Positive Solutions for Multiparameter Semipositone Discrete Boundary Value Problems via Variational Method

Abstract

We study the existence, multiplicity, and nonexistence of positive solutions for multiparameter semipositone discrete boundary value problems by using nonsmooth critical point theory and subsuper solutions method.

1. Introduction

Let and be the set of all integers and real numbers, respectively. For , define , , when .

In this paper, we consider the multiparameter semipositone discrete boundary value problem

(1.1)

where are parameters, is a positive integer, is the forward difference operator, , is a continuous positive function satisfying , and is continuous and eventually strictly positive with .

We notice that for fixed , whenever is sufficiently small. We call (1.1) a semipositone problem. Semipositone problems are derived from [1], where Castro and Shivaji initially called them nonpositone problems, in contrast with the terminology positone problems, put forward by Keller and Cohen in [2], where the nonlinearity was positive and monotone. Semipositone problems arise in bulking of mechanical systems, design of suspension bridges, chemical reactions, astrophysics, combustion, and management of natural resources; for example, see [3–6].

In general, studying positive solutions for semipositone problems is more difficult than that for positone problems. The difficulty is due to the fact that in the semipositone case, solutions have to live in regions where the nonlinear term is negative as well as positive. However, many methods have been applied to deal with semipositone problems, the usual approaches are quadrature method, fixed point theory, subsuper solutions method, and degree theory. We refer the readers to the survey papers [7, 8] and references therein.

Due to its importance, in recent years, continuous semipositone problems have been widely studied by many authors, see [9–15]. However, we noticed that there were only a few papers on discrete semipositone problems. One can refer to [16–18]. In these papers, semipositone discrete boundary value problems with one parameter were discussed, and subsuper solutions method and fixed point theory were used to study them. To the authors' best knowledge, there are no results established on semipositone discrete boundary value problems with two parameters. Here we want to present a different approach to deal with this topic. In [11], Costa et al. applied the nonsmooth critical point theory developed by Chang [19] to study the existence and multiplicity results of a class of semipositone boundary value problems with one parameter. We think it is also an efficient tool in dealing with the semipositone discrete boundary value problems with two parameters.

Our main objective in this paper is to apply the nonsmooth critical point theory to deal with the positive solutions of semipositone problem (1.1). More precisely, we define the discontinuous nonlinear terms

(1.2)

Now we consider the slightly modified problem

(1.3)

Just to be on the convenient side, we define , , , , where , ,

(1.4)

We will prove in Section 3 that the sets of positive solutions of (1.1) and (1.3) do coincide. Moreover, any nonzero solution of (1.3) is nonnegative.

Our main results are as follows.

Theorem 1.1.

Suppose that there are constants , , and such that when is large enough,

(1.5)

(1.6)

(1.7)

Then for fixed , there is a such that for , problem (1.3) has a nontrivial nonnegative solution. Hence problem (1.1) has a positive solution.

Remark 1.2.

By (1.6), there are constants such that for any ,

(1.8)

Equations (1.6) and (1.8) imply that

(1.9)

which shows that is superlinear at infinity.

Remark 1.3.

Equation (1.7) implies that is sublinear at infinity. Moreover, it is easy to know that

(1.10)

Hence is subquadratic at infinity.

Theorem 1.4.

Suppose that the conditions of Theorem 1.1 hold. Moreover, is increasing on . Then there is a such that for , problem (1.1) has at least two positive solutions for sufficiently small .

Theorem 1.5.

Suppose that the conditions of Theorem 1.1 hold. Moreover, is nondecreasing on . Then for fixed , problem (1.1) has no positive solution for sufficiently large .

2. Preliminaries

In this section, we recall some basic results on variational method for locally Lipschitz functional defined on a real Banach space with norm . is called locally Lipschitzian if for each , there is a neighborhood of and a constant such that

(2.1)

The following abstract theory has been developed by Chang [19].

Definition 2.1.

For given , the generalized directional derivative of the functional at in the direction is defined by

(2.2)

The following properties are known:

• (i) is subadditive, positively homogeneous, continuous, and convex;

• (ii)

• (iii) .

Definition 2.2.

The generalized gradient of at , denoted by , is defined to be the subdifferential of the convex function at , that is,

(2.3)

The generalized gradient has the following main properties.

• (1) For all , is a nonempty convex and -compact subset of ;

• (2) for all .

• (3) If are locally Lipschitz functional, then

  

  (2.4)

• (4) For any ,

• (5) If is a convex functional, then coincides with the usual subdifferential of in the sense of convex analysis.

• (6) If is Gâteaux differential at every point of of a neighborhood of and the Gâteaux derivative is continuous, then

• (7) The function

  

  (2.5)

  exists, that is, there is a such that .

• (8) .

• (9) If has a minimum at , then .

Definition 2.3.

is a critical point of the locally Lipschitz functional if .

Definition 2.4.

is said to satisfy Palais-Smale condition (PS) condition for short) if any sequence such that is bounded and has a convergent subsequence.

Lemma 2.5 (see [19, Mountain Pass Theorem]).

Let be a real Hilbert space and let be a locally Lipschitz functional satisfying (PS) condition. Suppose that and that the following hold.

• (i) There exist constants and such that if .

• (ii) There is an such that and .

Then possesses a critical value . Moreover, can be characterized as

(2.6)

where

(2.7)

Next we give the definitions of the subsolution and the supersolution of the following boundary value problem:

(2.8)

Definition 2.6.

If satisfies the following conditions:

(2.9)

then is called a subsolution of problem (2.8).

Definition 2.7.

If satisfies the following conditions:

(2.10)

then is called a supersolution of problem (2.8).

Lemma 2.8.

Suppose that there exist a subsolution and a supersolution of problem (2.8) such that in . Then there is a solution of problem (2.8) such that in .

Remark 2.9.

If (2.8) is replaced by (1.1), then we have similar definitions and results as Definitions 2.6, 2.7, and Lemma 2.8

3. Proof of Main Results

Let be the class of the functions such that . Equipped with the usual inner product and the usual norm

(3.1)

is an -dimensional Hilbert space. Define the functional on as

(3.2)

where , and

(3.3)

Clearly, is a locally Lipschitz function and is a locally Lipschitz functional on . By a simple computation, we obtain

(3.4)

By [19, Theorem 2.2], the critical point of the functional is a solution of the inclusion

(3.5)

where .

Remark 3.1.

We can show that for , for . For fixed and sufficiently small , . Then .

Remark 3.2.

If , then the above inclusion becomes

(3.6)

It is clear that is a positive definite matrix. Let be the largest and smallest eigenvalue of , respectively. Denote by . Let . Notice that for and for . Then

(3.7)

Similarly, for . Hence

(3.8)

Lemma 3.3.

If u is a solution of (1.3), then . Moreover, either in , or everywhere.

Proof.

It is not difficult to see that for . In fact, no matter that or , the former inequality holds. Hence .

If is a solution of (1.3), then we have

(3.9)

So . Hence . If , then

(3.10)

Therefore . It follows that everywhere.

Lemma 3.4.

If (1.6) and (1.7) hold, then for large , where .

Proof.

Notice that is equivalent to if . To prove that for large , it suffices to show that

(3.11)

By (1.6), for large , we have

(3.12)

Hence, if is large, then

(3.13)

Taking inferior limit on both sides of the above inequality, we have

(3.14)

Since is superlinear and is sublinear, . Then . Moreover, since is subquadratic and is superlinear, . Therefore, . From the above results, we can conclude that .

Lemma 3.5.

If (1.6) and (1.7) hold, then satisfies (PS) condition.

Proof.

Notice that . Let . From [19, Theorem 2.2], for any given , we have . Then

(3.15)

Therefore

(3.16)

By Lemma 3.4, there is a constant such that for . Suppose that is a sequence such that is bounded and as . Then by Properties (3) and (7) in Definition 2.2, there are and such that and

(3.17)

It implies that

(3.18)

Hence

(3.19)

This implies that is bounded. Since is finite dimensional, has a convergent subsequence in .□

Lemma 3.6.

For fixed , there exist and such that if , then for .

Proof.

By (1.5) and (1.7), there are such that

(3.20)

(3.21)

The equivalence of norm on implies that there exists such that , where . Let and . Let . It follows from (3.20) and (3.21) that there is such that if , then

(3.22)

Lemma 3.7.

There is an such that and .

Proof.

It follows from Remark 1.2 that for . By the equivalence of the norms on , there exists such that , where . Let be the eigenfunction to the principal eigenvalue of

(3.23)

with and . Let

(3.24)

Clearly . Since , for ,

(3.25)

Hence there is a such that . Let . Then and . The second condition of Mountain Pass theorem is verified.□

Proof of Theorem 1.1.

Clearly, . Lemma 3.5 implies that satisfies (PS) condition. It follows from Lemmas 3.6, 3.7, and 2.5 that has a nontrivial critical point such that . By Lemma 3.3 and Remark 3.2, is a positive solution of (1.1). The proof is complete.□

Proof of Theorem 1.4.

We will apply the subsuper solutions method to prove the multiplicity results.

Firstly, we will prove that there exists such that if , then the following boundary value problem

(3.26)

has a positive solution . In fact, since is increasing on and eventually strictly positive, for and some . Let be the eigenfunction to the principal eigenvalue of

(3.27)

with and .

Notice that and (see [20]). Let be a constant such that . For , , we have .

We will verify that is a subsolution of (3.26) for large. Notice that

(3.28)

On the other hand, for , we have , which implies that

(3.29)

Then for , . Next, for , we have for some and for some . Hence . Since is increasing and eventually strictly positive, there is a such that if and ,

(3.30)

Hence for , . Notice that . Then . So we have

(3.31)

that is, is a subsolution of (3.26).

Now we look for the supersolution of (3.26). Let be a solution of

(3.32)

Then , where

(3.33)

Clearly, for , . Define , where is large enough so in and

(3.34)

This is possible since is a sublinear function. So

(3.35)

which shows that is a supersolution of (3.26). Therefore, by Lemma 2.8, there is a solution of (3.26) such that .

Secondly, we will prove that is a subsolution of (1.1). Since and , it follows that

(3.36)

which implies that is a subsolution of (1.1).

Lastly, we will look for the supersolution of (1.1) and prove the existence of positive solution of (1.1). Let be as in (3.32). Notice that is sublinear. Define , where is independent of and large enough so that in and

(3.37)

Let be so small that

(3.38)

Then

(3.39)

Hence is a supersolution of (1.1). Thus, by Remark 2.9, problem (1.1) has a solution such that for and small, which is positive for .

Now we are going to find the second positive solution of problem (1.1). Notice that and are independent of . Since is positive on , by the definition of we have . Then for ,

(3.40)

where . On the other hand, by Lemma 3.6, we can take appropriate such that if , then for . Hence by Theorem 1.1, . So and , which shows that and are two different positive solutions of (1.1). The proof is complete.

Proof of Theorem 1.5.

Just to be on the contradiction side, let be a positive solution of (1.1). Since is superlinear and increasing, , there are such that for , . Hence for and , , where is the same as that of the proof of Lemma 3.7. If is large enough, then . Therefore for large and . Multiplying both sides of

(3.41)

by and summing it from to , we get

(3.42)

Multiplying both sides of (1.1) by and summing it from to , we have

(3.43)

It is easy to see that

(3.44)

Hence

(3.45)

For , we obtain a contradiction. So for a given , (1.1) has no positive solution if is large. The proof is complete.□

Example 3.8.

We give an example to illustrate the result of Theorem 1.1. Let and . Clearly, and satisfy the conditions of Theorem 1.1. Then problem (1.1) has at least a positive solution.