- Open Access
Multiplicity of positive solutions of superlinear semi-positone singular Neumann problems
© Li et al.; licensee Springer 2014
- Received: 5 February 2014
- Accepted: 12 September 2014
- Published: 8 October 2014
Neumann boundary value problems have been studied by many authors. We are mainly interested in the semi-positone case. This paper deals with the existence and multiplicity of positive solutions of a superlinear semi-positone singular Neumann boundary value problem.
The proof of our main results relies on a nonlinear alternative of Leray-Schauder type, the method of upper and lower solutions and on a well-known fixed point theorem in cones.
We obtained the existence of at least two different positive solutions.
- positive solutions
- Neumann problem
By the semi-positone case of (1.1), we mean that may change sign and satisfies where is a constant.
It is well known that the existence of positive solutions of boundary value problems has been studied by many authors in – and references therein. They mainly considered the case of and . In , the authors studied positive solutions of Neumann boundary problems of second order impulsive differential equations in the positone case, based on a nonlinear alternative principle of Leray-Schauder type and a well-known fixed point theorem in cones. This paper attempts to study the existence and multiplicity of positive solutions of second order superlinear singular Neumann boundary value problems in the semi-positone case. The techniques we employ here involve a nonlinear result of Leray-Schauder, the well-known fixed point theorem in cones and the method of upper and lower solutions. We prove that problem (1.1) has at least two different positive solutions. Moreover, we do not take the restrictions or .
Throughout this paper, we assume that the perturbed part satisfies the following hypotheses:
(H1): , , , , .
(H2): There exists a constant such that for all and .
We also use to denote the unique solution of (1.2) with , . In Section 3, we state and prove the main results of this paper.
For the reader’s convenience we introduce some results of Green’s functions. Let , , .
where m and n are linearly independent, and m, n and ω satisfy the following lemma.
is increasing and , ;
is decreasing and , ;
, , ;
, , ;
is a positive constant.
is continuous in Q;
is symmetrical on Q;
has continuous partial derivatives on , ;
For each fixed , satisfies for , . Moreover, for .
- (v)For , has discontinuity point of the first kind, and
Next we state the theorem of fixed points in cones, which will be used in Section 3.
, and , ; or
, and , .
Then T has a fixed point in.
for and .
T is well defined and maps X into K. Moreover, T is continuous and completely continuous.
In this section we establish the existence and multiplicity of positive solutions to (1.1). Since we are mainly interested in the attractive-superlinear nonlinearities in the semi-positone case, we assume that the hypotheses of the following theorem are satisfied.
Suppose that (H1) and (H2) hold. Furthermore, assume the following:
andis non-increasing andis non-decreasing in.
(H4): There existssuch that.
Then problem (1.1) has at least one positive solutionwith.
Before we present the proof of Theorem 3.1, we state and prove some facts.
It is easy to see that and .
Since , and using (3.1), we have .
This implies that is a strict lower solution to (3.3). □
has at least one positive solutionwith.
The existence is proved using the Leray-Schauder alternative principle together with a truncation technique.
where and , . is non-increasing.
This is a contradiction and the claim is proved. □
From this claim, the nonlinear alternative of Leray-Schauder guarantees that problem (3.6) (with ) has a fixed point, denoted by , in , i.e., problem (3.4) has a positive solution with . (In fact, it is easy to see that with .)
Suppose that (H1)-(H5) hold, thenis an upper solution of problem (3.3).
By Lemma 3.2 we know that is a solution to equation (3.4).
This implies that is an upper solution of problem (3.3). □
Suppose that (H1)-(H5) hold, then ().
Let , we will prove . If this is not true for , there exists such that , , . Then .
This is a contradiction and completes the proof of Lemma 3.4. □
Proof of Theorem 3.1
has a solution , , .
As a result, we will only concentrate our study on (3.14).
By Lemmas 3.1-3.4 and the upper and lower solutions method, we know that (3.3) has a solution with . Thus we have , .
where the uniform continuity of on is used. Therefore, u is a positive solution of (3.14).
This is a contradiction and completes the proof of Theorem 3.1. □
here. Then problem (3.17) has a positive solution.
Clearly, (H1)-(H3) and (H5) are satisfied.
This implies that there exists such that , so (H4) is satisfied.
Since . Thus all the conditions of Theorem 3.1 are satisfied, so the existence is guaranteed. □
Next we will find another positive solution to problem (1.1) by using Theorem 2.1.
Suppose that conditions (H1)-(H5) hold. In addition, it is assumed that the following two conditions are satisfied:
(H6): for some continuous non-negative functionsandwith the properties thatis non-increasing andis non-decreasing.
(H7): There existssuch that.
Then, besides the solution u constructed in Theorem 3.1, problem (1.1) has another positive solutionwith.
where is as in (2.2).
For each , we have . Since is continuous, it follows from Lemma 2.4 that the operator is well defined, is continuous and completely continuous.
This implies , i.e., (3.20) holds.
Let us consider again example (3.17) in Corollary 3.1 for the superlinear case, i.e., , and is continuous, is chosen such that (3.18) holds, here . Then problem (3.17) has a positive solution . Clearly, (H1)-(H6) are satisfied.
Thus all the conditions of Theorem 3.2 are satisfied, so the existence is guaranteed.
The authors express their thanks to the referee for his valuable suggestions. The work was supported by the National Natural Science Foundation of China (No: 11171350).
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