Existence and Nonexistence of Positive Solutions for Singular http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq1_HTML.gif -Laplacian Equation in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq2_HTML.gif

  • Caisheng Chen1Email author,

    Affiliated with

    • Zhenqi Wang1 and

      Affiliated with

      • Fengping Wang1

        Affiliated with

        Boundary Value Problems20102010:607453

        DOI: 10.1155/2010/607453

        Received: 15 August 2010

        Accepted: 10 December 2010

        Published: 15 December 2010

        Abstract

        We study the existence and nonexistence of solutions for the singular quasilinear problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq7_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq9_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq10_HTML.gif behave like http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq11_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq12_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq13_HTML.gif at the origin. We obtain the existence by the upper and lower solution method and the nonexistence by the test function method.

        1. Introduction

        In this paper, we study through the upper and lower solution method and the test function method the existence and nonexistence of solution to the singular quasilinear elliptic problem
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ1_HTML.gif
        (1.1)

        with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq14_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq15_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq16_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq17_HTML.gif are the locally Hölder continuous functions, not identically zero and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq18_HTML.gif ) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq19_HTML.gif are locally Lipschitz continuous functions.

        The study of this type of equation in (1.1) is motivated by its various applications, for instance, in fluid mechanics, in Newtonian fluids, in flow through porous media, and in glaciology; see [1]. The equation in (1.1) involves singularities not only in the nonlinearities but also in the differential operator.

        Many authors studied this kind of problem for the case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq20_HTML.gif ; see [27]. In these works, the nonlinearities have sublinear and suplinear growth at infinity, and they behave like a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq21_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq22_HTML.gif , or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq23_HTML.gif ) at the origin. Roughly speaking, in this case we say that the nonlinearities are concave and convex or "slow diffusion and fast diffusion''; see [8].

        When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq24_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq25_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq26_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq27_HTML.gif , by using the lower and upper solution method, Santos in [5] finds a real number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq28_HTML.gif , such that the problem (1.1) has at least one solution if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq29_HTML.gif .

        For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq30_HTML.gif , the existence and multiplicity of solution of singular elliptic equation like (1.1) in a bounded domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq31_HTML.gif with the zero Dirichlet data have been widely studied by many authors, for example, the authors [913] and references therein. Assunção et al. in [14] studied the multiplicity of solution for the singular equations in (1.1) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq32_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq33_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq34_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq35_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq36_HTML.gif . Similar consideration can be found in [1520] and references therein. We note that the variation method is widely used in the above references.

        Recently, Chen et al. in [21, 22], by using a variational approach, got some existence of solution for (1.1) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq37_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq38_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq39_HTML.gif . For the case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq40_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq41_HTML.gif , the problem for the existence of solution for (1.1) is still open. It seems difficult to consider the case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq42_HTML.gif by variational method.

        The main aim of this work is to study the existence and nonexistence of solution for (1.1), where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq43_HTML.gif is sublinear and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq44_HTML.gif is suplinear. We will use the upper and lower solution method. To the best of our knowledge, there is little information on upper and lower solution method for the problem (1.1). So it is necessary to establish this technique in unbounded domain. To obtain the existence, the assumption http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq45_HTML.gif (see (2.17) below) is essential. By this, an upper solution for (1.1) is obtained.

        We also obtain a sufficient condition on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq46_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq47_HTML.gif to guarantee the nonexistence of nontrivial solution for the problem (2.21). (see Theorem 2.5 below). It must be particularly pointed out that our primary interest is in the mixed case in which http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq48_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq49_HTML.gif satisfying
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ2_HTML.gif
        (1.2)
        while http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq50_HTML.gif satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ3_HTML.gif
        (1.3)

        This paper is organized as follows. In Section 2, we state the main results and present some preliminaries which will be used in what follows. We also introduce the precise hypotheses under which our problem is studied. In Section 3, we give the proof of some lemmas and the existence. The proof of nonexistence is given in Section 4.

        2. Preliminaries and Main Results

        Let us now introduce some weighted Sobolev spaces and their norms. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq51_HTML.gif be a bounded domain in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq52_HTML.gif with smooth boundary http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq53_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq54_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq55_HTML.gif , we define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq56_HTML.gif as being the subspace of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq57_HTML.gif of the Lebesgue measurable function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq58_HTML.gif , satisfying
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ4_HTML.gif
        (2.1)
        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq59_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq60_HTML.gif , we define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq61_HTML.gif (resp., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq62_HTML.gif as being the closure of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq63_HTML.gif (resp., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq64_HTML.gif ) with respect to the norm defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ5_HTML.gif
        (2.2)

        For the weighted Sobolev space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq65_HTML.gif , we have the following compact imbedding theorem which is an extension of the classical Rellich-Kondrachov compact theorem.

        Theorem 2.1 ((compact imbedding theorem) [13]).

        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq66_HTML.gif is an open bounded domain with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq67_HTML.gif boundary and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq68_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq69_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq70_HTML.gif . Then, the imbedding http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq71_HTML.gif is compact if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq72_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq73_HTML.gif .

        We now consider the existence of positive solutions for problem (1.1). Our main tool will be the upper and lower solution method. This method, in the bounded domain situation, has been used by many authors, for instance, [10, 12, 13]. But for the unbounded domain, we need to establish this method and then to construct an upper solution and a lower solution for (1.1). We now give the definitions of upper and lower solutions.

        Definition 2.2 (see [10, 12]).

        A function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq74_HTML.gif is said to be a weak lower solution of the equation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ6_HTML.gif
        (2.3)
        if
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ7_HTML.gif
        (2.4)
        or
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ8_HTML.gif
        (2.5)

        for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq75_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq76_HTML.gif .

        Similarly, a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq77_HTML.gif is said to be a weak upper solution of (2.3) if
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ9_HTML.gif
        (2.6)
        or
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ10_HTML.gif
        (2.7)

        for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq78_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq79_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq80_HTML.gif .

        A function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq81_HTML.gif is said to be a weak solution of (2.3) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq82_HTML.gif is a weak lower solution and weak upper solution of (2.3).

        A function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq83_HTML.gif is said to be less than or equal to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq84_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq85_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq86_HTML.gif .

        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq87_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq88_HTML.gif , we define the weighted Sobolev space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq89_HTML.gif as being the closure of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq90_HTML.gif with respect to the norm http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq91_HTML.gif defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ11_HTML.gif
        (2.8)

        The following lemma will be basic in our approach.

        Lemma 2.3.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq92_HTML.gif be Lipschitz continuous and nondecreasing in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq93_HTML.gif and locally Hölder continuous in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq94_HTML.gif . Moreover, assume that there exist the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq95_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ12_HTML.gif
        (2.9)
        Then, there exist a minimal weak solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq96_HTML.gif and a maximal weak solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq97_HTML.gif of (2.3) satisfying
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ13_HTML.gif
        (2.10)

        and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq98_HTML.gif .

        Proof.

        Denote http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq99_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq100_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq101_HTML.gif be a pair of upper and lower solutions of (2.3) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq102_HTML.gif , a.e. in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq103_HTML.gif . We consider the boundary value problem
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ14_HTML.gif
        (2.11)

        By Theorem  1.1 in [10], one concludes that there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq104_HTML.gif which is a weak solution of (2.11) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq105_HTML.gif a.e. in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq106_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq107_HTML.gif .

        We define its extension by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ15_HTML.gif
        (2.12)
        Similarly, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq108_HTML.gif be a weak solution of the boundary value problem
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ16_HTML.gif
        (2.13)
        and its extension is defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ17_HTML.gif
        (2.14)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq109_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq110_HTML.gif . By Theorem  2.4 in [12], we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ18_HTML.gif
        (2.15)
        for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq111_HTML.gif . In view of (2.15), the pointwise limits
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ19_HTML.gif
        (2.16)

        exist and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq112_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq113_HTML.gif .

        Similar to the proof Theorem  1.1 in [10] and the proof of Theorem  7.5.1 in [23], it is not difficult to get from Theorem 2.1 that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq114_HTML.gif is the maximal weak solution and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq115_HTML.gif the minimal solution of (2.3), which satisfies (2.10) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq116_HTML.gif . This ends the proof of Lemma 2.3.

        Our main results read as follows.

        Theorem 2.4 (existence).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq117_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq118_HTML.gif . Assume the following.

        The nonnegative functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq120_HTML.gif are Lipschitz continuous and nondecreasing, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq121_HTML.gif . Additionally, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq122_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq123_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq124_HTML.gif .

        The nonnegative functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq126_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq127_HTML.gif are locally Hölder continuous. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq128_HTML.gif . If

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ20_HTML.gif
        (2.17)

        then there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq129_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq130_HTML.gif , and the problem (1.1) admits a weak solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq131_HTML.gif .

        Theorem 2.5 (nonexistence).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq132_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq133_HTML.gif . Assume that

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq135_HTML.gif ;

        there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq137_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ21_HTML.gif
        (2.18)

        the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq139_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq140_HTML.gif satisfy

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ22_HTML.gif
        (2.19)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq141_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ23_HTML.gif
        (2.20)
        Then the problem
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ24_HTML.gif
        (2.21)

        has no nontrivial solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq142_HTML.gif .

        Remark 2.6.

        If assumption (2.19) holds, then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ25_HTML.gif
        (2.22)

        with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq143_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq144_HTML.gif .

        In fact, for this case, there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq145_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq146_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ26_HTML.gif
        (2.23)
        for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq147_HTML.gif . Therefore,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ27_HTML.gif
        (2.24)

        So, condition (2.19) implies (2.22).

        3. Proof of Existence

        Before proofing the existence, we present some preliminary lemmas which will be useful in what follows.

        Lemma 3.1.

        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq148_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq149_HTML.gif is local Hölder continuous and satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ28_HTML.gif
        (3.1)
        Then the problem
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ29_HTML.gif
        (3.2)

        has a weak solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq150_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq151_HTML.gif .

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq152_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq153_HTML.gif . Denote
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ30_HTML.gif
        (3.3)
        Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq154_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq155_HTML.gif . It is easy to verify that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ31_HTML.gif
        (3.4)
        This shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq156_HTML.gif (resp., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq157_HTML.gif ) is a lower (resp., upper) solution of (3.2). Then by Lemma 2.3, there exists a weak solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq158_HTML.gif for problem (3.2) satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq159_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ32_HTML.gif
        (3.5)

        Lemma 3.2.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq160_HTML.gif . If
        1. (1)
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ33_HTML.gif
          (3.6)
           
        1. (2)
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ34_HTML.gif
          (3.7)
           

        one has http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq161_HTML.gif .

        Proof.
        1. (1)
          Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq162_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq163_HTML.gif . By the Hölder inequality, we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ35_HTML.gif
          (3.8)
           
        1. (2)

          If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq164_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq165_HTML.gif , we take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq166_HTML.gif and then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq167_HTML.gif .

           
        Note that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ36_HTML.gif
        (3.9)

        This implies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq168_HTML.gif and ends the proof of Lemma 3.2.

        Corollary 3.3.

        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq169_HTML.gif satisfies the conditions in Lemma 3.2, then the problem (3.2) admits a solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq170_HTML.gif .

        Lemma 3.4.

        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq171_HTML.gif is nondecreasing and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq172_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq173_HTML.gif . Additionally, let the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq174_HTML.gif be locally Hölder continuous and satisfy
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ37_HTML.gif
        (3.10)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq175_HTML.gif . Then the problem
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ38_HTML.gif
        (3.11)

        has a weak solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq176_HTML.gif .

        Proof.

        We first consider the problem
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ39_HTML.gif
        (3.12)

        By Lemma 3.1, there is a solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq177_HTML.gif for (3.12) satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq178_HTML.gif . In order to get the existence of solution for (3.11), we chose a pair of upper-lower solution of the equation in (3.11) by means of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq179_HTML.gif .

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq180_HTML.gif . It is easy to verify that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq181_HTML.gif is an upper solution of
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ40_HTML.gif
        (3.13)
        if and only if
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ41_HTML.gif
        (3.14)
        or
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ42_HTML.gif
        (3.15)

        By the assumption on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq182_HTML.gif , we know that there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq183_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq184_HTML.gif . So, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq185_HTML.gif . Then we take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq186_HTML.gif so that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq187_HTML.gif is an upper solution of (3.13).

        We now construct a lower solution of (3.13). Consider the boundary value problem
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ43_HTML.gif
        (3.16)

        for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq188_HTML.gif .

        By Theorem  3.1 in [12], there exists a solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq189_HTML.gif for (3.16). We define an extension by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq190_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq191_HTML.gif . Then, by Theorem  2.4 in [12] and Díaz-Saá's inequality in [24], we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ44_HTML.gif
        (3.17)
        Setting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq192_HTML.gif and performing some standard computations, we see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq193_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ45_HTML.gif
        (3.18)

        and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq194_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq195_HTML.gif . Then, our result follows from Lemma 2.3.

        We now give the proof of Theorem 2.4.

        Proof of Theorem 2.4.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq196_HTML.gif be a solution of the problem
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ46_HTML.gif
        (3.19)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq197_HTML.gif . We see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq198_HTML.gif is an upper solution of the equation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ47_HTML.gif
        (3.20)
        if and only if
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ48_HTML.gif
        (3.21)
        or
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ49_HTML.gif
        (3.22)
        Since
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ50_HTML.gif
        (3.23)
        we have a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq199_HTML.gif , such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ51_HTML.gif
        (3.24)
        Denote
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ52_HTML.gif
        (3.25)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq200_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq201_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq202_HTML.gif and there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq203_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq204_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq205_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq206_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq207_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq208_HTML.gif . A simple computation shows that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ53_HTML.gif
        (3.26)
        Thus
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ54_HTML.gif
        (3.27)
        Hence, for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq209_HTML.gif , there exists a unique http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq210_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq211_HTML.gif . That is
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ55_HTML.gif
        (3.28)
        Now defining http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq212_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ56_HTML.gif
        (3.29)
        This shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq213_HTML.gif is an upper solution of (3.20). Noting that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ57_HTML.gif
        (3.30)

        we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq214_HTML.gif is an upper solution of (1.1). Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq215_HTML.gif be a solution of (3.11). Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq216_HTML.gif is a lower solution of (1.1). We now show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq217_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq218_HTML.gif .

        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq219_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq220_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq221_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq222_HTML.gif , then for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq223_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq224_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq225_HTML.gif . Without loss of generality, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq226_HTML.gif .

        From the proof of Lemma 3.4 and the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq227_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq228_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq229_HTML.gif . Further, by (3.17), we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq230_HTML.gif . Letting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq231_HTML.gif , we obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq232_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq233_HTML.gif .

        By Lemma 2.3, there exists a solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq234_HTML.gif for the problem (1.1). We then complete the proof of Theorem 2.4.

        Remark 3.5.

        The nonlinear term http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq235_HTML.gif can be regarded as a perturbation of the nonlinear term http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq236_HTML.gif .

        4. Proof of Nonexistence

        In order to prove the nonexistence of nontrivial solution of the problem (2.21), we use the test function method, which has been used in [25] and references therein. Some modification has been made in our proof. The proof is based on argument by contradiction which involves a priori estimate for a nonnegative solution of (2.21) by carefully choosing the special test function and scaling argument.

        Proof of Theorem 2.5.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq237_HTML.gif be defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ58_HTML.gif
        (4.1)

        and put http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq238_HTML.gif , by which the parameters http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq239_HTML.gif will be determined later. It is not difficult to verify that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq240_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq241_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq242_HTML.gif .

        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq243_HTML.gif is a solution to problem (2.21). Without loss of generality, we can assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq244_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq245_HTML.gif (otherwise, we consider http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq246_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq247_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq248_HTML.gif ). Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq249_HTML.gif be a parameter ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq250_HTML.gif will also be chosen below).

        By the Young inequality, we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ59_HTML.gif
        (4.2)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq251_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq252_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq253_HTML.gif satisfy (2.18) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq254_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq255_HTML.gif .

        Multiplying the equation in (2.21) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq256_HTML.gif and integrating by parts, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ60_HTML.gif
        (4.3)
        Then applying the Young inequality with parameter http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq257_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ61_HTML.gif
        (4.4)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq258_HTML.gif .

        Similarly, let us multiply the equation in (2.21) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq259_HTML.gif and integrate by parts:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ62_HTML.gif
        (4.5)
        By (4.4),
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ63_HTML.gif
        (4.6)
        Now, we apply the Hölder inequality to the integral on the right-hand side of (4.6):
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ64_HTML.gif
        (4.7)

        with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq260_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq261_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq262_HTML.gif .

        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq263_HTML.gif , we chose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq264_HTML.gif so small that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq265_HTML.gif . Then, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ65_HTML.gif
        (4.8)

        with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq266_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq267_HTML.gif .

        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq268_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq269_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq270_HTML.gif . Then we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ66_HTML.gif
        (4.9)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq271_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq272_HTML.gif .

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq273_HTML.gif . Then,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ67_HTML.gif
        (4.10)
        Similarly,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ68_HTML.gif
        (4.11)
        Then it follows from (4.5)–(4.11) that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ69_HTML.gif
        (4.12)
        with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq274_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ70_HTML.gif
        (4.13)
        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq275_HTML.gif , it follows from (4.12) that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ71_HTML.gif
        (4.14)

        This implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq276_HTML.gif , a.e. in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq277_HTML.gif . That is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq278_HTML.gif is a trivial solution for (2.21).

        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq279_HTML.gif , then (4.12) gives that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ72_HTML.gif
        (4.15)
        By (4.5), we derive
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ73_HTML.gif
        (4.16)
        Reasoning as in the first part of the proof, we infer that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_Equ74_HTML.gif
        (4.17)

        Letting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq280_HTML.gif in (4.17), we obtain (4.14). Thus, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq281_HTML.gif , a.e. in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F607453/MediaObjects/13661_2010_Article_943_IEq282_HTML.gif . Then the proof of Theorem 2.5 is completed.

        Declarations

        Acknowledgments

        The authors wish to express their gratitude to the referees for useful comments and suggestions. The work was supported by the Fundamental Research Funds for the Central Universities (Grant no. 2010B17914) and Science Funds of Hohai University (Grants no. 2008430211 and 2008408306).

        Authors’ Affiliations

        (1)
        Department of Mathematics, Hohai University

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        © Caisheng Chen et al. 2010

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