Abstract
This paper is devoted to the existence of periodic solutions for the onedimensional pLaplacian equation
where
Keywords:
periodic solutions; pLaplacian; Fučík spectrum; LeraySchauder degree; Borsuk theorem1 Introduction and main results
In this paper, we are concerned with the existence of solutions for the following periodic boundary value problem:
where
Existence and multiplicity of solutions of the periodic problems driven by the pLaplacian have been obtained in the literature by many people (see [15]). Many solvability conditions for problem (1.1) were established by using the asymptotic
interaction at infinity of the ratio
admits at least one nontrivial 2πperiodic solution (see [10] for
By [6], it follows that
Then they applied the Sturm’s comparison theorem and LeraySchauder degree theory to prove that problem (1.1) is solvable if the following relations hold:
uniformly for a.e.
Clearly, in this case, we have
we can see that the conditions on the ratio
where
(1) There exist
(2)
Here, the potential function G is nonresonant with respect to
In this paper, we want to obtain the solvability of problem (1.1) by using the asymptotic
interaction at infinity of both the ratios
For related works on resonant problems involving the Fučík spectrum, we also refer the interested readers to see [1219] and the references therein.
Our main result for problem (1.1) now reads as follows.
Theorem 1.1Assume that
(i) There exist constants
(ii) There exists
(iii) There exist constants
hold uniformly for a.e.
Then problem (1.1) admits a solution.
Remark If
For convenience, we introduce some notations and definitions.
2 Proof of the main result
Denote by deg the LeraySchauder degree. To prove Theorem 1.1, we need the following results.
Lemma 2.1[20]
Let Ω be a bounded open region in a real Banach spaceX. Assume that
Lemma 2.2 (Borsuk Theorem [20])
Assume thatXis a real Banach space. Let Ω be a symmetric bounded open region with
Proof of Theorem 1.1 Take
where
By (1.3) and the regularity arguments, it follows that
In what follows, we shall prove that there exists
Set
and
By (1.3), there exists
Then there exist
In addition, using (1.3) and the regularity arguments, there exists
Clearly,
Note that for
We now distinguish three cases:
(i)
(ii)
(iii)
In the following, it will be shown that each case leads to a contradiction.
Case (i). Let
Then, as
In addition, as shown in [11], we have
and
By (1.4) and (2.4), it follows that
Thus,
Here,
Now we prove that there exist
In fact, if not, we assume, by contradiction, that there exists a subsequence of
Combing with (2.5),
A contradiction. Hence, (2.9) holds.
For any
and
Denote
By (2.5), we obtain
Using (1.3), for a.e.
Thus,
By (1.4) and (2.2), we get
In view of (2.11), we obtain that
holds uniformly for a.e.
holds uniformly for a.e.
On the other hand, for
we obtain by (1.5)(1.6) that
Using
We claim that there exists subinterval
or subinterval
Indeed, if not, we assume that
Then by (1.7), it follows that
Case (ii). In this case, we have
Using similar arguments as in Case (i), by (1.4) and (2.4) it follows that
holds uniformly for a.e.
we obtain by (1.5) that
Using
We shall show that there exists subinterval
In fact, if not, we assume that
Taking 1 as test function in problem (2.23), we get
By
Case (iii). In this case,
In a word, (2.3) cannot hold, and hence by (2.2) there exists
Note that, for each
has a unique solution
Let
Consider the following homotopy:
for
From the invariance property of LeraySchauder degree, it follows that
Hence, problem (1.1) has a solution. The proof is complete. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
The first author sincerely thanks Professor Yong Li and Doctor Yixian Gao for their many useful suggestions and the both authors thank Professor ZhiQiang 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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