Open Access

Existence of Weak Solutions for a Nonlinear Elliptic System

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq1_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq2_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq3_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq4_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq5_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq6_HTML.gif on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq8_HTML.gif to the local tangential shear stress https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq9_HTML.gif

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq10_HTML.gif indicates the amount of slip. When https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq12_HTML.gif implies partial slip. As https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq14_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq15_HTML.gif defined in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq16_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ2_HTML.gif
(12)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ3_HTML.gif
(13)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ4_HTML.gif
(14)

Here we assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq17_HTML.gif is a bounded domain in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq18_HTML.gif with a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq19_HTML.gif boundary. We assume also that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq20_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq21_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq22_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq23_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq24_HTML.gif are given functions, while https://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; https://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 https://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 https://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 ( https://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, https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq31_HTML.gif and https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq35_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq36_HTML.gif defined in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq37_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq38_HTML.gif let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq39_HTML.gif and https://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

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

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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq43_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq45_HTML.gif such that

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq46_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq47_HTML.gif such that

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

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

https://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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq51_HTML.gif -Laplacian is as follows:

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq52_HTML.gif and https://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:

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

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

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

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

( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq63_HTML.gif ) If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq64_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq65_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq66_HTML.gif moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ15_HTML.gif
(210)
( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq67_HTML.gif ) If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq68_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq69_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq70_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ16_HTML.gif
(211)
and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq71_HTML.gif denotes the conjugate of r, namely, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq72_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq73_HTML.gif moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ17_HTML.gif
(212)
( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq74_HTML.gif ) If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq75_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq76_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ18_HTML.gif
(213)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq77_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq78_HTML.gif . Moreover
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq79_HTML.gif be a closed convex set ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq80_HTML.gif ), and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq82_HTML.gif . Then there exists
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ21_HTML.gif
(216)
in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq84_HTML.gif if
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ22_HTML.gif
(217)

whenever https://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:

https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ24_HTML.gif
(32)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq87_HTML.gif and a given https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq90_HTML.gif satisfies the following properties.

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

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq95_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq96_HTML.gif a.e. in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq97_HTML.gif , then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq98_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq99_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ27_HTML.gif
(35)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq100_HTML.gif . Here
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ28_HTML.gif
(36)

Lemma 3.3.

If https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq103_HTML.gif and write
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq105_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ30_HTML.gif
(38)
By Hölder's inequality,
https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq106_HTML.gif . Therefore we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq107_HTML.gif whenever https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq109_HTML.gif is monotone.

To show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq110_HTML.gif is coercive on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq111_HTML.gif , fix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq112_HTML.gif . Then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq113_HTML.gif is coercive on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq114_HTML.gif .

Finally, we show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq115_HTML.gif is weakly continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq116_HTML.gif . Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq118_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq119_HTML.gif . Select a subsequence { https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq120_HTML.gif } such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq121_HTML.gif a.e. in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq122_HTML.gif . Then it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ33_HTML.gif
(311)
a.e. in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq123_HTML.gif . Moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ34_HTML.gif
(312)
Thus we have that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ35_HTML.gif
(313)
weakly in https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ36_HTML.gif
(314)

weakly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq125_HTML.gif . Hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq126_HTML.gif is weakly continuous on https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq129_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq130_HTML.gif , and Theorem https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq131_HTML.gif ]. Since https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq135_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ37_HTML.gif
(315)

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

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq137_HTML.gif and choose nonnegative sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq138_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq139_HTML.gif in https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ38_HTML.gif
(316)
Because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq141_HTML.gif , we obtain
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq142_HTML.gif be bounded and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq143_HTML.gif . There is a weak solution https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq145_HTML.gif , assume that there exists another solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq146_HTML.gif Then we have that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ40_HTML.gif
(318)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq147_HTML.gif . If we take https://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.

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

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

( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq152_HTML.gif ) when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq153_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ42_HTML.gif
(320)
Here the Hölder inequality for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq154_HTML.gif , namely,
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq156_HTML.gif in (3.2), we have
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ45_HTML.gif
(323)
Young's inequality with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq157_HTML.gif implies
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ47_HTML.gif
(325)
Setting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq158_HTML.gif and subtracting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq159_HTML.gif from both sides, we obtain that
https://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 https://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:

( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq161_HTML.gif ) when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq162_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ49_HTML.gif
(327)
( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq163_HTML.gif ) when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq164_HTML.gif ,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq165_HTML.gif , there is a closed subset https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq166_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq167_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq168_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq169_HTML.gif uniformly on https://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
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ52_HTML.gif
(41)
and for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq172_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ53_HTML.gif
(42)
and for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq173_HTML.gif
https://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, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq175_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq176_HTML.gif satisfy
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq177_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ56_HTML.gif
(45)
Moreover, there exists a polynomial https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq178_HTML.gif that is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq179_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq180_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq181_HTML.gif in (4.3), we obtain
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ60_HTML.gif
(49)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq182_HTML.gif satisfies https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq184_HTML.gif depends on https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq186_HTML.gif ).

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

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

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

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq189_HTML.gif is a bounded continuous function, so
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ63_HTML.gif
(412)
We next estimate III:
https://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 https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq191_HTML.gif ).

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

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

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

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

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

for some polynomial https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq198_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ71_HTML.gif
(420)
Moreover, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq199_HTML.gif a.e. in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq200_HTML.gif , then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq201_HTML.gif determined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_Equ73_HTML.gif
(422)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq202_HTML.gif By virtue of (4.6), https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq204_HTML.gif independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq205_HTML.gif such that
https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708389/MediaObjects/13661_2009_Article_874_IEq206_HTML.gif )]. Therefore, arguments after [11,  equation ( https://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.