Positivity of the infimum eigenvalue for equations of p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq1_HTML.gif-Laplace type in R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq2_HTML.gif

  • In Hyoun Kim1 and

    Affiliated with

    • Yun-Ho Kim2Email author

      Affiliated with

      Boundary Value Problems20132013:214

      DOI: 10.1186/1687-2770-2013-214

      Received: 29 June 2013

      Accepted: 21 August 2013

      Published: 2 October 2013

      Abstract

      We study the following elliptic equations with variable exponents

      div ( ϕ ( x , | u | ) u ) = λ f ( x , u ) in  R N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equa_HTML.gif

      Under suitable conditions on ϕ and f, we show the existence of positivity of the infimum of all eigenvalues for the problem above, and then give an example to demonstrate our main result.

      MSC:35D30, 35J60, 35J92, 35P30, 47J10.

      Keywords

      p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq1_HTML.gif-Laplacian variable exponent Lebesgue-Sobolev spaces weak solution eigenvalue

      1 Introduction

      The variable exponent problems appear in a lot of applications, for example, elastic mechanics, electro-rheological fluid dynamics and image processing, etc. The study of variable mathematical problems involving p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq1_HTML.gif-growth conditions has attracted interest and attention in recent years. We refer the readers to [14] and references therein.

      In this paper, we are concerned with the eigenvalue problem of a class of equations of p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq1_HTML.gif-Laplacian type
      div ( ϕ ( x , | u | ) u ) = λ f ( x , u ) in  R N , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ1_HTML.gif
      (E)

      where the function ϕ ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq3_HTML.gif is of type | t | p ( x ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq4_HTML.gif with continuous nonconstant function p : R N ( 1 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq5_HTML.gif and f : R N × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq6_HTML.gif satisfies a Carathéodory condition. Recently, the authors in [5] obtained the positivity of the infimum of all eigenvalues for the p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq1_HTML.gif-Laplacian type subject to the Dirichlet boundary condition. As far as the authors know, there are no results concerned with the eigenvalue problem for a more general p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq1_HTML.gif-Laplacian type problem in the whole space R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq7_HTML.gif.

      When ϕ ( x , t ) = | t | p ( x ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq8_HTML.gif, the operator involved in (E) is called the p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq1_HTML.gif-Laplacian, i.e., Δ p ( x ) u : = div ( | u | p ( x ) 2 u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq9_HTML.gif. The studies for p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq1_HTML.gif-Laplacian problems have been extensively performed by many researchers in various ways; see [511]. In particular, by using the Ljusternik-Schnirelmann critical point theory, Fan et al. [8] established the existence of the sequence of eigenvalues of the p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq1_HTML.gif-Laplacian Dirichlet problem; see [12] for Neumann problems. Mihăilescu and Rădulescu in [13] obtained the existence of a continuous family of eigenvalues in a neighborhood of the origin under suitable conditions.

      The p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq1_HTML.gif-Laplacian is a natural generalization of the p-Laplacian, where p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq10_HTML.gif is a constant. There are a bunch of papers, for instance, [1418] and references therein. But the p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq1_HTML.gif-Laplace operator possesses more complicated nonlinearities than the p-Laplace operator, for example, it is nonhomogeneous, so a more complicated analysis has to be carefully carried out. Some properties of the p-Laplacian eigenvalue problems may not hold for a general p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq1_HTML.gif-Laplacian. For example, under some conditions, the infimum of all eigenvalues for the p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq1_HTML.gif-Laplacian might be zero; see [8]. The purpose of this paper is to give suitable conditions on ϕ and f to satisfy the positivity of the infimum of all eigenvalues for (E) still. This result generalizes Benouhiba’s recent result in [6] in some sense.

      This paper is organized as follows. In Section 2, we state some basic results for the variable exponent Lebesgue-Sobolev spaces, which are given in [19, 20]. In Section 3, we give sufficient conditions on ϕ and f to obtain the positivity of the infimum eigenvalue for the problem (E) above. Also, we present an example to illustrate our main result.

      2 Preliminaries

      In this section, we state some elementary properties for the variable exponent Lebesgue-Sobolev spaces, which will be used in the next section. The basic properties of the variable exponent Lebesgue-Sobolev spaces can be found from [19, 20].

      To make a self-contained paper, we first recall some definitions and basic properties of the variable exponent Lebesgue spaces L p ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq11_HTML.gif and the variable exponent Lebesgue-Sobolev spaces W 1 , p ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq12_HTML.gif.

      Set
      C + ( R N ) = { h C ( R N ) : inf x R N h ( x ) > 1 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equb_HTML.gif
      For any h C + ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq13_HTML.gif, we define
      h + = sup x R N h ( x ) and h = inf x R N h ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equc_HTML.gif
      For any p C + ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq14_HTML.gif, we introduce the variable exponent Lebesgue space
      L p ( x ) ( R N ) : = { u : u  is a measurable real-valued function,  R N | u ( x ) | p ( x ) d x < } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equd_HTML.gif
      endowed with the Luxemburg norm
      u L p ( x ) ( R N ) = inf { λ > 0 : R N | u ( x ) λ | p ( x ) d x 1 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Eque_HTML.gif

      The dual space of L p ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq11_HTML.gif is L p ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq15_HTML.gif, where 1 / p ( x ) + 1 / p ( x ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq16_HTML.gif. The variable exponent Lebesgue spaces are a special case of Orlicz-Musielak spaces treated by Musielak in [21].

      The variable exponent Sobolev space W 1 , p ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq12_HTML.gif is defined by
      W 1 , p ( x ) ( R N ) = { u L p ( x ) ( R N ) : | u | L p ( x ) ( R N ) } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equf_HTML.gif
      where the norm is
      u W 1 , p ( x ) ( R N ) = u L p ( x ) ( R N ) + u L p ( x ) ( R N ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ2_HTML.gif
      (2.1)
      Definition 2.1 The exponent p ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq17_HTML.gif is said to be log-Hölder continuous if there is a constant C such that
      | p ( x ) p ( y ) | C log | x y | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ3_HTML.gif
      (2.2)

      for every x , y R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq18_HTML.gif with | x y | 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq19_HTML.gif.

      Smooth functions are not dense in the variable exponent Sobolev spaces, without additional assumptions on the exponent p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq1_HTML.gif. Zhikov [22] gave some examples of Lavrentiev’s phenomenon for the problems with variable exponents. These examples show that smooth functions are not dense in variable exponent Sobolev spaces. However, when p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq1_HTML.gif satisfies the log-Hölder continuity condition, smooth functions are dense in variable exponent Sobolev spaces, and there is no confusion in defining the Sobolev space with zero boundary values, W 0 1 , p ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq20_HTML.gif, as the completion of C 0 ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq21_HTML.gif with respect to the norm u W 1 , p ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq22_HTML.gif (see [23, 24]).

      Lemma 2.2 [19, 20]

      The space L p ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq11_HTML.gif is a separable, uniformly convex Banach space, and its conjugate space is L p ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq15_HTML.gif, where 1 / p ( x ) + 1 / p ( x ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq23_HTML.gif. For any u L p ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq24_HTML.gif and v L p ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq25_HTML.gif, we have
      | R N u v d x | ( 1 p + 1 p ) u L p ( x ) ( R N ) v L p ( x ) ( R N ) 2 u L p ( x ) ( R N ) v L p ( x ) ( R N ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equg_HTML.gif

      Lemma 2.3 [19]

      Denote
      ρ ( u ) = R N | u | p ( x ) d x for all  u L p ( x ) ( R N ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equh_HTML.gif
      Then
      1. (1)

        ρ ( u ) > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq26_HTML.gif (=1; <1) if and only if u L p ( x ) ( R N ) > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq27_HTML.gif (=1; <1), respectively;

         
      2. (2)

        if u L p ( x ) ( R N ) > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq27_HTML.gif, then u L p ( x ) ( R N ) p ρ ( u ) u L p ( x ) ( R N ) p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq28_HTML.gif;

         
      3. (3)

        if u L p ( x ) ( R N ) < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq29_HTML.gif, then u L p ( x ) ( R N ) p + ρ ( u ) u L p ( x ) ( R N ) p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq30_HTML.gif.

         

      Lemma 2.4 [11]

      Let q L ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq31_HTML.gif be such that 1 p ( x ) q ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq32_HTML.gif for almost all x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq33_HTML.gif. If u L q ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq34_HTML.gif with u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq35_HTML.gif, then
      1. (1)

        if u L p ( x ) q ( x ) ( R N ) > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq36_HTML.gif, then u L p ( x ) q ( x ) ( R N ) q | u | q ( x ) L p ( x ) ( R N ) u L p ( x ) q ( x ) ( R N ) q + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq37_HTML.gif;

         
      2. (2)

        if u L p ( x ) q ( x ) ( R N ) < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq38_HTML.gif, then u L p ( x ) q ( x ) ( R N ) q + | u | q ( x ) L p ( x ) ( R N ) u L p ( x ) q ( x ) ( R N ) q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq39_HTML.gif.

         

      Lemma 2.5 [23]

      Let Ω R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq40_HTML.gif be an open, bounded set with Lipschitz boundary, and let p C + ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq41_HTML.gif with 1 < p p + < N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq42_HTML.gif satisfy the log-Hölder continuity condition (2.2). If q L ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq43_HTML.gif with q > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq44_HTML.gif satisfies
      q ( x ) p ( x ) : = N p ( x ) N p ( x ) for all  x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equi_HTML.gif
      then we have
      W 1 , p ( x ) ( Ω ) L q ( x ) ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equj_HTML.gif

      and the imbedding is compact if inf x Ω ( p ( x ) q ( x ) ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq45_HTML.gif.

      Lemma 2.6 [25]

      Suppose that p : R N R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq46_HTML.gif is a Lipschitz function with 1 < p p + < N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq42_HTML.gif. Let q L ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq31_HTML.gif and p ( x ) q ( x ) p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq47_HTML.gif for almost all x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq33_HTML.gif. Then there is a continuous embedding W 1 , p ( x ) ( R N ) L q ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq48_HTML.gif.

      3 Main result

      In this section, we shall give the proof of the existence of the positive eigenvalue for the problem (E), by applying the basic properties of the spaces L p ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq11_HTML.gif and W 1 , p ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq12_HTML.gif, which were given in the previous section.

      Throughout this paper, let p C + ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq14_HTML.gif satisfy the log-Hölder continuity condition (2.2) and X : = W 0 1 , p ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq49_HTML.gif with the norm
      u X = inf { λ > 0 : R N | u ( x ) λ | p ( x ) d x 1 } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equk_HTML.gif

      which is equivalent to norm (2.1).

      Definition 3.1 We say that u X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq50_HTML.gif is a weak solution of the problem (E) if
      R N ϕ ( x , | u | ) u ( x ) φ ( x ) d x = λ R N f ( x , u ) φ ( x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equl_HTML.gif

      for all φ X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq51_HTML.gif.

      Denote
      Ω 1 = { x R N : 1 < p ( x ) < 2 } , Ω 2 = { x R N : p ( x ) 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equm_HTML.gif

      (we allow the case that one of these sets is empty). Then it is obvious that R N = Ω 1 Ω 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq52_HTML.gif. We assume that:

      (H1) p , q C + ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq53_HTML.gif, p ( x ) < N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq54_HTML.gif, and 1 < p p + < q q + < p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq55_HTML.gif.

      1. (HJ1)

        ϕ : R N × [ 0 , ) [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq56_HTML.gif satisfies the following conditions: ϕ ( , η ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq57_HTML.gif is measurable on R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq7_HTML.gif for all η 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq58_HTML.gif and ϕ ( x , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq59_HTML.gif is locally absolutely continuous on [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq60_HTML.gif for almost all x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq33_HTML.gif.

         
      2. (HJ2)
        There are a function a L p ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq61_HTML.gif and a nonnegative constant b such that
        | ϕ ( x , | v | ) v | a ( x ) + b | v | p ( x ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equn_HTML.gif

        for almost all x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq33_HTML.gif and for all v R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq62_HTML.gif.

         
      3. (HJ3)
        There exists a positive constant c such that the following conditions are satisfied for almost all x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq33_HTML.gif:
        ϕ ( x , η ) c η p ( x ) 2 and η ϕ η ( x , η ) + ϕ ( x , η ) c η p ( x ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ4_HTML.gif
        (3.1)
        for almost all η ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq63_HTML.gif. In case x Ω 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq64_HTML.gif, assume that condition (3.1) holds for almost all η ( 1 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq65_HTML.gif, and in case x Ω 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq66_HTML.gif, assume that for almost all η ( 1 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq65_HTML.gif instead
        ϕ ( x , η ) c and η ϕ η ( x , η ) + ϕ ( x , η ) c . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ5_HTML.gif
        (3.2)
         
      4. (HJ4)
        For all x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq33_HTML.gif and all ξ R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq67_HTML.gif, the estimate holds
        0 a ( x , ξ ) ξ p + Φ 0 ( x , | ξ | ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equo_HTML.gif

        where a ( x , ξ ) = ϕ ( x , | ξ | ) ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq68_HTML.gif.

         
      Let us put
      Φ 0 ( x , t ) = 0 t ϕ ( x , η ) η d η http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equp_HTML.gif
      and define the functional Φ : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq69_HTML.gif by
      Φ ( u ) = R N Φ 0 ( x , | u ( x ) | ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equq_HTML.gif
      Then Φ C 1 ( X , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq70_HTML.gif [5], and its Gateaux derivative is
      Φ ( u ) , φ : = R N ϕ ( x , | u ( x ) | ) u ( x ) φ ( x ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ6_HTML.gif
      (3.3)
      Let f : R N × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq71_HTML.gif be a real-valued function. We assume that the function f satisfies the Carathéodory condition in the sense that f ( , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq72_HTML.gif is measurable for all t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq73_HTML.gif and f ( x , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq74_HTML.gif is continuous for almost all x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq33_HTML.gif. Denote
      γ ( x ) = r ( x ) r ( x ) q ( x ) for almost all  x R N , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equr_HTML.gif

      where q is given in (H1) and q ( x ) < r ( x ) < p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq75_HTML.gif. We assume that

      (F1) For all ( x , t ) R N × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq76_HTML.gif, f ( x , t ) t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq77_HTML.gif, and there is a nonnegative measurable function m with m L γ ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq78_HTML.gif such that
      | f ( x , t ) | m ( x ) | t | q ( x ) 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equs_HTML.gif
      Denoting F ( x , t ) = 0 t f ( x , s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq79_HTML.gif, it follows from (F1) that
      (F1 ) 0 F ( x , t ) m ( x ) q ( x ) | t | q ( x ) for all  ( x , t ) R N × R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equt_HTML.gif
      Define the functional Ψ , I λ : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq80_HTML.gif by
      Ψ ( u ) = R N F ( x , u ) d x and I λ ( u ) = Φ ( u ) λ Ψ ( u ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equu_HTML.gif
      Then it is easy to check that Ψ C 1 ( X , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq81_HTML.gif, and its Gateaux derivative is
      Ψ ( u ) , φ = R N f ( x , u ) φ ( x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ7_HTML.gif
      (3.4)
      for any u , φ X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq82_HTML.gif. Let us consider the following quantity:
      λ = inf u X { 0 } R N Φ 0 ( x , | u | ) d x R N F ( x , u ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ8_HTML.gif
      (3.5)

      For the case of ϕ ( x , | t | ) = | t | p ( x ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq83_HTML.gif and f ( x , t ) = m ( x ) | t | q ( x ) 2 t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq84_HTML.gif, where m ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq85_HTML.gif satisfies a suitable condition, Benouhiba [6] proved that λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq86_HTML.gif. In this section, we shall generalize the conditions on f and ϕ to satisfy λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq87_HTML.gif still.

      The following lemma plays a key role in obtaining the main result in this section.

      Lemma 3.2 Assume that assumptions (HJ3)-(HJ4), (H1), and (F1) hold and satisfy
      (H2) q + 1 2 p < q , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equv_HTML.gif
      then the functionals Φ and Ψ satisfy the following relations:
      lim u X 0 Φ ( u ) Ψ ( u ) = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ9_HTML.gif
      (3.6)
      and
      lim u X Φ ( u ) Ψ ( u ) = . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ10_HTML.gif
      (3.7)
      Proof Applying Lemmas 2.2, 2.4 and 2.6, we get
      | Ψ ( u ) | = | R N F ( x , u ) d x | R N | m ( x ) q ( x ) | u | q ( x ) | d x 2 q m L γ ( x ) ( R N ) | u | q ( x ) L r ( x ) q ( x ) ( R N ) 2 q m L γ ( x ) ( R N ) ( u L r ( x ) ( R N ) q + + u L r ( x ) ( R N ) q ) 2 C q m L γ ( x ) ( R N ) ( u X q + + u X q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ11_HTML.gif
      (3.8)
      for some positive constant C. Let u in X with u X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq88_HTML.gif. Then it follows from (HJ3), (HJ4), (3.8) and Lemma 2.3(3) that
      | Φ ( u ) Ψ ( u ) | R N Φ 0 ( x , | u | ) d x 4 C q m L γ ( x ) ( R N ) u X q c p + u X p + 4 C q m L γ ( x ) ( R N ) u X q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ12_HTML.gif
      (3.9)
      Since q > p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq89_HTML.gif, we conclude that
      Φ ( u ) Ψ ( u ) as  u X 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equw_HTML.gif
      Next, we show that relation (3.7) holds. From (H2), there exists a positive constant δ such that q + ( 1 / 2 ) p < δ < q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq90_HTML.gif, and thus we have
      p > 2 ( q + δ ) > 2 ( q δ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ13_HTML.gif
      (3.10)
      Let ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq91_HTML.gif be a measurable function such that
      max { p ( x ) γ ( x ) p ( x ) + δ γ ( x ) , p ( x ) p ( x ) + δ q ( x ) } ( x ) min { p ( x ) γ ( x ) p ( x ) + δ γ ( x ) , p ( x ) p ( x ) + δ q ( x ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ14_HTML.gif
      (3.11)
      holds for almost all x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq33_HTML.gif and
      δ ( + + 1 ) < q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ15_HTML.gif
      (3.12)
      Then we have L ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq92_HTML.gif and 1 < ( x ) < γ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq93_HTML.gif. Let u X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq50_HTML.gif with u X > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq94_HTML.gif. Then it follows from (F1′) and Lemma 2.2 that
      | Ψ ( u ) | 1 q R N m ( x ) | u | δ | u | q ( x ) δ d x 2 q m | u | δ L ( x ) ( R N ) | u | q ( x ) δ L ( x ) ( R N ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equx_HTML.gif
      Therefore, without loss of generality, we may suppose that m | u | δ L ( x ) ( R N ) > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq95_HTML.gif. From the inequality above, by using Lemma 2.3, Lemma 2.2 and Lemma 2.4 in order, we have
      | Ψ ( u ) | 2 q ( R N m ( x ) | u | δ ( x ) ) 1 | u | q ( x ) δ L ( x ) ( R N ) 4 q m ( x ) L γ ( x ) ( x ) ( R N ) 1 | u | δ ( x ) L ( γ ( x ) ( x ) ) ( R N ) 1 | u | q ( x ) δ L ( x ) ( R N ) 4 q m L γ ( x ) ( R N ) α ( u L δ ( x ) ( γ ( x ) ( x ) ) ( R N ) δ + + u L δ ( x ) ( γ ( x ) ( x ) ) ( R N ) δ ) × ( u L ( q ( x ) δ ) ( x ) ( R N ) q + δ + u L ( q ( x ) δ ) ( x ) ( R N ) q δ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equy_HTML.gif

      where α = { + / if  m L γ ( x ) ( R N ) > 1 , 1 if  m L γ ( x ) ( R N ) 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq96_HTML.gif

      By Young’s inequality, we get
      | Ψ ( u ) | 4 q m L γ ( x ) ( R N ) α ( u L δ ( x ) ( γ ( x ) ( x ) ) ( R N ) 2 δ + + u L δ ( x ) ( γ ( x ) ( x ) ) ( R N ) 2 δ + u L ( q ( x ) δ ) ( x ) ( R N ) 2 ( q + δ ) + u L ( q ( x ) δ ) ( x ) ( R N ) 2 ( q δ ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equz_HTML.gif
      Using (3.11), we get that
      p ( x ) < δ ( x ) ( γ ( x ) ( x ) ) p ( x ) , p ( x ) < ( q ( x ) δ ) ( x ) p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equaa_HTML.gif
      holds for almost all x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq33_HTML.gif. Hence it follows from Lemma 2.6 that
      | Ψ ( u ) | 4 C q m L γ ( x ) ( R N ) α ( u X 2 δ + + u X 2 ( q + δ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ16_HTML.gif
      (3.13)
      for some positive constant C. Therefore, we obtain that
      | Φ ( u ) Ψ ( u ) | c p + u X p 4 C q m L γ ( x ) ( R N ) α ( u X 2 δ + + u X 2 ( q + δ ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equab_HTML.gif

      From (3.10), with the inequality above, we conclude that relation (3.7) holds. □

      Lemma 3.3 Assume that (HJ1)-(HJ3) and (H1) hold. Then Φ is weakly lower semi-continuous, i.e., u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq97_HTML.gif in X implies that Φ ( u ) lim inf n Φ ( u n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq98_HTML.gif.

      Proof Suppose that u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq97_HTML.gif in X as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq99_HTML.gif. Since (HJ3) implies that Φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq100_HTML.gif is strictly monotone on X, we have that Φ is convex, and so,
      Φ ( u n ) Φ ( u ) + Φ ( u ) , u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equac_HTML.gif
      for any n. Then we get that
      lim inf n Φ ( u n ) Φ ( u ) + lim inf n Φ ( u ) , u n u = Φ ( u ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equad_HTML.gif

      The proof is complete. □

      Lemma 3.4 Assume that (H1) and (F1) hold. For any K [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq101_HTML.gif and all u X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq50_HTML.gif, the following estimate holds:
      | x | K F ( x , u ) d x 2 C q ( | x | K m ( x ) d x ) 1 γ 1 ( u X q + + u X q ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ17_HTML.gif
      (3.14)

      where γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq102_HTML.gif is either γ + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq103_HTML.gif or γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq104_HTML.gif.

      Proof Applying Lemmas 2.2, 2.4 and 2.6, we get
      | x | K F ( x , u ) d x | x | K m ( x ) q ( x ) | u | q ( x ) d x 2 q m L γ ( x ) ( { x R N : | x | K } ) | u | q ( x ) L r ( x ) q ( x ) ( { x R N : | x | K } ) 2 q ( | x | K m ( x ) d x ) 1 γ 1 ( u L r ( x ) ( R N ) q + + u L r ( x ) ( R N ) q ) 2 C q ( | x | K m ( x ) d x ) 1 γ 1 ( u X q + + u X q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equae_HTML.gif

      for some positive constant C. □

      Lemma 3.5 Assume that (H1) and (F1) hold. For almost all x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq33_HTML.gif and all t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq73_HTML.gif, the following estimate holds:
      F ( x , t ) 1 q ( m ( x ) γ ( x ) γ + | t | r ( x ) ( γ + ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ18_HTML.gif
      (3.15)

      Proof Since q ( x ) ( γ ( x ) ) = r ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq105_HTML.gif, estimate (3.15) is obtained from (F1′) and Young’s inequality. □

      Lemma 3.6 Assume that (H1) and (F1) hold. Then Ψ is weakly-strongly continuous, i.e., u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq97_HTML.gif in X implies that Ψ ( u n ) Ψ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq106_HTML.gif.

      Proof Let { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq107_HTML.gif be a sequence in X such that u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq97_HTML.gif in X. Then { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq107_HTML.gif is bounded in X. By Lemma 3.4, for each ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq108_HTML.gif, there is a positive constant K ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq109_HTML.gif such that
      | x | K ε F ( x , u n ) d x ε and | x | K ε F ( x , u ) d x ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ19_HTML.gif
      (3.16)
      holds for each n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq110_HTML.gif. It follows from Lemma 3.5 that the Nemytskij operator
      u F ( x , u ( x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equaf_HTML.gif
      is continuous from L r ( x ) ( B K ε ( 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq111_HTML.gif into L 1 ( B K ε ( 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq112_HTML.gif; see Theorem 1.1 in [26]. This together with Lemma 2.5 yields that
      | x | < K ε F ( x , u n ) d x | x | < K ε F ( x , u ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ20_HTML.gif
      (3.17)

      Using (3.16) and (3.17), we deduce that Ψ ( u n ) Ψ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq113_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq114_HTML.gif. The proof is complete. □

      We are in a position to state the main result about the existence of the positive eigenvalue for the problem (E).

      Theorem 3.7 Assume that (HJ1)-(HJ4), (H1), (H2), and (F1) hold. Then λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq115_HTML.gif is a positive eigenvalue of the problem (E). Moreover, the problem (E) has a nontrivial weak solution for any λ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq116_HTML.gif.

      Proof It is trivial by (3.5) that λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq117_HTML.gif. Suppose to the contrary that λ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq118_HTML.gif. Let { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq107_HTML.gif be a sequence in X { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq119_HTML.gif such that
      lim n Φ ( u n ) Ψ ( u n ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equag_HTML.gif
      As in (3.9), we have
      | Φ ( u n ) Ψ ( u n ) | C u n X p + q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equah_HTML.gif
      for some positive constant C. Since p + < q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq120_HTML.gif, we obtain that u n X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq121_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq114_HTML.gif. Hence it follows from Lemma 3.2 that
      lim n Φ ( u n ) Ψ ( u n ) = , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equai_HTML.gif

      which contradicts with the hypothesis. Hence we get λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq122_HTML.gif. The analogous argument as that in the proof of Theorem 4.5 in [5] proves that λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq115_HTML.gif is an eigenvalue of the problem (E); see also Theorem 3.1 in [6].

      Finally, we show that the problem (E) has a nontrivial weak solution for any λ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq116_HTML.gif. Notice that u is a weak solution of (E) if and only if u is a critical point of I λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq123_HTML.gif. Assume that λ > λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq124_HTML.gif is fixed. Let u X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq50_HTML.gif with u X > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq94_HTML.gif. With the help of (HJ3) and (HJ4), it follows from proceeding as in the proof of relation (3.13) in Lemma 3.2 that
      I λ ( u ) c p + u X p λ 4 C q m L γ ( x ) ( R N ) α ( u X 2 δ + + u X 2 ( q + δ ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equaj_HTML.gif
      Since p > 2 ( q + δ ) > 2 δ ( + / ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq125_HTML.gif, the inequality above implies that I λ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq126_HTML.gif as u X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq127_HTML.gif for λ > λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq124_HTML.gif, that is, I λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq123_HTML.gif is coercive. Also since the functional I λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq123_HTML.gif is weakly lower semi-continuous by Lemmas 3.3 and 3.6, we deduce that there exists a global minimizer u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq128_HTML.gif of I λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq123_HTML.gif in X. Since λ > λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq124_HTML.gif, we verify by definition (3.5) that there is an element ω in X { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq119_HTML.gif such that Φ ( ω ) / Ψ ( ω ) < λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq129_HTML.gif. Then I λ ( ω ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq130_HTML.gif. So we obtain that
      I λ ( u 0 ) = inf v X { 0 } I λ ( v ) < 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equak_HTML.gif

      Consequently, we conclude that u 0 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq131_HTML.gif. This completes the proof. □

      Now, we consider an example to demonstrate our main result in this section.

      Example 3.8 Let p C ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq132_HTML.gif with 2 p ( x ) < N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq133_HTML.gif satisfy the log-Hölder continuity condition (2.2). Suppose that a L 2 p ( x ) ( R N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq134_HTML.gif, and there is a positive constant a 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq135_HTML.gif such that a ( x ) a 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq136_HTML.gif for almost all x R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq33_HTML.gif. Let us consider
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equal_HTML.gif
      In this case, put
      ϕ ( x , | v | ) = ( a ( x ) + | v | 2 ) p ( x ) 2 2 and Φ 0 ( x , | v | ) = 1 p ( x ) ( a ( x ) + | v | 2 ) p ( x ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equam_HTML.gif
      for all v R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq137_HTML.gif. Denote the quantities
      λ = inf u X { 0 } R N 1 p ( x ) ( a ( x ) + | u | 2 ) p ( x ) 2 d x R N m ( x ) p ( x ) | u | q ( x ) d x and λ = inf u X { 0 } R N ( a ( x ) + | u | 2 ) p ( x ) 2 d x R N m ( x ) | u | q ( x ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equan_HTML.gif
      If conditions (H1)-(H2) hold, then we have
      1. (i)

        0 < λ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq138_HTML.gif,

         
      2. (ii)

        λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq115_HTML.gif is a positive eigenvalue of the problem (E0),

         
      3. (iii)

        the problem (E0) has a nontrivial weak solution for any λ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq116_HTML.gif,

         
      4. (iv)

        λ is not an eigenvalue of (E0) for λ < λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq139_HTML.gif.

         
      Proof It is clear that conditions (HJ1)-(HJ4) and (F1) hold. From the definitions of λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq140_HTML.gif and λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq115_HTML.gif, we know that
      q p + λ λ q + p λ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equao_HTML.gif
      and thus λ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq141_HTML.gif. Also, from the same argument as that in Theorem 3.7, we have λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq87_HTML.gif, and thus λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq142_HTML.gif. Applying Theorem 3.7, the conclusions (ii) and (iii) hold. Let λ < λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq139_HTML.gif. Suppose that λ is an eigenvalue of the problem (E0). Then there is an element v X { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq143_HTML.gif such that
      R N ( a ( x ) + | v | 2 ) p ( x ) 2 d x λ R N m ( x ) | v | q ( x ) d x = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equap_HTML.gif
      By the definition of λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq140_HTML.gif, we get that
      λ R N m ( x ) | v | q ( x ) d x R N ( a ( x ) + | v | 2 ) p ( x ) 2 d x = λ R N m ( x ) | v | q ( x ) d x < λ R N m ( x ) | v | q ( x ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equaq_HTML.gif

      a contradiction. □

      Declarations

      Acknowledgements

      The first author was supported by the Incheon National University Research Grant in 2012, and the second author was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2012R1A1A2042187) and a 2013 Research Grant from Sangmyung University.

      Authors’ Affiliations

      (1)
      Department of Mathematics, Incheon National University
      (2)
      Department of Mathematics Education, Sangmyung University

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