- Open Access
Existence of periodic solutions for a class of p-Laplacian equations
© Chang and Qiao; licensee Springer 2013
- Received: 26 September 2012
- Accepted: 5 April 2013
- Published: 19 April 2013
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.
- periodic solutions
- Fučík spectrum
- Leray-Schauder degree
- Borsuk theorem
where (), . A solution u of problem (1.1) means that u is and is absolutely continuous such that (1.1) is satisfied for a.e. .
There exist such that for all ;
, with .
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.
- (i)There exist constants such that(1.3)
- (ii)There exists such that(1.4)
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 .
Denote by deg the Leray-Schauder degree. To prove Theorem 1.1, we need the following results.
Lemma 2.1 
Let Ω be a bounded open region in a real Banach space X. Assume that is completely continuous and . Then the equation has a solution in Ω if .
Lemma 2.2 (Borsuk Theorem )
Assume that X is a real Banach space. Let Ω be a symmetric bounded open region with . Assume that is completely continuous and odd with . Then is odd.
changes sign in ;
In the following, it will be shown that each case leads to a contradiction.
Here, , .
A contradiction. Hence, (2.9) holds.
holds uniformly for a.e. .
Then by (1.7), it follows that . A contradiction. Combining (2.12)-(2.13) with (2.15)-(2.18), we obtain a contradiction.
By for a.e. , it follows that for a.e. , which is contrary to that . Hence, (2.22) holds. Clearly, (2.21)-(2.22) contradict (2.19).
Case (iii). In this case, and is uniformly bounded. Similar arguments as in Case (ii) imply a contradiction.
Hence, problem (1.1) has a solution. The proof is complete. □
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).
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