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New existence of hyperbolic orbits for a class of singular Hamiltonian systems
Boundary Value Problems volume 2015, Article number: 86 (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.
1 Introduction and main results
In this paper, we consider the following Hamiltonian systems with prescribed energy:
with
where \(u\in C^{2}(R,R^{N})\), \(V\in C^{1}(R^{N}\setminus\{0\},R)\) has a singularity at the origin. \(\nabla V(x)\) denotes the gradient with respect to the x variable, \((\cdot,\cdot):R^{N}\times R^{N} \rightarrow R\) denotes the standard Euclidean inner product in \(R^{N}\) and \(\cdot\) is the induced norm. The parabolic and hyperbolic orbits are defined as follows.
Definition 1
(see [1])
If \(u(t)\) is a solution for problem (1.1)(1.2) satisfying
then \(u(t)\) is called a parabolic orbit.
If \(u(t)\) satisfies
then \(u(t)\) is called a hyperbolic orbit.
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])
Assume that \(N\geq 3\) and the following conditions hold:
 (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
$$\begin{aligned} V(x)\leq\frac{C}{x^{\nu1}} \quad\textit{and}\quad \bigl\nabla V(x)\bigr\leq \frac{C}{x^{\nu}} \quad\textit{for } x\geq1. \end{aligned}$$
Then, for any given \(H>0\) and \(\theta_{+}\neq\theta_{}\), there exists a solution \(u(t)\) of (1.1)(1.2) such that
where \(\theta_{+}\), \(\theta_{}\in S^{N1}=\{x\in R^{N}\midx=1\}\) are the asymptotic directions for the solution \(u(t)\).
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
Suppose \(N\geq3\) and \(V\in C^{1}(R^{N}\setminus\{0\},R)\) satisfies (A_{1}), (A_{2}) and the following conditions:
 (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
$$\begin{aligned} x^{\beta}\biglV(x)\bigr\leq M_{0} \quad\textit{for all } x\geq \sigma_{1}. \end{aligned}$$
Then, for any \(H>0\), there exists at least one hyperbolic orbit for problem (1.1)(1.2).
Remark 1
Compared with condition (A_{4}), there is no condition on \(\nabla V(x)\) in (V_{4}), and we can give an example which satisfies (V_{4}), but not condition (A_{4}), such as
where \(Q(x)\in C^{1}(R^{N},R)\) such that \(V(x)\) (≤0) is of \(C^{1}\) class. It is easy to see that (1.3) also satisfies (A_{1}), (A_{2}), (V_{1}), (V_{2}) and (V_{3}).
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
Suppose \(N\geq3\) and \(V\in C^{1}(R^{N}\setminus\{0\},R)\) satisfies (A_{1}), (A_{2}), (V_{1}), (V_{2}), (V_{4}) and the following condition:
 (V_{5}):

there exist \(\kappa>2\), \(\rho_{0}>0\) and \(\sigma_{2}>0\) such that
$$\begin{aligned} x^{\kappa}\bigl\nabla V(x)\bigr\leq\rho_{0} \quad \textit{for all } x\geq\sigma_{2}. \end{aligned}$$
Then, for any \(H>0\), there exists at least one hyperbolic orbit for problem (1.1)(1.2) which possesses any given asymptotic directions at infinity.
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
For any given two unite vectors (directions) \(e_{\pm}\in S^{N1}\), we set
with the norm
Here we use r to denote the Euclidean length of \(q(0)\) and \(q(1)\). Furthermore, for convenience, we let \(r>1\) in the following proof. Let \(L^{\infty}([0,1],R^{N})\) be a space of measurable functions from \([0,1]\) into \(R^{N}\) and essentially bounded with the following norm:
Moreover, let \(f: \Lambda_{r}\rightarrow R\) be the functional defined by
Then one can easily check that \(f\in C^{1}(\Lambda_{r},R)\) and
Our way to get the hyperbolic orbit is to approach it with a sequence of approximate solutions. Firstly, we prove the existence of the approximate solutions, then we study the limit procedure. We consider the following approximate problems:
with
where \(T_{r}\) is a suitable number depending on the critical points of f and r which will be given in the following lemma.
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])
Let \(f(q)= \frac{1}{2}\int^{1}_{0}\dot{q}(t)^{2}\,dt\int^{1}_{0}(HV(q(t)))\,dt\) and \(\tilde{q}\in\Lambda_{r}\) be such that \(f^{\prime}(\tilde{q})=0\), \(f(\tilde{q})>0\). Set
Then \(\tilde{u}(t)=\tilde{q}(t/T)\) is a nonconstant Tperiodic solution for problem (1.1)(1.2).
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])
Suppose that in the domain \(D\subset R^{N}\), we have a solution \(\phi(t)\) for the following differential equation:
where \(x^{(k)}=d^{k}x/dt^{k}\), \(k=0,1,\ldots,n\), \(x^{(0)}=x\). Then \(\phi(tt_{0})\) with \(t_{0}\) being a constant is also a solution.
Next, we introduce Gordon’s strong force condition.
Lemma 3.3
(Strong force condition [10])
V is said to satisfy Gordon’s strong force condition if there exist a neighborhood \(\mathcal{N}\) of 0 and a function \(U\in C^{1}(R^{N}\setminus\{0\},R)\) such that

(i)
\(\lim_{x\to0}U(x)=\infty\);

(ii)
\(V(x)\geqU'(x)^{2}\) for every \(x\in\mathcal{N}\setminus\{0\}\).
It has been shown that if V satisfies Gordon’s strong force condition, then
Lemma 3.4
Suppose (A_{2}), (V_{1}) and (V_{2}) hold, then V satisfies Gordon’s strong force condition.
Proof
Let \(\phi(r)=V(r\widetilde{e})r^{2}\), where \(r=x\), \(\widetilde{e}=x/x\), then we have
It follows from (V_{1}) and (V_{2}) that there exists a constant \(\delta>0\) such that
Then we get
It follows from the definition of ϕ and (V_{2}) that there exists a constant \(C_{1}>0\) such that
We set \(U(x)=\sqrt{C_{1}}\lnx\), then by an easy calculation we obtain
which proves this lemma. □
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
We set
where
and \(\operatorname{deg}(\tilde{\gamma})\) denotes the Brouwer degree of \(\tilde{\gamma}\). We show that f satisfies the \((PS)^{+}\) condition on \(\Lambda_{r}\). Specifically, let \(c>0\), \(\{q_{j}\}\subset\Lambda_{r}\) such that \(f(q_{j})\rightarrow c\) and \(f'(q_{j})\rightarrow0\). By the definition of f, \(H>0\) and \(V\leq 0\), we can deduce that \(\\dot{q}_{j}\_{L^{2}}\) is bounded. Then there exists a constant \(C_{0}>0\) such that
where \([q_{j}]=\int^{1}_{0}q_{j}(t)\,dt\). If \([q_{j}]\) is unbounded, we obtain that \(\q_{j}\_{L^{\infty}}\) is also unbounded. It follows from (V_{4}) and (3.2) that
Then it is easy to see that
as \(j\rightarrow+\infty\), which is a contradiction. Then we can deduce that \([q_{j}]\) is bounded. Together with the boundedness of \(\\dot{q}_{j}\_{L^{2}}\), we obtain that \(q_{j}\) is bounded in \(\Lambda_{r}\), which implies that there exists a subsequence of \(\{q_{j}\}\), still denoted by \(\{q_{j}\}\), such that \(q_{j}\rightharpoonup q\). Moreover, \(q_{j}\rightarrow q\) uniformly on \([0,1]\). It follows from Lemmas 3.3 and 3.4 that \(q\notin\partial \Omega_{r}=\{q\in E_{R} \exists t'\in[0,1]\mbox{ s.t. }q(t')=0\}\). Otherwise, if q has collision, which means \(q\in \partial\Omega_{r}\), we can prove that
To prove this fact, there are two cases needed to be discussed.
Case 1. If \(q=\mathrm{constant}\), it follows from \(q\in \partial\Omega_{r}\) that \(q\equiv0\), which is a contradiction since \(q(0)=q(1)=r\).
Case 2. If \(q\neq \mathrm{constant}\), we have \(\int^{1}_{0}\dot{q}(t)^{2}\,dt>0\). Then by the weaklylowersemicontinuity of norm, we have
which implies that
By Lemmas 3.3, 3.4 and (3.1), we can deduce that (3.5) holds, which is a contradiction. Then we can see that \(q\in \Omega_{r}\) has no collision. By a standard argument, we can see \(q_{j}\rightarrow q\) in \(\Omega_{r}\), which proves the \((PS)^{+}\) condition.
Then the minimax value is introduced by
It follows from \(f(q)\geq0\) that \(b_{r}\geq0\). But it follows from the definition of f that the necessary condition for \(b_{r}=0\) is \(\gamma(\xi)\equiv \mathrm{constant}\), which contradicts with the definition of \(\Omega_{r}\). Via the standard minimax argument (similar to [5]), we can see that \(b_{r}\) is a critical value of f and there exists \(q_{r}\in\Omega_{r}\) such that
Let
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
Suppose that \(u_{r}(t): (T_{r},T_{r} )\rightarrow H^{1}\) is the solution obtained in Lemma 3.5, then \(\min_{t\in (T_{r},T_{r} )}u_{r}(t)\) is bounded from above uniformly. Specifically, there is a constant \(M>0\) independent of r such that
Proof
Since \(f^{\prime}(q_{r})=0\) and \(q_{r}\not\equiv0\), we can obtain that \(\langle f^{\prime}(q_{r}),q_{r}\rangle=0\), which implies that
Then we obtain that
It can be convinced that there exists \(\hat{t}\in [T_{r},T_{r} ]\) such that
which implies that
It follows from (V_{3}) and (V_{4}) that there exists a constant \(M_{1}>0\) independent of r such that
Then we finish the proof of this lemma. □
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
Suppose that \(u_{r}(t)\) is the solution for (2.1)(2.3) obtained in Lemma 3.5. Then there exists a constant \(m>0\) independent of r such that
Proof
Since \(u_{r}(t)\) is a solution for problem (2.1)(2.3), then we can deduce that
By hypothesis (V_{1}), we can find \(m\in(0,1)\) independent of r such that
for any \(t\in\{t\in (T_{r},T_{r} ) \mid\max_{t\in (T_{r},T_{r} )}u_{r}(t)\leq m\}\), which implies that \(u_{r}(t)\) is concave when \(u_{r}(t)\leq m\) and \(u_{r}(t)\) cannot take a local minimum such that \(\max_{t\in (T_{r},T_{r} )}u_{r}(t)\leq m\), which implies that
If not, we can assume that there exists \(\overline{t}\in (T_{r},T_{r} )\) such that \(u_{r}(\overline{t})< m\), then we can easily check that \(u_{r}(t)\) takes a local minimum at some \(\tilde{t}\) with \(u_{r}(\tilde{t})< m\), which is a contradiction. Then we obtain the conclusion. □
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
Suppose that \(r>\max\{M,\sigma_{1},1\}\), where M is defined in Lemma 4.1 and \(u_{r}(t)\) is the solution for problem (2.1)(2.3) obtained in Lemma 3.5. Set
and
where L is a constant independent of r such that \(M< L< r\). Then we have that
Proof
By the definition of \(u_{r}(t)\) we have
Then, by (A_{2}) and the definitions of \(t_{+}\) and \(t_{}\), we obtain
and
Since \(V\in C^{1}(R^{N}\setminus\{0\},R)\), it follows from Lemma 5.1 and (V_{4}) that there exists a constant \(M_{2}>0\) independent of r such that
which implies that
Combining (5.1) with the above inequality, we obtain that
Then we have
The limit for \(t_{}+T_{r}\) can be obtained in a similar way. The proof is completed. □
Lemma 5.3
Suppose that \(u_{r}(t)\) is the solution for problem (2.1)(2.3) obtained in Lemma 3.5. Then there exists a constant \(M_{3}>0\) independent of \(r>1\) such that
Proof
Firstly, we define the function \(\xi(t)\) on \([1,+\infty)\) as a solution of
where \(\sigma_{1}\) is defined in (V_{4}) and \(e_{0}\in S^{N1}\). And \(\tau_{r}>1\) is a real number such that \(\xi(\tau_{r})=r\). We can define \(\xi(t)\) in \((\infty,0]\) and \(\tau_{r}\) in a similar way. Then we can fix \(\varphi(t)\in\Omega_{1}\) (defined in Lemma 3.5) such that \(\tilde{\gamma}_{r}(t)\in\Omega_{r}\), where
Subsequently, we set \(u_{r}(t)=\tilde{\gamma}_{r}(\frac{t+a}{2a})\) for any \(a>0\), then it is easy to see that \(u_{r}(t)=\gamma_{r}(t)\) if \(\tau_{\pm r}=\pm a\). Similar to [9], we can deduce that for \(a>0\)
We divide the interval \([\tau_{r},\tau_{r}]\) into three parts \([\tau_{r},0]\cup[0,1]\cup[1,\tau_{r}]\), then we can estimate \(I_{[\tau_{r},\tau_{r}]}\) by three integrals. Firstly, we estimate \(I_{[1,\tau_{r}]}\). By (V_{4}), we have
for some \(M_{4}>0\) independent of r. Similarly, we can get
Since \(I_{[0,1]}\) is independent of r, we obtain that
for some \(M_{3}>0\) independent of r. Then by (5.4) and the definition of \(b_{r}\), we have
Then we finish the proof of this lemma. □
By Lemma 4.1, we can see that
where M is defined in Lemma 4.1. In the following proof, we set a translation as
and
Remark 4
By Lemma 3.2, \(u_{r}^{*}\) is also a solution for problem (2.1)(2.3).
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
Step 1: We will show that there exists a subsequence \(\{u_{r_{j}}^{*} \}\) of \(\{u_{r}^{*} \}\) convergent to \(u_{\infty}\) in \(C _{\mathrm{loc}}(R,R^{N})\). By the definition of \(u_{r}^{*}\), we can deduce that \(u_{r}^{*}\) is a solution of problem (2.1)(2.3). Since \(L>M\), we can deduce that \(t_{+}\geq t^{*}\geq t_{}\). Then Lemma 5.2 shows that
It follows from (2.2) that
which implies that
Then by (5.3) we have
which implies that
for each \(r>1\) and \(t_{1}, t_{2} \in [T_{r}+t^{*},T_{r}+t^{*} ]\), which shows that \(\{u_{r}^{*}\}\) is equicontinuous.
Subsequently, we show that \(u^{*}_{r}\) is uniformly bounded on any compact set of R. Take \(a,b \in R\) such that \(a < b\). When r is large enough, by Lemma 5.2, we can see that \([a,b]\subseteq[T_{r}+t^{*},T_{r}+t^{*}]\). Then, for any \(t\in[a,b]\), it follows from (5.8) and (5.5) that
which implies that
Then we have shown that \(u^{*}_{r}\) is uniformly bounded on any compact set of R and uniformly equicontinuous on R. By the ArzelàAscoli theorem, it follows from inequalities (5.9) and (5.10) that there is a subsequence \(\{u_{r_{j}}^{*}\}_{j>0}\) converging to \(u_{\infty}\) in \(C_{\mathrm{loc}}(R,R^{N})\).
Step 2: We will show that \(u_{\infty}\) is a solution of problem (1.1)(1.2). By Lemma 3.5 and the definition of \(u^{*}_{r_{j}}\), we have
with
for each \(j >0\) and \(t\in (T_{r}+t^{*},T_{r}+t^{*} )\). Take \(a,b \in R\) such that \(a < b\). By Lemma 4.1, \(u^{*}_{r_{j}}\) has no collision uniformly on \([a,b]\). So \(\ddot{u}_{r_{j}}(t)\) is continuous on \([a,b]\) and \(\ddot{u}_{r_{j}}(t) \rightarrow\nabla V(t,u_{\infty}(t))\) uniformly on \([a,b]\). It follows that \(\ddot{u}_{r_{j}}\) is a classical derivative of \(\dot{u}_{r_{j}}\) in \((a,b)\) for each \(j> 0\). Moreover, since \(\dot{u}_{r_{j}} \rightarrow\dot{u}_{\infty}\) uniformly on \([a,b]\), we get
with
for all \(t \in[a,b]\). Since a and b are arbitrary, we conclude that \(u_{\infty}\) satisfies (1.1)(1.2).
Step 3: We need to show the hyperbolicity of \(u_{\infty}(t)\). We prove this conclusion in an indirect way. First, we show that \(u_{\infty}(t)\rightarrow+\infty\) as \(t\rightarrow+\infty\). Otherwise, there exists a sequence denoted by \(t_{n}\) such that \(t_{n}\rightarrow+\infty\) as \(n\rightarrow+\infty\) and
for some \(M_{\infty}>0\). By (5.1) and (5.2), we obtain that
Lemma 5.3 shows that
Together with (5.7), we have
Subsequently, let \(L>\max\{M, M_{\infty}, \sigma_{1}\}\) be a constant independent of r in the proof of Lemma 5.2. It follows from (5.11) that \(t_{}<+\infty\). But
which implies that
This contradicts (5.12). Then we obtain that \(u_{\infty}(t)\rightarrow+\infty\) as \(t\rightarrow+\infty\). The proof for \(t\rightarrow\infty\) is similar. Then we finish the proof of Theorem 1. □
6 Proof of Theorem 2
Since condition (V_{5}) is stronger than (V_{3}), we can obtain the existence of hyperbolic orbits similar to the proof of Theorem 1. Subsequently, we give the estimate of the asymptotic directions of hyperbolic orbits at infinity. Similar to Felmer and Tanaka [9], we set
Using the motion and energy equations, we have
and
Lemma 6.1
(see[9])
Assume that \(u_{r}\) is a solution for problem (2.1)(2.3) obtained in Lemma 3.5. For any \(\eta\in(0,1)\), there exists \(L_{\eta}\geq m\) such that if
for some \(t_{0}\in(T_{r},T_{r})\), then we have for \(t\in[t_{0},T_{r}]\)

(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
Let \(u_{r}\) be a solution for problem (2.1)(2.3) obtained in Lemma 3.5 satisfying (6.1) and \(u_{r}(t)\geq\sigma_{2}\) with \(t\geq t_{0}\) for certain \(t_{0}\in(T_{r},T_{r})\) with \(\eta\in(0,\frac{1}{2})\) and \(L_{\eta}\) be as in Lemma 6.1. Then, for \(t\geq t_{0}\), there exist \(M_{6}, M_{7}>0\) independent of η, \(u_{r}(t)\) and \(t_{0}\) such that
Proof
By Lemma 5.1, (iii) of Lemma 6.1 and (V_{5}), we can estimate \(A(t)\) as follows.
for some \(M_{8}>0\) independent of r. Since we have
then it follows from (iii) of Lemma 6.1, (6.2) and (6.3) that
By energy equation and the definition of \(t_{0}\), we have
which implies that for some \(M_{6}, M_{7}>0\) independent of r
which proves this lemma. □
By Lemmas 5.3, 6.1 and 6.2, similar to [9], we have the following lemma.
Lemma 6.3
(see [9])
For any \(\varepsilon>0\), there exists \(M_{9}>0\) such that for \(r>M_{9}\)
where \(e_{+}\) is the given direction defined in \(\Lambda_{r}\) and \(t^{*}\) is defined as (5.5).
Let \(t_{1}\geq t^{*}\) such that \(u_{r}(t_{1})=L_{\eta}\). Then, for any \(\varepsilon>0\), we get
for all \(t\geq t_{1}\), which implies that
Similarly, we can get
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.
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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).
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Wu, DL., Zhang, S. New existence of hyperbolic orbits for a class of singular Hamiltonian systems. Bound Value Probl 2015, 86 (2015). https://doi.org/10.1186/s136610150349x
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DOI: https://doi.org/10.1186/s136610150349x