Periodic and subharmonic solutions for a class of second-order p-Laplacian Hamiltonian systems

In this paper, the periodic and subharmonic solutions are investigated for a class of second-order non-autonomous ordinary differential equations with a p-Laplacian. With the perturbation technique and the dual least action principle, some existence results are given of solutions to the convex p-Laplacian systems.


Introduction
In this paper, we consider the second-order ordinary differential equations with a p-Laplacian In what follows we always suppose that F : R × R N → R, (t, x) → F(t, x) is measurable and T-periodic in t for every x ∈ R N , continuously differentiable and convex in x for a.e. t ∈ R. When p = , () reduces to the second-order Hamiltonian system x(t) + ∇F t, x(t) = , a.e. t ∈ R.
Since F is T-periodic in t, it is natural to seek T-periodic solutions of () and (). Also, F is kT -periodic for k ∈ N, one can search the kT -periodic solutions, which is called subharmonic solutions. By a subharmonic solution, it means a kT periodic solution with k ≥  an integer, that is, the minimal period is strictly greater than T. When k = , it is a periodic solution or harmonic. Clearly, a solution x of () over [, kT] verifying x() = x(kT) andẋ() =ẋ(kT) can be extended by kT -periodicity over R to give a kT -periodic solution. So, it is fine to study the periodic boundary value of problem of () over [, kT] to present conditions for the existence of the periodic and subharmonic solutions of (). For the study of the subharmonic solutions, we refer to [-] for a few examples.
The dual action principle was firstly introduced by Clarke [] and developed by Clarke and Ekeland [-], which is from the spirit of optimal control theory and convex analy-http://www.boundaryvalueproblems.com/content/2014/1/260 sis; see []. Following this use and the direct variational method, the periodic solutions are obtained for the Hamiltonian system of () in [, , , ] and the references therein. Mawhin and Willem [] presented the existence results of solutions to the more general systems by using such a principle and the perturbation technique argument. In , Tian and Ge [] generalized the existence results to the p-Laplacian system (). By transforming the variable, they found a first-order Hamiltonian system equivalent to the secondorder p-Laplacian system (). Then by applying the Clarke duality, the perturbation technique, and the least action principle, they obtained the existence result of the periodic solution. The authors [] also discussed the multi-point boundary value problem of a second-order differential equation with a p-Laplacian.
Motivated by the work listed above, we aim to discuss the periodic and subharmonic solutions to the p-Laplacian systems of (). The proofs are depending on the dual least action principle and the perturbation arguments. To the best of our knowledge, there is no paper discussing the subharmonics solutions of p-Laplacian system by using the dual least action principle. This paper is a first try. The existence of the periodic solutions obtained in this paper slightly improves the result in []. New arguments are considered for the posterior estimates of the periodic solutions and the existence of subharmonic solutions are presented, which extend those in [, ].
Other existence results for periodic and subharmonic solutions of the p-Laplacian differential equation using other variational methods can be found in [, -].
The paper is organized as follows. In Section , we establish the variational structure of () and transfer the existence of the solution into the existence of a critical point of the corresponding functional. The dual action is mainly discussed here. In Section , sufficient conditions are presented to guarantee the existence of the periodic solution of (). We also estimate the prior bounds of all the periodic solutions of (). The existence of the subharmonic solutions are given in the last section.

Preliminary and dual action
Let k ≥  be any integer and p >  a constant. The Sobolev space W ,p kT (, kT; R N ) is the space of functions x : [, kT] → R N with x ∈ L p (, kT; R N ) having a weak derivativeẋ ∈ L p (, kT; R N ) and x() = x(kT). The norm over W ,p kT is defined by Here, we recall that kT] x(t) .
Consider the spaces X and Y defined by Here q is a constant such that  p +  q = . It is easy to verify that X and Y are reflexive Banach spaces and X * = Y . For Similarly, we can defineh  andh  . We denote byỸ the subspace of Y bỹ We easily find the following inequality.
Let u  = x, αu  = p (ẋ), then the second-order p-Laplacian system () can be changed to the first-order ones, for u = (u  , u  ) ∈ R N . Then the system () can be written by By the kT -periodic property, we have the Hamiltonian action of () on X given by where (·, ·) is the inner product in R N . Because the first part of ψ is indefinite, the dual least process is applied to discuss the existence of the periodic solutions of (). Let  the set of all convex lower semi-continuous functions from R N to (-∞, ∞] whose effective domain (not equal to ∞) is nonempty. Then the Fenchel transform H * (t, ·) of H(t, ·) ∈  is defined by Similarly, we also can define (A  ) There exist positive constants α, δ, and positive functions β, γ ∈ L q (, kT; [, +∞)) such that Furthermore, we have F * (t, ·) ∈ C  (R N ) and From the definition of H  and Lemma ., we have the following result.
Let v = -Ju and by duality, we have So the dual action can be defined on Y by X is the functional we needed since the critical points of X on Y coincide with the solutions of (). Because X (v + c) = X (v), it suffices to find a critical point inỸ . Similarly to the discussion of the related lemmas in reference [], we have the following results.
Lemma . Suppose the condition (A  ) holds. Then X is continuously differential onỸ .
For any h ∈Ỹ , we have Remark . There are some extended versions of the inequalities in Lemmas . and .; see [, ].

Periodic solutions
In this section, we discuss the kT -periodic solution of (). Here we note that the H(t, u) is Theorem . Suppose the following conditions are satisfied.
(A  ) There exists l ∈ L pq (, kT; R N ) such that for all x ∈ R N and a.e. t ∈ [, kT], one has (A  ) There exists α ∈ (, (kT) -max{p,q}/q ), γ ∈ L max{p,q} (, kT; R N ) such that for all x ∈ R N and a.e. t ∈ [, kT], one has Then the problem () has at least one solution u = (u  , u  ) ∈ X such that u  is the kTperiodic solutions of () and minimizes the dual action Proof From the discussion in Section , we can see that if the problem () has one solution u = (u  , u  ) ∈ X, then u  ∈ W ,p kT is for the kT -periodic solutions of (). Now we need to prove the problem () has at least one kT -periodic solution in X. The proof is divided into three parts.
Step : Existence of a solution for the perturbed problem. Choose  >  such that  < α +  < min (kT) -p , (kT) - , and for any  < <  , let Clearly, H  (t, u  ) is strictly convex and continuously differentiable in u  for a.e. t ∈ [, kT].

From (A  ) and (A  ), we have
Because the function g(s) = p s p - α |l(t)|s p- , s > , attains its minimum at s = (p -)|l(t)|/α , we have From Lemma . and Lemma ., we find that the perturbed dual action is continuously differentiable onỸ and if v ∈Ỹ is a critical point of X , the function u defined by u (t) = ∇H * t,v (t) http://www.boundaryvalueproblems.com/content/2014/1/260 i.e., where we rewrite u = (u  , u  ) ∈ X. Meanwhile, from Lemma ., we have which together with () and Lemma . implies that with δ  > , δ  > . Thus every minimizing sequence of X onỸ is bounded by () and Lemma .. From the continuity of H and the definition of H * , we can see that the second term of X is weakly lower semi-continuous onỸ . Meanwhile the first part of X is weakly continuous. So X is weakly lower semi-continuous. This implies that X has a minimum at some v ∈Ỹ . So () has a solution u = ∇H * (t,v (t)).
Since u X is bounded, there is a sequence n →  (n → ∞) with n ∈ (,  ) and a function u ∈ X such that u n u as n → ∞.
So {v n } converges weakly to v(t) = -J(u(t) -ū). From (), we have the integrated form Because u n converges weakly to u in X, u n converges uniformly to u in C ∞ kT (see Proposition . in []). So, let n → ∞ and we have that is, u is a solution of () in X.
Finally, we show v = -J(u(t) -ū) minimizes the dual action X onỸ . Because v n is a minimum of X n and H * n (t, v) ≤ H * (t, v), we have By the duality between u n andv n and the definition of H n (t, v), we have Moreover, as v n converges weakly to v inỸ , Jv n converges weakly to Jv. Letting n → ∞ we obtain, byv(t) = ∇H(t, u(t)), where m is an integer such that  ≤ m ≤ p.
When the parameter α is smaller, we can obtain the prior bound for all the solutions of the p-Laplacian system ().
for a.e. t ∈ [, kT] and x ∈ R N , then each solution of () satisfies the inequalities Proof We still set u  = x, αu  = p (ẋ), and the equalities () hold. Easily we find It follows from That is, Integrating over [, kT] and using Lemma ., we have Meanwhile, by the convexity of F, we have which completes the proof. as |x| → +∞ uniformly in t ∈ R. Then for each k ∈ R\{}, the system () has a kT-periodic solution x k such that x k ∞ → +∞ and such that the minimal period T k of x k tends to ∞ when k → +∞. http://www.boundaryvalueproblems.com/content/2014/1/260

Subharmonic solutions
We notice that Inequalities () and () imply that { x k n ∞ } is bounded, which is a contraction. The proof is complete.