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New periodic solutions of singular Hamiltonian systems with fixed energies
Journal of Inequalities and Applications volume 2014, Article number: 400 (2014)
Abstract
By using the variational minimizing method with a special constraint and the direct variational minimizing method without constraint, we study second-order Hamiltonian systems with a singular potential and , which may have an unbounded potential well, and prove the existence of non-trivial periodic solutions with a prescribed energy. Our results can be regarded as complements of the well-known theorems of Benci-Gluck-Ziller-Hayashi and Ambrosetti-Coti Zelati and so on.
MSC:35A15, 47J30.
1 Introduction
Seifert [1] in 1948 and Rabinowitz [2, 3] in 1978 and 1979 studied classical second-order Hamiltonian systems without singularity, based on their work, Benci [4, 5] and Gluck and Ziller [6] and Hayashi [7] used a Jacobi metric and very complicated geodesic methods and algebraic topology to study the periodic solutions with a fixed energy of the following system:
They proved a very general theorem.
Theorem 1.1 Suppose , if
is bounded and non-empty, then (1.1)-(1.2) has a periodic solution with energy h.
Furthermore, if
then (1.1)-(1.2) has a nonconstant periodic solution with energy h.
For the existence of multiple periodic solutions for (1.1)-(1.2) with compact energy surfaces, we can refer to Groessen [8] and Long [9] and the references therein.
In the 1987 paper of Ambrosetti and Coti Zelati [10], Clark-Ekeland’s dual action principle, Ambrosetti-Rabinowitz’s mountain pass theorem etc. were used to study the existence of T-periodic solutions of the second-order equation
where
is such that
here is a bounded and convex domain, and they got the following result.
Theorem 1.2 Suppose that
-
(i)
;
-
(ii)
for some and for all x near Γ (superquadraticity near Γ);
-
(iii)
for some and for all .
Let be the greatest eigenvalue of and . Then has for each a periodic solution with minimal period T.
For systems, a natural interesting problem is if
is unbounded: can we get a nonconstant periodic solution for the system (1.1)-(1.2)?
In 1987, Offin [11] firstly generalized Theorem 1.1 to some non-compact cases under and complicated geometrical assumptions on potential wells, but it seems to be difficult to verify this for concrete potentials under the geometrical conditions.
In 1988, Rabinowitz [12] studied multiple periodic solutions for classical Hamiltonian systems with potential , where is -periodic in positions and is T-periodic in t.
In 1990, using Clark-Ekeland’s dual variational principle and Ambrosetti-Rabinowitz’s mountain pass lemma, Coti Zelati et al. [13] studied Hamiltonian systems with convex potential wells, they got the following result.
Theorem 1.3 Let Ω be a convex open subset of containing the origin O. Let be such that
(V1) , .
(V2) , .
(V3) , s.t. , .
(V4) , , or
(V4)′ , .
Then, for every , (1.1) has a solution with minimal period T.
In Theorems 1.2 and 1.3, the authors assumed the convex conditions for potentials and potential wells so that they can apply Clark-Ekeland’s dual variational principle; we notice that Theorems 1.1-1.3 essentially made the following assumption:
So all the potential wells are bounded.
For singular Hamiltonian systems with a fixed energy , Ambrosetti and Coti Zelati in [14, 15] used Ljusternik-Schnirelmann theory on a manifold to get the following theorem.
Theorem 1.4 (Ambrosetti and Coti Zelati [14])
Suppose satisfies , and
-
(A1) , ;
-
(A2) , ;
-
(A3) , s.t. , ;
-
(A4) , , s.t. , ;
-
(A5) , .
Then (1.1)-(1.2) has at least one nonconstant periodic solution.
Besides Ambrosetti-Coti Zelati, many other mathematicians [16–34] studied singular Hamiltonian systems, here we only mention a related recent paper of Carminati, Sere and Tanaka [16]. They used complex variational and topological methods to generalize Pisani’s results [17], and they got the following theorem.
Theorem 1.5 Suppose , and satisfies , and
-
(B1) , ;
-
(B2) , ;
-
(B3) , ;
-
(A4) , , s.t. , .
Then (1.1)-(1.2) has at least one periodic solution with the given energy h and whose action is at most with
Theorem 1.6 Suppose , , and satisfies , and (B1), (A4) and
(B2)′ ;
(B3)′ , .
Then (1.1)-(1.2) has at least one periodic solution with the given energy h whose action is at most .
By using the variational minimizing method with a special constraint, we obtain the following result.
Theorem 1.7 Suppose and , and satisfies (A1)-(A3) and
(A4)′ , s.t. , ;
(A5)′ , .
Then for any , (1.1)-(1.2) has at least one nonconstant periodic solution with the given energy h.
Using the direct variational minimizing method, we get the following theorem.
Theorem 1.8 Suppose and , and satisfies
(B1)′ , ;
(P1)′ , ;
(A3)′ , , s.t. , ;
(A4) , , s.t. , .
Then for any , (1.1)-(1.2) has at least one nonconstant periodic solution with the given energy h.
Corollary 1.9 Suppose and
Then for any , (1.1)-(1.2) has at least one nonconstant periodic solution with the given energy h.
Remark In Theorem 1.8, the assumption on regularity for potential V is weaker than Theorems 1.1-1.6. Comparing Theorem 1.5 with Theorem 1.8, our (B1)′ is also weaker than (B1), and (A3)′ is also different from (B2)-(B3) and (B3)′.
2 A few lemmas
Let
Then the standard norm is equivalent to
Let
Lemma 2.1 ([14])
Let
If (A1) holds, then F is a manifold with codimension 1 in . Let
and let be such that and . Set
If (A2) holds, then is a nonconstant T-periodic solution for (1.1)-(1.2). Moreover, if (A2) holds, then on F and , if and only if u is constant.
Let and be such that and . Set
Then is a nonconstant T-periodic solution for (1.1)-(1.2). Furthermore, if , , then on Λ and , if and only if u is a nonzero constant.
Lemma 2.3 (Sobolev-Rellich-Kondrachov [35, 36])
and the imbedding is compact.
Let .
-
(1)
If , then we have the Friedrics-Poincaré inequality:
-
(2)
If , then we have Wirtinger’s inequality:
and Sobolev’s inequality:
Lemma 2.5 (Eberlein-Shmulyan [37])
A Banach space X is reflexive if and only if any bounded sequence in X has a weakly convergent subsequence.
Definition 2.6 (Tonelli [35])
Let X be a Banach space, .
-
(i)
If for any strongly converges to , we have
then we call lower semi-continuous at .
-
(ii)
If for any weakly converges to , we have
then we call weakly lower semi-continuous at .
Using the famous Ekeland variational principle, Ekeland proved the following.
Lemma 2.7 (Ekeland [38])
Let X be a Banach space, be a closed (weakly closed) subset, let be the geodesic distance between two points and in X, be the geodesic distance between x and the set F. Suppose that Φ defined on X is Gateaux-differentiable and lower semi-continuous (or weakly lower semi-continuous) and assume restricted on F is bounded from below. Then there is a sequence such that
Let X be a Banach space, be a closed subset. Suppose that Φ defined on X is Gateaux-differentiable, if sequence is such that
then has a strongly convergent subsequence.
Then we say that f satisfies the condition at the level c for the closed subset .
We notice that if , then the above condition is the classical Cerami-Palais-Smale condition [40].
We can give a weaker condition than the condition.
Definition 2.9 Let X be a Banach space, be a weakly closed subset. Suppose that Φ defined on X is Gateaux-differentiable, if sequence such that
then has a weakly convergent subsequence.
Then we say that f satisfies the condition.
Lemma 2.10 (Gordon [18])
Let V satisfy the so-called Gordon strong force condition:
There exists a neighborhood of O and a function such that:
-
(i)
;
-
(ii)
for every .
Let
Then we have
Let
Then we have
By Lemmas 2.7 and 2.10, it is easy to prove the following.
Lemma 2.11 Let X be a Banach space, let be a weakly closed subset. Suppose that Φ defined on F is Gateaux-differentiable and weakly lower semi-continuous and bounded from below on F. If Φ satisfies the condition or the condition, and suppose that
then Φ attains its infimum on F.
The next lemma is a variant on the classical Tonelli’s theorem, whose proof is easy, so we omit its proof.
Lemma 2.12 Let X be a Banach space, be a weakly closed subset. Suppose that is defined on an open subset and is Gateaux-differentiable on Λ and weakly lower semi-continuous and bounded from below on , if ϕ is coercive, that is, as , and suppose that
then ϕ attains its infimum on .
3 The proof of Theorem 1.7
By the symmetrical condition (A5)′, it is easy to prove that the critical point of the functional f on is also the critical point of the functional f on Λ.
Let
Lemma 3.1 Assume (A4)′ holds, then for any weakly convergent sequence , we have
Proof Similar to the proof of Zhang [19]. □
Lemma 3.2 is a weakly closed subset in .
Proof Let be a weakly convergent sequence, we use the embedding theorem to find which uniformly converges to .
Now we claim , and then it is obvious that . In fact, if , by , and the condition (A4)′ we have
So satisfies Gordon’s strong force condition, and by his lemma, we have
The condition (A4)′ implies
Hence
This is a contradiction. □
Lemma 3.3 is weakly lower semi-continuous on
Proof For any , then by Sobolev’s embedding theorem and functional analysis, we have uniform convergence:
-
(i)
If , then by , we have
It’s well known that the norm is weakly lower semi-continuous, we have
Hence
-
(ii)
If , then by our assumption on V which satisfies Gordon’s strong force condition, we have
-
(1)
(1) If , then
Then similar to the proof in [19], we have
So in this case we have
-
(2)
(2) If , then by the weakly lower semi-continuity for norm, we have
So by Gordon’s lemma, we have
□
-
(1)
Lemma 3.4 The functional has a positive lower bound on F.
Proof By the definitions of and F and the assumption (A2), we have
□
By the definitions of the functional and its domain , and the conditions on the energy and the potential , it is easy to prove the following lemma.
Lemma 3.5 The functional is coercive.
Furthermore, we claim that
since otherwise, attains the infimum 0, then by the symmetry of , we have , which contradicts the definition of . Now by Lemmas 3.1-3.4 and Lemmas 2.11 and 2.12, we know attains the infimum on F, furthermore we know that the minimizer is nonconstant.
4 The proof of Theorem 1.8
In order to prove the Cerami-Palais-Smale type condition and get a nonconstant periodic solution in non-symmetrical case, we need to add a topological condition, we know that there are winding numbers (degrees) in the planar case, so we define
Lemma 4.1 If , then .
Proof By V satisfying Gordon’s strong force condition, we have
-
(1)
If , then by Sobolev’s embedding theorem, we have
Then by , we have such that
and is an equivalent norm of and
So in this case, we have
-
(2)
If , then by the weakly lower semi-continuity for the norm, we have
So by Gordon’s lemma, we have
□
Lemma 4.2 Under the assumptions of Theorem 1.8,
satisfies the condition on , that is, if satisfies
then has a strongly convergent subsequence in .
Proof Since makes sense, we know
We claim is bounded. In fact, by , we have
By (A3)′ we have
By (4.2) and (4.3) we have
where , , . So .
Then we claim is bounded.
We notice that
If is unbounded, then there is a subsequence, still denoted by s.t. . Since
we have
By Friedrics-Poincaré’s inequality and the condition (P1), we have
So , which contradicts , hence is bounded, and is bounded. Furthermore, similar to the proof of Ambrosetti and Coti Zelati [15], strongly converges to . □
It is easy to prove the following.
Lemma 4.3 Under the assumption (B1)′, on Λ, that is, f has a lower bound.
Lemma 4.4 Under the assumptions of Theorem 1.8, is weakly lower semi-continuous on the closure of Λ.
Now we can prove our Theorem 1.8, in fact, by Lemma 4.1, we know that the infimum of f on is equal to the infimum of f on the closure of . Furthermore, we can prove the infimum of f on is greater than zero, otherwise if it is zero, the corresponding minimizer must be constant, then the winding number is zero, which is a contradiction. Now by the above lemmas, especially Lemma 2.11, we know that f attains the positive infimum on and the corresponding minimizer must be nonconstant.
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The authors would like to thank the editor and the referees for their many valuable comments. This paper was partially supported by NSF of China and the Grant for the Advisors of PhD students.
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Li, F., Hua, Q. & Zhang, S. New periodic solutions of singular Hamiltonian systems with fixed energies. J Inequal Appl 2014, 400 (2014). https://doi.org/10.1186/1029-242X-2014-400
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DOI: https://doi.org/10.1186/1029-242X-2014-400