Abstract
In this paper, we consider a class of nonperiodic damped vibration problems with superquadratic nonlinearities. We study the existence of nontrivial ground state homoclinic orbits for this class of damped vibration problems under some conditions weaker than those previously assumed. To the best of our knowledge, there has been no work focused on this case.
MSC: 49J40, 70H05.
Keywords:
nonperiodic damped vibration problems; ground state homoclinic orbits; superquadratic nonlinearity1 Introduction and main results
We shall study the existence of ground state homoclinic orbits for the following nonperiodic damped vibration system:
where M is an antisymmetric
If
This is a classical equation which can describe many mechanic systems such as a pendulum. In the past decades, the existence and multiplicity of periodic solutions and homoclinic orbits for (1.2) have been studied by many authors via variational methods; see [118] and the references therein.
The periodic assumptions are very important in the study of homoclinic orbits for
(1.2)
since periodicity is used to control the lack of compactness due to the fact that
(1.2)
is set on all ℝ. However, nonperiodic problems are quite different from the ones
described in periodic cases. Rabinowitz and Tanaka [11] introduced a type of coercive condition on the matrix
and obtained the existence of a homoclinic orbit for nonperiodic (1.2) under the AmbrosettiRabinowitz (AR) superquadratic condition:
where
We should mention that in the case where
where
(L_{1}) There is a constant
(L_{2}) There is a constant
which were firstly used in [15]. It is not hard to check that the matrixvalued function
We define an operator
Since M is an antisymmetric
(J_{1})
To state our main result, we still need the following assumptions:
(H_{1})
(H_{2})
(H_{3}) For some
(H_{4})
(H_{5}) For all
Our main results read as follows.
Theorem 1.1If (L_{1})(L_{2}), (J_{1}) and (H_{1})(H_{5}) hold, then (1.1) has at least one nontrivial homoclinic orbit.
Theorem 1.2Let ℳ be the collection of solutions of (1.1), then there is a solution that minimizes the energy functional
over ℳ, where the spaceEis defined in Section 2. In addition, if
uniformly int, then there is a nontrivial homoclinic orbit that
minimizes the energy functional over
Remark 1.1 Although the authors [21] have studied (1.1) with superquadratic nonlinearities, our superquadratic condition (H_{4}) is weaker than (1.4) in [21]. Moreover, we study the ground state homoclinic orbit of (1.1). To the best of our knowledge, there has been no result published concerning the ground state homoclinic orbit of (1.1).
Example 1.1
(1)
(2)
where
The rest of the present paper is organized as follows. In Section 2, we establish the variational framework associated with (1.1), and we also give some preliminary lemmas, which are useful in the proofs of our main results. In Section 3, we give the detailed proofs of our main results.
2 Preliminary lemmas
In the following, we use
It is well known that W is continuously embedded in
Since M is an antisymmetric
Let
where
By a similar proof of Lemma 3.1 in [15], we can prove that if conditions (L_{1}) and (L_{2}) hold, then
Therefore, it is easy to prove that the spectrum
and the corresponding system of eigenfunctions
By (J_{1}), we may let
Then one has the orthogonal decomposition
with respect to the inner product
where
and
where a and b are defined in (J_{1}).
For problem (1.1), we consider the following functional:
Then I can be rewritten as
Let
for any
The following abstract critical point theorem plays an important role in proving our
main result. Let E be a Hilbert space with the norm
Let
with
For
where
The variant weak linking theorem is as follows.
Lemma 2.1 ([26])
The family of
where
(a)
(b)
(c)
(d)
Then, for almost all
where
In order to apply Lemma 2.1, we shall prove a few lemmas. We pick
It is easy to see that
Thus we get
Lemma 2.2Under assumptions of Theorem 1.1, then
Proof By the definition of
which is due to
Therefore, Lemma 2.2 implies that condition (b) holds. To continue the discussion, we still need to verify condition (d), that is, the following two lemmas.
Lemma 2.3Under assumptions of Theorem 1.1, there are two
positive constants
Proof By (H_{1}), (H_{3}), (2.4) and the Sobolev embedding
theorem, for all
where C is a positive constant. It implies the conclusion if we take
Lemma 2.4Under assumptions of Theorem 1.1, then there is
an
where
Proof Suppose by contradiction that there exist
Therefore,
It follows from
There are renamed subsequences such that
We claim that
Case 1. If
Case 2. If
Therefore, Cases 1 and 2 imply that (2.6) holds. Therefore, by (2.5), (2.6) and the
facts
that is,
which contradicts (2.5). The proof is finished. □
Therefore, Lemmas 2.3 and 2.4 imply that condition (d) of Lemma 2.1 holds. Applying Lemma 2.1, we soon obtain the following fact.
Lemma 2.5Under assumptions of Theorem 1.1, for almost
all
where the definition of
Lemma 2.6Under assumptions of Theorem 1.1, for almost
all
Proof Let
and
By Lemma 2.5 and the fact that
That is,
It follows from (2.7), (2.8) and the fact
The proof is finished. □
Applying Lemma 2.6, we soon obtain the following fact.
Lemma 2.7Under assumptions of Theorem 1.1, for
every
Lemma 2.8Under assumptions of Theorem 1.1, then
where
Proof This follows from (H_{5}) if we take
Lemma 2.9The sequences given in Lemma 2.7 are bounded.
Proof Write
Let
Case 1. If
which together with Lemmas 2.3 and 2.7 and
It is a contradiction.
Case 2. If
Since
it follows from the definition of I that
Take
Thus (2.9) holds.
Let
Therefore, (2.9) implies that
It follows from
Note that Lemmas 2.3 and 2.7 and (H_{4}) imply that
It follows from the fact
for all sufficiently large n. We take
for all sufficiently large n. By (H_{1}) and (H_{3}), we have
For all sufficiently large n, by (2.13) and (2.14), it follows from
This implies that
Therefore,
3 Proofs of the main results
Proof of Theorem 1.1 From Lemma 2.7, there are sequences
Hence, in the limit,
Thus
Similar to (2.7) and (2.8), we know
It follows from
Therefore,
Proof of Theorem 1.2 By Theorem 1.1,
If u is a solution of (1.1), then by Lemma 2.8 (take
Thus
By Lemma 2.9, the sequence
Hence, in the limit,
Thus
It follows from
Now suppose that
It follows from (H_{1}) that for any
Let
where
Note that
which together with (3.3), Hölder’s inequality and the Sobolev embedding theorem implies
Similarly, we have
From (3.5) and (3.6), we get
which means
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The main idea of this paper was proposed by GWC and GWC prepared the manuscript initially and JW performed a part of steps of the proofs in this research. All authors read and approved the final manuscript.
Acknowledgements
The authors thank the referees and the editors for their helpful comments and suggestions. Research was supported by the Tianyuan Fund for Mathematics of NSFC (Grant No. 11326113) and the Key Project of Natural Science Foundation of Educational Committee of Henan Province of China (Grant No. 13A110015).
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