Open Access

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

Boundary Value Problems20102010:607453

DOI: 10.1155/2010/607453

Received: 15 August 2010

Accepted: 10 December 2010

Published: 15 December 2010

Abstract

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

1. Introduction

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

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

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

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

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

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

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

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

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

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

2. Preliminaries and Main Results

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

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

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

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

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

Definition 2.2 (see [10, 12]).

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

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

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

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

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

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

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

The following lemma will be basic in our approach.

Lemma 2.3.

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

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

Proof.

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

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

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

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

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

Our main results read as follows.

Theorem 2.4 (existence).

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

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

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

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

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

Theorem 2.5 (nonexistence).

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

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

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

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

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

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

Remark 2.6.

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

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

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

So, condition (2.19) implies (2.22).

3. Proof of Existence

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

Lemma 3.1.

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

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

Proof.

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

Lemma 3.2.

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

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

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

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

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

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

Corollary 3.3.

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

Lemma 3.4.

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

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

Proof.

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

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

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

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

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

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

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

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

We now give the proof of Theorem 2.4.

Proof of Theorem 2.4.

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

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

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

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

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

Remark 3.5.

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

4. Proof of Nonexistence

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

Proof of Theorem 2.5.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Declarations

Acknowledgments

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

Authors’ Affiliations

(1)
Department of Mathematics, Hohai University

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© Caisheng Chen et al. 2010

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