- Open Access
New periodic solutions for a class of singular Hamiltonian systems
Boundary Value Problems volume 2014, Article number: 42 (2014)
We use the variational minimizing method to study the existence of new nontrivial periodic solutions with a prescribed energy for second order Hamiltonian systems with singular potential , which may have an unbounded potential well.
MSC:34C15, 34C25, 58F.
1 Introduction and main results
For singular Hamiltonian systems with a fixed energy ,
Theorem 1.1 (Ambrosetti-Coti Zelati )
(A3) , s.t.
(A4) , , s.t.
Then (1.1)-(1.2) have at least one non-constant periodic solution.
After Ambrosetti-Coti Zelati, a lot of mathematicians studied singular Hamiltonian systems. Here we only mention a related recent paper of Carminati-Sere-Tanaka , in which they used complex variational and geometrical and topological methods to generalize Pisani’s results . They got the following theorems.
Theorem 1.2 Suppose , and satisfies (A0), (A4), and
(B2) , ;
(B3) , .
Then (1.1)-(1.2) have at least one periodic solution with the given energy h and whose action is at most with
Theorem 1.3 Suppose , and satisfies (B1), (A4), and
(B3′) , .
Then (1.1)-(1.2) have at least one periodic solution with the given energy h and whose action is at most .
Using variational minimizing methods, we get the following theorem.
Theorem 1.4 Suppose satisfies
(V1) , , , s.t.
Then for any , (1.1)-(1.2) have at least one non-constant periodic solution with the given energy h.
2 A few lemmas
Then the standard norm is equivalent to
By symmetry condition (V3), similar to Ambrosetti-Coti Zelati , let
We define the equivalent norm in :
Let and be such that and . Set
Then is a non-constant T-periodic solution for (1.1)-(1.2). Furthermore, if , , then on Λ and , if and only if u is a nonzero constant.
If such that and , then we find that is a non-constant T-periodic solution for (1.1)-(1.2).
Lemma 2.2 (Gordon )
Let V satisfy the so-called Gordon Strong Force condition: There exist a neighborhood of 0 and a function such that:
for every .
Then we have
Then we have
Lemma 2.3 Let X be a Banach space, and let be a weakly closed subset. Suppose that is defined on an open subset and for any . Let for . Assume and is weakly lower semi-continuous on , and that it is coercive on :
Then ϕ attains its infimum in .
Proof We set
in fact, by the assumptions, it is obvious that . Now if , then there exists such that . Then we know that is bounded, since ϕ is coercive. By the Eberlein-Schmulyan theorem, has a weakly convergent subsequence. Finally, by the definition for c and the assumption for the weakly lower semi-continuity for , we know . This is a contradiction.
Now we know that there exists minimizing sequence such that . Furthermore by the coercivity of ϕ we know that is bounded; then has a weakly convergent subsequence. We claim the weak limit , since otherwise by the assumption. On the other hand, by the definition of the infimum c and the assumption for the weak lower semi-continuity for on , we know . This is a contradiction. So the weak limit and . □
3 The proof of Theorem 1.4
Lemma 3.1 Assume (V1) hold, then for any weakly convergent sequence , we have
Proof Notice that (V1) imply Gordon’s strong force condition. By the weak limit and V satisfying Gordon’s strong force condition, we have
By in the Hilbert space , we know that is bounded.
If , then by Sobolev’s embedding theorem, we have the uniform convergence property:
By the symmetry of , we have , then we have Sobolev’s inequality:
Then we have
So in this case we have
If , then we have the following. By the weakly lower semi-continuity for the norm, we have
So, by Gordon’s lemma, we have
Lemma 3.2 is weakly lower semi-continuous on .
Proof For any , by Sobolev’s embedding theorem, we have uniform convergence:
If , then by , we have
By the weakly lower semi-continuity for norm, we have
If , then by Λ satisfying Gordon’s strong force condition, we have
If , then
Then we have
So in this case we have
If . By the weakly lower semi-continuity for norm, we have
So by Gordon’s lemma, we have
Lemma 3.3 is a weakly closed subset of .
Proof By Sobolev’s embedding theorems, the proof is obvious. □
Lemma 3.4 The functional is coercive on .
Proof By the definition of and the assumption (V2), we have
Lemma 3.5 The functional attains the infimum on ; furthermore, the minimizer is non-constant.
Proof By Lemma 2.2 and Lemmas 3.1-3.3, we know that the functional attains the infimum in ; furthermore, we claim that
since otherwise, attains the infimum 0, then by the symmetry of , we have , which contradicts the definition of . Now we know that the minimizer is non-constant. □
Ambrosetti A, Coti Zelati V: Closed orbits of fixed energy for singular Hamiltonian systems. Arch. Ration. Mech. Anal. 1990, 112: 339-362. 10.1007/BF02384078
Ambrosetti A, Coti Zelati V: Periodic Solutions for Singular Lagrangian Systems. Springer, Berlin; 1993.
Carminati C, Sere E, Tanaka K: The fixed energy problem for a class of nonconvex singular Hamiltonian systems. J. Differ. Equ. 2006, 230: 362-377. 10.1016/j.jde.2006.01.021
Tanaka K: Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2000, 17: 1-33. 10.1016/S0294-1449(99)00102-X
Pisani L: Periodic solutions with prescribed energy for singular conservative systems involving strong forces. Nonlinear Anal. TMA 1993, 21: 167-179. 10.1016/0362-546X(93)90107-4
Gordon WB: Conservative dynamical systems involving strong forces. Trans. Am. Math. Soc. 1975, 204: 113-135.
This study was supported by the Scientific Research Fund of Sichuan Provincial Education Department (11ZA172) and the Scientific Research Fund of Science Technology Department of Sichuan Province (2011JYZ010).
The authors declare that they have no competing interests.
XW proved the main theorem, SS participated in the proof and helped to draft the manuscript. Both authors read and approved the final manuscript.