 Research
 Open access
 Published:
New periodic solutions with a prescribed energy for a class of Hamiltonian systems
Boundary Value Problems volumeÂ 2017, ArticleÂ number:Â 30 (2017)
Abstract
We consider a class of second order Hamiltonian systems with a \(C^{2}\) potential function. The existence of new periodic solutions with a prescribed energy is established by the use of constrained variational methods.
1 Introduction
In this paper, we examine the existence of periodic solutions for second order Hamiltonian systems
with a fixed energy. The first major result in this direction we would like to highlight can be derived from the work of Benci [1], GluckZiller [2], and Hayashi [3], which is based on the earlier work of Seifert [4] in 1948 and following the highly influential papers of Rabinowitz [5, 6] in 1978 and 1979. Utilizing the Jacobi metric and a very involved interplay between geodesic methods and algebraic topology, the following general theorem is established.
Theorem 1.1
Suppose \(V\in C^{1}(\mathbb{R}^{n},\mathbb{R})\). If the potential well
is bounded and nonempty, then the system (1.1)(1.2) has a periodic solution with energyÂ h. Furthermore, if
then the system (1.1)(1.2) has a nonconstant periodic solution with energyÂ h.
For the existence of multiple periodic solutions for (1.1)(1.2) with compact energy surfaces, we can refer the reader to Groessen [7] and Long [8] and the references therein.
For the weakly attractive potential V defined on an open subset Î© of \(\mathbb{R}^{n}\), Ambrosetti and Coti Zelati [9] (TheoremÂ 16.7) proved the following.
Theorem 1.2
Suppose \(V\in C^{2}(\Omega,\mathbb{R})\) satisfies
 \((V10)\) :

\(3\langle V^{\prime}(x),x\rangle+\langle V''(x)x,x\rangle\neq0\), \(\forall x\in\Omega\);
 \((V11)\) :

\(\langle V^{\prime}(x),x\rangle>0\), \(\forall x\in \Omega\);
 \((V12)\) :

\(\exists\alpha\in(0,2)\), such that \(\langle V^{\prime}(x), x\rangle\geq\alpha V(x)\), \(\forall x\in\Omega\);
 \((V13)\) :

\(\exists\beta\in(0,2)\) and \(r>0\) such that \(\langle V^{\prime}(x), x\rangle\leq\beta V(x)\), \(\forall0<\vert x\vert <r\);
 \((V14)\) :

\(G_{\infty}\geq0\); where \(G_{\infty}=\lim_{\vert x\vert \rightarrow\infty }\inf G(x)\), \(G(x)=[V(x)+\frac{1}{2}\langle V^{\prime}(x),x\rangle]\).
Then \(\forall h<0\), the system (1.1)(1.2) (referred to as \((P_{h})\)) has at least one nonconstant weak periodic solution with the given energyÂ h.
Using a simpler constrained variational minimizing method, we obtain the following result.
Theorem 1.3
Suppose \(V\in C^{2}(\mathbb{R}^{n},\mathbb{R})\) and \(h \in\mathbb{R}\) satisfy
 \((V_{1})\) :

\(V(q)=V(q)\);
 \((V_{2})\) :

\(\langle V^{\prime}(q),q\rangle>0\), \(\forall q\neq0\);
 \((V_{3})\) :

\(3\langle V^{\prime}(q),q\rangle+\langle V''(q)q,q\rangle>0\), \(\forall q\neq0\);
 \((V_{4})\) :

\(\exists\mu_{1}>0\), \(\mu_{2}\geq0\), such that \(\langle V^{\prime}(q), q\rangle\geq\mu_{1} V(q)\mu_{2}\);
 \((V_{5})\) :

\(\lim_{\vert q\vert \rightarrow\infty }\sup[V(q)+\frac{1}{2}\langle V^{\prime}(q),q\rangle]\leq A\);
 \((V_{6})\) :

\(\frac{\mu_{2}}{\mu_{1}}< h< A\).
Then the system (1.1)(1.2) has at least one nonconstant periodic solution with the given energyÂ h.
Remark 1.4
Comparing TheoremÂ 16.7 of Ambrosetti and Coti Zelati [9] with our TheoremÂ 1.3, we notice that our condition \((V_{2})\) corresponds to their \((V11)\), our condition \((V_{3})\) corresponds to their \((V10)\), our condition \((V_{4})\) corresponds to their \((V12)\) and \((V13)\), our conditions \((V_{5})\) and \((V_{6})\) correspond to their \((V14)\). Since the potential in TheoremÂ 16.7 of Ambrosetti and Coti Zelati has a singularity, but the potential in TheoremÂ 1.3 has no singularity, the two theorems are essentially different.
Remark 1.5
Take for \(V(x)\) the following \(C^{\infty}\) function:
Then \(V(x)\) satisfies (\(V_{1}\))(\(V_{5}\)) in TheoremÂ 1.3 if we take \(\mu_{1}=\mu_{2}>0\) and \(A=1\), but \((V_{6})\) does not hold.
Proof of TheoremÂ 1.3
We verify (\(V_{1}\))(\(V_{5}\)) by calculation:
(1) It is obvious for \((V_{1})\).
(2) For \((V_{2})\) and \((V_{3})\), we notice that
(3) For \((V_{4})\), we set
We will prove \(w(x)>\mu_{1}\); in fact,
From \(w^{\prime}(x)=0\), we have \(x=\frac{1}{1+\mu_{1}}\) or 0 or \(\frac{1}{1+\mu_{1}}\).
It is easy to see that \(w(x)\) is strictly increasing on \((\infty ,\frac{1}{1+\mu_{1}}]\) and \([0,\frac{1}{1+\mu_{1}}]\), but strictly decreasing on \([\frac {1}{1+\mu_{1}},0]\) and \([\frac{1}{1+\mu_{1}},+\infty)\). We notice that
and
So
When we take \(\mu_{2}=\mu_{1}>0\), \((V_{4})\) holds.
(4) For \((V_{5})\), we have
â€ƒâ–¡
Corollary 1.6
Given \(a>0\), \(n\in\mathbb{N}\), define \(V(x)=a\vert x\vert ^{2n}+e^{\frac{1}{\vert x\vert }}\), \(x\neq0\); \(V(0)=0\). Then, for \(h>1\), the system (1.1)(1.2) has at least one nonconstant periodic solution with the given energyÂ h.
Remark 1.7
The potential \(V(x)=e^{\frac{1}{\vert x\vert }}\), \(\forall x\neq0\); \(V(0)=0\) in RemarkÂ 1.5 is noteworthy since the potential function is nonconvex and bounded which satisfies neither of the conditions of Theorems 1.1, Offinâ€™s geometrical conditions [10], nor BergPasquottoVandervorstâ€™s complex topological assumptions [11]. For this potential, the potential well \(\{x\in\mathbb{R}^{n}:V(x)\leq h\}\) is a bounded set if \(h<1\), but for \(h\geq1\) it is \(\mathbb{R}^{n}\)Â  an unbounded set. We also notice that the symmetrical condition on the potential simplified our TheoremÂ 1.2 and its proof. It would be interesting to obtain nonconstant periodic solutions when the symmetrical condition is deleted.
2 A few lemmas
Let
denotes the periodic functional space of period 1. Then the standard \(H^{1}\) norm is
Lemma 2.1
[12]
For \(u\in H^{1}\), define
For \(u,v\in H^{1}\) and \(s \in\mathbb{R}\), let
Then
and
therefore, if \((V_{3})\) holds, then on M, \(g'(u)\neq0\), which implies M is a \(C^{1}\) manifold with codimension 1 in \(H^{1}\).
Let
and \(\widetilde{u}\in M\) such that \(f^{\prime}(\widetilde{u})=0\) and \(f(\widetilde{u})>0\). Set
If \((V_{2})\) holds, then \(\widetilde{q}(t)=\widetilde{u}(t/T)\) is a nonconstant Tperiodic solution for (1.1)(1.2).
When the potential is even, then by Palaisâ€™ symmetrical principle [13] and LemmaÂ 2.1 we have the following.
Lemma 2.2
[12]
Let
and suppose (\(V_{1}\))(\(V_{3}\)) hold. If \(\widetilde{u}\in F\) is such that \(f^{\prime}(\widetilde{u})=0\) and \(f(\widetilde{u})>0\), then \(\widetilde{q}(t)=\widetilde{u}(\frac{t}{T})\) is a nonconstant Tperiodic solution for (1.1)(1.2); in addition, we have
Wirtingerâ€™s inequality [14] implies
from which it follows that \((\int^{1}_{0}\vert \dot{u}\vert ^{2}\,dt )^{1/2}\) is an equivalent norm for the space \(H^{1}\).
Lemma 2.3
Let X be a Banach space and \(F\subset X\) a weakly closed subset. Suppose Î¦ defined on F is Gateauxdifferentiable, weakly lower semicontinuous and bounded from below onÂ F. Suppose further that Î¦ satisfies the following \((\mathit{WPS})_{\inf\Phi ,F}\) condition:

If \(\{x_{n}\}\subset F\) such that \(\Phi(x_{n}) \rightarrow c\) and \(\Vert \Phi'(x_{n})\Vert \rightarrow0\), then \(\{x_{n}\}\) has a weakly convergent subsequence.
Then Î¦ attains its infimum onÂ F.
Proof
By Ekelandâ€™s variational principle [15, 16], we know that there is a sequence \(\{x_{n}\}\subset F\) satisfying
Since Î¦ satisfies the \((\mathit{WPS})_{\inf\Phi,F}\) condition, \(\{x_{n}\}\) has a weakly convergent subsequence which as a weak limit x. Because \(F\subset X\) is a weakly closed subset, we have \(x\in F\). Finally, by the weakly lower semicontinuous assumption on Î¦, we conclude that Î¦ attains its infimum onÂ F.â€ƒâ–¡
3 The proof of Theorem 1.3
We prove TheoremÂ 1.3 by the following sequence of lemmas. In the following, f and F are defined as in (2.1) and (2.2), respectively.
Lemma 3.1
If (\(V_{1}\))(\(V_{6}\)) hold, then, for any given \(c>0\), f satisfies the \((\mathit{PS})_{c,F}\) condition; that is, if \(\{u_{n}\}\subset F\) satisfies
then \(\{u_{n}\}\) has a strongly convergent subsequence.
Proof
We first prove that under our assumptions the constrained set \(F\neq\emptyset\). For any given \(u\in H^{1}\) satisfying \(u(t)\neq0\), \(\forall t\in[0,1]\) and for \(a>0\), let
By the assumption \((V_{3})\), we have
and so \(g_{u}\) is strictly increasing. Since \(V\in C^{2}\), we know that, for any given \(a>0\),
is uniformly continuous on \([0,1]\).
Hence by \((V_{5})\), we have
By \((V_{4})\), we notice that
Since \(\frac{\mu_{2}}{\mu_{1}}< h< A\), we see that the equation \(g_{u}(a)=h\) has a unique solution \(a(u)\) with \({a(u)u\in M}\).
By \(f(u_{n})\rightarrow c\), we have
and by \((V_{4})\) we see that
Condition \((V_{6})\) provides \(h>\frac{\mu_{2}}{\mu_{1}}\). Then (3.6) and (3.8) imply \(\int^{1}_{0}\vert \dot{u_{n}}(t)\vert ^{2}\,dt\) is bounded and \(\Vert u_{n}\Vert =\Vert \dot{u}_{n}\Vert _{L^{2}}\) is bounded.
We know that \(H^{1}\) is a reflexive Banach space, so \(\{u_{n}\}\) has a weakly convergent subsequence; furthermore, by the embedding theorem the weakly convergent subsequence also uniformly converges to some \(u\in H^{1}\). The standard argument can show that \(\{u_{n}\}\) has a subsequence which converges under the \(H^{1}\) norm. We omit the details of this standard demonstration.â€ƒâ–¡
Lemma 3.2
\(f(u)\) is weakly lower semicontinuous onÂ F.
Proof
For any \(u_{n}\subset F\) with \(u_{n}\rightharpoonup u\), by Sobolevâ€™s embedding theorem we have the uniform convergence
Since \(V\in C^{1}(\mathbb{R}^{n},\mathbb{R})\), we have
By the weakly lower semicontinuity of the norm, we see that
and so
Then
â€ƒâ–¡
Lemma 3.3
F is a weakly closed subset in \(H^{1}\).
Proof
This follows easily from Sobolevâ€™s embedding theorem and \(V\in C^{1}(\mathbb{R}^{n},\mathbb{R})\).â€ƒâ–¡
Lemma 3.4
The functional \(f(u)\) has a positive lower bound onÂ F.
Proof
By the definitions of \(f(u)\), F, and the assumption \((V_{2})\), we have
We claim further that
otherwise, \((V_{2})\) implies \(u(t)=\mathit{const}\), and by the symmetrical property \(u(t+1/2)=u(t)\) we have \(u(t)=0\), \(\forall t\in\mathbb{R}\). But assumptions \((V_{4})\) and \((V_{6})\) imply
which contradicts the definition of F since \(V(0)=h \) if we have \(0\in F\). Now by LemmasÂ 3.13.4 and LemmaÂ 2.3, we see that \(f(u)\) attains the infimum on F and we know that the minimizer is nonconstant.â€ƒâ–¡
References
Benci, V: Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems. Ann. Inst. Henri PoincarÃ©, Anal. Non LinÃ©aire 1, 401412 (1984)
Gluck, H, Ziller, W: Existence of periodic motions of conservative systems. In: Bombieri, E (ed.) Seminar on Minimal Submanifolds. Princeton University Press, Princeton (1983)
Hayashi, K: Periodic solutions of classical Hamiltonian systems. Tokyo J. Math. 6, 473486 (1983)
Seifert, H: Periodische Bewegungen mechanischer Systeme. Math.Â Z. 51, 197216 (1948)
Rabinowitz, PH: Periodic solutions of Hamiltonian systems. Commun. Pure Appl. Math. 31, 157184 (1978)
Rabinowitz, PH: Periodic solutions of a Hamiltonian systems on a prescribed energy surface. J.Â Differ. Equ. 33, 336352 (1979)
Van Groesen, EWC: Analytical minimax methods for Hamiltonian break orbits with a prescribed energy. J.Â Math. Anal. Appl. 132, 112 (1988)
Long, Y: Index Theory for Symplectic Paths with Applications. BirkhÃ¤user, Basel (2002)
Ambrosetti, A, Coti Zelati, V: Periodic Solutions of Singular Lagrangian Systems. BirkhÃ¤user, Basel (1993)
Offin, D: A class of periodic orbits in classical mechanics. J.Â Differ. Equ. 66, 90117 (1987)
Berg, J, Pasquotto, F, Vandervorst, R: Closed characteristics on noncompact hypersurfaces in \(\mathbb{R}^{2n}\). Math. Ann. 343, 247284 (2009)
Ambrosetti, A, Coti Zelati, V: Closed orbits of fixed energy for singular Hamiltonian systems. Arch. Ration. Mech. Anal. 112, 339362 (1990)
Palais, R: The principle of symmetric criticality. Commun. Math. Phys. 69, 1930 (1979)
Mawhin, J, Willem, M: Critical Point Theory and Applications. Springer, Berlin (1989)
Ekeland, I: On the variational principle. J.Â Math. Anal. Appl. 47, 324353 (1974)
Ekeland, I: Nonconvex minimization problems. Bull. Am. Math. Soc. (N.S.) 1(3), 443474 (1979)
Acknowledgements
The authors sincerely thank the editor and the referees for their many valuable comments and suggestions. Shiqing Zhang and Fengying Li were partially supported by NSFC (11671278). Ying Lv was partially supported by NSFC (11601438).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that no competing interests exist.
Authorsâ€™ contributions
The authors contributed equally to this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Li, F., Lv, Y. & Zhang, S. New periodic solutions with a prescribed energy for a class of Hamiltonian systems. Bound Value Probl 2017, 30 (2017). https://doi.org/10.1186/s1366101707615
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366101707615