Structure of positive solution sets of differential boundary value problems
© Xian and Xunxia; licensee Springer. 2013
Received: 26 September 2012
Accepted: 8 March 2013
Published: 22 April 2013
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.
where f is ϕ-superlinear at ∞ and maybe negative and p is a positive continuous function, is a parameter.
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:
(A1) ϕ is an odd, increasing homeomorphism on R with concave on .
(A2) For each , there exists such that , and (note that (A2) implies the existence of such that , and ).
(A3) is continuous.
uniformly for .
From , we have the following Lemmas 2.1 and 2.2.
Further, as and as .
Lemma 2.2 Let ϕ be as in Lemma 2.1. Then there exist and such that for , .
The proof is complete. □
Let for each , where and are defined as that in Lemma 2.2 and Lemma 2.3, respectively. Then is also a cone of E. From Lemma 2.3, we know that is completely continuous.
From [, Lemma 29.1], we have Lemma 2.4.
Lemma 2.4 Let X be a compact metric space. Assume that A and B are two disjoint closed subsets of X. Then either there exist a connected component of X meeting both A and B or where , are disjoint compact subsets of X containing A and B, respectively.
Let U be an open and bounded subset of the metric space . We set , whose boundary is denoted by . Consider a map , such that is compact and . Such a map h will also be called an admissible homotopy on U. If h is an admissible homotopy, for every and every , one has that and it makes sense to evaluate .
Lemma 2.5 If h is an admissible homotopy on , the is constant for all .
so , which is a contradiction. Then (2.3) holds. The proof is complete. □
3 Main results
Now we give our main results of this paper.
where denotes the projection of onto the λ-axis.
Proof We divide our proof into four steps.
where . Obviously, is a completely continuous operator.
So, and .
which contradicts to (3.10) and (3.11). Therefore, there must exist such that is bounded.
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. □
Corollary 3.1 Let (A1) to (A4) hold. Then there exists such that problem (1.1) has a positive solution for with as .
This paper is supported by Innovation Project of Jiangsu Province postgraduate training project (CXLX12_0979).
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