Existence of Weak Solutions for a Nonlinear Elliptic System

  • Ming Fang1Email author and

    Affiliated with

    • Robert P Gilbert2

      Affiliated with

      Boundary Value Problems20092009:708389

      DOI: 10.1155/2009/708389

      Received: 3 April 2009

      Accepted: 31 July 2009

      Published: 26 August 2009

      Abstract

      We investigate the existence of weak solutions to the following Dirichlet boundary value problem, which occurs when modeling an injection molding process with a partial slip condition on the boundary. We have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq1_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq2_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq3_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq4_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq5_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq6_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq7_HTML.gif .

      1. Introduction

      Injection molding is a manufacturing process for producing parts from both thermoplastic and thermosetting plastic materials. When the material is in contact with the mold wall surface, one has three choices: (i) no slip (which implies that the material sticks to the surface) (ii) partial slip, and (iii) complete slip [15]. Navier [6] in 1827 first proposed a partial slip condition for rough surfaces, relating the tangential velocity http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq8_HTML.gif to the local tangential shear stress http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq9_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ1_HTML.gif
      (11)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq10_HTML.gif indicates the amount of slip. When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq11_HTML.gif , (1.1) reduces to the no-slip boundary condition. A nonzero http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq12_HTML.gif implies partial slip. As http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq13_HTML.gif , the solid surface tends to full slip.

      There is a full description of the injection molding process in [3] and in our paper [7]. The formulation of this process as an elliptic system is given here in after.

      Problem 1.

      Find functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq14_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq15_HTML.gif defined in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq16_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ2_HTML.gif
      (12)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ3_HTML.gif
      (13)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ4_HTML.gif
      (14)

      Here we assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq17_HTML.gif is a bounded domain in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq18_HTML.gif with a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq19_HTML.gif boundary. We assume also that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq20_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq21_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq22_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq23_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq24_HTML.gif are given functions, while http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq25_HTML.gif is a given positive constant related to the power law index; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq26_HTML.gif is the pressure of the flow, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq27_HTML.gif is the temperature. The leading order term http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq28_HTML.gif of the PDE (1.3) is derived from a nonlinear slip condition of Navier type. Similar derivations based on the Navier slip condition occur elsewhere, for example, [8, 9], [10, equation ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq29_HTML.gif )] .

      The mathematical model for this system was established in [7]. Some related papers, both rigorous and formal, are [3, 1113]. In [11, 13], existence results in no-slip surface, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq30_HTML.gif , are obtained, while in [3, 7], Navier's slip conditions, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq31_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq32_HTML.gif , are investigated, and numerical, existence, uniqueness, and regularity results are given. Although the physical models are two dimensional, we shall carry out our proofs in the case of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq33_HTML.gif dimension.

      In Section 2, we introduce some notations and lemmas needed in later sections. In Section 3, we investigate the existence, uniqueness, stability, and continuity of solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq34_HTML.gif to the nonlinear equation (1.3). In Section 4, we study the existence of weak solutions to Problem 1.

      Using Rothe's method of time discretization and an existence result for Problem 1, one can establish existence of week solutions to the following time-dependent problem.

      Problem 2.

      Find functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq35_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq36_HTML.gif defined in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq37_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ5_HTML.gif
      (15)

      The proof is only a slight modification of the proofs given in [11, 13] and is omitted here.

      2. Notations and Preliminaries

      2.1. Notations

      In this paper, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq38_HTML.gif let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq39_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq40_HTML.gif denote the usual Sobolev space equipped with the standard norm. Let

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ6_HTML.gif
      (21)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq41_HTML.gif . The conjugate exponent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq42_HTML.gif is

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ7_HTML.gif
      (22)

      We assume that the boundary values http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq43_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq44_HTML.gif for Problem 1 can be extended to functions defined on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq45_HTML.gif such that

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ8_HTML.gif
      (23)

      We further assume that there exist positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq46_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq47_HTML.gif such that

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ9_HTML.gif
      (24)

      Finally, we assume that for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq48_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq49_HTML.gif a.e. in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq50_HTML.gif indicates

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ10_HTML.gif
      (25)

      For the convenience of exposition, we assume that

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ11_HTML.gif
      (26)

      Next, we recall some previous results which will be needed in the rest of the paper.

      2.2. Preliminaries

      An important inequality (e.g., see [11, page 550] ) in the study of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq51_HTML.gif -Laplacian is as follows:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ12_HTML.gif
      (27)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq52_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq53_HTML.gif are certain constants.

      To establish coercivity condition, we will use the following inequality:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ13_HTML.gif
      (28)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq54_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq55_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq56_HTML.gif .

      Using the Sobolev Embedding Theorem and Hölder's Inequality, we can derive the following results (for more details, see [11, Lemma  3.4] and [13, Lemma  4.2]).

      Lemma 2.1.

      The following statements hold

      ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq57_HTML.gif ) For any positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq58_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq59_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq60_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq61_HTML.gif then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ14_HTML.gif
      (29)

      moreover, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq62_HTML.gif

      ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq63_HTML.gif ) If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq64_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq65_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq66_HTML.gif moreover,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ15_HTML.gif
      (210)
      ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq67_HTML.gif ) If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq68_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq69_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq70_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ16_HTML.gif
      (211)
      and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq71_HTML.gif denotes the conjugate of r, namely, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq72_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq73_HTML.gif moreover,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ17_HTML.gif
      (212)
      ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq74_HTML.gif ) If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq75_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq76_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ18_HTML.gif
      (213)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq77_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq78_HTML.gif . Moreover
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ19_HTML.gif
      (214)

      The existence proof will use the following general result of monotone operators [14, Corollary  III.1.8, page 87] and [15, Proposition  17.2].

      Proposition 2.2.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq79_HTML.gif be a closed convex set ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq80_HTML.gif ), and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq81_HTML.gif be monotone, coercive, and weakly continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq82_HTML.gif . Then there exists
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ20_HTML.gif
      (215)

      The uniqueness proof is based on a supersolution argument (similar definition can be found in [15, Chapter  3]).

      Definition 2.3.

      A function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq83_HTML.gif is a weak supersolution of the equation
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ21_HTML.gif
      (216)
      in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq84_HTML.gif if
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ22_HTML.gif
      (217)

      whenever http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq85_HTML.gif is nonnegative.

      3. A Dirichlet Boundary Value Problem

      We study the following Dirichlet boundary value problem:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ23_HTML.gif
      (31)

      Definition 3.1.

      We say that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq86_HTML.gif is a weak solution to (3.1) if
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ24_HTML.gif
      (32)

      for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq87_HTML.gif and a given http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq88_HTML.gif .

      Theorem 3.2.

      Assume that conditions (2.1)–(2.6) are satisfied. Then there exists a unique weak solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq89_HTML.gif to the Dirichlet boundary value problem (3.1) in the sense of Definition 3.1. In addition, the solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq90_HTML.gif satisfies the following properties.

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq91_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ25_HTML.gif
      (33)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq92_HTML.gif is a constant independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq93_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq94_HTML.gif ;

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq95_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq96_HTML.gif a.e. in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq97_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ26_HTML.gif
      (34)

      The idea behind the existence proof is related to [15, 16]. We will first consider the following Obstacle Problem.

      Problem 3.

      Find a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq98_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq99_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ27_HTML.gif
      (35)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq100_HTML.gif . Here
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ28_HTML.gif
      (36)

      Lemma 3.3.

      If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq101_HTML.gif is nonempty, then there is a unique solution p to the Problem 3 in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq102_HTML.gif .

      Proof of Lemma 3.3.

      Our proof will use Proposition 2.2.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq103_HTML.gif and write
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ29_HTML.gif
      (37)

      It follows from the proof in [15, Proposition  17.2] that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq104_HTML.gif is a closed convex set.

      Next we define a mapping http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq105_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ30_HTML.gif
      (38)
      By Hölder's inequality,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ31_HTML.gif
      (39)

      Here we used Assumption (2.6), that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq106_HTML.gif . Therefore we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq107_HTML.gif whenever http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq108_HTML.gif . Moreover, it follows from inequality (2.7) that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq109_HTML.gif is monotone.

      To show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq110_HTML.gif is coercive on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq111_HTML.gif , fix http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq112_HTML.gif . Then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ32_HTML.gif
      (310)

      Inequality (2.8) is used to arrive at the last step. This implies that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq113_HTML.gif is coercive on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq114_HTML.gif .

      Finally, we show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq115_HTML.gif is weakly continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq116_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq117_HTML.gif be a sequence that converges to an element http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq118_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq119_HTML.gif . Select a subsequence { http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq120_HTML.gif } such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq121_HTML.gif a.e. in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq122_HTML.gif . Then it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ33_HTML.gif
      (311)
      a.e. in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq123_HTML.gif . Moreover,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ34_HTML.gif
      (312)
      Thus we have that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ35_HTML.gif
      (313)
      weakly in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq124_HTML.gif . Since the weak limit is independent of the choice of the subsequence, it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ36_HTML.gif
      (314)

      weakly in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq125_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq126_HTML.gif is weakly continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq127_HTML.gif . We may apply Proposition 2.2 to obtain the existence of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq128_HTML.gif .

      Our uniqueness proof is inspired by [15, Lemmas http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq129_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq130_HTML.gif , and Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq131_HTML.gif ]. Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq132_HTML.gif does not satisfy condition (3.4) of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq133_HTML.gif operator in [15], we need to prove the following lemma, which is equivalent to [15, Lemma  3.11]. Then uniqueness can follow immediately from [15, Lemma  3.22].

      Lemma 3.4.

      If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq134_HTML.gif is a supersolution of (2.16) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq135_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ37_HTML.gif
      (315)

      for all nonnegative http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq136_HTML.gif .

      Proof.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq137_HTML.gif and choose nonnegative sequence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq138_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq139_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq140_HTML.gif . Equation (2.6) and Hölder inequality imply that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ38_HTML.gif
      (316)
      Because http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq141_HTML.gif , we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ39_HTML.gif
      (317)

      and the lemma follows.

      Similar to [15, Corollary  17.3, page 335], one can also obtain the following Corollary.

      Corollary 3.5.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq142_HTML.gif be bounded and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq143_HTML.gif . There is a weak solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq144_HTML.gif to (3.1) in the sense of Definition 3.1.

      Proof of Theorem 3.2.

      The existence result is given in Corollary 3.5, and we now turn to proof of uniqueness. For a given http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq145_HTML.gif , assume that there exists another solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq146_HTML.gif Then we have that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ40_HTML.gif
      (318)

      for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq147_HTML.gif . If we take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq148_HTML.gif in above equation, from inequality (2.7), we have the following.

      ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq149_HTML.gif ) when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq150_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ41_HTML.gif
      (319)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq151_HTML.gif is a positive constant;

      ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq152_HTML.gif ) when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq153_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ42_HTML.gif
      (320)
      Here the Hölder inequality for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq154_HTML.gif , namely,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ43_HTML.gif
      (321)

      is applied to the last inequality.

      Poincaré's inequality implies that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq155_HTML.gif a.e. We complete the uniqueness proof.

      Next we prove (3.3). Taking http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq156_HTML.gif in (3.2), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ44_HTML.gif
      (322)
      From (2.4), and the Hölder inequality, we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ45_HTML.gif
      (323)
      Young's inequality with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq157_HTML.gif implies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ46_HTML.gif
      (324)

      and (3.3) follows immediately from (2.3) and (2.6).

      Finally, we prove (3.4). From weak solution definition (3.2), we know that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ47_HTML.gif
      (325)
      Setting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq158_HTML.gif and subtracting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq159_HTML.gif from both sides, we obtain that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ48_HTML.gif
      (326)

      Denote the right-hand side by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq160_HTML.gif . Similar to arguments in the uniqueness proof, we arrive at the folloing:

      ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq161_HTML.gif ) when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq162_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ49_HTML.gif
      (327)
      ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq163_HTML.gif ) when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq164_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ50_HTML.gif
      (328)
      Egorov's Theorem implies that for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq165_HTML.gif , there is a closed subset http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq166_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq167_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq168_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq169_HTML.gif uniformly on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq170_HTML.gif . Application of the absolute continuity of the Lebesgue Integral implies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ51_HTML.gif
      (329)

      Theorem 3.2 is proved.

      4. Nonlinear Elliptic Dirichlet System

      Definition 4.1.

      We say that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq171_HTML.gif is a weak solution to Problem 1 if
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ52_HTML.gif
      (41)
      and for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq172_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ53_HTML.gif
      (42)
      and for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq173_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ54_HTML.gif
      (43)

      Theorem 4.2.

      Assume that (2.1)–(2.6) hold. Then there exists a weak solution to Problem 1 in the sense of Definition 4.1.

      We shall bound the critical growth, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq174_HTML.gif , on the right-hand side of (4.2).

      Lemma 4.3.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq175_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq176_HTML.gif satisfy
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ55_HTML.gif
      (44)
      and (4.3). Then, under the conditions of Theorem 4.2, for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq177_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ56_HTML.gif
      (45)
      Moreover, there exists a polynomial http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq178_HTML.gif that is independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq179_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq180_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ57_HTML.gif
      (46)

      Proof.

      We first show (4.5). Letting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq181_HTML.gif in (4.3), we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ58_HTML.gif
      (47)

      After some straightforward computations this yields exactly (4.5).

      We now show (4.6). We denote the four terms on the right-hand side of equation (4.5) by I, II, III, and IV, respectively. Under the conditions of Lemma 4.3, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ59_HTML.gif
      (48)
      Part (iii) of Lemma 2.1 and Sobolev's imbedding theorems indicate
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ60_HTML.gif
      (49)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq182_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq183_HTML.gif .

      According to Sobolev's imbedding theorems, the integrability of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq184_HTML.gif depends on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq185_HTML.gif . We estimate II in three different cases.

      Case 1 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq186_HTML.gif ).

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ61_HTML.gif
      (410)

      Case 2 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq187_HTML.gif ).

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ62_HTML.gif
      (411)

      Case 3 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq188_HTML.gif ).

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq189_HTML.gif is a bounded continuous function, so
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ63_HTML.gif
      (412)
      We next estimate III:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ64_HTML.gif
      (413)

      The estimate of the first term used Hölder inequality and Sobolev's imbedding theorems. The argument of the second estimate is similar to that of I.

      Recall that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq190_HTML.gif . Similar to II, we estimate IV in three different cases.

      Case 1 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq191_HTML.gif ).

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ65_HTML.gif
      (414)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq192_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ66_HTML.gif
      (415)

      Case 2 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq193_HTML.gif ).

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ67_HTML.gif
      (416)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq194_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ68_HTML.gif
      (417)

      Case 3 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq195_HTML.gif ).

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ69_HTML.gif
      (418)
      These estimates lead to
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ70_HTML.gif
      (419)

      for some polynomial http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq196_HTML.gif .

      Proof of Theorem 4.2.

      Using Theorem 3.2, let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq197_HTML.gif then for (3.2) there exists a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq198_HTML.gif satisfying
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ71_HTML.gif
      (420)
      Moreover, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq199_HTML.gif a.e. in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq200_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ72_HTML.gif
      (421)
      Next, using Lemma 4.3, we can define a linear functional http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq201_HTML.gif determined by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ73_HTML.gif
      (422)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq202_HTML.gif By virtue of (4.6), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq203_HTML.gif is well defined, and there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq204_HTML.gif independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq205_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ74_HTML.gif
      (423)

      We notice that (4.2) is the same as [11,  equation ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq206_HTML.gif )]. Therefore, arguments after [11,  equation ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq207_HTML.gif )] can be used to complete the proof of Theorem 4.2.

      Declarations

      Acknowledgments

      The project is partially supported by NSF/STARS Grant (NSF-0207971) and Research Initiation Awards at the Norfolk State University. The second author's work has been supported in part by NSF Grants OISE-0438765 and DMS-0920850. The project is also partially supported by a grant at Fudan University.

      Authors’ Affiliations

      (1)
      Department of Mathematics, Norfolk State University
      (2)
      Department of Mathematical Sciences, University of Delaware

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      Copyright

      © M. Fang and R.P. Gilbert 2009

      This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.