Abstract
By properly constructing a functional and by using the critical point theory, we establish the existence of homoclinic solutions for a class of subquadratic secondorder Hamiltonian systems. Our result generalizes and improves some existing ones. An example is given to show that our theorem applies, while the existing results are not applicable.
Keywords:
homoclinic solutions; critical point theory; Hamiltonian systems; nontrivial solution1 Introduction
Consider the following secondorder Hamiltonian system:
where
By now, the existence and multiplicity of homoclinic solutions for secondorder Hamiltonian
systems have been extensively investigated in many papers (see, e.g., [117] and the references therein) via variational methods. More precisely, many authors
studied the existence and multiplicity of homoclinic solutions for (HS); see [517]. Some of them treated the case where
Recently, Zhang and Yuan [15] obtained the existence of a nontrivial homoclinic solution for (HS) by using a standard
minimizing argument. In this paper,
Theorem 1.1 (See [[15], Theorem 1.1])
Assume thatLandWsatisfy the following conditions:
(H1)
(H2)
Then (HS) possesses at least one nontrivial homoclinic solution.
In [1517], the authors considered the case where
where
In this paper, our aim is to revisit (HS) and study the subquadratic case which is not included in [1517]. Now, we state our main result.
Theorem 1.2Let the above condition (H1) hold. Moreover, assume that the following conditions hold:
(H3)
(H4)
Then (HS) possesses at least one nontrivial homoclinic solution.
Remark 1.1 Obviously, the condition (H2) is a special case of (H3)(H4). If (H2) holds, so do
(H3)(H4); however, the reverse is not true.
Remark 1.2 We still consider the function
Due to
so
Example 1.1 Consider the following secondorder Hamiltonian system with
where
Let
Remark 1.3 It is easy to see that (H2) in Theorem 1.1 is not satisfied, so we cannot obtain the existence of homoclinic solutions for the Hamiltonian system (2) by Theorem 1.1. On the other hand, W does not satisfy the conditions (W2) and (W5) of [17], then we cannot obtain the existence of homoclinic solutions for the Hamiltonian system (2) by Theorem 1.1 in [17].
The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proof of Theorem 1.2.
2 Preliminary results
In order to establish our result via the critical point theory, we firstly describe some properties of the space on which the variational associated with (HS) is defined. Like in [15], let
Then the space E is a Hilbert space with the inner product
and the corresponding norm
for all
and
respectively. In particular, for
here
Lemma 2.1There exists a constant
Proof From (H1), we can imply that there exists a constant
for all
So, the lemma is proved. □
Lemma 2.2 ([[9], Lemma 1])
Suppose thatLsatisfies (H1). Then the embedding ofEin
Lemma 2.3Suppose that (H1) and (H4) are satisfied. If
Proof Assume that
Since
for all
On the other hand, by Lemma 2.2,
Now, we introduce more notation and some necessary definitions. Let E be a real Banach space,
Lemma 2.4 ([[18], Theorem 2.7])
LetEbe a real Banach space, and let us have
is a critical value ofI.
3 Proof of Theorem 1.2
Now, we are going to establish the corresponding variational framework to obtain homoclinic
solutions of (HS). Define the functional
Lemma 3.1Under the assumptions of Theorem 1.2, we have
which yields that
Moreover, Iis a continuously Fréchetdifferentiable functional defined onE, i.e.,
Proof We firstly show that
Combining (5) and (8), we show that
where
It is easy to check that
Thus, it is sufficient to show that this is the case for
which is defined for all
It is obvious that
for all
Moreover, for any
where
as
By Lemma 2.3 and the Hölder inequality, we obtain that
as
Lastly, we check that critical points of I are classical solutions of (HS) satisfying
which yields that
It follows from
Hence, q satisfies
Lemma 3.2Under the assumptions of Theorem 1.2, Isatisfies the (PS) condition.
Proof In fact, assume that
for every
We firstly prove that
Combining (13) and (14), we obtain that
Since
Hence,
as
So,
Now, we can give the proof of Theorem 1.2.
Proof of Theorem 1.2 By (5) and (8), for every
Since
On the other hand, take
where
which yields that
Competing interests
The authors declare that they have no competing interests.
Author’s contributions
The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.
Acknowledgements
This work is supported by the Research Foundation of Education Bureau of Hunan Province, China (No.11C0594). The authors would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work.
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