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Multiplicity of positive solutions of superlinear semi-positone singular Neumann problems
Boundary Value Problems volume 2014, Article number: 217 (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.
We will be concerned with the existence and multiplicity of positive solutions of the superlinear singular Neumann boundary value problem in the semi-positone case
Here the type of perturbations may be singular near and is superlinear near . From the physical point of view, has an attractive singularity near if
and the superlinearity of means that
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 .
In Section 2, we perform a study of the sign of the Green’s function of the corresponding linear problems
In detail, we construct the Green’s function and give a sufficient condition to ensure is positive. This fact is crucial for our arguments. We denote
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 , , .
Considering the homogeneous boundary value problem
and let be the Green’s function of problem (2.1). Then can be written as
where m and n are linearly independent, and m, n and ω satisfy the following lemma.
Suppose that (H1) holds and problem (2.1) has only zero solution, then there exist two functionsandsatisfying:
is increasing and , ;
is decreasing and , ;
, , ;
, , ;
is a positive constant.
The Green’s functiondefined by (2.2) has the following properties:
is continuous in Q;
is symmetrical on Q;
has continuous partial derivatives on , ;
For each fixed , satisfies for , . Moreover, for .
For , has discontinuity point of the first kind, and
Suppose that conditions in Lemma 2.1hold andis continuous. Then the problem
has a unique solution, which can be written as
Next we state the theorem of fixed points in cones, which will be used in Section 3.
Let X be a Banach space and K (⊂X) be a cone. Assume that, are open subsets of X with, , and let
be a continuous and compact operator such that either
, and , ; or
, and , .
Then T has a fixed point in.
In applications below, we take with the supremum norm and define
One may readily verify that K is a cone in X. Now suppose that is continuous and define an operator by
for and .
T is well defined and maps X into K. Moreover, T is continuous and completely continuous.
3 Main results
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:
(H3): There exist continuous, non-negative functionsandsuch that
andis non-increasing andis non-decreasing in.
(H4): There existssuch that.
(H5): There exists a constant, such 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.
First, it is easy to see that we can take and such that
Suppose that (H1)-(H5) hold, thenis a strict lower solution to the problem
It is easy to see that and .
Since , and using (3.1), we have .
By assumption (H5), we have
This implies that is a strict lower solution to (3.3). □
Suppose that (H1)-(H5) hold. Then the problem
has at least one positive solutionwith.
The existence is proved using the Leray-Schauder alternative principle together with a truncation technique.
Since (H4) holds, we have
Consider the family of problems
where and , . is non-increasing.
Problem (3.5) is equivalent to the following fixed point problem in
where is defined by
Then we have, for all x,
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).
If , then
If , then
Since , we have
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 .
Since , , and is non-increasing, we have
This is a contradiction and completes the proof of Lemma 3.4. □
Proof of Theorem 3.1
To show (1.1) has a positive solution, we will show
has a solution , , .
If this is true, then is a positive solution of (1.1) since
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 , .
By the fact that is a bounded and equi-continuous family on , the Arzela-Ascoli theorem guarantees that has a subsequence , which converges uniformly on to a function . Then u satisfies , for all x. Moreover, satisfies the integral equation
Letting , we arrive at
where the uniform continuity of on is used. Therefore, u is a positive solution of (3.14).
Finally, it is not difficult to show that . Assume otherwise: note that . By Lemma 2.4, for all x, and . Hence, for all x,
Then we have for all x,
This is a contradiction and completes the proof of Theorem 3.1. □
Let us consider the following boundary value problem
where, andis continuous, is chosen such that
here. Then problem (3.17) has a positive solution.
We will apply Theorem 3.1 with and
Clearly, (H1)-(H3) and (H5) are satisfied.
Since , , then there exists such that
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.
Let and K be a cone in X defined by (2.5). Let
and define the operator by
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.
First we show
In fact, if , then and for . So we have
This implies , i.e., (3.20) holds.
Next we show
To see this, let , then and for . As a result, it follows from (H6) and (H7) that, for ,
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.
Since , then (H7) is satisfied for R large enough because when ,
Thus all the conditions of Theorem 3.2 are satisfied, so the existence is guaranteed.
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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).
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.