Analysis on Existence of Positive Solutions for a Class Second Order Semipositone Differential Equations

Abstract

In this paper, we study the existence of positive solutions of the following second-order semipositone system (see equation 1). By applying a well-known fixed-point theorem, we prove that the problem admits at least one positive solution, if is bounded below.

1. Introduction

This paper is focused on the existence of positive solutions of a second-order semipositone system where is a positive constant and satisfies the following assumption: is continuous, and where is continuous and on [0,1].

The second-order elliptic systems have a strong physical meaning in quantum mechanics models [1, 2], in semiconductor theory [3], or in a time- and space-dependent mathematical model of nuclear reactors in a closed container [4]. To the best of our knowledge, existence and multiplicity of nontrivial solutions of BVP(1) have been widely studied by using the variational method [5], bifurcation techniques [6, 7], or fixed-point theorems [8–11]. In general, in order to ensure the positivity of the solutions of Equation (1), one of the crucial assumptions is that the nonlinearity is nonnegative. Of course, the natural question is whether Equation (1) has a positive solution or not if satisfies the assumption .

On the other hand, many authors have been interested in finding the relations between the positivity of solutions and the changing sign of the nonlinearity in order to prove the existence of the positive solutions. We refer the readers to [12–16] and the references.

Inspired by these references, the purpose of this paper is to find some new conditions, which are used to study the existence and multiplicity of positive solutions of the semipositone Equation (1). The main tool is the following well-known fixed-point theorem.

Lemma 1 [17]. Let be a Banach space and be a cone in . Assume and are open bounded subsets of with . Let be a completely continuous operator such that (a), for , and(b)there exists a such thatThen, has a fixed point in . The same conclusion remains valid if (a) holds on and (b) holds on .

The paper is organized as follows: in Section 2, we give some preliminaries, which are about the properties of the Green functions, the notations of some sets, etc.; in Section 3, we give the main results and the corresponding proof. In Section 4, some examples are given to illustrate the main results.

2. Preliminary

Let be the Green function of linear boundary value problem where .

Lemma 2 [18]. Let , then can be expressed by (i)(ii)(iii).

Lemma 3 [18]. The function has the following properties: (i)(ii)(iii)where , if ; , if ; and , if .

Lemma 4. For the function , there exists a such that

Proof. (i)For , we haveLet Since is positive and continuous on [0,1] and , we have In the similar way, we also have Choosing . Then, for any , we have Since , then for any , we have Therefore, there exists a such that (ii)For , it is obvious that (iii)For , we have Using the similar discussion of Case (i), it follows that there exists a such that For convenience, let denote the Green function for . Then, Equation (1) can be rewritten as Furthermore, let , where is the unique solution of the linear boundary value problem Then, we rewrite (17) as where From the above discussion, then we have the following lemma.

Lemma 5. Assume that holds. Then, is a positive solution of (1) if only if is a positive solution of the following problem: with . Here, denotes the Heaviside function of a single real variable: Let denote the Banach space with the norm .
Define a cone by where . Define an operator by

Lemma 6. Assume that (F₀) holds. Then, , and is completely continuous.

Proof. For any , from Lemma 3, it follows that which implies that .
Now, we show that is completely continuous.
First, we show that maps the bounded set into itself. Since and are continuous, for any given , let Then, for , we have which implies that is uniformly bounded.
Second, for , we have which implies that the operator is equicontinuous.
Thus, by applying the Arzela-Ascoli theorem [17], we obtain that is relatively compact, namely, the operator is compact.
Finally, we claim that is continuous. Assume that which converges to uniformly on [0,1]. By Lebesgue’s dominated convergence theorem and letting , we have So, is continuous on . The proof is completed.
At the end of this section, let Define the height functions In addition, we need to select some suitable open bounded sets. For any , let From [19, 20], we can conclude the lemma below.

Lemma 7. (i), are open relative to (ii)(iii) if and only if (iv)If , then , for

3. Main Results

Theorem 8. Assume that (F₀) holds. In addition, the function satisfies the following assumption:
(F₁) There exists a such that and Then, we have (i)If , then (1) has at least one partly positive solution , namely, (ii)If , then (1) has at least one positive solution , which satisfies

Proof. For any , it is obvious that Then, from (F₁) it follows that which implies that (a) of Lemma 1 holds.
Let From [21], we have that Then, there exists a with such that where satisfies Let ; now we prove that . On the contrary, if there exists a pair of and such that , then from (iv) of Lemma 7, it follows that Furthermore, for , we have which contradicts with the statement (iii) of Lemma 7. So (b) holds.
Since , from Lemma 7, we have . Therefore, by Lemma 1, we can get that has at least one positive fixed-point . Hence, the inequalities hold, On the other hand, since , we have . (i)Since we have (ii)From Lemmas 3 and 4, we have which implies that Therefore, (1) has one positive solution .

Theorem 9. Assume that (F₀) holds. In addition, the function satisfies the following assumptions:
(F₂) There exists a such that and (F₃) There exists a such that (F₄) , uniformly for .
Then, (1) has at least two positive solution, which satisfies

Proof. From (F₂) and Theorem 8, it follows that there exists a solution and Now, we apply Lemma 1 to prove the existence of another solution .
Since and , then we can define the operator For any , it is obvious that Then, we have which implies that (a) of Lemma 1 holds.
Since , uniformly for , there exists a such that where satisfies Let , now we prove that . On the contrary, if there exists a pair of , then from (iv) of Lemma 7, it follows that Furthermore, we have which contradicts with the statement (iii) of Lemma 7. So (b) holds.

Therefore, from Lemma 1, we can get that has at least one positive fixed-point . Hence, the inequalities hold

On the other hand, since , we have .

Finally, since

(1) has at least two positive solutions.

4. Examples

Example 10. Let us consider the following system: where is continuous, and It is obvious that holds. Via some computations, we have Choosing . Then, we have Furthermore, we have Choosing . Then, It is easy to get which implies that and hold.

Finally, it is obvious that

So holds.

Therefore, by Theorem 9, (61) has two positive solutions.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All the authors contributed equally and significantly in writing this article. All the authors read and approved the final manuscript.

Acknowledgments

The authors were supported by the Fundamental Research Funds for the Central Universities (B200202003), the Guizhou Provincial Science and Technology Fund (QKH-JICHU[2017], Grant no. 1408) and the Scientific Research Starting Foundation for Doctorate Research from Shandong Jiaotong University.