Existence results for a class of nonlocal problems involving p-Laplacian

  • Yang Yang1Email author and

    Affiliated with

    • Jihui Zhang2

      Affiliated with

      Boundary Value Problems20112011:32

      DOI: 10.1186/1687-2770-2011-32

      Received: 7 January 2011

      Accepted: 11 October 2011

      Published: 11 October 2011

      Abstract

      This paper is concerned with the existence of solutions to a class of p-Kirchhoff type equations with Neumann boundary data as follows:

      - M Ω u p d x p - 1 Δ p u = f ( x , u ) , in Ω ; u υ = 0 , on Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equa_HTML.gif

      By means of a direct variational approach, we establish conditions ensuring the existence and multiplicity of solutions for the problem.

      Keywords

      Nonlocal problems Neumann problem p-Kirchhoff's equation

      1. Introduction

      In this paper, we deal with the nonlocal p-Kirchhoff type of problem given by:
      - M Ω u p d x p - 1 Δ p u = f ( x , u ) , in Ω ; u υ = 0 , on Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equ1_HTML.gif
      (1.1)
      where Ω is a smooth bounded domain in R N , 1 < p < N, ν is the unit exterior vector on ∂Ω, Δ p is the p-Laplacian operator, that is, Δ p u = div(|∇u|p−2u), the function M : R+R+ is a continuous function and there is a constant m0 > 0, such that
      ( M 0 ) M ( t ) m 0 for all t 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equb_HTML.gif
      f ( x , t ) : Ω ¯ × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq1_HTML.gif is a continuous function and satisfies the subcritical condition:
      f ( x , t ) C ( t q - 1 + 1 ) , for some p < q < p * = N p N - p , N 3 ; + , N = 1 , 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equ2_HTML.gif
      (1.2)

      where C denotes a generic positive constant.

      Problem (1.1) is called nonlocal because of the presence of the term M, which implies that the equation is no longer a pointwise identity. This provokes some mathematical difficulties which makes the study of such a problem particulary interesting. This problem has a physical motivation when p = 2. In this case, the operator M(∫Ω|∇u|2dxu appears in the Kirchhoff equation which arises in nonlinear vibrations, namely
      u t t - M ( Ω u 2 d x ) Δ u = f ( x , u ) , in Ω × ( 0 , T ) ; u = 0 , on Ω × ( 0 , T ) ; u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equc_HTML.gif

      P-Kirchhoff problem began to attract the attention of several researchers mainly after the work of Lions [1], where a functional analysis approach was proposed to attack it. The reader may consult [28] and the references therein for similar problem in several cases.

      This work is organized as follows, in Section 2, we present some preliminary results and in Section 3 we prove the main results.

      2. Preliminaries

      By a weak solution of (1.1), then we say that a function u ε W1,p(Ω) such that
      M Ω u p d x p - 1 Ω u p - 2 u φ d x = Ω f ( x , u ) φ d x , for all φ W 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equd_HTML.gif
      So we work essentially in the space W1,p(Ω) endowed with the norm
      u = Ω ( u p + u p ) d x 1 p , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Eque_HTML.gif
      and the space W1,p(Ω) may be split in the following way. Let W c = 〈1〉, that is, the subspace of W1,p(Ω) spanned by the constant function 1, and W 0 = { z W 1 , p ( Ω ) , Ω z = 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq2_HTML.gif, which is called the space of functions of W1,p(Ω) with null mean in Ω. Thus
      W 1 , p ( Ω ) = W 0 W c . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equf_HTML.gif

      As it is well known the Poincaré's inequality does not hold in the space W1,p(Ω). However, it is true in W0.

      Lemma 2.1[8] (Poincaré-Wirtinger's inequality) There exists a constant η > 0 such that Ω z p d x η Ω z p d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq3_HTML.giffor all zW0.

      Let us also recall the following useful notion from nonlinear operator theory. If X is a Banach space and A : XX* is an operator, we say that A is of type (S+), if for every sequence {x n }n≥1X such that x n x weakly in X, and lim sup n A ( x n ) , x n - x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq4_HTML.gif. we have that x n x in X.

      Let us consider the map A : W1,p(Ω) → W1,p(Ω)* corresponding to −Δ p with Neumann boundary data, defined by
      A ( u ) , v = Ω u p - 2 u v d x , u , v W 1 , p ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equ3_HTML.gif
      (2.1)

      We have the following result:

      Lemma 2.2[9, 10]The map A : W1,p(Ω) → W1,p(Ω)* defined by (2.1) is continuous and of type (S+).

      In the next section, we need the following definition and the lemmas.

      Definition 2.1. Let E be a real Banach space, and D an open subset of E. Suppose that a functional J : DR is Fréchet differentiable on D. If x0D and the Fréchet derivative J' (x0) = 0, then we call that x0is a critical point of the functional J and c = J(x0) is a critical value of J.

      Definition 2.2. For JC1(E, R), we say J satisfies the Palais-Smale condition (denoted by (PS)) if any sequence {u n } ⊂ E for which J(u n ) is bounded and J'(u n ) → 0 as n → ∞ possesses a convergent subsequence.

      Lemma 2.3[11]Let X be a Banach space with a direct sum decomposition X = X1X2, with k = dimX2 < ∞, let J be a C1function on X, satisfying (PS) condition. Assume that, for some r > 0,
      J ( u ) 0 f o r u X 1 , u r ; J ( u ) 0 f o r u X 2 , u r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equg_HTML.gif

      Assume also that J is bounded below and inf X J < 0. Then J has at least two nonzero critical points.

      Lemma 2.4[12]Let X = X1X2, where X is a real Banach space and X2 ≠ {0}, and is finite dimensional. Suppose JC1(X, R) satisfies (PS) and

      (i) there is a constant α and a bounded neighborhood D of 0 in X2such that J| ∂D ≤ α and,

      (ii) there is a constant β > α such that J X 1 β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq5_HTML.gif,

      then J possesses a critical value c ≥ β, moreover, c can be characterized as
      c = inf h Γ max u D ¯ J ( h ( u ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equh_HTML.gif

      where Γ = { h C ( D ¯ , X ) h = i d o n D } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq6_HTML.gif.

      Definition 2.3. For JC1(E, R), we say J satisfies the Cerami condition (denoted by (C)) if any sequence {u n } ⊂ E for which J(u n ) is bounded and (1 ||u n ||) J'(u n )|| → 0 as n → ∞ possesses a convergent subsequence.

      Remark 2.1 If J satisfies the (C) condition, Lemma 2.4 still holds.

      In the present paper, we give an existence theorem and a multiplicity theorem for problem (1.1). Our main results are the following two theorems.

      Theorem 2.1 If following hold:

      (F0) 0 lim u 0 p F ( x , u ) u p < m 0 p - 1 η a . e . x Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq7_HTML.gif, where F ( x , u ) = 0 u f ( x , s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq8_HTML.gif, η appears in Lemma 2.1 ;

      (F1) lim u p F ( x , u ) u p 0 a . e . x Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq9_HTML.gif;

      (F2) lim u Ω F ( x , u ) d x = - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq10_HTML.gif.

      Then the problem (1.1) has least three distinct weak solutions in W1,p(Ω).

      Theorem 2.2 If the following hold:

      (M1) The function M that appears in the classical Kirchhoff equation satisfies M ^ ( t ) ( M ( t ) ) p - 1 t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq11_HTML.giffor all t ≥ 0, where M ^ ( t ) = 0 t [ M ( s ) ] p - 1 d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq12_HTML.gif;

      (F3) f ( x , u ) u > 0 f o r a l l u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq13_HTML.gif;

      (F4) lim u p F ( x , u ) u p = 0 a . e . x Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq14_HTML.gif;

      (F5) lim u ( f ( x , u ) u - p F ( x , u ) ) = - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq15_HTML.gif.

      Then the problem (1.1) has at least one weak solution in W1,p(Ω).

      Remark 2.2 We exhibit now two examples of nonlinearities that fulfill all of our hypotheses
      f ( x , u ) = m 0 p - 1 2 η u p - 2 u - u q - 2 u , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equi_HTML.gif
      hypotheses (F0), (F1), (F2) and (1.2) are clearly satisfied.
      f ( x , u ) = a r c t a n u + u 1 + u 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equj_HTML.gif

      hypotheses (F3), (F4) and (F5) and (1.2) are clearly satisfied.

      3. Proofs of the theorems

      Let us start by considering the functional J : W1,p(Ω) → R given by
      J ( u ) = 1 p M ^ Ω u p d x - Ω F ( x , u ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equk_HTML.gif

      Proof of Theorem 2.1 By (F0), we know that f(x, 0) = 0, and hence u(x) = 0 is a solution of (1.1).

      To complete the proof we prove the following lemmas.

      Lemma 3.1 Any bounded (PS) sequence of J has a strongly convergent subsequence.

      Proof: Let {u n } be a bounded (PS) sequence of J. Passing to a subsequence if necessary, there exists uW1,p(Ω) such that u n u. From the subcritical growth of f and the Sobolev embedding, we see that
      Ω f ( x , u n ) ( u n - u ) d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equl_HTML.gif
      and since J'(u n )(u n u) → 0, we conclude that
      M Ω u n p d x p - 1 Ω u n p - 2 u n ( u n - u ) d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equm_HTML.gif
      In view of condition (M0), we have
      Ω u n p - 2 u n ( u n - u ) d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equn_HTML.gif

      Using Lemma 2.2, we have u n u as n → ∞. □

      Lemma 3.2 If condition (M0), (F1) and (F2) hold, then lim u J ( u ) = + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq16_HTML.gif.

      Proof: If there are a sequence {u n } and a constant C such that ||u n || → ∞ as n → ∞, and J(u n ) ≤ C (n = 1, 2 ···), let v n = u n u n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq17_HTML.gif, then there exist v0W1,p(Ω) and a subsequence of {v n }, we still note by {v n }, such that v n v0 in W1,p(Ω) and v n v0 in L p (Ω).

      For any ε > 0, by (F1), there is a H > 0 such that F ( x , u ) ε p u p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq18_HTML.gif for all |u|H and a.e. x ∈ Ω, then there exists a constant C > 0 such that F ( x , u ) ε p u p + C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq19_HTML.gif for all uR, and a.e. x ∈ Ω, Consequently
      C u n p J ( u n ) u n p = 1 u n p 1 p M ^ Ω u n p d x - Ω F ( x , u n ) d x 1 p m 0 p - 1 Ω v n p d x - ε p Ω v n p d x - C Ω u n p = 1 p m 0 p - 1 - 1 p m 0 p - 1 + ε p Ω v n p d x - C Ω u n p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equo_HTML.gif
      It implies Ω|v0| p dx ≥ 1. On the other hand, by the weak lower semi-continuity of the norm, one has
      v 0 lim inf n v n = 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equp_HTML.gif
      Hence Ω v 0 p d x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq20_HTML.gif, so |v0(x)| = constant ≠ 0 a.e. x ∈ Ω. By (F2), lim u n Ω F ( x , u n ) d x - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq21_HTML.gif. Hence
      C J ( u n ) = 1 p M ^ Ω u n p d x - Ω F ( x , u n ) d x - Ω F ( x , u n ) d x + a s n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equq_HTML.gif

      This is a contradiction. Hence J is coercive on W1,p(Ω), bounded from below, and satisfies the (PS) condition. □

      By Lemma 3.1 and 3.2, we know that J is coercive on W1,p(Ω), bounded from below, and satisfies the (PS) condition. From condition (F0), we know, there exist r > 0, ε > 0 such that
      0 F ( x , u ) m 0 p - 1 p η - ε u p , f o r u r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equr_HTML.gif
      If uW c , for ||u|| ≤ ρ1, then |u| r, we have
      J ( u ) = 1 p M ^ Ω u p d x - Ω F ( x , u ) d x = - Ω F ( x , u ) d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equs_HTML.gif
      If uW0, then from condition (F0) and (1.2), we have
      F ( x , u ) m 0 p - 1 p η - ε u p + C u q , f o r u R , q ( p , p * ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equt_HTML.gif
      Noting that
      Ω u p d x η Ω u p d x , u W 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equu_HTML.gif
      we can obtain
      J ( u ) = 1 p M ^ Ω u p d x - Ω F ( x , u ) d x 1 p m 0 p - 1 Ω u p d x - m 0 p - 1 p η Ω u p d x + ε Ω u p d x - C Ω u q d x C ε u p - C C 1 u q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equv_HTML.gif

      Choose ||u|| = ρ2 small enough, such that J(u) ≥ 0 for ||u|| ≤ ρ2 and uW0.

      Now choose ρ = min{ρ1, ρ2}, then, we have
      J ( u ) 0 f o r u W c , u ρ ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equw_HTML.gif
      J ( u ) 0 f o r u W 0 , u ρ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equx_HTML.gif

      If inf{J(u), uW1,p(Ω)} = 0, then all uW c with ||u|| ≤ ρ are minimum of J, which implies that J has infinite critical points. If inf{J(u), uW1,p(Ω)} < 0 then by Lemma 2.3, J has at least two nontrivial critical points. Hence problem (1.1) has at least two nontrivial solutions in W1,p(Ω), Therefore, problem (1.1) has at least three distinct solutions in W1,p(Ω). □

      Proof of Theorem 2.2. We divide the proof into several lemmas.

      Lemma 3.3 If condition (F3) and (F5) hold, then J W c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq22_HTML.gifis anticoercive. (i.e. we have that J(u) → -∞, as |u| → ∞, uR.)

      Proof: By virtue of hypothesis (F5), for any given L > 0, we can find R1 = R1(L) > 0 such that
      F ( x , u ) 1 p L + 1 p f ( x , u ) u , f o r a . e . x Ω , u > R 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equy_HTML.gif
      Thus, using hypothesis (F3), we have
      F ( x , u ) 1 p L - C , f o r a . e . x Ω u R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equz_HTML.gif
      So
      Ω F ( x , u ) d x 1 p L Ω - C Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equaa_HTML.gif
      Since L > 0 is arbitrary, it follows that
      Ω F ( x , u ) d x , a s u , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equab_HTML.gif
      and so
      J ( u ) W C = - Ω F ( x , u ) d x - , a s u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equac_HTML.gif

      This proves that J W c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq22_HTML.gif is anticoercive. □

      Lemma 3.4 If hypothesis (F4) holds, then J W 0 - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq23_HTML.gif.

      Proof: For a given 0 < ε < m 0 p - 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq24_HTML.gif, we can find C ε > 0 such that F ( x , u ) ε p η u p + C ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq25_HTML.gif for a.e. x ∈ Ω all uR. Then
      J ( u ) u W 0 = 1 p M ^ Ω u p d x - Ω F ( x , u ) d x 1 p m 0 p - 1 Ω u p d x - m 0 p - 1 p η Ω u p d x - C Ω - C Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equad_HTML.gif

      then J W 0 - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq23_HTML.gif. □

      Lemma 3.5 If condition (F4) (F5) hold, then J satisfies the (C) condition.

      Proof: Let {u n }n ≥1W1,p(Ω) be a sequence such that
      J ( u n ) M 1 , n 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equ4_HTML.gif
      (3.1)
      with some M1 > 0 and
      ( 1 + u n ) J ( u n ) 0 , in W 1 , p ( Ω ) * a s n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equ5_HTML.gif
      (3.2)
      We claim that the sequence {u n } is bounded. We argue by contradiction. Suppose that ||u|| → +∞, as n → ∞, we set v n = u n u n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq17_HTML.gif, ∀n ≥ 1. Then ||v n || = 1 for all n ≥ 1 and so, passing to a subsequence if necessary, we may assume that
      v n v in W 1 , p ( Ω ) ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equae_HTML.gif
      v n v in L p ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equaf_HTML.gif
      from (3.2), we have ∀hW1,p(Ω)
      M Ω u n p d x p - 1 Ω v n p - 2 v n h d x - Ω f ( x , u n ) h u n p - 1 d x ε n 1 + u n h u n p - 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equ6_HTML.gif
      (3.3)

      with ε n ↓ 0.

      In (3.3), we choose h = v n vW1,p(Ω), note that by virtue of hypothesis (F4), we have
      f ( x , u n ) u n p - 1 0 in L p ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equag_HTML.gif

      where 1 p + 1 p = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq26_HTML.gif.

      So we have
      M Ω u n p d x p - 1 Ω v n p - 2 v n ( v n - v ) d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equah_HTML.gif
      Since M(t) > m0 for all t ≥ 0, so we have
      Ω v n p - 2 v n ( v n - v ) d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equai_HTML.gif
      Hence, using the (S+) property, we have v n v in W1,p(Ω) with ||v|| = 1, then v ≠ 0. Now passing to the limit as n → ∞ in (3.3), we obtain
      Ω v p - 2 v h d x 0 , h W 1 , p ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equaj_HTML.gif

      then v = ξ ∈ R. Then |u n (x)| → +∞ as n → +∞. Using hypothesis (F5), we have f(x, u n (x))u n (x) - pF(x, u n (x)) → -∞ for a.e x ∈ Ω.

      Hence by virtue of Fatou's Lemma, we have
      Ω f ( x , u n ) u n - p F ( x , u n ) d x - , a s n + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equ7_HTML.gif
      (3.4)
      From (3.1), we have
      M ^ Ω u n p d x - p Ω F ( x , u n ) d x - p M 1 , n 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equ8_HTML.gif
      (3.5)
      From (3.2), we have
      M Ω u n p d x p - 1 Ω u n p - 2 u n h d x - Ω f ( x , u n ) h d x ε n h 1 + u n h W 1 , p ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equak_HTML.gif
      With ε n ↓ 0. So choosing h = u n W1,p(Ω), we obtain
      - M ( Ω u n p d x ) p - 1 Ω u n p d x + Ω f ( x , u n ) u n d x - ε n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equ9_HTML.gif
      (3.6)
      Adding (3.5) and (3.6), noting that M ^ ( t ) ( M ( t ) ) p - 1 t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_IEq11_HTML.gif for all t ≥ 0, we obtain
      Ω ( f ( x , u n ) u n - p F ( x , u n ) ) d x - M 2 , n 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-32/MediaObjects/13661_2011_Article_78_Equ10_HTML.gif
      (3.7)

      comparing (3.4) and (3.7), we reach a contradiction. So {u n }in bounded in W1,p(Ω). Similar with the proof of Lemma 3.1, we know that J satisfied the (C) condition. □

      Sum up the above fact, from Lemma 2.4 and Remark 2.1, Theorem 2.2 follows from the Lemma 3.3 to 3.5.

      Declarations

      Acknowledgements

      The authors would like to thank the referees for their valuable comments and suggestions.

      This study was supported by NSFC (No. 10871096), the Fundamental Research Funds for the Central Universities (No. JUSRP11118).

      Authors’ Affiliations

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

      References

      1. Lions JL: On some equations in boundary value problems of mathematical physics. In Contemporary developments in Continuum Mechanics and Partial Differential equations (Proc. Internat. Sympos., Inst. Mat., Univ. fed. Rio de Janeiro, Riio de Janeiro, 1977), North-Holland Mathematics Studies. Volume 30. North-Holland, Amsterdam; 1978:284-346.
      2. Alves CO, Corrêa FJSA, Ma TF: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput Math Appl 2005, 49(1):85-93. 10.1016/j.camwa.2005.01.008View ArticleMathSciNetMATH
      3. Ma TF, Rivera JEM: Positive solutions for a nonlinear elliptic transmission problem. Appl Math Lett 2003, 16(2):243-248. 10.1016/S0893-9659(03)80038-1View ArticleMathSciNetMATH
      4. Corrêa FJSA, Figueiredo GM: On an elliptic equation of p-Kirchhoff type via variational methods. Bull Austral Math Soc 2006, 74: 263-277. 10.1017/S000497270003570XView ArticleMathSciNetMATH
      5. Perera K, Zhang ZT: Nontrivial solutions of Kirchhoff-type problems via the Yang-index. J Differ Equ 2006, 221(1):246-255. 10.1016/j.jde.2005.03.006View ArticleMathSciNetMATH
      6. Zhang ZT, Perera K: Sign-changing solutions of Kirchhoff type problems via invariant sets of descent flow. J Math Anal Appl 2006, 317(2):456-463. 10.1016/j.jmaa.2005.06.102View ArticleMathSciNetMATH
      7. Mao AM, Zhang ZT: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal 2009, 70: 1275-1287. 10.1016/j.na.2008.02.011View ArticleMathSciNetMATH
      8. Corrêa FJSA, Nascimento RG: On a nonlocal elliptic system of p-Kirchhoff type under Neumann boundary condition. Math Comput Model 2009, 49: 598-604. 10.1016/j.mcm.2008.03.013View ArticleMATH
      9. Gasiński L, Papageorgiou NS: Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems. Chapman and hall/CRC Press, Boca Raton; 2005.
      10. Gasiński L, Papageorgiou NS: Nontrivial solutions for a class of resonant p-Laplacian Neumann problems. Nonlinear Anal 2009, 71: 6365-6372. 10.1016/j.na.2009.06.039View ArticleMathSciNetMATH
      11. Brezis H, Nirenberg L: Remarks on finding critical points. Commun Pure Appl Math 1991, 44: 939-963. 10.1002/cpa.3160440808View ArticleMathSciNetMATH
      12. Rabinowitz PH: Minimax methods in critical point theory with applications to differential equations. In CBMS Regional Conference Series in Mathematics. Volume 65. American Mathematical Soceity, Providence; 1986.

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      © Yang and Zhang; licensee Springer. 2011

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