In this paper, we first obtain some results on the structure of positive solution sets of differential boundary value problems. Then by using the results, we obtain an existence result for differential boundary value problems. The method used to show the main result is the global bifurcation theory.
Keywords:structure of positive solution sets; differential boundary value problems; bifurcation theory
This paper considers the differential boundary value problem
Equations of form (1.1) occur in the study for the p-Laplacian equation, non-Newtonial fluid theory and the turbulent flow of a gas in a porous medium. The case where
i.e., perturbations of the p-Laplacian, has received much attention in the recent literature. Also, problem (1.1) with has been studied by several authors in recent years (see  and the references therein). Here, we are interested in the case when may be negative (the so-called semipositone case) (see  and its references for a review). As pointed out by Lions in , semi-positone problems are mathematically very challenging. During the last ten years, finding positive solutions to semi-positone problems has been actively pursued and significant progress on semi-positone problems has taken place; see [4-8] and the references therein. For instance, Hai et al. considered the existence positive solution of (1.1). Under some super-linear conditions on the non-linear term f, they proved that there exists such that (1.1) has one positive solution for . The main method in  used to show the main result are the fixed-point theorems.
The main purpose of this paper is going to study the structure of the positive set of (1.1). Rabinowitz  gave the first important results on the structure of the solution sets of non-linear equations and obtained by the degree theoretic method. Amamn  studied the structure of the positive solution set of non-linear equations; the reader is referred to [12,13] for other results concerning the structure of solution sets of non-linear equations. In our paper, we will study the existence results for an unbounded connected component of a positive solution set for the differential boundary value problem of (1.1). This paper generalizes some results from the literature . The paper is arranged as follows. In Section 2, we will give some preliminary lemmas. The main results will be given in Section 3.
2 Some lemmas
For convenience, we make the following assumptions:
here C is a constant such that
We know that C exists and is unique for every (see ). Then if and only if u is a solution of
From , we have the following Lemmas 2.1 and 2.2.
Proof By integrating, it follows that (3.1) has the unique solution given by
On the other hand, using the inequality
The proof is complete. □
From [, Lemma 29.1], we have Lemma 2.4.
Lemma 2.4LetXbe a compact metric space. Assume thatAandBare two disjoint closed subsets ofX. Then either there exist a connected component ofXmeeting bothAandBorwhere, are disjoint compact subsets ofXcontainingAandB, respectively.
LetUbe an open and bounded subset of the metric space. We set, whose boundary is denoted by. Consider a map, such thatis compact and. Such a maphwill also be called an admissible homotopy onU. Ifhis an admissible homotopy, for everyand every, one has thatand it makes sense to evaluate.
Assume by contradiction that
From (2.3) and (2.4), we have
3 Main results
For convenience, let us introduce the following symbols. For any r,
Now we give our main results of this paper.
Proof We divide our proof into four steps.
Step 1. Let
Step 2. Let
From Lemma 2.6, we have
This implies that T has no bifurcation point on . From step 1,we have , then for each , denote by the connected component of the metric space emitting from . Now we will show that, there must exist such that is unbounded. Assume on the contrary that is bounded for each . Take a bounded open neighborhood in for each such that
and , where denotes the closure of in the metric space . Let denote the boundary of in the metric space . Obviously, is a compact subset. Assume that . From the maximal connectedness of , there is no connected subset of meeting both and . From Lemma 2.4, there exist compact disjoint subsets and of such that and , and . Let and be the -neighborhood of in the metric space . Set
Obviously, the collection of the subsets
From (3.5) and (3.8), we have
Then from Lemma 2.5, we have
Obviously, . For each , denote by the connected component of the metric space , which passes the point p. Now we shall prove that there must exist a such that is an unbounded connected component of the metric space . On the contrary, assume that is bounded for each . Then, for each , in the same way as in the construction of in (3.6) we can show that there exists a neighborhood of in such that
then by (3.12) we have
From (3.12) and (3.13), we see that . Obviously, and . Note the unboundedness of , then . Now we have , which is a contradiction of the connectedness of . Therefore, there must exist such that is an unbounded connected component of .
The proof is complete. □
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read, checked and approved the final manuscript.
This paper is supported by Innovation Project of Jiangsu Province postgraduate training project (CXLX12_0979).
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