We are interested in the following singular boundary value problem:
where is a parameter and is the Stieltjes integral. The function and w may be singular at and/or , and . Some a priori estimates and the existence, multiplicity and nonexistence of positive solutions are obtained. Our proofs are based on the method of global continuous theorem, the lower-upper solutions methods and fixed point index theory. Furthermore, we also discuss the interval of parameter μ such that the problem has a positive solution.
Keywords:singularity; global continuous theorem; solution of boundedness; fixed point index; positive solution
We are concerned with the second order nonlocal boundary value problem:
Integral boundary conditions and multi-point boundary conditions for differential equations come from many areas of applied mathematics and physics [1-7]. Recently, singular boundary value problems have been extensively considered in a lot of literature [1,2,5,8], since they model many physical phenomena including gas diffusion through porous media, nonlinear diffusion generated by nonlinear sources, chemically reacting systems as well as concentration in chemical or biological problems. In all these problems, positive solutions are very meaningful.
where hg are nonnegative functions, subjected to the nonlocal boundary conditions
involving a Stieltjes integral with a signed measure, that is, A has bounded variation. They dealt with many boundary conditions given in the literature in a unified way by utilizing the fixed point index theory in cones.
Recently, many researchers were interested in the global structure of positive solutions for the nonlinear boundary value problem (see, e.g., [3,6,7]). In 2009, Ma and An  considered the problem (1.1). Assume that
They obtained the following main result:
Theorem 1.1 (, Theorem 4.1])
In this paper, we first obtain the global structure of positive solutions by the use of global continuous theorem, and some a priori estimates. Applying the analysis technique, we construct the lower and upper solutions. Those combined with the fixed point index theory, the existence, multiplicity and nonexistence of positive solutions to (1.1) in the case are investigated. Finally, we discuss the interval of parameter μ such that the problem (1.1) has positive solutions. The proof of the method which is based on the construction of some bounds of the solution together with global continuous theorem and fixed point index is of independent interest, and is different from the other papers.
This paper is arranged as follows. We will give some hypotheses and lemmas in Section 2. In Section 3, new criteria of the existence, multiplicity and nonexistence of a positive solution are obtained. Moreover, an example is given to illustrate our result.
2 Preliminaries and lemmas
Throughout this paper, we suppose that the following conditions hold:
Lemma 2.1 ( Global continuation theorem)
LetXbe a Banach space and letKbe an order cone inX. Consider the equation
Lemma 2.2 ()
3 Main results
This, together with (3.1) and (3.2), implies
On the other hand, multiplying (1.1) by ψ and integrating by parts, we obtain that
This is a contradiction. □
and let be the positive eigenfunction corresponding to (see ). It is easy to see that . Multiplying (3.4) by and integrating by parts, we get that
This completes the proof. □
Lemma 3.3Assume that (H0)-(H3) hold. Then we have
Hence, we find that
Theorem 3.1Assume that the conditions (H0)-(H3) hold and. Then there exists a constantsuch that the problem (1.1) has at least two positive solutions for, and at least one positive solution for, and no positive solution for.
From Lemma 2.1 and Remark 2.2, we can find that there exists an unbounded continuum emanating from in the closure of the set of positive solutions in and for all . Meanwhile, Lemma 3.1 and Lemma 3.3 respectively imply that is bounded (, ) and unbounded (, and ). Therefore, we conclude that the set of (3.5) is nonempty. Those combined with Lemma 3.2 follows that is well defined and . From the definition of , it is easy to see that the problem (1.1) has at least two positive solutions for . Again, since the continuum is a compact connected set and T is a completely continuous operator, the problem (1.1) has at least one positive solution at .
Next, we only show that the problem (1.1) has no positive solution for any . Suppose on the contrary that there exists some () such that the problem (1.1) has a positive solution corresponding to . Then we will prove that the problem (1.1) has at least two positive solutions for any which contradicts the definition of (3.5).
For the sake of obtaining the contradiction, we divide the proof into four steps.
Step 1. Constructing a modified boundary value problem.
Indeed, using (3.6), we have
Let u be a positive solution of (3.9), then we claim that
Suppose this fails, that is, . Clearly, we only show that , for . Comparing the boundary conditions (3.8) and (3.9), the only following three cases need to be considered. Case I. There exists such that and , for and some ; Case II. There exists such that and for . Case III. There exists such that , , and . See the three Figures 1, 2 and 3.
This together with (3.11) and (3.12) leads to
This is a contradiction.
This contradicts (3.13).
Therefore, we conclude that the claim (3.10) holds.
Then is completely continuous and u is a positive solution of (3.9) if and only if on K. From the definition of , it implies that there exists such that , for all . Consequently, we get from Lemma 2.2 that
Step 4. We conclude that the problem (1.1) has at least two positive solutions corresponding to μ.
Since the problem (1.1) is equivalent to the problem (3.9) on , we get that the problem (1.1) has a positive solution in . Without loss of generality, we may suppose that T has no fixed point on (otherwise the proof is completed). Then is well defined and (3.15) implies
On the other hand, from Lemma 3.2, we choose such that the problem (1.1) has no positive solution in K. By apriori estimate in , there exists such that for all possible positive solutions of (1.1) with , we know that . Define by
Hence, by the additivity property and (3.16), we have
Then we conclude that the problem (1.1) has at least two positive solutions corresponding to μ. □
Let f attain its maximum at the point of . If , then adopting the similar method as in , Theorem 1], we get that for , the problem (1.1) has at least two positive solutions and such that by the use of compression of conical shells in , Corollary 20.1]. Consequently, we know that .
(ii) If u is a positive solution of the equation (1.1) corresponding to μ, then we have
i.e., from (3.17),
Corollary 3.1Assume that (H1)-(H3) hold. Consider the followingm-point boundary value problem
whereμis a positive parameter, , , and. Then there exists a constantsuch that the problem (3.18) has at least two positive solutions for, and at least one positive solution for, and no positive solution for.
Proof In the boundary condition of (1.1), if we let
Then the boundary condition of (1.1) reduces to the m-point boundary condition of (3.18). Applying the method of Theorem 3.1, we get the conclusion. □
Example 3.1 We consider the following singular boundary value problem:
Thus, it implies that (2.3) holds. It is easy to verify that the conditions (H2) and (H3) hold. Therefore, by Theorem 3.1, we obtain that there exists a constant such that the problem (3.19) has at least two positive solutions for , and at least one positive solution for , and no positive solution for .
The author declares that they have no competing interests.
The author typed, read and approved the final manuscript.
The author would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work. The work was supported partly by NSCF of Tianyuan Youth Foundation (No. 11126125), K. C. Wong Magna Fund of Ningbo University, Subject Foundation of Ningbo University (No. xkl11044) and Hulan’s Excellent Doctor Foundation of Ningbo University.
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