- Open Access
Existence and multiplicity of symmetric positive solutions for nonlinear boundary-value problems with p-Laplacian operator
Boundary Value Problems volume 2014, Article number: 44 (2014)
In this paper, we establish the existence and multiplicity of symmetric positive solutions for a class of p-Laplacian fourth-order differential equations with integral boundary conditions. Our proofs use the Leray-Schauder nonlinear alternative and Krasnoselkii’s fixed-point theorem in cones.
In this paper, we are concerned with the existence of symmetric positive solutions of the following fourth-order boundary-value problem with integral boundary conditions:
where , is the p-Laplacian operator, i.e., , and with .
Now, let us list the following conditions which are to be used in our theorems:
(H1) is symmetric on and on any subinterval of ;
(H2) and for , is symmetric in t and even υ, i.e., f satisfies and ;
(H3) are symmetric functions on and , , where
Boundary-value problems with integral boundary conditions for ordinary differential equations represent a very interesting and important class of problems. They include two-, three-, multi-point and nonlocal boundary-value problems as special cases. For an overview of the literature on integral boundary-value problems and symmetric solutions, see [1–7] and the references therein.
In , Zhang and Liu considered the following fourth-order boundary-value problems with p-Laplacian operator:
where , , , , may be singular at and/or 1 and .
In , Zhang and Ge considered the existence and nonexistence of positive solutions of the following fourth-order boundary-value problems with integral boundary conditions:
where w may be singular at and (or) , , and are nonnegative.
In , Ma considered the existence of a symmetric positive solution for the fourth-order nonlocal boundary-value problem (BVP). The author obtained at least one symmetric positive solution by using the fixed-point index in cones. We have
where , is continuous, symmetric on , and maybe singular at and . is continuous and is symmetric on , for all .
Motivated by the above works, we consider the existence of one and multiple symmetric positive solutions for the BVP (1.1)-(1.2).
The organization of the paper is as follows. In Section 2, we present some necessary lemmas that will be used to prove our main results. In Section 3, we use the Leray-Schauder nonlinear alternative to get the existence of at least one symmetric positive solution for the nonlinear BVP (1.1)-(1.2). In Section 4, we use the Krasnoselkii fixed-point theorem to get the existence of multiple symmetric positive solutions for the nonlinear BVP (1.1)-(1.2).
In this paper, a symmetric positive solution u of (1.1)-(1.2) means a solution of (1.1)-(1.2) satisfying and , .
To state and prove the main results of this paper, we will make use of the following lemmas.
Lemma 2.1 Assume that (H3) holds. Then for any , the BVP
has a unique solution u and u can be expressed in the form
Proof First suppose that is a solution of the BVP (2.1)-(2.2). We have
It is easy to see by integration of both sides of (2.1) on that
Integrating again, we get
Letting in (2.5), we find
Substituting , we obtain
Substituting (2.8) into (2.7) we have
where is defined in (2.4).
Next let u be as in (2.9), then
Taking the derivative of (2.10), we get
and it is easy to verify that . The proof is complete. □
Lemma 2.2 Assume that (H3) is satisfied. Then for any , the BVP
has a unique solution v
Lemma 2.3 
If (H3) holds, then, for all , the following results are true.
, , ;
, , ;
where , , are defined by (2.4) and (2.14), respectively.
Lemma 2.4 If and then
To obtain the existence of symmetric positive solutions of the BVP (1.1)-(1.2), the following Leray-Schauder nonlinear alternative and Krasnoselkii fixed-point theorem are useful.
Lemma 2.5 
Let E be a Banach space with closed and convex. Assume U is an open subset of P with and is a continuous and compact map. Then either
T has a fixed point in , or
there exist and such that .
Lemma 2.6 
Let P be a cone of a real Banach space E, and be two bounded open sets in E such that . Let operator be completely continuous. Suppose that one of the two conditions
for , for ,
for , for ,
is satisfied. Then T has at least one fixed point in .
Let the space equipped with the norm be our Banach space. Define P to be cone in E by
Assume that u is a solution of the BVP (1.1)-(1.2). Then from Lemma 2.1, we get
From Lemma 2.2, we have
3 The existence of one symmetric positive solution
In order to state the following results we need to introduce the notation:
Theorem 3.1 Assume that (H1)-(H3) are satisfied and is continuous, , and there exist nonnegative functions such that
and there exist such that or . Then there exists a constant such that for any , the BVP (1.1)-(1.2) has at least one nontrivial symmetric positive solution .
Proof It is easy to see that the BVP (1.1)-(1.2) has a solution if and only if u is a fixed point of the operator equation
For all , we have by
which implies Tu is concave on .
On the other hand, using (H1)-(H3) and Lemma 2.3 we have
for all . In a similar way .
It follows that for . Noticing that is symmetric on , is symmetric on and is symmetric on and even in υ we have
i.e., , . Therefore, is symmetric on . So and . By applying the Arzela-Ascoli theorem, we can see that is relatively compact. In view of Lebesgue convergence theorem, it is obvious that T is a continuous operator. Hence, is completely continuous operator. By a similar argument in  we may proceed; we omit the details here.
, for all , and , , we know that , . Thus , . Let
Suppose , such that . Then
By Lemma 2.3
Choose . Then when , we have
Consequently, . This contradicts period, by (i) of Lemma 2.5, T has a fixed point , since , then when , the BVP (1.1)-(1.2) has a nontrivial symmetric positive solution . The proof is complete. □
Theorem 3.2 Assume that (H1)-(H3) are satisfied and is continuous, , and
Then there exists a constant such that for any , the BVP (1.1)-(1.2) has at least one nontrivial symmetric positive solution .
Proof Let such that . By (3.2), there exists such that
Let . Then for any , we have
From Theorem 3.1, we know that the BVP (1.1)-(1.2) has at least one nontrivial symmetric positive solution . The proof is complete. □
Corollary 3.1 Assume that (H1)-(H3) are satisfied and is continuous, , and
Then there exists a constant , such that for any , the BVP (1.1)-(1.2) has at least one nontrivial solution .
Example 3.1 We consider the following fourth-order BVP.
Let , in (1.1) and . Then
It is obvious that is continuous, symmetric on the interval and even υ, we have
It follows from a direct calculation that
Then by Theorem 3.1 we know that the BVP (3.3)-(3.4) has a nontrivial symmetric positive solution for any .
4 The existence of multiple symmetric positive solutions
In this section, we impose growth conditions on f which allows us to apply Lemma 2.6 to establish the existence of two symmetric positive solutions of the BVP (1.1)-(1.2), and we begin by introducing some notation:
Theorem 4.1 Assume that (H1)-(H3) are satisfied and f satisfies the following conditions for .
There exist numbers such that for and .
There exist numbers such that for .
Then the BVP (1.1)-(1.2) has at least two nontrivial symmetric positive solution .
Proof Let the operator T be defined by (3.1).
We first show that
For , we obtain , which implies . Hence for , by Lemma 2.3,
Note that . Thus, implies
By (4.1) and (4.2), we obtain
Next we show that
For , we obtain , which implies . Hence for , by Lemma 2.3,
By (4.4), we obtain
Applying Lemma 2.6 to (4.3) and (4.5) shows that T has a fixed point with . Also, let and note that . So for . Then and , and we obtain . Applying Theorem 4.1(i) for , we have for . From Lemma 2.6, T has a fixed point with .
Then the BVP (1.1)-(1.2) has two nontrivial symmetric positive solutions with . The proof is complete. □
Lomtatidze A, Malaguti L: On a nonlocal boundary value problems for second order nonlinear singular differential equations. Georgian Math. J. 2000, 7: 133-154.
Gallardo JM: Second order differential operators with integral boundary conditions and generation of semigroups. Rocky Mt. J. Math. 2000, 30: 1265-1292. 10.1216/rmjm/1021477351
Zhanga X, Feng M, Ge W: Symmetric positive solutions for p -Laplacian fourth-order differential equations with integral boundary conditions. J. Comput. Appl. Math. 2008, 222: 561-573. 10.1016/j.cam.2007.12.002
Zhang X, Liu L: Eigenvalue of fourth-order m -point boundary value problem with derivatives. Comput. Math. Appl. 2008, 56: 172-185. 10.1016/j.camwa.2007.08.048
Ma R, Xu L: Existence of positive solutions of nonlinear fourth-order boundary value problem. Appl. Math. Lett. 2010, 23: 537-543. 10.1016/j.aml.2010.01.007
Avery RI, Henderson J: Three symmetric positive solutions for a second-order boundary value problem. Appl. Math. Lett. 2000, 13: 1-7.
Feng H, Pang H, Ge W: Multiplicity of symmetric positive solutions for a multipoint boundary value problem with a one-dimensional p -Laplacian. Nonlinear Anal. 2008, 69: 3050-3059. 10.1016/j.na.2007.08.075
Zhang X, Liu L: Positive solutions of fourth-order four-point boundary value problems with ϕ -Laplacian operator. J. Math. Anal. Appl. 2007, 336: 1414-1423. 10.1016/j.jmaa.2007.03.015
Zhang X, Ge W: Positive solutions for a class of boundary-value problems with integral boundary conditions. Comput. Math. Appl. 2009, 58: 203-215. 10.1016/j.camwa.2009.04.002
Ma H: Symmetric positive solutions for nonlocal boundary value problems of fourth order. Nonlinear Anal. 2008, 68: 645-651. 10.1016/j.na.2006.11.026
Guo D, Lashmikanthan V: Nonlinear Problems in Abstract Cones. Academic Press, San Diego; 1988.
The authors would like to thank the referees for their useful comments and suggestions.
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.