Existence and multiplicity of symmetric positive solutions for nonlinear boundary-value problems with p-Laplacian operator
© Senlik and Hamal; licensee Springer. 2014
Received: 1 October 2013
Accepted: 7 February 2014
Published: 20 February 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.
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 ;
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.
where , , , , may be singular at and/or 1 and .
where w may be singular at and (or) , , and are nonnegative.
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.
where is defined in (2.4).
and it is easy to verify that . The proof is complete. □
Lemma 2.3 
, , ;
, , ;
where , , are defined by (2.4) and (2.14), respectively.
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 
T has a fixed point in , or
there exist and such that .
Lemma 2.6 
for , for ,
for , for ,
is satisfied. Then T has at least one fixed point in .
3 The existence of one symmetric positive solution
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 .
which implies Tu is concave on .
for all . In a similar way .
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.
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. □
Then there exists a constant such that for any , the BVP (1.1)-(1.2) has at least one nontrivial symmetric positive solution .
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. □
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.
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
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 .
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. □
The authors would like to thank the referees for their useful comments and suggestions.
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