Multiplicity of symmetric brake orbits of asymptotically linear symmetric reversible Hamiltonian systems

1.   Introduction

In this paper, let $J = \left(\begin{array}{cc}0 & -I\\I & 0\end{array}\right)$ and $N = \left(\begin{array}{cc}-I & 0\\0&I\end{array}\right)$ with $I$ being the identity matrix on ${{\bf R}}^n$. Denote by $\mathcal{L}_s({{\bf R}}^{2n})$ the space of all symmetric matrices in ${{\bf R}}^{2n}$ and ${{{\rm{Sp}}}}(2n)$ the symplectic group of $2n\times 2n$ matrices.

We call $B\in C(S^1, \mathcal{L}_s({{\bf R}}^{2n}))$ satisfies condition (BS1) if $B(-t) = NB(t)N$ for all $t\in {{\bf R}}$ and $B$ is $\frac{1}{2}$-periodic, where $S^1 = {{\bf R}}/{{\bf Z}}$.

Let $H\in C^1({{\bf R}}\times{{\bf R}}^{2n}, {{\bf R}})$ and $H'$ denote the gradient of $H$ with respect to the last $2n$ variables. We assume $H$ satisfies the following conditions:

(H1)$H'(t, x) = B_0(t)x+o(|x|)$ as $|x|\rightarrow \; \; 0$ uniformly in $t$,

(H2)$H'(t, x) = B_\infty x+o(|x|)$ as $|x|\rightarrow \; \; \infty$ uniformly in $t$,

(H3) $H(-t, Nx) = H(t, x) = H(t, -x) = H(t+\frac{1}{2}, -x)$, $\forall (t, x)\in ({{\bf R}}\times {{\bf R}}^{2n})$.

In ^[1] of 2008, the second author of this paper considered the multiplicity of 1-periodic brake orbits of asymptotically linear symmetric reversible Hamiltonian systems. Since condition (H3) holds, it is natural to consider 1-periodic solution of the asymptotically linear Hamiltonian systems

We call the above 1-periodic solutions symmetric brake orbits. If $H(-t, Nx) = H(t, x)$ for all $(t, x)\in ({{\bf R}}\times {{\bf R}}^{2n})$, we say $H$ is reversible.

Note that conditions (H1)-(H3) yield that both $B_0$ and $B_\infty$ belong to $C(S^1, \mathcal{L}_s({{\bf R}}^{2n}))$ and satisfy the above (BS1) condition.

In 1948, Seifert firstly studied brake orbits in Hamiltonian system in ^[2]. For the existences and multiple existence results and more details on brake orbits one can refer the paper ^[3,4] and the references therein. In ^[1] the second author of this paper obtained multiple existence of brake orbits of the asymptotically linear Hamiltonian systems (1.1)-(1.2) under certain conditions. In this paper we will study multiplicity of symmetric brake orbits of Hamiltonian systems (1.1)-(1.2).

In ^[5], the difference between $B_0$ and $B_\infty$, i.e., the different behaviors of $H$ at zero and infinity plays an important role in the study of 1-periodic solutions of (1.1). For this reason we also define the relative Morse index to measure the "true" difference between $B_0$ and $B_\infty$ under symmetric brake orbit boundary value. We shall study the relation between the relative Morse index and the Maslov-type index $i_{\sqrt{-1}}^{L_0}$ and $\nu_{\sqrt{-1}}^{L_0}$ defined below. As application, we obtain a multiplicity of symmetric brake orbits (1.1)-(1.2).

For any symplectic path ${{\gamma}}$ in ${{{\rm{Sp}}}}(2n)$, the Maslov-type index for symmetric brake orbits boundary values of ${{\gamma}}$ is defined in ^[6] to be a pair of integers $(i_{L_0}({{\gamma}}), \nu_{L_0}({{\gamma}}))\in {{\bf Z}}\times \{0, \, 1, \, 2\, ,..., n\}$.

For any continuous path $B$ in $\mathcal{L}_s({{\bf R}}^{2n})$ satisfying condition (BS1), as in ^[6], we define

where the symplectic path ${{\gamma}}_B$ is the fundamental solution of the following linear Hamiltonian system

We will briefly introduce such Malov-type index theory in Section 2.

In order to consider the multiplicity of symmetric brake orbits, define

Equip $\tilde{E}$ with the usual $W^{1/2, 2}$ norm. Then $\tilde{E}$ is a Hilbert space with the associated inner product $\langle, \rangle$. We define two self adjoint liner operators $\tilde{{A}}$, $\tilde{{B}}$ from $\tilde{E}$ to $\tilde{E}$ by

where $B$ is a continuous path in $\mathcal{L}_s({{\bf R}}^{2n})$ satisfying condition (BS1).

As in ^[5], we denote by $M^+(\cdot)$, $M^-(\cdot)$, and $M^0(\cdot)$ the positive definite, negative definite, and null subspaces of the self adjoint linear operator defining it respectively.

Definition 1.1. Let $B_1$ and $B_2$ be continuous paths in $\mathcal{L}_s({{\bf R}}^{2n})$ satisfying condition (BS1). We define the relative Morse index of $\tilde{{B}}_1$ and $\tilde{{B}}_2$ by

where we also denote by $\tilde{{B}}_1$ and $\tilde{{B}}_2$ the operators defined by (1.6) respectively. We call $I(\tilde{{B}}_1, \tilde{{B}}_2)$ the relative Morse index, following ^[5]. By Theorem 1.2 below this relative More index is well defined. Note that such definition of relative Morse index is different from those defined in ^[7], ^[8], and ^[9] etc, which is difference of Morse index or spectral flow or definition from Garlerkin approximation, the definition if $\dim\ker(A-B_1)$.

Using the iteration theory of Maslov-type index theory and result in ^[1], we obtain the relation between the Maslov-type index ($i_{\sqrt{-1}}^{L_0}$, $\nu_{\sqrt{-1}}^{L_0}$) and the relative Morse index as the following

Theorem 1.1. Let $B_1$ and $B_2$ be be continuous paths in $\mathcal{L}_s({{\bf R}}^{2n})$ satisfying condition (BS1). We have

As application, we obtain the main result of this paper.

Theorem 1.2. Suppose that $H$ satisfies (H1), (H2), (H3), and $\nu_{\sqrt{-1}}^{L_0}(B_0) = \nu_{\sqrt{-1}}^{L_0}(B_\infty) = 0$. Then (1.1)-(1.2) has at least $|i_{\sqrt{-1}}^{L_0}(B_1)-i_{\sqrt{-1}}^{L_0}(B_\infty)|$ pairs of nontrivial 1-periodic symmetric brake orbits.

Organization: In Section 2, we will briefly introduce the Maslov-type index and its iteration theory for symplectic path under brake orbit boundary value. Based on this index theory we give he proof of Theorem 1.1. As application, in Section 3, we give he proof of Theorem 1.2.

Throughout this paper, let ${{\bf N}}$, ${{{\bf Z}}}$, ${{{\bf R}}}$, ${{{\bf C}}}$ and ${\bf U}$ denote the set of natural integers, integers, rational numbers, real numbers, complex numbers and the unit circle in ${{{\bf C}}}$, respectively.

2.   Maslov-type index theory and Relative Morse index

In this section we will prove Theorem 1.1. We first simply recall the Maslov-type index and its iteration theory for brake orbits.

As we know, in 1984, Conley and Zehnder in their celebrated paper ^[10] introduced an index theory for the non-degenerate symplectic paths in the real symplectic matrix group ${{{\rm{Sp}}}}(2n)$ for $n\geq2$. Since then, there are tremendous works about this kind of index theory developed or generalized in various directions. In 2006, combined with the Maslov index formulated in ^[11], Long, Zhang and Zhu developed an index theory called $\mu$-index in ^[12] and obtained an result on the existence of multiple brake orbits. The difference of the Maslove-type $\mu$-index in ^[12] and the Maslov-type L-index in that paper is constant n (half of the dimension). In ^[13], Liu and Zhang established the Bott-type iteration formulas and some precise iteration formula of the L-index theory and proved the multiplicity of brake orbits on every $C^2$ compact convex symmetric hypersurface in ${{\bf R}}^{2n}$.

Set

where we omit $T$ from the notation of $\mathcal P_T$ if $[0, T]$ is replaced by $[0, +\infty)$. Let $J$ be the standard almost complex in $({{\bf R}}^{2n}, \omega_0)$ and $J$ is a compatible with $\omega_0$, i.e.,

A $n$-dimensional subspace $\Lambda\subseteq {{{\bf R}}}^{2n}$ is called a Lagrangian subspace if $\omega_0(x, y) = 0$, for any $x, y\in\Lambda$. Let $F = {{{\bf R}}}^{2n}\bigoplus{{{\bf R}}}^{2n}$ be equipped with symplectic form $(-\omega_0)\bigoplus\omega_0$. Then $\mathcal{J} = (-J)\bigoplus J$ is an almost complex structure on $F$ and $\mathcal{J}$ is compatible with $(-\omega_0)\bigoplus\omega_0$. Denote by $\mathcal {L}ag(F)$ the set of Lagrangian subspaces of $F$. Then for any $M\in {\rm{Sp}} (2n)$, its graph

Denote by $L_0 = \{0\} \times{{{\bf R}}}^n$ and $L_1 = {{{\bf R}}}^n \times \{0\}$ the two fixed Lagrangian subspaces of ${{{\bf R}}}^{2n}$ and let

Then both $V_j, Gr(M)|_{V_j}\in \mathcal{L}ag(F)$ for $M\in {\rm Sp}(2n)$ and $j = 0, 1$.

Denote by $\mu_F^{CLM}(V, W, [a, b])$ the Maslov-type index for (ordered) pair of paths of Lagrangian subspaces $(V, W)$ in $F$ on $[a, b]$, which is defined by Cappel, Lee and Miller in ^[11].

Definition 2.1. (cf. ^[6,12,13]) For $\gamma\in\mathcal{P}_\tau(2n)$, define

For $j = 0, 1$, we define $(i_{L_j}(\gamma), \nu_{L_j}(\gamma)) = (i_{\omega}^{L_j}(\gamma), \nu_{\omega}^{L_j}(\gamma))$ if $\omega = 1$. Note that, for any continuous path $\Psi\in{{\cal P}}_\tau$, the following Maslov-type indices of $\Psi$ is defined by (cf ^[1,12])

When there is no confusion we will omit the intervals in the above definitions. Hence we have

For $B\in \mathcal{C}(S_T, \mathcal L_s{({{\bf R}}}^{2n}))$, the fundamental solution $\gamma_B$ of the linear Hamiltonian system

satisfies $\gamma_B\in \mathcal P_T(2n)$, and is called the associated symplectic path of $B$. For $\omega\in{\bf U}$, we define the Maslov-type indices of $B$ via the restriction $\gamma_B|_{[0, T/2]}\in \mathcal P_{T/2}(2n)$ :

In 1956, Bott in ^[14] established the famous iteration formula of the Morse index for closed geodesics on Riemannian manifolds. For convex Hamiltonian systems, Ekeland developed the similar Bott-type iteration index formulas for the Ekeland index theory (cf. ^[15] of 1990). In 1999 (cf. ^[16]), Long established the Bott-type iteration formulas for the Maslov-type index theory. Motivated by the above results, in ^[13] of Liu and Zhang in 2014, the following Bott-type iteration formulas for the $L_0$-index was established.

Definition 2.2. (cf. ^[13]) Given an $\tau > 0$, a positive integer $k$ and a path $\gamma\in \mathcal{P}_\tau(2n)$, the $k$-th iteration $\gamma^k$ of $\gamma$ in brake orbit boundary sense is defined by $\tilde\gamma|_{[0, k\tau]}$ with

where $\gamma(2\tau): = N\gamma(\tau)^{-1}N\gamma(\tau)$. $

Theorem 2.1. (cf. ^[13] of Liu and Zhang in 2014) Suppose $\gamma\in\mathcal{P}_{\tau}(2n)$, for the iteration symplectic paths $\gamma^k$, when $k$ is odd, there hold

when $k$ is even, there hold

where $\omega_k = e^{\pi\sqrt{-1}/k}$, and $(i_{\omega}, \nu_{\omega})$ is the $\omega$-index pair defined by Long(cf. ^[16]).

Proof of Theorem 1.1. For any $B\in \mathcal{L}_s({{\bf R}}^{2n})$ satisfying condition (BS1), as in ^[1], we define

Equip $E$ with the usual $W^{1/2, 2}$ norm. Then $E$ is a Hilbert space with the associated inner product $\langle, \rangle$. We define two self adjoint liner operators $A$, $B$ from $E$ to $E$ by

We also define

Equip $\hat{E}$ with the usual $W^{1/2, 2}$ norm. We define two self adjoint liner operators $\hat{A}$, $\hat{B}$ from $\hat{E}$ to $\hat{E}$ by

Then $\tilde{{E}}$ and $\hat{E}$ are both subspaces of $E$ and $A$ invariant, we have both the $A$ orthogonal and $B$ orthogonal decomposition

Since $B$ satisfies condition (BS1), one can verify the following orthogonal decomposition

Then we have the orthogonal decomposition

where $M^*(A-B)\subset E$, $M^*(\tilde{{A}}-\tilde{{B}})\subset \tilde{{E}}$, $M^*(\hat{A}-\hat{B})\subset \hat{E}$. So by the definitions of $I(B_1, B_2)$, $I(\tilde{{B}}_1, \tilde{{B}}_2)$, $I(\hat{B}_1, \hat{B}_2)$ we have

where $I(B_1, B_2)$ and $I(\hat{B}_1, \hat{B}_2)$ are defined similarly as (1.3).

Since $B$ satisfies condition (BS1), one has $B\in C(S^{1/2}, \mathcal{L}_s({{\bf R}}^{2n}))$. Thus

So by Theorem 1.2 of ^[1] and (2.8) we have

Also by Theorem 1.2 of ^[1] and (2.8) we have

Since both $B_1$ and $B_2$ satisfy condition (BS1), for $j = 1, 2$, by Theorem 2.3 we have

By (2.11)–(2.15) one has

Thus Theorem 1.1 holds by the definitions of $(i_{\sqrt{-1}}^{L_0}({{\gamma}}_{B}), \nu_{\sqrt{-1}}^{L_0}({{\gamma}}_{B}))$ in (1.3).

3.   Proof of Theorem 1.2.

In this section we prove Theorem 1, 2.

We study the 1-periodic brake orbit solution of Hamiltonian system (1.1)-(1.2)

It is well know that $x$ is a solution of (1.1)-(1.2) if and only if it is a critical point of the functional $f$ defined on $\tilde{{E}}$ as follows

where $\tilde{{E}}$ is defined by (1.5), $\tilde{{A}}$ is defined in (1.6), $\tilde{{\Phi}}(x) = \int_0^1-H(t, x)dt$. It is easy to check that $\tilde{{\Phi}}'(x)$ is compact.

In ^[17], Benci proved the following important abstract theorem:

Theorem 3.1. Let $f\in C^1(E, {{\bf R}})$ have the form (3.1) and satisfy

$({\rm{f1}})$ Every sequence $\{u_j\}$ such that $f(u_j)\rightarrow c < \tilde{{\Phi}}(0)$ and $||f'(u_j)||\rightarrow 0$ as $j\rightarrow +\infty$ is bounded.

$({\rm{f2}})$ $\tilde{{\Phi}}(u) = \tilde{{\Phi}}(-u)$, $u\in \tilde{{E}}$.

$({\rm{f3}})$ There are two closed subspaces of $\tilde{{E}}$, $E^+$ and $E^-$, and a constant $\rho > 0$ such that

$({\rm{a}})$ $f(u) > 0$ for $u\in E^+$, where $c_0 < c_\infty < \tilde{{\Phi}}(0)$ be two constants.

$({\rm{b}})$ $f(u) < c_\infty < \tilde{{\Phi}}(0)$ for $u\in E^-\cap S_\rho$, $(S_\rho = \{u\in E|\; ||u|| = \rho\}).$

Then the number of pairs of nontrivial critical points of $f$ is greater than or equalto $\dim(E^+\cap E^-)-{\rm cod}(E^-+E^+)$. More over, the corresponding critical values belong toto $[c_0, c_\infty]$.

Proof of Theorem 1.2. We take the method in ^[1,5] to prove this theorem.

We set $\tilde{{E}}^+ = M^+(\tilde{{A}}-\tilde{{B}}_\infty)$ and $\tilde{{E}}^- = M^-(\tilde{{A}}-B_0)$. By Definition 1.1 and Theorem 1.2, we have

Here ${\tilde{{B}}}_0$ and ${\tilde{{B}}}_\infty$ are compact operators from $\tilde{{E}}$ to $\tilde{{E}}$ defined by (1.6). Since $0$ is an isolated eigenvalue of $\tilde{{A}}$ with $n$-dimensional eigenspace $\tilde{{E}}_0$, by (4-4$'$) of ^[17], there exist two real numbers $\alpha < 0$ and $\beta > 0$ such that

Define

and let $g_\infty(x) = \int_0^1 V_\infty (t, x)dt$ and $g_0(x) = \int_0^1 V_0 (t, x)dt$, then we have

By (H1)-(H2) and the same arguments in the proof of Lemma 5.5 of ^[17], we get

So by definition of $g_0$ and (3.9), we have

By (3.3) and (3.10) we have

Since $\alpha < 0$, there exist a constant $\rho > 0$ and ${{\gamma}}_1 < 0$ such that

Setting $c_\infty = \frac{{{\gamma}}_1}{2}+\tilde{{\Phi}}(0)$, (f3)(b) of Theorem 3.1 is satisfied.

By (H2) for there exist $M > 0$ such that

Thus

Then by (3.4) and (3.14), for every $u\in \tilde{{E}}^+$, we get

This implies that $f$ is bounded from below on $\tilde{{E}}^+$ and we can set

Thus (f3)(a) of Theorem 3.1 is satisfied.

Since $\nu_1({\tilde{{B}}}_\infty) = 0$, $M^0(\tilde{{A}}-{\tilde{{B}}}_\infty) = 0$. Now we prove that (f1) is satisfied. other wise we can suppose $||u_j||\to +\infty$ as $j\to +\infty$, then by (3.6) and (3.8) we have

But by (4-4$'$) of ^[17] there exists a real number $\alpha' > 0$ such that

Hence by (3.17) we have

which contradicts (3.16). This proves (f1) in Theorem 3.1.

(H3) implies (f2) of Theorem 3.1 holds. Hence by Theorem 3.1, (1.1)-(1.2) has at least $i_{\sqrt{-1}}^{L_0}({\tilde{{B}}}_0)-i_{\sqrt{-1}}^{L_0}({\tilde{{B}}}_\infty)$ pairs of nontrivial solutions whenever $i_{\sqrt{-1}}^{L_0}({\tilde{{B}}}_0)-i_{\sqrt{-1}}^{L_0}({\tilde{{B}}}_\infty) > 0$. If $i_{\sqrt{-1}}^{L_0}({\tilde{{B}}}_0)-i_{\sqrt{-1}}^{L_0}({\tilde{{B}}}_\infty) > 0$, we replace $f$ by $-f$ and let $E^+ = M^-(\tilde{{A}}-{\tilde{{B}}}_\infty)$ and $E^- = M^+(\tilde{{A}}-{\tilde{{B}}}_0)$. By almost the same proof we can show that (f1)-(f3) of Theorem 3.1 hold. And by Theorems 1.2 and 3.1 (1.1)-(1.2) has at least $i_{\sqrt{-1}}^{L_0}({\tilde{{B}}}_\infty)-i_{\sqrt{-1}}^{L_0}({\tilde{{B}}}_0)$ pairs of nontrivial brake orbit solution. The proof of Theorem 1.3 is completed.

Similarly to Theorems 1.4 and 1.5 of ^[5] or ^[1]), we have

Remark 3.1. If $\nu_{\sqrt{-1}}^{L_0}({\tilde{{B}}}_\infty) > 0$, we can prove (f1) of Theorem 3.1 under other additional conditions while we can prove (f2) and (f3) are satisfied under (H1)-(H3) by the same proof of Theorem 1.3.

Suppose the following condition:

(H4) $V_\infty '(t, x)$ is bounded and $V(t, x)\to +\infty$ as $|x|\to +\infty$, uniformly in $t$.

By the proof of Theorem 5.2 of ^[17] and Theorem 4.1 of ^[18] (f1) holds.

Suppose the following conditions:

(H5) There is $r > 0$ and $p\in (1, 2)$ such that

(H6) $\overline{\lim}_{|x|\to \infty}|x|^{-1}|V_\infty'(t, x)|\le c < \frac{1}{2}.$

(H7) There are constant $a_1 > 0$ and $a_2 > 0$ such that $V_\infty(t, x)\ge a|z|^p-a_2$.

By the proof of Theorem 4.11 of ^[18] (f1) holds.

Then under either additional condition (H4) or (H5)-(H7), (1.1)-(1.2) has at least $i_{\sqrt{-1}}^{L_0}({\tilde{{B}}}_0)-i_{\sqrt{-1}}^{L_0}({\tilde{{B}}}_\infty)-\nu_{\sqrt{-1}}^{L_0}({\tilde{{B}}}_\infty)$ pairs of nontrivial solutions whenever $i_{\sqrt{-1}}^{L_0}({\tilde{{B}}}_0)-i_{\sqrt{-1}}^{L_0}({\tilde{{B}}}_\infty)-\nu_{\sqrt{-1}}^{L_0}({\tilde{{B}}}_\infty) > 0$.

Acknowledgments

This work is partially supported by the NSFC Grants 11790271 and 11171341, National Key R & D Program of China 2020YFA0713301, and LPMC of Nankai University. The authors sincerely thanks the referees for their careful reading and valuable comments and suggestions.

Conflict of interest

The authors declare there is no conflict of interest.