Existence of Weak Quasi-periodic Solutions for a Second Order Hamiltonian System with Damped Term via a Pde Approach

In this paper, we investigate the existence of weak quasi-periodic solutions for the second order Hamiltonian system with damped term: ¨ u(t) + q(t) ˙ u(t) + DW(u(t)) = 0, t ∈ R, = −K(x) + F(x) + H(x) for all x ∈ R n and W is concave and satisfies the Lipschitz condition. Under some reasonable assumptions on q, K, F, H, we obtain that system has at least one weak quasi-periodic solution. Motivated by Berger et al. (1995) and Blot (2009), we transform the problem of seeking a weak quasi-periodic solution of system (HSD) into a problem of seeking a weak solution of some partial differential system. We construct the variational functional which corresponds to the partial differential system and then by using the least action principle, we obtain the partial differential system has at least one weak solution. Moreover, we present two propositions which are related to the working space and the variational functional, respectively.


Introduction
Assume that ω = (ω 1 , . . ., ω m ) is a list of linearly independent real numbers over the rationals.Definition 1.1 ([20]).u : R → R n is said to be quasi-periodic with m basic frequencies if there exists a function x → Φ(x) ∈ R n which is Lipschitz continuous for x ∈ R m and periodic of period 1 in each of its arguments, and m real numbers ω 1 , . . ., ω m linearly independent over the rationals, such that u(t) = Φ(ω 1 t, . . ., ω m t).Any such choice of ω 1 , . . ., ω m will be called a set of basic frequencies for u(t).
In this paper, we are concerned with the second order Hamiltonian system with damped term: ü(t) + q(t) u(t) + DW(u(t)) = 0, t ∈ R, (HSD) where DW denotes the gradient of W, q : R → R is a quasi-periodic function with module of frequencies generated by ω = (ω 1 , . . ., ω m ), and satisfies the following condition: (Q) q(t) = ∑ m j=1 q j (t), where q j (t) is continuous on R and 1 ω j -periodic, j = 1, . . ., m; and W : R n → R, W(x) = −K(x) + F(x) + H(x) for all x ∈ R n and satisfies the following condition: (W) W ∈ C 1 (R n , R), and there exists a positive constant L such that |DW(x) − DW(y)| ≤ L|x − y|, for all x, y ∈ R n .

Next, we present the definition of weak ω-quasi-periodic solution for system (HSD).
For the purpose, we need to recall some function spaces which can be seen in [15], [16], [6] and [7] for more details. Define Then the mean value of f is the limit (when it exists) lim T→∞

2T
T −T f (t)dt.A fundamental property of Bohr almost periodic function u is that such function has convergent mean, that is, the limit lim T→∞ which is a complex vector and is called Fourier-Bohr coefficient of u.Let and Mod(u) the Z-module generated by Λ(u).Denote Z ω by the Z-module generated by Then u is a weak ω-quasi-periodic solution of system (HSD).
Hamiltonian system is a very important model in physics and it has also extensively appeared in other subjects such as life science, social science, bioengineering, space science and so on.Hence, the theory of Hamiltonian system has been focused on for a long time by mathematicians and physicists.Especially, over the past 40 years, the existence and multiplicity of various solutions have attracted lots of mathematicians.Since Hamiltonian system possesses the variational structure, variational method becomes a very effective tool to deal with those problems on the existence and multiplicity of solutions for Hamiltonian systems.There have been many contributions on periodic solutions, subharmonic solutions and homoclinic solutions (for example, see [10,14,17,[21][22][23][24][25]30] and reference therein).For the investigation about almost periodic solutions of Hamiltonian system, there are less works.Joël Blot and co-authors made some important contributions and had a list of papers (see [1][2][3][4][5][6][7]).We refer the reader to [8, 9, 11-13, 18, 28, 29] for some other known results.Next we only recall two works which have a direct relationship with our problem investigated in this paper.
In 1995, via a PDE approach and the least action principle, Berger and Zhang [11] investigated the existence of quasi-periodic solutions of fixed frequencies for the nondissipative second order Duffing equation: where u : R → R, a > 0, b > 0 and f : R → R is a quasi-periodic function with frequencies ω.
For system case, in 2009, Blot [7] investigated the existence of ω-quasi-periodic solution for the second order Hamiltonian system without the damped term: via a PDE viewpoint which is partially similar to [11], where u : R → R n , W ∈ C 2 (R n ) and is concave, and C * is defined by (2.3) below and e ∈ B 1,2 In order to obtain the ω-quasi-periodic solution of system (HS), the author first investigated the existence of weak ω-quasi-periodic solution for system (HS) via a PDE approach.To be precise, the author transformed the problem into seeking a weak solution of the partial differential system: where . Furthermore, in order to obtain weak solution of system (1.2), the author first investigated the existence of weak solution for the partial differential system: Then by careful analysis, as k → +∞, the sequence {U k } which consists of weak solutions of system (1.3) converges to a weak solution of system (1.2).Following ideas in [11] and [7], in this paper, when W satisfies some reasonable growth conditions, we investigate the existence of weak ω-quasi-periodic solutions for system (HSD) via a similar PDE approach.
Next, we transform the problem of seeking a weak ω-quasi-periodic solution of (HSD) into a problem of seeking a weak solution of partial differential system (PDS*) below.
By (Q), define q : R m → R by q(x) = ∑ m j=1 qj (x j ), where qj : R → R defined by qj (x j ) := q j ( x j 2πω j ) which satisfies qj (2πω j t) = q j 2πω j t 2πω j = q j (t).
It is easy to verify that qj is 2π-periodic.
, where Qj : R → R defined by Qj (x j ) := Q j ( Then we have which implies that Consider the second order elliptic partial differential system: where Then it is easy to verify that U is 1-periodic in each of its arguments if U is 2π-periodic in each of its arguments.Hence, if U is a weak 2πperiodic solution of system (PDS), then u(t) = U(2πωt) = U(tω) is a weak ω-quasi-periodic solution of system (HSD).Furthermore, in order to obtain a 2π-periodic solution of system (PDS), following the idea of Blot [7], we seek a weak solution U of the Dirichlet boundary value problem where Ω := (−π, π) m ⊂ R m , and the solution can be extendable into a 2π-periodic solution U of system (PDS) on R m .
We organize our paper as follows.In Section 2, we introduce the working spaces.In Section 3, we present the variational functional which corresponds to system (PDS*) and then by using the least action principle, give two existence theorems.Finally, we present two propositions which are related to the working space and the variational functional, respectively.

Working spaces
In this section, we present the working spaces which were established in [6] and [7].
Let T m := R m /2πZ m be the m-dimensional torus and Ω : and |U| 2 is locally Lebesgue-integrable on R m and with the inner product and where (2.1) Define For U ∈ L 2 , define Then 2), it is easy to obtain that Then H 1 ω is a Hilbert space and H 2 ω is also a Hilbert space. Let with the inner product and the norm ) is a Hilbert space.Following Blot [7], one can extend a function U ∈ H 1 0 (Ω) to a function Ũ ∈ H 1 and by a trace theorem, one can give sense to U = 0 on ∂Ω if U ∈ H 1 0 (Ω) so that H 1 0 (Ω) ⊂ H 1 and for U ∈ H 1 0 (Ω), the following inequality holds: there exists C * > 0 such that ) is also a Hilbert space.We refer the reader for more details about the above working spaces to [6] and [7].

Main results
In [26], Wu and Chen directly construct a variational functional which corresponds to the second order Hamiltonian system like (HSD) in order to investigate the existence of periodic solutions.Motivated by [26], we define a functional J : When (Q) and (W) hold, a standard argument can be made easily so that J is of class C 1 and for all V ∈ CL ω H 1 0 (Ω).Remark 3.1.When n = 1, in [19] and [27], there have been more general functionals which correspond to more general partial differential equations.In some sense, when n = 1, the functional J (U) can be seen as a special case of those in [19] and [27] if we choose a i,j (x, u) ≡ e Q(x) , i, j = 1, . . ., m, where the details of a i,j (x, u) can be seen in [19] and [27].Lemma 3.2.Assume that J (U * ) = 0 for some U * ∈ CL ω H 1 0 (Ω).Then u * (t) := U * (2πωt) is a weak ω-quasi-periodic solution of system (HSD).
Next, we present two propositions which are related to the working space and the variational functional, respectively.Proposition 3.7.For U ∈ CL ω H 1 0 (Ω), (2.3) also holds. Proof.
Proof.For an arbitrary V ∈ CL ω H 1 0 (Ω), there exists a sequence {V k } ⊂ H 1 0 (Ω) such that V k − V H 1 ω → 0. Then by Hölder's inequality and (2.1), we have where M * = max x∈ Ω e Q(x) .Note that J (U * ), V = 0 for all V ∈ H 1 0 (Ω).Hence, by (3.11), we have Remark 3.9.Proposition 3.7 and Proposition 3.8 are maybe useful for one to seek the critical points of the functional J by using those abstract critical point theorems with Palais-Smale condition.We try to do such things by using Ekeland variational principle so that the restriction on concavity of W can be deleted.However, we come across a difficulty whether the embedding CL ω H 1 0 (Ω) → L 2 (Ω) is compact.Note that the embedding H 1 0 (Ω) → L 2 (Ω) is compact.So maybe one can reduce our problem from CL ω H 1 0 (Ω) to H 1 0 (Ω) by Proposition 3.8.However, a new difficulty whether (PS) sequence of J is bounded in H 1 0 (Ω) appears.This is a problem that is worthy of consideration.