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Ground state periodic solutions for Duffing equations with superlinear nonlinearities
Advances in Difference Equations volume 2014, Article number: 139 (2014)
In this paper, we study a general second order differential equation with superlinear nonlinearity. We obtain ground state and geometrically distinct periodic solutions of this equation by a generalized Nehari manifold approach. In particular, our result extends some existing ones.
1 Introduction and main result
In the past decades, many authors have studied the autonomous equation
where and . For example, many authors have obtained the existence and multiplicity of periodic solutions by various methods, such as a generalized form of the Poincaré-Birkhoff theorem, critical point theory, phase-plane analysis combined with shooting methods or fixed point theorems of planar homeomorphisms, and continuation methods based on degree theory; see [1–7] and the references therein. Some authors [8, 9] have obtained the existence of infinitely many periodic and subharmonic solutions of (1.1) by using the Poincaré-Birkhoff theorem or Moser’s twist theorem .
where g satisfies some strong force condition near . If , then (1.2) becomes
Torres  has proved (1.3) with having one positive or negative solution.
However, in some cases in mathematical physics, the global nonnegative of (i.e., , ) is not satisfied, thus it is necessary for us to study the case that is not uniformly nonnegative for all . Therefore, we shall study the existence of infinitely many T-periodic solutions of the following general second order differential equation:
Here, we need not assume that .
We are interested in the following case:
(V1) is T-periodic and 0 belongs to a spectral gap of , .
Let , we assume that g satisfies the following assumptions:
(G1) is a Carathéodory function and it is T-periodic in t, besides, for some and .
(G2) as uniformly in .
(G3) as uniformly in .
(G4) is strictly increasing on for all .
Let . By (V1), we have the decomposition , where and are the positive and negative spectral subspaces of in E, respectively. The corresponding functional of (1.4) is
The following set has been introduced by Pankov :
By definition, ℳ contains all nontrivial critical points of Φ.
First, we consider ground state T-periodic solutions of (1.4), that is, solutions corresponding to the least energy of the action functional of (1.4):
Theorem 1.1 If (V1) and (G1)-(G4) hold, then (1.4) has at least one ground state T-periodic solution.
Remark 1.1 By the Schauder fixed point theorem, Esmailzadeh and Nakhaie-Jazar  obtained a periodic solution for the Mathieu-Duffing type equation
But it does not exclude the trivial solution, which always exists. Obviously, the nonlinearity with satisfies (G1)-(G4), thus our Theorem 1.1 implies (1.5) admits a ground state 2Ω-periodic solution if 0 belongs to a spectral gap of , that is, .
Remark 1.2 Torres  proved (1.4) with and has one positive or negative solution, where is a Carathéodory function and T-periodic in t. But we consider a more general equation (1.4) than the equation in , and we obtain a ground state T-periodic solution of (1.4) by a Nehari manifold approach. Therefore, our result extends the result in .
Now, we consider the multiplicity of solutions of (1.4). For and , let be defined by . We note that if x is a solution of (1.4), then so are all elements of the orbit of x under the action of ℤ, . Two solutions and are said to be geometrically distinct if and are disjoint.
Theorem 1.2 If (V1), (G1)-(G4), andis odd in x hold, then (1.4) admits infinitely many pairs of geometrically distinct T-periodic solutions.
Example 1.1 As simple applications of Theorems 1.1 and 1.2, we consider the following examples:
where , and with T-periodic in t. It is not hard to check that the above functions all satisfy assumptions (G1)-(G4).
The rest of our paper is organized as follows. In Section 2, we establish the variational framework associated with (1.4), and we also give some preliminary lemmas, which are useful in the proofs of our results, and then we give the detailed proofs of our Theorems 1.1 and 1.2.
2 Variational frameworks and preliminary lemmas
Throughout this paper we denote by the usual norm and C for generic constants.
Let under the usual norm and the corresponding inner product defined by
Thus E is a Hilbert space. We will seek solutions of (1.4) as critical points of the functional Φ associated with (1.4) and given by
Let , then and the derivatives are given by
which implies that (1.4) is the corresponding Euler-Lagrange equation for Φ. Therefore, we have reduced the problem of finding a nontrivial solution of (1.4) to that of seeking a nonzero critical point of the functional Φ on E.
In what follows, we always assume that (V1) and (G1)-(G4) are satisfied. Obviously, (G1) and (G2) imply that for each there is such that
By (V1), we have the decomposition , where and are the positive and negative spectral subspaces of in E, respectively. Let
Obviously, the quadratic part of Φ, is positive on and negative on . Moreover, we may define an new inner product on E with corresponding norm such that
Therefore, Φ can be rewritten as
Let . We define for the following subspaces of E:
and the convex subset
2.1 Proof of Theorem 1.1
Lemma 2.1for all.
Proof This follows immediately from (G2) and (G4). □
Lemma 2.2 ()
Letbe numbers withand. Then
Lemma 2.3 If, thenfor any, . Hence x is the unique global maximum of.
Proof We rewrite Φ by
Since , we have
which together with , and Lemma 2.2 implies that
So the proof is finished. □
Lemma 2.4 The following statements hold true:
There issuch that, where.
Proof (a) First, for , we have and
by (2.1), hence the second inequality follows if is sufficiently small.
Second, since for every , there is such that . Therefore, by virtue of Lemma 2.3, and the first inequality follows.
(b) For , by Lemma 2.1, we have
from which the conclusion follows. □
Lemma 2.5 Ifis a compact subset, then there existssuch thatonfor every.
Proof Without loss of generality, we may assume that for every . Suppose by contradiction that there exist and , , such that for all j and as . Passing to a subsequence, we may assume that , . Set , then
Hence and therefore , for a subsequence, , and a.e. . Therefore, , hence , it follows from (G3) and the Fatou lemma that
which contradicts (2.6). □
Lemma 2.6 For each, the setconsists of precisely one pointwhich is the unique global maximum of.
Proof By Lemma 2.3, it suffices to show that . Since , we may assume that , . By Lemma 2.5, there exists such that on . By Lemma 2.4(a), for small . Therefore, . It is easy to see that Φ is weakly upper semicontinuous on , therefore, for some . This is a critical point of , so for all . Consequently, , as required. □
Lemma 2.7 Φ is coercive on ℳ, that is, as, .
Proof Arguing by contradiction, suppose there exists a sequence such that and for some . Let . Then and a.e. as after passing to a subsequence. Suppose
Then it follows from (2.1) that for each . By Lemma 2.4(b), . Hence, by Lemma 2.3, we obtain
This yields a contradiction if . Hence,
Since in (), we have . Then
it follows again from (G3) and the Fatou lemma that
Hence we have
which is a contradiction. This contradiction establishes the lemma. □
Lemma 2.8 The map, (see Lemma 2.6) is continuous.
Proof Let , it suffices to show that for any sequence with , we have for some subsequence.
Without loss of generality, we may assume that for all j, so that . By Lemma 2.5 and Lemma 2.6, there exists such that
Therefore, by Lemma 2.7, . Passing to a subsequence, we may assume that
where by Lemma 2.4(b). Therefore, we have
Note that , where . It follows from Lemma 2.6 that
which together with Fatou’s lemma and the weak lower semicontinuity of the norm implies that
However, Lemma 2.6 implies that . Hence all inequalities above must be equalities and it follows that , besides, due to Lemma 2.6, so we have . □
We now consider the functional
which is continuous by Lemma 2.8. Here, we should mention that Lemmas 2.9 and 2.10 are due to Szulkin and Weth .
Lemma 2.9, and
Proof For , we put , so we have . Let . Choose such that for , and put . We may write with . Then , and the function , , is continuous by Lemma 2.8.
Note that , which together with Lemma 2.6 and the mean value theorem implies that
where . Note that . Similarly, we have
where . Therefore, combining (2.12) and (2.13), we conclude that
Hence, is linear (and continuous) in z and depends continuously on w. So the assertion follows from Proposition 1.3 in . □
Next we consider the unit sphere
We note that the restriction of the map to has an inverse given by
We also consider the restriction of to .
Lemma 2.10 The following statements hold true:
is a Palais-Smale sequence for Ψ if and only ifis a Palais-Smale sequence for Φ.
We have. Moreover, is a critical point of Ψ if and only ifis a critical point of Φ, and the corresponding critical values coincide.
Proof (a) is a direct consequence of Lemma 2.9.
To prove (b), let be a sequence such that , and let . Since for every j we have an orthogonal splitting
and , we have and using (a), we also have the following relation:
If as , it follows from Lemma 2.4(b) that as . On the other hand, if as , it follows from Lemma 2.7 that is bounded, and hence as . Hence, is a Palais-Smale sequence for Ψ if and only if is a Palais-Smale sequence for Φ.
The proof of (c) is similar to that of (b) and is omitted. □
Now, we complete the proof of Theorem 1.1.
Proof of Theorem 1.1 From Lemma 2.4(a), we know that . Moreover, if satisfies , then is a minimizer of Ψ and therefore a critical point of Ψ, so that is a critical point of Φ by Lemma 2.10. It remains to show that there exists a minimizer of . By Ekeland’s variational principle , there exists a sequence such that and as . Put , then and as by Lemma 2.10(b). By Lemma 2.7, is bounded and hence and a.e. after passing to a subsequence. If
then by (2.1), the Hölder’s inequality, and Sobolev’s imbedding theorem, we have
for some . It follows that as . Therefore,
Therefore, , which contradicts Lemma 2.4(b). This contradiction shows that (2.15) cannot hold. Note that in , so and . Particularly, we see that , which yields .
On the other hand, by Lemma 2.1, the Fatou lemma and the boundedness of , we get
which implies that . Therefore, we conclude that . □
2.2 Proof of Theorem 1.2
In order to prove Theorem 1.2, we still need the following lemmas. In what follows, we always assume that (V1), (G1)-(G4) and the nonlinearity is odd in x with are satisfied.
Lemma 2.11 The mapdefined in (2.14) is Lipschitz continuous.
Proof For , we have, by Lemma 2.4(b),
The proof is completed. □
Remark 2.1 It is easy to see that both maps , are equivariant with respect to the ℤ-action given by for . So, by Lemma 2.10(c), the orbits consisting of critical points of Φ are in one-to-one correspondence with the orbits consisting of critical points of Ψ.
To continue the proof, we need the following notation. For we put
Note that for every d due to Lemma 2.7. We may choose a subset ℱ of K such that and each orbit has a unique representative in ℱ. By Remark 2.1, it suffices to show that the set ℱ is infinite. Suppose to the contrary that
Proof We can choose and such that for all j and
Let . After passing to a subsequence, we have , and either for almost all j or . If the first case holds, we have
If the second case holds, we have , thus , where due to the definitions of K and . Therefore, this lemma is proven. □
Lemma 2.13 Let. Ifare two Palais-Smale sequences for Ψ, then eitherasor, wheredepends on d but not on the particular choice of Palais-Smale sequences.
Proof Let and for . Then both sequences are bounded Palais-Smale sequences for Φ by Lemma 2.7 and the definition of Ψ. Let p is the parameter in (G1). We distinguish two cases.
Case 1. If
Note that and are bounded Palais-Smale sequences for Φ, it follows from (2.1), Hölder’s inequality, and Sobolev’s inequality that
for all , where is arbitrary, , , , and do not depend on the choice of ε. Therefore, by (2.17) and (2.18), we have as . Similarly, as . Therefore,
it follows from Lemma 2.11 that
Case 2. If
Since and are bounded, we may pass to a subsequence such that
where for by Lemma 2.4(b). Note that in , thus by (2.19), , thus . We first consider the case where and , so that and
Then by (2.14), the definition of and the weak lower semicontinuity of the norm, we have
where and . Since , an elementary geometric argument and the inequalities above imply that
where κ is defined in Lemma 2.12. It remains to consider the case where either or . If , then , and
The case is treated similarly. The proof is finished. □
It is well known (see , Lemma II.3.9) that Ψ admits a pseudo-gradient vector field, i.e., there exists a Lipschitz continuous map (where is the tangent bundle) with for all and
Let be the corresponding (Ψ-decreasing) flow defined by
and , are the maximal existence times of the trajectory in negative and positive direction. Note that Ψ is strictly decreasing along trajectories of η.
For deformation type arguments, the following lemma is crucial.
Lemma 2.14 For everythe limitexists and is a critical point of Ψ.
Proof Fix and put .
Case 1: . For , by (2.20), (2.21), and Lemma 2.10(c), we have
Since , this implies that exists and then it must be a critical point of Ψ (otherwise the trajectory could be continued beyond ).
Case 2: . To prove that exists, it clearly suffices to establish the following property:
We suppose by contradiction that (2.22) is not satisfied. Then there exists (where is given in Lemma 2.13) and a sequence with and for every n. Choose the smallest such that
Then by (2.20) and (2.21), we have
Note that Ψ is strictly decreasing along trajectories of η, and it follows that
as , thus and there exist such that , where . Similarly we find a largest for which and then satisfying . As and , and are two Palais-Smale sequences such that
which contradicts Lemma 2.13, hence (2.22) is true. Therefore, exists, and obviously it must be a critical point of Ψ. □
In the following, for a subset and , we put
Lemma 2.15 Let. Then for everythere existssuch that
Proof By (2.16), (a) is obviously satisfied for small enough. Without loss of generality, we may assume and . In order to find ε such that (b) holds, we let
We claim that . Indeed, suppose by contradiction that there exists a sequence such that . Passing to a subsequence, using the finiteness condition (2.16) and the ℤ-invariance of Ψ, we may assume for some . Let . Then and
which contradicts Lemma 2.13. Hence . Let
and choose such that (a) holds. By Lemma 2.14 and (a), we know that the only way (b) can fail is that
In this case we let
Then by (2.20), (2.21), and (2.26), we have
which together with (2.20), (2.21), and (2.25) imply that
Therefore, , thus , which contradicts our assumption (2.27). □
Now, we complete the proof of Theorem 1.2.
In particular, if there does not exist a finite i, we set . Finally, we set .
For the usual Krasnoselskii genus, let A and B are closed and symmetric subsets, then we have the following properties (see ):
Mapping property: If there exists an odd map , then .
Monotonicity property: If , then .
Continuity property: If A is compact, then and there is a such that is a closed and symmetric subset and , where is defined in (2.24).
We consider the nondecreasing sequence of Lusternik-Schnirelman values for Ψ defined by
Obviously, . Next, we claim that
To prove this claim, we let and let . By Lemma 2.12, we know or 1 (depending on whether is empty or not). By the continuity property 4 of the genus, there exists such that
where and . Choose such that Lemma 2.15 holds, then for every , there exists such that . Thus, we may define the following entrance time map:
which satisfies for every . Note that is not a critical value of Ψ by Lemma 2.15. By for all , (2.21) and the definition of Ψ, we know e is a continuous (and even) map. Thus, by (2.21), we have
is odd and continuous. Therefore, by the properties 1-3 of the genus and the definition of , we have
it follows from (2.29) that
It follows from the definition of and of that if and if . Since , we get (2.28) holds.
Therefore, (2.28) implies that there is an infinite sequence of pairs of geometrically distinct critical points of Ψ with , which contradicts (2.16). The proof is finished. □
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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).
The authors declare that they have no competing interests.
The main idea of this paper was proposed by G-WC and G-WC 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.
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Chen, G., Wang, J. Ground state periodic solutions for Duffing equations with superlinear nonlinearities. Adv Differ Equ 2014, 139 (2014). https://doi.org/10.1186/1687-1847-2014-139
- second order differential equations
- ground state periodic solutions
- spectral gap
- generalized Nehari manifold approach