Nonlinear Analysis: Theory, Methods & Applications
Structure of positive solution sets of semi-positone singular boundary value problems☆
Introduction
Consider the differential boundary value problem where is a parameter, and and satisfy:
(H1) , for all , , and where , , , , , , is nonincreasing, and
The boundary value problem (1.1λ) has an integral boundary condition. Problems with a boundary condition of this type arise naturally in thermal conduction problems [1], semiconductor problems [2], hydrodynamic problems [3], and problems with integral boundary conditions have been studied by many authors; see [4], [5], [6], [7], [8], [9] and the references therein. When in (1.2), the nonlinear term in (1.1λ) is allowed to take negative values, and so, the boundary value problem (1.1λ) is a so-called semi-positone boundary value problem. The study of semi-positone problems was formally introduced by Castro and Shivaji [10]. From an application viewpoint one is usually interested in the existence of positive solutions for semi-positone problems. Significant progress on semi-positone problems has been made in the last ten years; see [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] and the references therein. The boundary value problem (1.1λ) is also singular since the nonlinear term in (1.1λ) may be infinite at or or . In the past twenty years singular boundary value problems have been studied extensively; see [18], [19], [22] and the references therein.
The main purpose of this paper is to study the structure of the positive solution set of (1.1λ). Rabinowitz [23] gave the first important results on the structure of the solution sets of nonlinear equations obtained by the degree theoretic method. H. Amann [24] studied the structure of the positive solution set of nonlinear equations; the reader is referred to [25], [26] for other results concerning the structure of solution sets of nonlinear equations. In our paper we will show that an unbounded connected component of the positive solution set bifurcates from or . To show our main results, we will first study the nonsingular approximation of the boundary value problem (1.1λ). Using global bifurcation theories we show that an unbounded connected component of the positive solution set of the nonsingular approximation of the boundary value problem (1.1λ) bifurcates from the trivial solution . Then, by using some results concerning the connected component of limit sets (see Lemma 2.14, Lemma 2.15 of this paper), we obtain an unbounded connected component of the positive solution set of the boundary value problem (1.1λ). As an application of the main results, we will also give some existence results for positive solutions of the singular semi-positone boundary value problem (1.1λ). This paper generalizes some results from the literature [11], [12], [13], [14], [15], [16], [17], [18], [19], [20].
The paper is arranged as follows. In Section 2, we will give some preliminary results. The main results will be given in Section 3. To illustrate the main results of this paper, in the last section an example will be given.
Section snippets
Some lemmas
Let and , the usual real Banach space of continuous functions with the maximum norm . Let denote the set of all positive integers and It follows from (H1) that . Let . Then is a cone of . Let for each and . Then is also a cone of . In the sequel
Main results
Now we give the main results of this paper. Let We will study the structure of the set . Let be a connected component of . In the sequel we say that comes from when if there exists a sequence such that as .
Theorem 3.1 Suppose (H1) , (H2) (H3) (or (H4) ) holds, . Then possesses an unbounded connected component which comes from and tends to .
Proof First we consider the case when
References (31)
Second-order boundary value problems with integral boundary conditions
Nonlinear Anal.
(2009)- et al.
Sign-changing solutions to second-order integral boundary value problems
Nonlinear Anal.
(2008) Existence and nonexistence results for positive solutions of an integral boundary value problem
Nonlinear Anal.
(2006)- et al.
Existence results for semipositone boundary value problems
Acta Math. Sci.
(2001) Positive solutions for semipositone conjugate boundary value problems
J. Math. Anal. Appl.
(2000)- et al.
Discrete semipositone higher-order equations
Comput. Math. Appl.
(2003) - et al.
Nonpositone discrete boundary value problems
Nonlinear Anal.
(2000) Multiplicity results for positive solutions of some semi-positone three-point boundary value problems
J. Math. Anal. Appl.
(2004)Positive solutions for singular -point boundary value problems with positive parameter
J. Math. Anal. Appl.
(2004)Some global results for nonlinear eigenvalues
J. Funct. Anal.
(1971)
On bifurcation from infinity
J. Differential Equations
Positive solutions of generalized Emden–Fowler equation
Nonlinear Anal.
A property of connected components and its applications
Topol. Appl.
Structure of a class of singular boundary value problem with superlinear effect
J. Math. Anal. Appl.
Global structure of positive solutions for nonlocal boundary value problems involving integral conditions
Nonlinear Anal.
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This paper was supported by NSFC10971179, the Natural Science Foundation of Jiangsu Education Committee (09KJB110008) and the Qing Lan Project.