ) | p ( t ) d t + 0 T | u k ( t ) | p ( t ) d t - 0 T ( F ( t , u k ( t ) ) , u k ( t ) ) d t = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equab_HTML.gif
        Hence, we have
        c k = φ ( u k ) = 0 T 1 p ( t ) | u ˙ k ( t ) | p ( t ) d t + 0 T 1 p ( t ) | u k ( t ) | p ( t ) d t - 0 T F ( t , u k ( t ) ) d t , (1)  1 p - 0 T | u ˙ k ( t ) | p ( t ) d t + 1 p - 0 T | u k ( t ) | p ( t ) d t - 0 T F ( t , u k ( t ) ) d t , (2)  = 0 T ( F ( t , u k ( t ) ) , u k ( t ) ) d t - 0 T F ( t , u k ( t ) ) d t , (3)  (4)  http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ17_HTML.gif
        (3.14)
        since c k → ∞, we conclude
        u k as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equac_HTML.gif
        by (3.14). In fact, if not, going to a subsequence if necessary, we may assume that
        u k C 7 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equad_HTML.gif

        for all k ∈ ℕ and some positive constant C7.

        Combining (A') and (3.14), we have
        c k 0 T ( F ( t , u k ( t ) ) , u k ( t ) ) d t - 0 T F ( t , u k ( t ) ) d t , ( C 7 + 1 ) max 0 s C 7 a ( s ) 0 T b ( t ) d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equae_HTML.gif

        which contradicts c k → ∞. This completes the proof of Theorem 3.1.

        Proof of Theorem 3.2. To prove {u n } has a convergent subsequence in space W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif is the same as that in the proof of Theorem 3.1, thus we omit it. It is obvious that ϕ is even and ϕ(0) = 0 under condition (B5), and so we only need to verify other conditions in Lemma 2.13.

        Proposition 3.1. Under the condition (B5), there exist two positive constants σ and α such that ϕ(u) ≥ α for all u W ̃ T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq63_HTML.gif and ||u|| = σ.

        Proof. In view of condition (B5), there exist two positive constants ε and δ such that
        0 < ε < C 1 and 0 < δ < ε , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equaf_HTML.gif
        where C1 is the same as in (3.3), and
        | F ( t , x ) | ( M + ε ) | x | r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ18_HTML.gif
        (3.15)

        for a.e. t ∈ [0, T] and |x| ≤ δ.

        Let σ:= δ/C1 and ||u|| = σ, since σ < 1, we have
        u p + I ( u ) and u C 1 u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ19_HTML.gif
        (3.16)

        by Lemmas 2.4 and 2.8.

        Combining (3.15) and (3.16), we have
        φ ( u ) = 0 T 1 p ( t ) | u ( t ) | p ( t ) d t + 0 T 1 p ( t ) | u ˙ ( t ) | p ( t ) d t - 0 T F ( t , u ( t ) ) d t (1) 1 p + 0 T ( | u ( t ) | p ( t ) + | u ˙ ( t ) | p ( t ) ) d t - ( M + ε ) 0 T | u ( t ) | r 1 d t (2) 1 p + u p + - ( M + ε ) T C 1 r 1 u r 1 (3) = 1 p + - ( M + ε ) T C 1 r 1 σ r 1 - p + σ p + , (4) (5) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equag_HTML.gif
        so we can choose σ small enough, such that
        1 p + - ( M + ε ) T C 1 r 1 σ r 1 - p + 1 2 p + and α : = 1 2 p + σ p + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equah_HTML.gif

        and this completes the proof of Proposition 3.1.

        Proposition 3.2. For any finite dimensional subspace W of W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif, there is r2 = r2(W) > 0 such that ϕ(u) ≤ 0 for u W \ B r 2 ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq64_HTML.gif, where B r 2 ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq65_HTML.gif is an open ball in W of radius r2 centered at 0.

        Proof. The proof of Proposition 3.2 is the same as the proof of the condition (A2) in the proof of Theorem 3.1.

        We have proved the functional ϕ satisfies all the conditions of Lemma 2.13, ϕ has an unbounded sequence of critical values c k = ϕ(u k ) by Lemma 2.13. Arguing as in the proof of Theorem 3.1, system (1.1) has infinite solutions {u k } in W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif for every positive integer k such that ||u k || → +∞, as k → ∞. The proof of Theorem 3.2 is complete.

        Proof of Theorem 3.3. First, we show that ϕ satisfies the (PS) c * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq55_HTML.gif for every c ∈ ℝ. Suppose n j → ∞, u n j Y n j , φ ( u n j ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq57_HTML.gif and ( φ | Y n j ) ( u n j ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq58_HTML.gif, then { u n j } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq59_HTML.gif is a bounded sequence, otherwise, { u n j } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq59_HTML.gif would be unbounded sequence, passing to a subsequence, still denoted by { u n j } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq59_HTML.gif such that u n j 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq66_HTML.gif and u n j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq67_HTML.gif. Note that
        0 T ( 1 - γ p ( t ) ) ( | u ˙ n j ( t ) | p ( t ) + | u n j ( t ) | p ( t ) ) d t = φ ( u n j ) , u n j - γ φ ( u n j ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ20_HTML.gif
        (3.17)
        However, from (3.17), we have
        - γ φ ( u n j ) ( 1 - γ p - ) u n j p - - ( φ | Y n j ) ( u n j ) u n j , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equai_HTML.gif
        thus ||u n || is a bounded sequence in W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif. Going, if necessary, to a subsequence, we can assume that u n j u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq68_HTML.gif in W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif. As X = n j Y n j ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq69_HTML.gif, we can choose v n j Y n j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq70_HTML.gif such that v n j u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq71_HTML.gif. Hence
        lim n j φ ( u n j ) , u n j - u = lim n j φ ( u n j ) , u n j - v n j + lim n j φ ( u n j ) , v n j - u = lim n j ( φ | Y n j ) ( u n j ) , u n j - v n j = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equaj_HTML.gif

        In view of (3.4) and (3.5), we can also conclude u n j u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq72_HTML.gif, furthermore, we have φ ( u n j ) φ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq73_HTML.gif.

        Let us prove ϕ'(u) = 0 below. Taking arbitrarily ω k Y k , notice when n j k we have
        φ ( u ) , ω k = φ ( u ) - φ ( u n j ) , ω k + φ ( u n j ) , ω k (1) = φ ( u ) - φ ( u n j ) , ω k + ( φ | Y n j ) ( u n j ) , ω k . (2) (3) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equak_HTML.gif
        Going to limit in the right side of above equation reaches
        φ ( u ) , ω k = 0 , ω k Y k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equal_HTML.gif

        so ϕ'(u) = 0, this shows that ϕ satisfies the (PS) c * http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq55_HTML.gif for every c ∈ ℝ.

        For any finite dimensional subspace W W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq74_HTML.gif, there exists ε1 > 0 such that
        meas { t [ 0 , T ] : a ( t ) | u ( t ) | γ ε 1 u γ } ε 1 , u W \ { 0 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ21_HTML.gif
        (3.18)
        Otherwise, for any positive integer n, there exists u n W \ {0} such that
        meas { t [ 0 , T ] : a ( t ) | u n ( t ) | γ 1 n u n γ } < 1 n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equam_HTML.gif
        Set v n ( t ) : = u n ( t ) u n W \ { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq75_HTML.gif, then ||v n || = 1 for all n ∈ ℕ and
        meas { t [ 0 , T ] : a ( t ) | v n ( t ) | γ 1 n } < 1 n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ22_HTML.gif
        (3.19)

        Since dimW < ∞, it follows from the compactness of the unit sphere of W that there exists a subsequence, denoted also by {v n }, such that {v n } converges to some v0 in W. It is obvious that ||v0|| = 1.

        By the equivalence of the norms on the finite dimensional space W, we have v n v0 in L p - ( 0 , T ; N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq76_HTML.gif, i.e.,
        0 T | v n - v 0 | p - d t 0 as n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ23_HTML.gif
        (3.20)
        By (3.20) and Hölder inequality, we have
        0 T a ( t ) | v n - v 0 | γ d t ( 0 T a ( t ) p - p - - γ d t ) p - - γ p - ( 0 T | v n - v 0 | p - d t ) γ p - (1) = a p - - γ p - ( 0 T | v n - v 0 | p - d t ) γ p - 0 , as n . (2) (3) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ24_HTML.gif
        (3.21)
        Thus, there exist ξ1, ξ2 > 0 such that
        meas { t [ 0 , T ] : a ( t ) | v 0 ( t ) | γ ξ 1 } ξ 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ25_HTML.gif
        (3.22)
        In fact, if not, we have
        meas { t [ 0 , T ] : a ( t ) | v 0 ( t ) | γ 1 n } = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equan_HTML.gif

        for all positive integer n.

        It implies that
        0 0 T a ( t ) | v 0 | γ + p - d t < T n v 0 p - C 6 p - T n v 0 p - 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equao_HTML.gif
        as n → ∞, where C6 is the same in (3.3). Hence v0 = 0 which contradicts that ||v0|| = 1. Therefore, (3.22) holds. Now let
        Ω 0 = { t [ 0 , T ] : a ( t ) | v 0 ( t ) | γ ξ 1 } , Ω n = { t [ 0 , T ] : a ( t ) | v n ( t ) | γ < 1 n } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equap_HTML.gif

        and Ω n c = [ 0 , T ] \ Ω n = { t [ 0 , T ] : a ( t ) | v n ( t ) | γ 1 n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq77_HTML.gif.

        By (3.19) and (3.22), we have
        meas  ( Ω n Ω 0 ) = meas  ( Ω 0 \ ( Ω n c Ω 0 ) meas  ( Ω 0 ) meas ( Ω n c Ω 0 ) ξ 2 1 n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equaq_HTML.gif
        for all positive integer n. Let n be large enough such that
        ξ 2 - 1 n 1 2 ξ 2 and 1 2 γ - 1 ξ 1 - 1 n 1 2 γ ξ 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equar_HTML.gif
        then we have
        0 T a ( t ) | v n - v 0 | γ d t Ω n Ω 0 a ( t ) | v n - v 0 | γ d t (1) 1 2 γ - 1 Ω n Ω 0 a ( t ) | v 0 | γ d t - Ω n Ω 0 a ( t ) | v n | γ d t (2) ( 1 2 γ - 1 ξ 1 - 1 n ) meas ( Ω n Ω 0 ) (3) ξ 1 2 γ . ξ 2 2 = ξ 1 ξ 2 2 γ + 1 > 0 (4) (5) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equas_HTML.gif

        for all large n, which is a contradiction to (3.21). Therefore, (3.18) holds.

        For any uZ k , let
        u p - : = 0 T | u ( t ) | p - d t 1 p - and γ k : = sup u Z k , u = 1 u p - , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equat_HTML.gif
        then we conclude γ k → 0 as k → ∞ as in the proof of Theorem 3.1.
        φ ( u ) = 0 T 1 p ( t ) | u ˙ ( t ) | p ( t ) d t + 0 T 1 p ( t ) | u ( t ) | p ( t ) d t - 0 T F ( t , u ( t ) ) d t (1) 1 p + u p + - 0 T a ( t ) | u ( t ) | γ d t (2) 1 p + u p + - ( 0 T a ( t ) p - p - - γ d t ) p - - γ p - u p - γ (3) 1 p + u p + - ( 0 T a ( t ) p - p - - γ d t ) p - - γ p - γ k γ u γ . (4) (5) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equ26_HTML.gif
        (3.23)
        Let ρ k : = ( 2 c p + γ k γ ) 1 p + - γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq78_HTML.gif, where c : = ( 0 T a ( t ) p - p - - γ d t ) p - - γ p - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq79_HTML.gif, it is obvious that ρ k → 0, as k → ∞. In view of (3.23), We conclude
        inf u Z k , u = ρ k φ ( u ) 1 2 p + ρ k p + > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equau_HTML.gif

        so the condition (A7) in Lemma 2.15 is satisfied.

        Furthermore, by (3.23), for any uZ k with ||u|| ≤ ρ k , we have
        φ ( u ) - c γ k γ u γ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equav_HTML.gif
        Therefore,
        - c γ k γ ρ k γ inf u Z k , u ρ k φ ( u ) 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equaw_HTML.gif
        so we have
        inf u Z k , u ρ k φ ( u ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equax_HTML.gif

        for ρ k , γ k → 0, as k → ∞.

        For any uY k \ {0} with ||u|| ≤ 1,
        φ ( u ) = 0 T 1 p ( t ) | u ˙ ( t ) | p ( t ) d t + 0 T 1 p ( t ) | u ( t ) | p ( t ) d t - 0 T a ( t ) | u ( t ) | γ d t (1) 1 p - u p - - 0 T a ( t ) | u ( t ) | γ d t (2) 1 p - u p - - ε 1 u γ meas ( Ω u ) (3) 1 p - u p - - ε 1 2 u γ , (4) (5) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equay_HTML.gif
        where ε1 is given in (3.18), and
        Ω u : = meas { t [ 0 , T ] : a ( t ) | u ( t ) | γ ε 1 u γ } ε 1 , u Y k \ { 0 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equaz_HTML.gif
        Choosing 0 < r k < min { ρ k , ( p - ε 2 2 ) 1 p - - γ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq80_HTML.gif, we conclude
        i k : = max u Y k , u = r k φ ( u ) < - 1 p - r k p - < 0 k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equba_HTML.gif

        i.e., the condition (A6) in Lemma 2.15 is satisfied. The proof of Theorem 3.3 is complete.

        4. Example

        In this section, we give three examples to illustrate our results.

        Example 4.1. In system (1.1), let F ( t , x ) = | x | 8 + T 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq81_HTML.gif and
        p ( t ) = 7 + t , 0 t T 2 , - t + T + 7 , T 2 < t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equbb_HTML.gif
        Choose
        β = 8 + T 2 , r = 2 , μ = μ = 8 + T 2 and Q = Q = 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equbc_HTML.gif

        so it is easy to verify that all the conditions (B1)-(B4) are satisfied. Then by Theorem 3.1, system (1.1) has infinite solutions {u k } in W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif for every positive integer k such that ||u k || → +∞, as k → ∞.

        Example 4.2. In system (1.1), let F(t, x) = |x|8 and
        p ( t ) = { 5 , 0 t T / 2 5 + sin 2 π t T , T / 2 < t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equbd_HTML.gif

        We choose β = 1 3 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq82_HTML.gif, r = 2, μ' = 8, r1 = 7, Q' = 1 and M = 1, so it is easy to verify that all the conditions of Theorem 3.2 are satisfied. Then by Theorem 3.2, so system (1.1) has infinite solutions {u k } in W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif for every positive integer k such that ||u k || → +∞, as k → ∞.

        Example 4.3. In system (1.1), let F(t, x) = a(t)|x|3 where
        a ( t ) = { T , t = 0 t , 0 < t T , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Eqube_HTML.gif
        and
        p ( t ) = { 5 , 0 t T / 2 5 + sin 2 π t T , T / 2 < t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_Equbf_HTML.gif

        It is easy to verify that all the conditions of Theorem 3.3 are satisfied. Then by Theorem 3.3, so system (1.1) has infinite solutions {u k } in W T 1 , p ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-33/MediaObjects/13661_2011_Article_79_IEq3_HTML.gif for every positive integer k.

        Declarations

        Acknowledgements

        The authors thank the anonymous referees for valuable suggestions and comments which led to improve this article. This Project is supported by the National Natural Science Foundation of China (Grant No. 11171351).

        Authors’ Affiliations

        (1)
        School of Mathematical Sciences and Computing Technology, Central South University

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