- Research Article
- Open Access
© Caisheng Chen et al. 2010
- Received: 15 August 2010
- Accepted: 10 December 2010
- Published: 15 December 2010
- Weak Solution
- Bounded Domain
- Nontrivial Solution
- Lower Solution
- Weighted Sobolev Space
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 . 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 ; see [2–7]. In these works, the nonlinearities have sublinear and suplinear growth at infinity, and they behave like a function ( , or ) at the origin. Roughly speaking, in this case we say that the nonlinearities are concave and convex or "slow diffusion and fast diffusion''; see .
When , , and , , by using the lower and upper solution method, Santos in  finds a real number , such that the problem (1.1) has at least one solution if .
For , the existence and multiplicity of solution of singular elliptic equation like (1.1) in a bounded domain with the zero Dirichlet data have been widely studied by many authors, for example, the authors [9–13] and references therein. Assunção et al. in  studied the multiplicity of solution for the singular equations in (1.1) with , , , and in . Similar consideration can be found in [15–20] 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 and , . For the case , , the problem for the existence of solution for (1.1) is still open. It seems difficult to consider the case by variational method.
The main aim of this work is to study the existence and nonexistence of solution for (1.1), where is sublinear and 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 (see (2.17) below) is essential. By this, an upper solution for (1.1) is obtained.
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.
Theorem 2.1 ((compact imbedding theorem) ).
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.
The following lemma will be basic in our approach.
By Theorem 1.1 in , one concludes that there exists which is a weak solution of (2.11) with a.e. in for .
Similar to the proof Theorem 1.1 in  and the proof of Theorem 7.5.1 in , it is not difficult to get from Theorem 2.1 that is the maximal weak solution and the minimal solution of (2.3), which satisfies (2.10) and . This ends the proof of Lemma 2.3.
Our main results read as follows.
Theorem 2.4 (existence).
Theorem 2.5 (nonexistence).
So, condition (2.19) implies (2.22).
Before proofing the existence, we present some preliminary lemmas which will be useful in what follows.
We now give the proof of Theorem 2.4.
Proof of Theorem 2.4.
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  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.
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).
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