The Existence and Multiplicity of Solutions for Singular Boundary Value Systems with p-Laplacian

Zengxia Cai

School of Science, Linyi University, Linyi, China.

DOI: 10.4236/jamp.2015.34052                

Abstract

This paper presents sufficient conditions for the existence of positive solutions for the fourth-order boundary value problem system with p-Laplacian operator. The existence of single or multiple positive solutions for the system is showed through the fixed point index theory in cones under some assumptions.

Keywords

Coupled Singular Boundary Value Problem, Positive Solution, Fixed Point Index Theorem

1. Introduction

In this paper, we are concerned with the existence and multiplicity of positive solutions for the system (BVP):

where, , and, is allow- ed to have singularity at.

Several papers ([1]-[4]) have studied the solution of fourth-order boundary value problems. But results about fourth-order differential eguations with p-Laplacian have rarely seen. Recently, several papers ([6]-[8]) have been devoted to the study of the coupled boundary value problem.

Motivated by the results mentioned above, here we establish some sufficient conditions for the existence of to (BVP) (1.1) under certain suitable weak conditions. The main results in this paper improve and generalize the results by others.

The following fixed-point index theorem in cones is fundamental.

Theorem A [9] Assume that is a Banach space, is a cone in, and,

, if is a completely operator and,.

1) If for, , then i;

2) If for, then i.

2. Preliminaries and Lemmas

In this paper, let and then is a Banach spa-

ce with the norm, , where, , then

is a cone of. In tnhis paper, i.e.,

Suppose is the Green function of the following boundary problem: z = 0, , , then

Obviously, , ,

Define a cone as follows and

define an integral operator by, where

Let us list the following assumptions for convenience.

is singular at or 1, and

Lemma 2.1 is a solution of BVP (1.1) if and only if has fixed points.

It is easy to see that if is a solution of BVP (1.1).

Lemma 2.2 Suppose that hold, then.

Lemma 2.3 Suppose that hold. Then is completely continuous.

Proof Firstly, assume is a bounded set, we have

Then is bounded, therefore is bounded.

Secondly, suppose, , then is bounded, we get

Due to the continuity of, by and above fomula together with Lebesgue Dominated Convergence

Theorem, then when. Therefore is continuous.

Lastly, since is continuous in, so it is uniformly continous. For all for all, when, we get

Then for all, we have

So is equicontinuous, by Arzela-Ascoli theorem we know is relatively compact.

Therefore, is completely continuous.

For convenience we denote

3. Main Results

Theorem 3.1 Suppose that holds. If the following conditions are satisfied:

; or

Then the system (1.1) has at least one positive solution,

Proof By Lemma 2.3, we know is completely continuous. By, there exists, when , , we have, where satisfies

. Let, when, we get

Hence,. Similarly, we have then, th-

erefore,. By Theorem A, i.

On the other hand, from, if, there exists, for satisfing, we get when. Set such that, let , when, , we get

, so

Hence,. then,

If, with the similar proofs of the condition, we get. Then

,. In either case, we always may set

, By Theorem A, i Through the additivity of the fixed point index we know that

Therefore it follows from the fixed-point theorem that has a fixed point, and thus, is a positive solution of BVP (1.1).

Theorem 3.2 Suppose that holds. If the following conditions are satisfied:

; or,

Then the system (1.1) has at least one positive solution,

Proof By lemma 2.3, we know is completely continuous. From, if, for sat- isfying, there exists, when, , we have.

Let, when, , we get

, then

Hence,. then

If, take satisfying, such taht. Similarly, we get

, then, In either case, we

always may set,. By Theorem A, i.

On the other hand, from, there exists such that, when, where satisfies. There are two cases to consider.

Case (i). Suppose that is bounded, then there exists M_i > 0 satisfying,

. Taking, let, when,

we get

Hence,. Similarly, we have, hence,

then,.

Case (ii). Suppose that is unbounded, since is continuous in , so there exists constant and two points such that, and. Then we get, i = 1,

2. Let, when, we get

Hence,. Similarly, we have, then,

so,. In either case, we always may set

,. By Theorem A, i. Through the additivity of the fixed point index we know that

Therefore it follows from the fixed-point theorem that has a fixed point, and thus, is a positive solution of BVP (1.1). This completes the proof.

Remark 3.1 Note that if is superlinear or sublinear, our conclusions hold. Limit conditions of in this paper are more weak and general.

Remak 3.2 When and, our results generalize and improve the results of [1]-[4].

Theorem 3.3 Suppose that holds. If the following conditions are satisfied:

where satisfies;

or, where satisfies then the system (1.1) has at least one positive solution,

Proof. Choosing such that and,

or From, there exists such that when . Let when, we get

Hence,. Similarly, we have, so, then

,. By Theorem A, i.

On the other hand, From (H₆), if, there exists, such that when. Set such that, let

, when, , we get

, then

Hence,. then,.

If, by, with the similar proofs of the condition, we get

. Then, In either case,

we always may set,. By Theorem A, i. Through the additivity of the fixed point index we know that

Therefore it follows from the fixed-point theorem that has a fixed point, and thus, is a positive solution of BVP (1.1). This completes the proof.

Theorem 3.4 Suppose that holds. If the following conditions are satisfied:

where satisfies,;

where satisfies, , then the system (1.1) has at least one positive solution,

The proofs are similar to that of Theorem 3.2 and are omitted.

Theorem 3.5 Assume that holds. If the following conditions are satisfied:

; or,

Then the system (1.1) has at least two positive solutions and satisfying .

Theorem 3.6 Assume that hold. then the system (1.1) has at least two positive solutions and satisfying.

Remark 3.3 Under suitable weak conditions, the multiplicity results for fourth-order singular boundary value problem with -Laplacian are established. Our results extend and improve the results of [5]-[8].

Conflicts of Interest

The authors declare no conflicts of interest.

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