Periodic Solutions for a Class of Second-order Hamiltonian Systems of Prescribed Energy

In this paper, the existence of non-constant periodic solutions for a class of conservative Hamiltonian systems with prescribed energy is obtained by the saddle point theorem.


Introduction and main results
Consider the second order Hamiltonian system ü(t) + ∇V(u) = 0, (1.1) where V : R N → R is a C 1 -map and ∇V(x) denotes the gradient with respect to the x variable, (•, •) : R N × R N → R denotes the standard inner product in R N and | • | is the induced norm.Furthermore, h stands for the total energy of system (1.1).
Hamiltonian systems have many applications in applied science.There are many papers [1-8, 10-12, 14, 15] which obtained the existence of periodic and connected orbits for (1.1).As we know, along with a classical solution of (1.1), the total energy is a constant.In 1978, under some constraint on the energy sphere, Rabinowitz [10] used variational methods to prove the existence of periodic solutions for a class of first order Hamiltonian systems with prescribed energy.After then, the prescribed energy problems have been studied by many mathematicians [1-4, 6, 7, 11] using geometric, topological or variational methods.In 1984, Benci [4] obtained the following theorem.
As shown in [4], condition (A 1 ) is necessary for the existence of periodic solutions of system (1.1)-(1.2).However, the periodic solution may be constant in Theorem A. The author needed the following condition to obtain the existence of non-constant periodic solutions, which is Furthermore, it is assumed that V is of C 2 class in Theorem A. Recently, Zhang [15] has proved the existence of non-constant periodic solutions for system (1.1)-(1.2) with V being only required to be of C 1 class.He got the following theorem.
In 2012, Che and Xue [6] proved the existence of periodic solutions for system (1.1)-(1.2) under some weaker assumptions.They considered the energy h to be a parameter and used monotonicity method to obtain the existence of periodic solutions.Then they obtained the following theorem.Subsequently, let V ∞ = lim inf |x|→+∞ V(x).

Theorem C ([6]
). Suppose that V ∈ C 1 (R N , R) satisfies (B 1 ) and the following conditions Then for all h ∈ µ 2 µ 1 , V ∞ , there exists a non-constant periodic solution of energy h.But condition (B 1 ) is still needed for proving the compactness condition.Motivated by these papers, we will obtain the existence of periodic solutions for system (1.1)-(1.2) under some different conditions.The following theorem is our main result.
Remark 1.2.In Theorem 1.1, the total energy could be negative if V(0) is smaller than zero which is different from Theorem B and Theorem C. Furthermore, there are functions satisfying (V 1 ), (V 2 ) but not the conditions (B 1 ) and (B 3 ).For example, let

Variational settings
Let us set H 1 = W 1,2 (R/Z, R N ).And we define the equivalent norm in H 1 as follows.
The maximum norm is defined by In order to deal with the prescribed energy situation, let f : H 1 → R be the functional defined by This functional has been used by van Groesen [14] to study the existence of brake orbits for smooth Hamiltonian systems with prescribed energy and by A. Ambrosetti and V. Coti Zelati [1,2] to study the existence of periodic solutions of singular Hamiltonian systems.It can easily be checked that f ∈ C 1 (H 1 , R) and In this paper, we still make use of the saddle point theorem introduced by Benci and Rabinowitz in [5] to look for the critical points of f .First, we recall that a functional I is said to satisfy the (PS) + condition, if any sequence {u n } ⊂ H 1 satisfying with any C > 0, implies a convergent subsequence.

Lemma 2.1 ([5]
).Let X be a Banach space and let f ∈ C(X, R) satisfy where e ∈ X 2 , e = 1, The following lemma shows that the critical points of f are non-constant periodic solutions after being scaled.Lemma 2.2.Let f be defined as in (2.1) and q ∈ H 1 such that f ( q) = 0, f ( q) > 0. Set Proof.The proof of this lemma is similar to Lemma 3.1 of [2].Here we sketch the proof for the readers' convenience.Since f ( q) = 0, we can deduce that f ( q), ν = 0 for all ν ∈ H 1 which can be written as Then we divide equation (2.3) by 1 0 (h − V( q(t)))dt which is positive since f ( q) > 0 and obtain that 1 0 This shows ũ(t) = q(t/T) satisfies (1.1).The conservation of energy for (2.4) shows that there exists a constant K such that By the definition of T, we integrate (2.5) on [0,1] and get that We finish the proof of this lemma.

Proof of Theorem 1.1
It is known that the deformation lemma can be proved when the usual (PS) + condition is replaced by (CPS) C condition (see Lemma 3.1 for the definition of (CPS) C ) which means that Lemma 2.1 holds under (CPS) C condition with positive level.Subsequently, we apply Lemma 2.1 to obtain the critical points of f under (CPS) C condition for any C > 0.
Lemma 3.1.Suppose that the conditions of Theorem 1.1 hold, then f satisfies (CPS) C condition which means that for all C > 0, and then sequence {u j } j∈N has a strongly convergent subsequence.
Since (∇V(u j (t)), u j (t)) → 0 as j → ∞ for all t ∈ Λ, there exists η > 0 such that for any j > η and t ∈ Λ we have Let N > η in (3.5), we can obtain which contradicts (3.6).Then we obtain (3.4).By Egorov's theorem, we can see that there exists 0) and (3.7), we can deduce that there exists l > 0 such that V(u j (t)) ≤ V(0) + ε 0 for j > l and t ∈ Λ 1 , where By uj L 2 → ∞ as j → ∞ and the definition of f , we can deduce that f (u j ) → +∞ as j → ∞, which contradicts (3.1).Then we get that uj L 2 is bounded.Next, we claim that |u j (0)| is still bounded.Otherwise, there is a subsequence, still denoted by {u j }, such that |u j (0)| → +∞ as j → +∞.Since uj L 2 is bounded, by Hölder's inequality, we can deduce that min 0≤t≤1 Then it follows from lim inf |x|→∞ V(x) = +∞ that there exist ζ > h and r > 0 such that for all |x| ≥ r.By the definition of f , it follows from (3.8) that ≤ 0 for j large enough.
which contradicts (3.2).Hence |u j (0)| is bounded, which implies that u j is bounded.Then there is a weakly convergent subsequence, still denoted by {u j }, such that u j u 0 in H 1 .The following proof is similar to that in [15].Then we have u j → u strongly in H 1 .Hence f satisfies (CPS) C condition.Subsequently, we use Lemma 2.1 to prove that the functional f possesses at least one critical point.Lemma 3.2.Suppose that the conditions of Theorem 1.1 hold, then functional f possesses at least one critical point in H 1 .
Proof.We set that For all u ∈ X 2 , by Poincaré-Wirtinger's inequality, we obtain that there exists a constant (3.9) Moreover, if u ∈ X 2 , the Sobolev's inequality shows that δ, then we can deduce from (3.9) and (3.10) that u ∞ ≤ δ.Thus we have When u ∈ ∂Q, there are two cases needed to be discussed.
which implies that f | ∂Q ≤ 0 for L large enough.
Case 2. If u ∈ P. For σ > 0, set Then there exists ε 1 > 0 such that meas (Γ ε 1 (u)) ≥ ε 1 (3.12) for all u ∈ P. Otherwise, there exists a sequence {u n } n∈N ⊂ P such that Furthermore, we claim that there exist constants τ 1 , τ 2 > 0 such that If not, we have meas Γ 1 n (v 0 ) = 0 for all n ∈ N. Then by Sobolev's embedding theorem, we have . Consequently, for n large enough, we have which contradicts (3.14).Then we obtain (3.12).By the definition of Γ ε 1 (u), we conclude that for any u ∈ P, we have V(u(t))dt − V(u(t))dt → −∞ as L → +∞, which implies (3.11).Together with Lemma 3.1, we can deduce from Lemma 2.1 that f possesses a critical value c.Hence there exists a u 0 ∈ H 1 such that Then we finish the proof of this lemma.
Finally, it follows from Lemma 2.2 that system (1.1)-(1.2) possesses at least one nonconstant periodic solution.Then we finish the proof of Theorem 1.1.