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

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 . https://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 ) https://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 ) https://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 ) https://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 , https://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 ) https://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 https://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 , ) https://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 https://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 ) https://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 ) https://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 https://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 https://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 ) https://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 ) https://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 ) https://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 ) https://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 ) https://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 https://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 ) https://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 ) https://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 ) https://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 ) https://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 ) https://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 } . https://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 ) https://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 ) . https://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 ) https://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 < } , https://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 } . https://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 ) https://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 ) https://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 https://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 ) https://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 ) } , https://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 ) . https://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 ( ) https://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 | https://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 https://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 https://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 ) https://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 ) https://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 ) https://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 ) https://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 ) https://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 ) https://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 ) https://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 https://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 ) https://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 ) https://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 ) . https://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 ) . https://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 https://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 https://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 https://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 + https://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 https://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 https://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 ) https://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 ) https://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 https://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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq34_HTML.gif with u 0 https://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 https://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 + https://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 https://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 https://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 https://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 ¯ ) https://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 https://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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq43_HTML.gif with q > 1 https://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 , https://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 ) , https://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 https://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 https://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 https://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 ) https://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 ) https://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 https://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 ) https://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 ) https://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 ) https://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 ) https://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 ) https://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 } , https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equl_HTML.gif

for all φ X https://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 } https://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 https://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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq53_HTML.gif, p ( x ) < N https://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 ) https://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 , ) https://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 ) https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq7_HTML.gif for all η 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq58_HTML.gif and ϕ ( x , ) https://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 , ) https://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 https://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 ) https://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 https://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 https://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 https://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 https://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 https://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 ) https://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 https://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 , ) https://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 https://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 , ) https://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 . https://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 https://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 https://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 | ) , https://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 https://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 https://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 https://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 . https://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 ) https://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 . https://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 https://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 ) https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq73_HTML.gif and f ( x , ) https://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 https://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 , https://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 ) https://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 https://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 https://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 ) https://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 . https://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 https://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 . https://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 https://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 ) . https://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 ) https://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 https://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 https://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 . https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq84_HTML.gif, where m ( x ) https://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 https://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 https://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 , https://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 ) = https://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 ) = . https://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 ) https://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 https://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 . https://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 + https://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 . https://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 https://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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equ13_HTML.gif
(3.10)
Let ( x ) https://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 ) } https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq33_HTML.gif and
δ ( + + 1 ) < q . https://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 ) https://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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq93_HTML.gif. Let u X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq50_HTML.gif with u X > 1 https://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 ) . https://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 https://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 ) , https://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 . https://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 ) ) . https://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 ) https://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 https://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 ) ) https://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 ) ) . https://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 https://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 ) https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq97_HTML.gif in X as n https://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 https://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 https://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 ) . https://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 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq101_HTML.gif and all u X https://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 ) , https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq102_HTML.gif is either γ + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq103_HTML.gif or γ https://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 ) https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq33_HTML.gif and all t R https://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 + ) ) . https://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 ) https://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 https://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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq106_HTML.gif.

Proof Let { u n } https://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 https://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 } https://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 https://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 https://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 https://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 https://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 ) ) https://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 ) ) https://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 ) ) https://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 . https://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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq113_HTML.gif as n https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq118_HTML.gif. Let { u n } https://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 } https://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 . https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq121_HTML.gif as n https://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 ) = , https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq123_HTML.gif. Assume that λ > λ https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq50_HTML.gif with u X > 1 https://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 ) ) . https://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 ( + / ) https://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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq126_HTML.gif as u X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq127_HTML.gif for λ > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq124_HTML.gif, that is, I λ https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq128_HTML.gif of I λ https://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 https://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 } https://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 https://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 https://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 . https://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 https://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 ) https://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 https://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 ) https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq33_HTML.gif. Let us consider
https://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 https://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 https://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 . https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq138_HTML.gif,

     
  2. (ii)

    λ https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq140_HTML.gif and λ https://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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equao_HTML.gif
and thus λ λ https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_IEq87_HTML.gif, and thus λ > 0 https://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 https://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 } https://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 . https://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 https://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 , https://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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© Kim and Kim; licensee Springer 2013

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