This paper is devoted to the existence of periodic solutions for the one-dimensional p-Laplacian equation
where (), . By using some asymptotic interaction of the ratios and with the Fučík spectrum of related to periodic boundary condition, we establish a new existence theorem of periodic solutions for the one-dimensional p-Laplacian equation.
Keywords:periodic solutions; p-Laplacian; Fučík spectrum; Leray-Schauder degree; Borsuk theorem
1 Introduction and main results
In this paper, we are concerned with the existence of solutions for the following periodic boundary value problem:
Existence and multiplicity of solutions of the periodic problems driven by the p-Laplacian have been obtained in the literature by many people (see [1-5]). Many solvability conditions for problem (1.1) were established by using the asymptotic interaction at infinity of the ratio with the Fučík spectrum for under periodic boundary condition (see e.g., [2,4,6-9]). In , Del Pino, Manásevich and Murúa firstly defined the Fučík spectrum for under periodic boundary value condition as the set consisting of all the pairs such that the equation
admits at least one nontrivial 2π-periodic solution (see  for ). Let
By , it follows that
Then they applied the Sturm’s comparison theorem and Leray-Schauder degree theory to prove that problem (1.1) is solvable if the following relations hold:
Clearly, in this case, we have , which is usually called that the nonlinearity f is nonresonant with respect to the Fučík spectrum . In , Anane and Dakkak obtained a similar result by using the property of nodal set for eigenfunctions. If f is resonant with respect to , i.e., there exists such that , uniformly for a.e. , together with the Landesman-Lazer type condition, Jiang  obtained the existence of solutions of (1.1) by applying the variational methods and symplectic transformations. In these works, either f is resonant or nonresonant with respect to , the solvability of problem (1.1) was assured by assuming that the ratio stays at infinity in the pointwise sense asymptotically between two consecutive curves of . Note that
we can see that the conditions on the ratio are more general than that on the ratio . Recently, Liu and Li  studied the nondissipative p-Laplacian equation
Here, the potential function G is nonresonant with respect to and the ratio is not required to stay at infinity in the pointwise sense asymptotically between two consecutive branches of and it may even cross at infinity multiple Fučík spectrum curves.
In this paper, we want to obtain the solvability of problem (1.1) by using the asymptotic interaction at infinity of both the ratios and with the Fučík spectrum for under periodic boundary condition. Here, . The goal is to obtain the existence of solutions of (1.1) by requiring neither the ratio stays at infinity in the pointwise sense asymptotically between two consecutive branches of nor the limits exist. We shall prove that problem (1.1) admits a solution under the assumptions that the nonlinearity f has at most -linear growth at infinity and the ratio has a limit as , while the ratio stays at infinity in the pointwise sense asymptotically between two consecutive branches of . Our result will complement the results in the literature on the solvability of problem (1.1) involving the Fučík spectrum.
Our main result for problem (1.1) now reads as follows.
Then problem (1.1) admits a solution.
Remark If , where with , and satisfy (1.7), e is continuous on ℝ and , then and . By Theorem 1.1, it follows that problem (1.1) admits a solution. It is easily seen that the result of  cannot be applied to this case. Note that one can also obtain the solvability of (1.1) in this case by the result of , while in Theorem 1.1 we do not require the pointwise limit at infinity of the ratio as in .
For convenience, we introduce some notations and definitions. () denotes the usual Sobolev space with inner product and norm , respectively. () denotes the space of m-times continuous differential real functions with norm
2 Proof of the main result
Denote by deg the Leray-Schauder degree. To prove Theorem 1.1, we need the following results.
Lemma 2.2 (Borsuk Theorem )
In what follows, we shall prove that there exists independent of such that for all possible solution of (2.1). Assume by contradiction that there exist a sequence of number and corresponding solutions of (2.1) such that
We now distinguish three cases:
In the following, it will be shown that each case leads to a contradiction.
Case (i). Let
In addition, as shown in , we have . Define
By (1.4) and (2.4), it follows that
A contradiction. Hence, (2.9) holds.
By (1.4) and (2.2), we get
In view of (2.11), we obtain that
we obtain by (1.5)-(1.6) that
Case (ii). In this case, we have
we obtain by (1.5) that
Taking 1 as test function in problem (2.23), we get
Let . Define the operator by . Denote . Clearly, is well defined for all . Owing to , there is a continuous curve , , , whose image is in and such that , . From the invariance property of Leray-Schauder degree under compact homotopies, it follows that the degree is constant for . Obviously, the operator is odd. By the Borsuk’s theorem, it follows that for all . Thus,
Consider the following homotopy:
From the invariance property of Leray-Schauder degree, it follows that
Hence, problem (1.1) has a solution. The proof is complete. □
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
The first author sincerely thanks Professor Yong Li and Doctor Yixian Gao for their many useful suggestions and the both authors thank Professor Zhi-Qiang Wang for many helpful discussions and his invitation to Chern Institute of Mathematics. The first author is partially supported by the NSFC Grant (11101178), NSFJP Grant (201215184) and FSIIP of Jilin University (201103203). The second author is partially sup- ported by NSFC Grant (11226123).
Aizicovici, S, Papageorgiou, NS, Staicu, V: Nonlinear resonant periodic problems with concave terms. J. Math. Anal. Appl.. 375, 342–364 (2011). Publisher Full Text
Liu, W, Li, Y: Existence of periodic solutions for p-Laplacian equation under the frame of Fučík spectrum. Acta Math. Sin. Engl. Ser.. 27, 545–554 (2011). Publisher Full Text
Manásevich, R, Mawhin, J: Periodic solutions for nonlinear systems with p-Laplacian-like operators. J. Differ. Equ.. 145, 367–393 (1998). Publisher Full Text
Reichel, W, Walter, W: Sturm-Liouville type problems for the p-Laplacian under asymptotic nonresonance conditions. J. Differ. Equ.. 156, 50–70 (1999). Publisher Full Text
Yang, X, Kim, Y, Lo, K: Periodic solutions for a generalized p-Laplacian equation. Appl. Math. Lett.. 25, 586–589 (2012). Publisher Full Text
Del Pino, M, Manásevich, R, Murúa, A: Existence and multiplicity of solutions with prescribed period for a second order quasilinear ODE. Nonlinear Anal.. 18, 79–92 (1992). Publisher Full Text
Jiang, M: A Landesman-Lazer type theorem for periodic solutions of the resonant asymmetric p-Laplacian equation. Acta Math. Sin.. 21, 1219–1228 (2005). Publisher Full Text
Boccardo, L, Drábek, P, Giachetti, D, Kućera, M: Generalization of Fredholm alternative for nonlinear differential operators. Nonlinear Anal.. 10, 1083–1103 (1986). Publisher Full Text
Drábek, P, Invernizzi, S: On the periodic boundary value problem for forced Duffing equation with jumping nonlinearity. Nonlinear Anal.. 10, 643–650 (1986). Publisher Full Text
Fonda, A: On the existence of periodic solutions for scalar second order differential equations when only the asymptotic behaviour of the potential is known. Proc. Am. Math. Soc.. 119, 439–445 (1993). Publisher Full Text
Habets, P, Omari, P, Zanolin, F: Nonresonance conditions on the potential with respect to the Fučík spectrum for the periodic boundary value problem. Rocky Mt. J. Math.. 25, 1305–1340 (1995). Publisher Full Text
Omari, P, Zanolin, F: Nonresonance conditions on the potential for a second-order periodic boundary value problem. Proc. Am. Math. Soc.. 117, 125–135 (1993). Publisher Full Text
Zhang, M: Nonresonance conditions for asymptotically positively homogeneous differential systems: the Fučík spectrum and its generalization. J. Differ. Equ.. 145, 332–366 (1998). Publisher Full Text
Zhang, M: The rotation number approach to the periodic Fučík spectrum. J. Differ. Equ.. 185, 74–96 (2002). Publisher Full Text