New existence of hyperbolic orbits for a class of singular Hamiltonian systems
 DongLun Wu^{1}Email author and
 Shiqing Zhang^{1}
Received: 20 March 2015
Accepted: 13 May 2015
Published: 30 May 2015
Abstract
A new existence of hyperbolic orbits is obtained for a class of singular Hamiltonian systems with prescribed energies by taking the limit for a sequence of approximate solutions. Furthermore, we show that the hyperbolic orbits possess the given directions at infinity.
Keywords
1 Introduction and main results
Definition 1
(see [1])
The solutions of Hamiltonian systems have been studied by many mathematicians (see [1–18] and the references therein). In 1922, Chazy showed that there are only seven possible final evolutions in the threebody problem. And the parabolic and hyperbolic orbits have been obtained for problem (1.1) when V is singular at the origin by [4, 9, 19] with variational methods. In [19], the authors obtained the existence of collisionfree parabolic orbits for a Newtonian nbody problem starting from any initial configuration and asymptotic to every minimizing normalized central configuration.
In this paper, we mainly consider the strong force case. For this case, in 2000, Felmer and Tanaka obtained the following theorem.
Theorem A
(see [9])
 (A_{1}):

\(V\in C^{1}(R^{N}\setminus\{0\},R)\),
 (A_{2}):

\(V(x)\leq0\) for all \(x\in R^{N}\setminus\{0\}\),
 (A_{3}):

there are constants \(\zeta>2\), \(\rho>0\) and \(d_{0}>0\) such that
 (i)
\(V(x)\geq\frac{d_{0}}{x^{\zeta}}\) for \(0<x\leq\rho\),
 (ii)
\((x,\nabla V(x))+2V(x)\rightarrow+\infty\) as \(x\rightarrow0\),
 (i)
 (A_{4}):

there exist \(\nu>2\) and \(C>0\) such that$$\begin{aligned} V(x)\leq\frac{C}{x^{\nu1}} \quad\textit{and}\quad \bigl\nabla V(x)\bigr\leq \frac{C}{x^{\nu}} \quad\textit{for } x\geq1. \end{aligned}$$
The proof of Theorem A depends on the difference of the given asymptotic directions. In this paper, we try to relax the conditions on the asymptotic directions and the growth condition (V_{4}). First, we consider problem (1.1)(1.2) on a bounded interval, and then we let the interval go to infinity to get hyperbolic orbits. The following theorems are our main results.
Theorem 1
 (V_{1}):

\(2 V(x)+(x,\nabla V(x))\rightarrow+\infty\) as \(x\rightarrow0\),
 (V_{2}):

\(V(x)\rightarrow\infty\) as \(x\rightarrow0\),
 (V_{3}):

\((\nabla V(x),x)\rightarrow0\) as \(x\rightarrow+\infty\),
 (V_{4}):

there exist constants \(\beta>2\), \(M_{0}>0\) and \(\sigma_{1}\geq1\) such that$$\begin{aligned} x^{\beta}\biglV(x)\bigr\leq M_{0} \quad\textit{for all } x\geq \sigma_{1}. \end{aligned}$$
Remark 1
In order to estimate the asymptotic direction of the hyperbolic orbit, we need to strengthen condition (V_{3}), which is the following theorem.
Theorem 2
 (V_{5}):

there exist \(\kappa>2\), \(\rho_{0}>0\) and \(\sigma_{2}>0\) such that$$\begin{aligned} x^{\kappa}\bigl\nabla V(x)\bigr\leq\rho_{0} \quad \textit{for all } x\geq\sigma_{2}. \end{aligned}$$
Remark 2
There is no restriction on the asymptotic directions of the hyperbolic solution at infinity in Theorems 1 and 2, which is different from Theorem A, and the restriction on the asymptotic directions is important in the proof of the blowup argument.
Remark 3
Since the total energy is a positive constant, to show a solution \(u(t)\) is a hyperbolic solution, we just need to show that \(u(t)\rightarrow\infty\) as \(t\rightarrow\pm\infty\).
The paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we obtain the existence of approximate solutions. In Section 4, we give some estimates of the approximate solutions. In Section 5, we give the proof of Theorem 1. In Section 6, we give the proof of Theorem 2.
2 Variational settings
3 Existence of approximate solutions
The approximate solutions are obtained by the minimax methods. It is known that the critical points of f correspond to the approximate solutions after an appropriate scaling of time. The following lemma shows this fact.
Lemma 3.1
(see [1])
In this paper, we need to let the time t tend to ±∞. So when we scale the time, we translate t to a suitable interval so that the function is still a solution after the translation, which can be guaranteed by the following lemma.
Lemma 3.2
(Translation property [13])
Next, we introduce Gordon’s strong force condition.
Lemma 3.3
(Strong force condition [10])
 (i)
\(\lim_{x\to0}U(x)=\infty\);
 (ii)
\(V(x)\geqU'(x)^{2}\) for every \(x\in\mathcal{N}\setminus\{0\}\).
Lemma 3.4
Suppose (A_{2}), (V_{1}) and (V_{2}) hold, then V satisfies Gordon’s strong force condition.
Proof
Subsequently, we look for the minimax type critical points of f.
Lemma 3.5
Suppose that the conditions of Theorem 1 hold, then for any \(r>1\) there exists at least one approximate solution on \(\Lambda_{r}\) for problem (2.1)(2.3), where \(T_{r}\) is defined in the proof.
Proof
Case 1. If \(q=\mathrm{constant}\), it follows from \(q\in \partial\Omega_{r}\) that \(q\equiv0\), which is a contradiction since \(q(0)=q(1)=r\).
Then, by Lemmas 3.1 and 3.2, we obtain that \(u_{r}(t)=q_{r}(\frac{t+T_{r}}{2T_{r}}): (T_{r},T_{r} )\rightarrow H^{1}\) is a nontrivial solution for problem (2.1)(2.3). The lemma is proved. □
4 Blowingup argument
In order to process the limit procedure, it is necessary to show that the minimum of \(u_{r}(t)\) has a uniform bound from above which guarantees that the asymptotic solutions cannot diverge to infinity as \(r\rightarrow+\infty\). Specifically, we obtain the following lemma.
Lemma 4.1
Proof
5 Proof of Theorem 1
In this section, we prove the existence of hyperbolic orbits by some estimates of asymptotic solutions. Firstly, we prove that the asymptotic solutions are uniformly collisionfree, which can be shown by the strong force condition.
Lemma 5.1
Proof
The proofs of the following two lemmas are similar to those in [9] and [18], we sketch the proofs for the reader’s convenience.
Lemma 5.2
Proof
Lemma 5.3
Proof
Lemma 5.4
Let \(u_{r} \in\Lambda_{r} \) be the solution of problem (2.1)(2.3) and \(u_{r}^{*}\) be defined as (5.6). Then there exists a subsequence \(\{u_{r_{j}}^{*} \}\) of \(\{u_{r}^{*} \}\) convergent to \(u_{\infty}\) in \(C _{\mathrm{loc}}(R,R^{N})\). Furthermore, \(u_{\infty}\) is a hyperbolic solution of problem (1.1)(1.2).
Proof
6 Proof of Theorem 2
Lemma 6.1
(see[9])
 (i)
\(\omega(t)<\eta\),
 (ii)
\(\frac{d}{dt}u_{r}(t)\geq\sqrt{1\eta^{2}}\dot {u}_{r}(t)\),
 (iii)
\(\frac{d}{dt}u_{r}(t)\geq\sqrt{2(1\eta ^{2})H}\),
 (iv)
\(u_{r}(t)\gequ_{r}(t_{0})+\sqrt{2(1\eta^{2})H}(tt_{0})\).
Lemma 6.2
Proof
By Lemmas 5.3, 6.1 and 6.2, similar to [9], we have the following lemma.
Lemma 6.3
(see [9])
From the above discussion, we can see that there exists at least one hyperbolic solution for (1.1)(1.2) with \(H>0\) which has the given asymptotic direction at infinity. Then we finish the proof of Theorem 2.
Declarations
Acknowledgements
The authors are very grateful to the referees for their very helpful comments and suggestions, which greatly improved the presentation of this paper. This work is supported by the National Natural Science Foundation of China (No. 11301358).
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.
Authors’ Affiliations
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