Ground state periodic solutions for Duffing equations with superlinear nonlinearities
© Chen and Wang; licensee Springer. 2014
Received: 24 December 2013
Accepted: 28 April 2014
Published: 9 May 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
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 .
Torres  has proved (1.3) with having one positive or negative solution.
Here, we need not assume that .
We are interested in the following case:
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 .
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.
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.
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.
2.1 Proof of Theorem 1.1
Lemma 2.1for all.
Proof This follows immediately from (G2) and (G4). □
Lemma 2.2 ()
Lemma 2.3 If, thenfor any, . Hence x is the unique global maximum of.
So the proof is finished. □
There issuch that, where.
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.
from which the conclusion follows. □
Lemma 2.5 Ifis a compact subset, then there existssuch thatonfor every.
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, .
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.
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 . □
which is continuous by Lemma 2.8. Here, we should mention that Lemmas 2.9 and 2.10 are due to Szulkin and Weth .
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.
Hence, is linear (and continuous) in z and depends continuously on w. So the assertion follows from Proposition 1.3 in . □
We also consider the restriction of to .
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.
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.
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 .
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.
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.
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.
The case is treated similarly. The proof is finished. □
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 .
Since , this implies that exists and then it must be a critical point of Ψ (otherwise the trajectory could be continued beyond ).
which contradicts Lemma 2.13, hence (2.22) is true. Therefore, exists, and obviously it must be a critical point of Ψ. □
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 .
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).
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. □
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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