New periodic solutions for a class of singular Hamiltonian systems
© Wang and Shen; licensee Springer. 2014
Received: 14 November 2013
Accepted: 10 February 2014
Published: 19 February 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
Theorem 1.1 (Ambrosetti-Coti Zelati )
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) , .
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
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 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 )
for every .
Then ϕ attains its infimum in .
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
- (1)If , then by Sobolev’s embedding theorem, we have the uniform convergence property:
- (2)If , then we have the following. By the weakly lower semi-continuity for the norm, we have
Lemma 3.2 is weakly lower semi-continuous on .
- (i)If , then by , we have
- (ii)If , then by Λ satisfying Gordon’s strong force condition, we have
- (1)If , then
- (2)If . By the weakly lower semi-continuity for norm, 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 .
Lemma 3.5 The functional attains the infimum on ; furthermore, the minimizer is non-constant.
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. □
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).
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