Prescribed-time cluster practical consensus for nonlinear multi-agent systems based on event-triggered mechanism

1.   Introduction and main result

In this paper, we are interested in the existence and some asymptotic properties of ground state homoclinic orbits for the following first-order Hamiltonian system

with the Hamiltonian function

Here, $z = (u, v)\in \mathbb{R}^{N}\times\mathbb{R}^{N} = \mathbb{R}^{2N}$, $\epsilon > 0$ is a parameter, $L$ is a symmetric $2N\times2N$ matrix-valued function, and

is the usual symplectic matrix with $I$ being the identity matrix in $\mathbb{R}^N$. As usual, we refer to a solution $z$ of system (1.1) as a homoclinic orbit if $z(t)\not\equiv 0$ and $z(t)\rightarrow 0$ as $|t|\rightarrow \infty$.

It is widely known that Hamiltonian systems are very important dynamical systems, which have many applications in several natural science areas such as relativistic mechanics, celestial mechanics, gas dynamics, chemical kinetics, optimization and control theory and so on. The complicated dynamical behavior of Hamiltonian systems has the attracted attention of many mathematicians and physicists ever since Newton wrote down the differential equations describing planetary motions and derived Kepler's ellipses as solutions. For more details on applications of Hamiltonian systems, one can refer to ^[1] and the monograph ^[2] of Mawhin and Willem. We also refer the readers to see ^[3] for the Stokes-Dirac structures of the port-Hamiltonian systems.

In the past few decades, Hamiltonian systems have attracted a considerable amount of interest due to many powerful applications in different fields; the literature related to these systems is extensive and encompasses several interesting lines of research on the topic of nonlinear analysis, including the existence, nonexistence, multiplicity and finer qualitative properties of homoclinic orbits. Here we cannot provide a complete and fully detailed list of references, but rather we limit ourselves to mentioning the works which are closely related to the content of the present paper.

A major breakthrough was the pioneering paper of Rabinowitz ^[4] from 1978 who, for the first time, obtained periodic solutions of system (1.1) by using variational methods. After the celebrated work of Rabinowitz ^[4], based on the dual action and mountain pass argument, Coti-Zelati et al. ^[5] obtained the existence and multiplicity of homoclinic orbits under the condition of strictly convexity. Later on, this result was further detailed by ^[6,7] in which the authors established the existence result for infinitely many homoclinic orbits. Without the convexity condition, Hofer and Wysocki ^[8] independently investigated the existence of homoclinic orbits by combining the Fredholm operator theory and the linking argument. Tanaka ^[9] employed a suitable subharmonic approach to obtain one homoclinic orbit by relaxing the convexity condition. In ^[10], Rashkovskiy studied the quantization process of Hamiltonian and non-Hamiltonian systems.

The main unusual feature of the first-order Hamiltonian system is that the associated energy functional is strongly indefinite. Generally speaking, for the strongly indefinite functionals refined variational methods like the Nehari manifold method and mountain pass theorem still do not apply. Some general critical point theories like the generalized linking theorem and other weaker versions for strongly indefinite functionals were subsequently developed by Kryszewski and Szulkin in ^[11] and Bartsch and Ding in ^[12]. Since then, based on the critical point theorems from ^[11,12] for strongly indefinite functionals, many scholars have gradually begun to investigate the existence and multiplicity of homoclinic orbits for non-autonomous Hamiltonian systems under some different conditions. More precisely, under the conditions that $H$ depends periodically on $t$ and has super-quadratic growth in $z$, Arioli and Szulkin ^[13], Chen and Ma ^[14], and Ding and Willem ^[15] obtained the existence result. Ding ^[16], Ding and Girardi ^[17], and Zhang et al. ^[18] studied the multiplicity result for homoclinic orbits. Concerning the asymptotic quadratic growth case, we refer to the work done by Szulkin and Zou ^[19] and Sun et al. ^[20]. Here we would like to emphasize that the periodicity condition is used to resolve the issue stemming from the lack of compactness since system (1.1) is set on the whole space $\mathbb{R}$.

On the other hand, without the condition of periodicity, the non-periodic problem is quite different due to the lack of compactness of Sobolev embeddings. In an early paper ^[21], Ding and Li utilized the coercive property of $L$ to establish a variational framework with compactness, and they proved the existence of homoclinic orbits for the super-quadratic growth case. Also under the framework of compactness, Zhang and Liu ^[22] studied the sub-quadratic growth case. Regarding the asymptotically quadratic case, based on the infinite-dimensional linking argument, Ding and Jeanjean ^[23] established a multiplicity result for homoclinic orbits. Moreover, they imposed a control on the size of $G$ with respect to the behavior of $L$ to recover sufficient compactness. For the existence and exponential decay of homoclinic orbits for system (1.1) with nonperiodic super-quadratic and lack of compactness, we refer the reader to ^[24]. We also mention the recent paper by Zhang et al. ^[25] in which the existence and decay of ground state homoclinic orbits for system (1.1) with asymptotic periodicity are explored. For other results related to the Hamiltonian systems with strongly variational structure, we refer the reader to ^{[26,27,28,29,30,31,32]} and the references therein.

It is worth pointing out that, in all of the works mentioned above, the authors were concerned mainly with the study of the existence and multiplicity of homoclinic orbits, and there are no papers considering the asymptotic properties of homoclinic orbits. Inspired by this fact and the work done by Alves and Germano ^[33] in which the authors investigated the existence and concentration of ground state solutions for the Schrödinger equation; in the present paper we aim to further study the existence and some asymptotic properties of ground-state homoclinic orbits for system (1.1) with Hamiltonian function (1.2). This is a very interesting issue that has motivated the present work.

To continue the discussion, we introduce the following notation

then, system (1.1) takes the following form

Before stating our results, we suppose the following conditions hold for $L$, $A$ and $G$.

($L$) $L$ is a constant symmetric $2N\times 2N$ matrix such that $\sigma(\mathscr{J}L)\cap i \mathbb{R} = \varnothing$, where $\sigma$ denotes the spectrum of operator $\mathscr{J}L$.

($A$) $A\in C(\mathbb{R}, \mathbb{R})$ and $0 < \inf_{ t\in {\mathbb{R}}}A(t)\leq A_{\infty} : = \lim\limits_{|t|\rightarrow \infty}A(t) < A(0) = \max\limits_{ t\in {\mathbb{R}}}A(t)$;

$(g_{1})$ $G_{z}(|z|) = g(|z|)z$, $g\in C(\mathbb{R}^{+}, \mathbb{R}^{+})$, and there exist $p > 2$ and $c_{0} > 0$ such that

$(g_{2})$ $g(s) = o(1)$ as $s\rightarrow0$, and $G(s)/s^{2}\to+\infty$ as $s\rightarrow +\infty$;

$(g_{3})$ $g(s)$ is strictly increasing in $s$ on $(0, +\infty)$.

Next we state the main result of this paper as follows.

Theorem 1.1. Assume that conditions $(L)$, $(A)$, and $(g_{1})$-$(g_{3})$ hold. Then we have the following results:

$\rm{(a)}$ there exists $\epsilon_{0} > 0$ such that system (1.1) has a ground state homoclinic orbit $z_{\epsilon}$ for each $\epsilon\in (0, \epsilon_{0})$;

$\rm{(b)}$ $|z_{\epsilon}|$ attains its maximum at $t_{\epsilon}$, then,

moreover, $z_{\epsilon}(t+t_{\epsilon})\to z$ as $\epsilon\to 0$, where $z$ is a ground state homoclinic orbit of the limit system

$\rm{(c)}$ additionally, if $A, g\in C^{1}$, and $g'(s)s = o(1)$ as $s\to0$, then there exist constants $c, C > 0$ such that

We would like to emphasize that since our problem is carried out in the whole space, then the strongly indefiniteness of energy functionals and the lack of compactness are two major difficulties that we encounter in order to guarantee the existence of homoclinic orbits. More precisely, one reason is that strongly indefinite functionals are unbounded from below and from above so that the classical methods from the calculus of variations do not apply. The other reason is that the lack of compactness leads to the energy functionals not satisfying the necessary compactness property.

Let us now outline the methods involved to prove Theorem 1.1. Indeed, based on the above reasons, first, we will take advantage of the method of the generalized Nehari manifold developed by Szulkin-Weth ^[34] to handle system (1.1), this is because such a strategy helps to overcome the difficulty caused by strongly indefinite features. Second, we must verify that the energy functional possesses the necessary compactness property at some energy level. This target will be accomplished by applying the energy comparison argument to establish some precise comparison relationships for the ground-state energy value between the original problem and certain auxiliary problems. Finally, combining the compactness analysis technique, Kato's inequality, and the sub-solution estimate, we can obtain the concentration property and decay of homoclinic orbits. Then Theorem 1.1 follows naturally.

This paper is organized as follows. In Section 2, we establish the functional analytic setting associated with system (1.1). In Section 3, we present some technical results, and obtain the existence result for ground-state homoclinic orbits for the autonomous system. Section 4 is devoted to proofs of Theorem 1.1.

2.   Functional analytic setting

Throughout the present paper, we will use the following notations:

● $\|\cdot\|_{s}$ denotes the norm of the Lebesgue space $L^{s}(\mathbb{R})$, $1\leq s\leq +\infty$;

● $(\cdot, \cdot)_{2}$ denotes the usual inner product of the space $L^{2}(\mathbb{R})$;

● $c$, $c_i$, $C_{i}$ represent various different positive constants.

In what follows, we will establish the variational framework to work for system (1.1).

Recall that $S = -(\mathscr{J}\frac{d}{dt}+L)$ is a self-adjoint operator on the space $L^{2}: = L^{2}(\mathbb{R}, \mathbb{R}^{2N})$ with the domain $\mathcal{D}(S) = H^{1}(\mathbb{R}, \mathbb{R}^{2N})$; according to the discussion in ^[13], we can know that, under the condition $(L)$, there exists $a > 0$ such that $(-a, a)\cap \sigma(S) = \emptyset$ (see also ^[16,19]). Therefore, the space $L^{2}$ has the following orthogonal decomposition

corresponding to the spectrum decomposition of $S$ such that $S$ is positive definite (resp. negative definite) in $L^{+}$ (resp. $L^{-}$).

We use $|S|$ to denote the absolute value of $S$, and $|S|^{1/2}$ denotes the square root of $|S|$. Let $E = \mathscr{D}(|S|^{1/2})$ be the domain of the self-adjoint operator $|S|^{1/2}$, which is a Hilbert space equipped with the following inner product

and the norm $\|z\|^{2} = (z, z)$. Evidently, $E$ possesses the following decomposition

which is orthogonal with respect to the two inner products $(\cdot, \cdot)_{2}$ and $(\cdot, \cdot)$. Moreover, by using the polar decomposition of $S$ we can obtain that

Furthermore, from ^[16] we have the embedding theorem, that is, $E$ embeds continuously into $L^{q}$ for each $q\geq 2$ and compactly into $L^{q}_{loc}$ for all $q\geq1$. Hence, there exists a constant $\gamma_{q} > 0$ such that for all $z\in E$

From the assumptions $(g_{1})$ and $(g_{2})$, we can deduce that for any $\epsilon > 0$, there exists a positive constant $C_{\epsilon}$ such that

Next, we define the corresponding energy functional of system (1.3) on $E$ by

Applying the polar decomposition of $S$, then the energy functional $I_{\epsilon}$ has another representation as follows

Evidently, according to condition $(L)$, we can see that $I_{\epsilon}$ is strongly indefinite. Furthermore, from conditions $(g_{1})$ and $(g_{2})$ we can infer that $I_{\epsilon}\in C^{1}(E, \mathbb{R})$, and we have

Making use of a standard argument we can check that critical points of $I_{\epsilon}$ are homoclinic orbits of system (1.1).

3.   The autonomous system

We shall make use of the techniques of the limit problem to prove the main results; in this section we introduce some related results for the autonomous system.

For any constant $\mu > 0$, in what follows we consider the autonomous system given by

Similarly, following the above comments, we define the energy functional $I_{\mu}$ corresponding to system (3.1) as follows

Evidently, we have

In order to obtain the ground state homoclinic orbits of system (3.1), we will use the method of the generalized Nehari manifold developed by Szulkin and Weth ^[34]. To do this, we first introduce the following generalized Nehari manifold

and we define the ground state energy value $c_{\mu}$ of $I_{\mu}$ on $\mathscr{N}_{\mu}$

Furthermore, for every $z\in E\backslash E^{-}$, we also need to define the subspace

and the convex subset

We note that $E(z)$ and $\widehat{E}(z)$ do not depend on $\mu$, but depend on the operator $S$.

The main result in this section is the following theorem:

Theorem 3.1. Assume that condition $(L)$ holds and let $(g_{1})$-$(g_{3})$ be satisfied, then, problem (3.1) has at least a ground state homoclinic orbit $z\in\mathscr{N}_{\mu}$ such that $I_{\mu}(z) = c_{\mu} > 0$.

3.1.   Technical results

In this subsection, we are going to prove some technical results which will be used in the proof of Theorem 3.1. The following result involves the translation that will be used frequently in this paper, the proof can be found in [33, Lemma 2.1].

Lemma 3.1. For all $u = u^{+}+u^{-}\in E$ and $y\in\mathbb{R}$, if $v(t): = u(t+y)$, then $v\in E$ with $v^{+}(t) = u^{+}(t+y)$ and $v^{-}(t) = u^{-}(t+y)$.

We give an important estimate, which plays a crucial role in the later proof.

Lemma 3.2. Let $z\in \mathscr{N}_{\mu}$, then, for each $v\in \mathcal{H}: = \{sz+w: s\geq -1, w\in E^{-}\}$ and $v\neq 0$, we have the following energy estimate

Hence $z$ is a unique global maximum of $I_{\mu}|_{\widehat{E}(z)}$.

Proof. We follow the similar ideas explored in [34, Proposition 2.3.] to prove this lemma. Observe that, for any $z\in \mathscr{N}_{\mu}$, we directly obtain

Let $v = sz+w\in\mathcal{H}$, then, $z+v = (1+s)z+w\in \widehat{E}(z)$. By an elemental computation, we can get

where

Using $(g_2)$ and $(g_3)$ and combining the arguments used in ^[35] (see also ^[36]), we can conclude that $\widetilde{g}(s, z, v) < 0$. Therefore, we have

Evidently, we know that $z$ is a unique global maximum of $I_{\mu}|_{\widehat{E}(z)}$.

Lemma 3.3. Assume that $(g_{1})$ and $(g_{2})$ are satisfied, then, we have the following two conclusions:

$\rm{(i)}$ there exists $\rho > 0$ such that $c_{\mu} = \inf_{\mathscr{N}_{\mu}}I_{\mu}\geq \inf_{S_{\rho}}I_{\mu} > 0$, where

$\rm{(ii)}$ for any $z\in \mathscr{N}_{\mu}$, then $\|z^{+}\|^{2}\geq \max\{\|z^{-}\|^{2}, 2c_{\mu}\} > 0$.

Proof. (ⅰ) Let $z\in E^{+}$, we can deduce from (2.1) and (2.2) that

Evidently, we can see that there is $\rho > 0$, for $\|z\| = \rho$ small enough such that $\inf_{S_{\rho}}I_{\mu} > 0$.

On the other hand, for each $z\in \mathscr{N}_{\mu}$, there exists a positive constant $s$ such that $s\|z\| = \rho$, then $sz\in \widehat{E}(z)\cap S_{\rho}$. From Lemma 3.2, one can derive that

Therefore, we have

which shows that the conclusion (ⅰ) holds.

(ⅱ) First we note that, from $(g_{3})$, it follows that

For each $z\in \mathscr{N}_{\mu}$, combining this with the definition of $c_{\mu}$, we have

Hence, we can derive that $\|z^{+}\|^{2}\geq \max\{\|z^{-}\|^{2}, 2c_{\mu}\} > 0$. The proof is finished.

Lemma 3.4. Assume that $\Omega\subset E^{+}\setminus \{0\}$ is a compact subset, thus, there exists $R > 0$ such that $I_{\mu} < 0$ on $E(z)\setminus B_{R}(0)$ for all $z\in \Omega$.

Proof. The proof follows as in [34, Lemma 2.5], here, we omit the details.

Following the result in ^[34] (see [34, Lemma 2.6]), we can establish the uniqueness of maximum point of $I_{\mu}$ on the set $\widehat{E}(z)$.

Lemma 3.5. For any $z\in E\backslash E^{-}$, then the set $\mathscr{N}_{\mu}\cap \widehat{E}(z)$ consists of precisely one point $\widetilde{m}_{\mu}(z)\neq0$, which is the unique global maximum of $I_{\mu}|_{\widehat{E}(z)}$.

Proof. On account of Lemma 3.2, it is sufficient to prove that $\mathscr{N}_{\mu}\cap \widehat{E}(z)\neq\emptyset$. Since $\widehat{E}(z) = \widehat{E}(z^{+})$, without loss of generality, we can suppose that $z\in E^{+}$ and $\|z\| = 1$. By Lemma 3.3-(ⅰ), we obtain that $I_{\mu}(sz) > 0$ for $s\in (0, +\infty)$ small enough. Lemma 3.4 yields that $I_{\mu}(sz) < 0$ for $sz\in \widehat{E}(z)\backslash B_{R}(0)$. Consequently, we can deduce that $0 < \sup I_{\mu}(\widehat{E}(z)) < \infty$. Because $\widehat{E}(z)$ is convex and the functional $I_{\mu}$ is weakly supper semi-continuous on $\widehat{E}(z)$, we conclude that there exists $\widehat{z}\in \widehat{E}(z)$ such that $I_{\mu}(\widehat{z}) = \sup I_{\mu}(\widehat{E}(z))$. This shows that $\widehat{z}$ is a critical point of $I_{\mu}|_{\widehat{E}(z)}$; therefore,

hence, $\widehat{z}\in \mathscr{N}_{\mu}$. So, $\widehat{z}\in \mathscr{N}_{\mu}\cap \widehat{E}(z)$. The proof is finished.

Combining Lemma 3.2 with Lemma 3.5, we obtain the following conclusion.

Lemma 3.6. For each $z\in E\backslash E^{-}$, then, there is a unique pair $(s^{\ast}, \varphi^{\ast})$ with $s^{\ast}\in (0, +\infty)$ and $\varphi^{\ast}\in E^{-}$ such that $s^{\ast}z+\varphi^{\ast}\in \mathscr{N}_{\mu}\cap \widehat{E}(z)$ and

Moreover, if $z\in\mathscr{N}_{\mu}$, then we have that $s^{*} = 1$ and $\varphi^{\ast} = z^{-}$.

Lemma 3.7. $I_{\mu}$ is coercive on $\mathscr{N}_{\mu}$, that is, $I_{\mu}(z)\to+\infty$ as $\|z\|\to+\infty$, $z\in\mathscr{N}_{\mu}$.

Proof. Seeking for a contradiction, assume that there exists $\{z_{n}\}\subset \mathscr{N}_{\mu}$ such that

Setting $w_{n}: = z_{n}/\|z_{n}\|$, we obtain that $\|z^{+}_{n}\|\geq \|z^{-}_{n}\|$ from Lemma 3.3(ⅱ), then, for every $n\in \mathbb{N}$, we get that $\|w^{+}_{n}\|^{2}\geq \|w^{-}_{n}\|^{2}$ and $\|w^{+}_{n}\|^{2}\geq \frac{1}{2}$. There exist $\{y_{n}\}\subset \mathbb{Z}$, $r > 0$ and $\delta > 0$ such that

If this is not true, then according to Lions' concentration-compactness principle, we can conclude that $w^{+}_{n}\rightarrow 0$ in $L^{q}(\mathbb{R})$ for $q > 2$. Combining (2.1) and (2.2), we know that, for every $s > 0$,

then we get

This yields a contradiction if $s > \sqrt{4\widehat{c}}$; hence we prove that (3.2) holds.

Let us define $\tilde{z}_{n}(t): = z_{n}(t+y_{n})$, and $\tilde{w}_{n}(t): = w_{n}(t+y_{n})$, then, $\tilde{w}_{n}^{+}\rightharpoonup \tilde{w}^{+}$, and (3.2) yields that $\tilde{w}^{+}\neq 0$. Since $\tilde{z}_{n}(t) = \tilde{w}_{n}(t)\|\tilde{z}_{n}\|$, it follows that $\tilde{z}_{n}(t)\rightarrow +\infty$ almost everywhere in $\mathbb{R}$ as $\|\tilde{z}_{n}\| = \|z_{n}\|\rightarrow +\infty$. Applying the Fatou's lemma, we can derive that

where $[\tilde{z}_{n}\neq 0]$ is the Lebesgue measure of the set $\{t\in \mathbb{R}: \tilde{z}_{n}(t)\neq 0\}$. Therefore

we get a contradiction. The proof is finished.

We want to utilize the method of the generalized Nehari manifold to prove the main result. To do this, we set $S^{+}: = \{z\in E^{+}: \|z\| = 1\}$ in $E^{+}$, and we define the following mapping

and the inverse of $m_{\mu}$ is

Following from the proof of [34, Lemma 2.8], we can see that $\widetilde{m}_{\mu}$ is continuous and $m_{\mu}$ is a homeomorphism.

We now consider the reduced functionals

which is continuous since $\widetilde{m}_{\mu}$ is continuous.

The following results establish some crucial properties involving the reduced functionals $\widetilde{\Phi}_{\mu}$ and $\Phi_{\mu}$, which play important roles in our arguments. And their proofs follow the proofs of [34, Proposition 2.9, Corollary 2.10].

Lemma 3.8. The following conclusions are true:

$\rm{(a)}$ $\widetilde{\Phi}_{\mu}\in C^{1}(E^{+}\backslash\{0\}, \mathbb{R})$ and for $z, v\in E^{+}$ and $z\neq0$,

$\rm{(b)}$ $\Phi_{\mu}\in C^{1}(S^{+}, \mathbb{R})$ and for each $z\in S^{+}$ and $v\in T_{z}(S^{+}) = \{u\in E^{+}:(z, u) = 0\}$,

$\rm{(c)}$ $\{z_n\}$ is a (PS)-sequence for $\Phi_{\mu}$ if and only if $\{\widetilde{m}_{\mu}(z_n)\}$ is a (PS)-sequence for $I_{\mu}$.

$\rm{(d)}$ We have

Moreover, $z\in S^{+}$ is a critical point of $\Phi_{\mu}$ if and only if $\widetilde{m}_{\mu}(z)\in \mathscr{N}_{\mu}$ is a critical point of $I_{\mu}$ and the corresponding critical values coincide.

3.2.   Proof of Theorem 3.1

Based on the above preliminaries, in this subsection we give the complete proof of Theorem 3.1, and further study the monotonicity and continuity of the ground-state energy $c_{\mu}$.

Proof of Theorem 3.1: According to Lemma 3.3, it is easy to see that $c_{\mu} > 0$. We note that, if $z\in \mathscr{N}_{\mu}$ with $I_{\mu}(z) = c_{\mu}$, then $m^{-1}_{\mu}(z)\in S^{+}$ is a minimizer of $\Phi_{\mu}$; hence, it is a critical point of $\Phi_{\mu}$. Consequently, Lemma 3.8 yields that $z$ is a critical point of $I_{\mu}$. We have to prove that there exists a minimizer $\tilde{z}\in \mathscr{N}_{\mu}$ such that $I_{\mu}(\tilde{z}) = c_{\mu}$. Actually, Ekeland's variational principle yields that there exists a sequence $\{v_{n}\}\subset S^{+}$ such that $\Phi_{\mu}(v_n)\to c_{\mu}$ and $\Phi'_{\mu}(v_n)\to 0$ as $n\to\infty$. For all $n\in\mathbb{N}$, setting $z_n = \widetilde{m}_{\mu}(v_n)\in \mathscr{N}_{\mu}$, then $I_{\mu}(z_n)\to c_{\mu}$ and $I'_{\mu}(z_n)\to 0$ by Lemma 3.8. By virtue of Lemma 3.7, we can see that $\{z_n\}$ is bounded in $E$. Moreover, it satisfies

If this is not true, then Lions' concentration-compactness principle implies that $z_n\to 0$ in $L^{q}(\mathbb{R})$ for any $q > 2$. Therefore, from (2.1) and (2.2), we can derive that

Then, we get

Since $c_{\mu} > 0$, obviously we get a contradiction. Thus, there exist $\{y_n\}\subset\mathbb{Z}$ and $\delta > 0$ such that

Let us define $\tilde{z}_n(t) = z_{n}(t+y_n)$, then, we have

Observe that $I_{\mu}$ is the invariant under translation since (3.1) is autonomous, then, we have $\|\tilde{z}_n\| = \|z_n\|$ and

Passing to a subsequence, we may suppose that $\tilde{z}_n \rightharpoonup \tilde{z}$ in $E$, $\tilde{z}_n \to \tilde{z}$ in $L_{loc}^{q}(\mathbb{R})$ for $q > 2$, and $\tilde{z}_n(t) \to \tilde{z}(t)$ almost everywhere on $\mathbb{R}$. According to (3.3) and (3.4), then we can derive that $\tilde{z}\neq 0$ and $I_{\mu}'(\tilde{z}) = 0$. This implies that $\tilde{z}\in \mathscr{N}_{\mu}$ and $I_{\mu}(\tilde{z})\geq c_{\mu}$. On the other hand, applying Fatou's lemma we can obtain

that is, $I_{\mu}(\tilde{z})\leq c_{\mu}$. Consequently, $I_{\mu}(\tilde{z}) = c_{\mu}$ and $\tilde{z}$ is a critical point of $I_{\mu}$, which implies that $\tilde{z}$ is a ground-state homoclinic orbit of problem (3.1). So, we have completed the proof of Theorem 3.1.

As a byproduct of the Theorem 3.1, we show the monotonicity and continuity of $c_{\mu}$.

Lemma 3.9. The function $\mu \mapsto c_{\mu}$ is decreasing and continuous on $(0, +\infty)$.

Proof. In what follows, let $z_{\mu_{1}}$ and $z_{\mu_{2}}$ be as ground state homoclinic orbits of $I_{\mu_{1}}$ and $I_{\mu_{2}}$, respectively. Assume that $\mu_{1} > \mu_{2}$. First of all, we want to verify that the function $\mu \mapsto c_{\mu}$ is decreasing. On account of Lemma 3.6 we can find that there exist $s_{1} > 0$ and $\varphi_{1}\in E^{-}$ such that

then we have

Combining the fact that

with the inequality $\mu_{2}-\mu_{1} < 0$, we can infer that

We finish the proof by demonstrating that the function $\mu \mapsto c_{\mu}$ is decreasing on $(0, +\infty)$.

In order to claim the continuity of $c_{\mu}$, we divide the proof into two steps:

Step 1: Let $\{\mu_{n}\}$ be a sequence such that $\mu_{1}\leq \mu_{2}\leq \cdots \leq \mu_{n}\leq \cdots \leq \mu$ and $\mu_{n}\to\mu$.

Claim 1: $c_{\mu_{n}}\to c_{\mu}$ as $n\to\infty$.

Indeed, let $z_{\mu}$ be the ground state homoclinic orbit of system (3.1). On account of Lemma 3.6, we can see that there exist $s_{n} > 0$ and $\varphi_{n}\in E^{-}$ such that

Note that, using $(g_3)$ and computing directly, we get

so, for all $n\in \mathbb{N}$ and $z\in E$, we have that $I_{\mu_{1}}(z)\geq I_{\mu_{n}}(z)$. Then by Lemma 3.4, it holds that there exists $R > 0$ such that

According to Lemma 3.3 and the monotonicity of $c_{\mu}$, we can obtain

consequently, it follows that

From (3.5) and (3.6), one can check that $\|s_{n}z_{\mu}+\varphi_{n}\|\leq R$; this shows that the sequence $\{s_{n}z_{\mu}+\varphi_{n}\}$ is bounded in $E$. Hence, it is easy to see that

then we get

On the other hand, since $c_{\mu}\leq c_{\mu_{n}}$ for all $n\in \mathbb{N}$, we can infer that

Step 2: Let $\{\mu_{n}\}$ be a sequence such that $\mu_{1}\geq \mu_{2}\geq \cdots \geq \mu_{n}\geq \cdots \geq \mu$ and $\mu_{n}\to\mu$.

Claim 2: $c_{\mu_{n}}\to c_{\mu}$ as $n\to\infty$.

In fact, let $z_{n}$ be the ground state homoclinic orbit of the system (3.1) with $\mu = \mu_{n}$, then, there exist $s_{n} > 0$ and $\varphi_{n}\in E^{-}$ such that

We can easily obtain that the sequence $\{z_{n}\}$ is bounded by Lemma 3.7. Moreover, we can find that there exist $\delta > 0$, $r > 0$ and $\{y_{n}\}\subset \mathbb{Z}$ such that for each $n\in \mathbb{N}$, we have

Otherwise, using Lions' concentration-compactness principle we can deduce that $z_{n}^{+}\rightarrow 0$ in $L^{q}(\mathbb{R})$ for all $q > 2$. Combining (2.1) with (2.2), it holds that

Therefore, we have

this shows that $\|z_{n}^{+}\|^{2}\rightarrow 0$, which is a contradiction to the inequality $\|z_{n}^{+}\|^{2}\geq 2c_{\mu_{n}} > 0$ from Lemma 3.3. So, (3.7) holds.

Setting $\tilde{z}_{n}(t): = z_{n}(t+y_{n})$, one can check that $\{\tilde{z}_{n}\}$ is bounded in $E$; passing to a subsequence, $\tilde{z}_{n}^{+}\rightharpoonup \tilde{z}^{+}\neq 0$ in $E$. Set $\mathcal{V}: = \{\tilde{z}_{n}^{+}\}\subset E^{+}\backslash \{0\}$; hence, $\mathcal{V}$ is bounded and the sequence does not weakly converge to zero in $E$. Then by Lemma 3.4, there exists $R > 0$ such that for every $z\in \mathcal{V}$, we obtain

Define $\tilde{\varphi}_{n}(t): = \varphi_{n}(t+y_{n})$, we have

In view of (3.8) and (3.9), we can conclude that $\|s_{n}\tilde{z}_{n}+\tilde{\varphi}_{n}\|\leq R$ for all $n\in \mathbb{N}$, then, $\|s_{n}z_{n}+\varphi_{n}\|\leq R$, which implies that the sequence $\{s_{n}z_{n}+\varphi_{n}\}$ is bounded in $E$, and $\int_{\mathbb{R}}G(|s_{n}z_{n}+\varphi_{n}|)\mathrm{d}t$ is also bounded. Thus, we obtain

Combining this with the fact that $c_{\mu}\geq c_{\mu_{n}}$ for all $n\in \mathbb{N}$, then we have

We have finished the proof of the lemma.

4.   The proof of Theorem 1.1

4.1.   Existence of ground state homoclinic orbits

In this subsection we will give the proof involving the existence result for ground state homoclinic orbits for system (1.1). As before, we define the associated generalized Nehari manifold

and the ground state energy value

Applying the same arguments explored in the Section $3$, we can show that for every $z\in E\backslash E^{-}$ the set $\mathscr{N}_{\epsilon}\cap \widehat{E}(z)$ is a singleton set, and the element of this set is the unique global maximum of $I_{\epsilon}|_{\widehat{E}(z)}$, that is, there exists a unique pair $t > 0$ and $\varphi \in E^{-}$ such that

Therefore, the following mapping is well-defined:

and the inverse of $m_{\epsilon}$ is

Accordingly, the reduced functional $\widetilde{\Phi}_{\epsilon}:E^{+}\backslash\{0\}\to\mathbb{R}$ and the restriction $\Phi_{\epsilon}: S^{+}\to\mathbb{R}$ can respectively be defined by

Moreover, from the above discussions in Section $3$, we can check that all related conclusions in Section $3$ hold for $I_{\epsilon}$, $c_{\epsilon}$, $\mathscr{N}_{\epsilon}$, $\widetilde{m}_{\epsilon}$, $m_{\epsilon}$, $\widetilde{\Phi}_{\epsilon}$ and $\Phi_{\epsilon}$, respectively.

Meanwhile, concerning the limit problem given by

for the sake of simplicity, we will use the notations $I_{0}$, $c_{0}$ and $\mathscr{N}_{0}$ to denote $I_{A(0)}$, $c_{A(0)}$ and $\mathscr{N}_{A(0)}$, respectively.

Next, we will state the relationship of the ground state energy value between system (1.3) and limit system (4.1), and this is very significant in our following arguments.

Lemma 4.1. The limit $\lim\limits_{\epsilon\rightarrow 0}c_{\epsilon} = c_{0}$ holds.

Proof. Let be $\epsilon_{n}\to 0$ as $n\to\infty$. Evidently, using Lemma 3.9 we obtain that $c_{0}\leq c_{\epsilon_{n}}$ for all $n\in\mathbb{N}$; thus, $c_{0}\leq\liminf_{n\to\infty}c_{\epsilon_{n}}$.

On the other hand, Theorem 3.1 shows that the limit system (4.1) has a ground state homoclinic orbit $z_{0}$. Then, according to Lemma 3.6, we can find that there are $s_{n}\in(0, +\infty)$ and $\varphi_{n}\in E^{-}$ such that $s_{n}z_{0}^{+}+\varphi_{n}\in \mathscr{N}_{\epsilon_{n}}$, and

As in the previous section, we can see that $\{s_{n}z_{0}^{+}+\varphi_{n}\}$ is bounded in $E$. Thus, without loss of generality, we assume that $s_{n}\rightarrow s_{0}$ and $\varphi_{n}\rightharpoonup \varphi$ in $E^{-}$. Therefore, we can deduce from the weakly lower semi-continuity of the norm and Fatou's Lemma that

Obviously, we can get

finishing the proof.

In view of the above discussion, we obtain that $I_{0}(s_{0}z_{0}^{+}+\varphi) = I_{0}(z_{0}) = c_{0}$; then, both $s_{0}z_{0}^{+}+\varphi$ and $z_{0}$ are the elements of $\mathscr{N}_{0}\cap \widehat{E}(z_{0})$. But, according to Lemma 3.6, there is only one element in $\mathscr{N}_{0}\cap \widehat{E}(z_{0})$, so we can conclude that $s_{0}z_{0}^{+}+\varphi = z_{0}$ and $s_{n}\rightarrow s_{0} = 1$, where $\varphi_{n}\rightharpoonup \varphi = z_{0}^{-}$.

As a byproduct of the Lemma 4.1, we can directly obtain the following result.

Lemma 4.2. Assume that condition $(A)$ holds, then, there is $\epsilon_{0} > 0$ such that $c_{\epsilon} < c_{A_{\infty}}$ for $\epsilon\in (0, \epsilon_{0})$.

Proof. From condition $(A)$ we can see that $A(0) > A_{\infty}$. Then from Lemma 3.9 we have that $c_{0} < c_{A_{\infty}}$. Observe that, Lemma 4.1 yields that there exists $\epsilon_{0} > 0$ small enough such that $c_{\epsilon} < c_{A_{\infty}}$ for all $\epsilon\in (0, \epsilon_{0})$. Therefore, we get that $c_{\epsilon} < c_{A_{\infty}}$ for $\epsilon\in (0, \epsilon_{0})$.

Using similar arguments as for the proof of Lemma 3.7, one can easily check the following lemma.

Lemma 4.3. The energy functional $I_{\epsilon}$ is coercive on $\mathscr{N}_{\epsilon}$ for each $\epsilon\geq 0$.

Next we give the proof involving the existence result for ground state homoclinic orbits for system (1.1).

Lemma 4.4. Assume that conditions $(L)$, $(A)$ and $(g_1)$-$(g_3)$ are satisfied, then, system (1.1) has a ground-state homoclinic orbit for each $\epsilon \in(0, \epsilon_{0})$.

Proof. Following the proof of Theorem 3.1 and using Lemma 3.8, we must prove that there exists $z\in\mathscr{N}_{\epsilon}$ such that $I_{\epsilon}(z) = c_{\epsilon}$. Indeed, applying Ekeland's variational principle, there exists $\{u_{n}\}\subset S^{+}$ such that $\Phi_{\epsilon}(u_n)\to c_{\epsilon}$ and $\Phi'_{\epsilon}(u_n)\to 0$. Put $z_n = \widetilde{m}_{\epsilon}(u_n)\in \mathscr{N}_{\epsilon}$ for all $n\in\mathbb{N}$. Then from Lemma 3.8 we have that $I_{\epsilon}(z_n)\to c_{\epsilon}$ and $I'_{\epsilon}(z_n)\to 0$. Furthermore, in view of Lemma 4.3, we can prove that $\{z_n\}$ is bounded. Then, up to a subsequence, we can suppose that $z_{n}\rightharpoonup z$ in $E$. Evidently, $I'_{\epsilon}(z) = 0$.

In what follows we need to show that $z\neq 0$ and $I_{\epsilon}(z) = c_{\epsilon}$. Combining the fact that $z_n\in \mathscr{N}_{\epsilon}$ with Lemma 3.3, we have

which yields that

As in the previous section, we can check that there exists a sequence $\{y_{n}\}\subset \mathbb{Z}$, $r > 0$ and $\delta > 0$ such that

Now, we need to prove that the sequence $\{y_{n}\}$ is bounded in $\mathbb{R}$. Arguing by contradiction we can suppose that $\{y_{n}\}$ is unbounded and $|y_{n}|\rightarrow +\infty$ as $n\rightarrow \infty$. Setting $w_{n}(t): = z_{n}(t+y_{n})$, then, $w_{n}\rightharpoonup w$ in $E$; we can obtain $w\neq 0$ from (4.2). By choosing the test function $\psi\in C_{0}^{\infty}(\mathbb{R})$, we get

Letting $n\rightarrow +\infty$, then we obtain

This shows that $w$ is a nontrivial solution of system (3.1) with $\mu = A_{\infty}$ and $w\in \mathscr{N}_{A_{\infty}}$.

Employing the Fatou's lemma we can derive that

Therefore, it follows that

However, Lemma 4.2 yields that $c_{\epsilon} < c_{A_{\infty}}$ when $\epsilon < \epsilon_{0}$, which leads to a contradiction. So, we can conclude that $\{y_{n}\}$ is bounded. Then for all $n\in \mathbb{N}$, there exists $r_{0} > 0$ such that $B_{r}(y_{n})\subset B_{r_{0}}(0)$, it holds that

Therefore, we obtain that $z_{n}\rightharpoonup z$ in $E$ with $z\neq 0$. By repeating the steps in (4.3) and (4.4), we know that $z\in\mathscr{N}_{\epsilon}$ is a nontrivial solution for system (1.1), thus, $c_{\epsilon}\leq I_{\epsilon}(z)$.

On the other hand, according to Fatou's lemma, we infer that

Thus, $c_{\epsilon} = I_{\epsilon}(z)$. Evidently, it is easy to see that $z$ is a ground state homoclinic orbit of system (1.1). We complete the proof.

Let

be the set of all nontrivial critical points of $I_{\epsilon}$. In order to describe some important properties of ground state homoclinic orbits, next, we get the following regularity result by taking advantage of the bootstrap argument (see ^[37] for the iterative steps), this result can also be found in [24, Lemma 2.3].

Lemma 4.5. If $z\in \mathcal{K}_{\epsilon}$ with $|I_{\epsilon}(z)|\leq C_{1}$ and $\|z\|_{2}\leq C_{2}$; then, $z\in W^{1, q}(\mathbb{R}, \mathbb{R}^{2N})$ for any $q > 2$, and $\|z\|_{W^{1, q}}\leq C_{q}$, where $C_{q}$ depends only on $C_{1}, C_{2}$ and $q$.

Below, we use $\mathscr{L}$ to denote the set of all ground state homoclinic orbits of system (1.1). Let $z\in\mathscr{L}$, then, $I_{\epsilon}(z) = c_{\mu}$; applying a standard argument we can show that $\mathscr{L}$ is bounded in $E$; therefore, $\|z\|_{2}\leq \widehat{c}$ for all $z\in\mathscr{L}$ and some $\widehat{c} > 0$. Hence, making use of Lemma 4.5, we see that, for each $q > 2$, there exists $C_{q}$ such that

Moreover, combining the Sobolev embedding theorem, we can show that there exists $C_{\infty} > 0$ such that

4.2.   Concentration of ground state homoclinic orbits

We now shall prove the concentration behavior of the maximum points of the ground state homoclinic orbit. Let $z_{\epsilon}$ be a ground state homoclinic orbit of system (1.1), which can be obtained by Lemma 4.4. Our aim is to show that if $t_{\epsilon}$ is a maximum point of $|z_{\epsilon}|$, then,

In other words, we must show that if $\epsilon_{n}\rightarrow 0$, up to a subsequence, $\epsilon_{n}t_{\epsilon_{n}}\rightarrow t_{0}$ for some $t_{0}\in \mathscr{A}$, where

denotes the set of the maximum points of $A(t)$.

Let $\{\epsilon_{n}\}\subset (0, \epsilon_{0})$ with $\epsilon_{n}\rightarrow 0$ as $n\to\infty$ and $z_{\epsilon_{n}}\in\mathscr{L}$; we write $z_{n}: = z_{\epsilon_{n}}$. Then, we have

Evidently, in view of Lemma 4.3, we can easily check that $\{z_{n}\}$ is bounded in $E$.

Lemma 4.6. There exist a sequence $\{y_{n}\}\subset \mathbb{Z}$ and two constants $r > 0$, $\delta > 0$ such that

Proof. Arguing by contradiction, we suppose that for any $r_{1} > 0$,

Then, according to Lions' concentration-compactness principle, we conclude that $z_{n}\rightarrow 0$ in $L^{q}(\mathbb{R})$ for all $q > 2$. Furthermore, by (2.1) and (2.2), we obtain

Therefore, it follows that

Evidently, this is impossible because $c_{\epsilon_{n}} > 0$ (see Lemma 3.3). We complete the proof.

Lemma 4.7. The sequence $\{\epsilon_{n}y_{n}\}$ is bounded, and $\lim\limits_{n\rightarrow \infty}\epsilon_{n}y_{n} = x_{0}\in \mathscr{A}$.

Proof. Setting $v_{n}(t): = z_{n}(t+y_{n})$, up to a subsequence, it is easy to see that $v_{n}\rightharpoonup v$ in $E$ with $v\neq 0$ from Lemma 4.6. In what follows, we want to prove that the sequence $\{\epsilon_{n}y_{n}\}$ is bounded. If this is not true, we can suppose that there is a subsequence $\{\epsilon_{n}y_{n}\}$ such that $|\epsilon_{n}y_{n}|\rightarrow +\infty$ as $n\rightarrow +\infty$. Since $z_{n}$ is the ground state homoclinic orbit of system (1.1), $v_{n}$ solves the following system

and the energy

Furthermore, for every $\phi\in E$, we have

Since $A(\epsilon_{n}t+\epsilon_{n}y_{n})\to A_{\infty}$, given that $v_n\rightharpoonup v$ and $\phi\in C^{\infty}_{0}(\mathbb{R})$, we get

Thereby, $v$ is a nontrivial homoclinic orbit of system (3.1) with $\mu = A_{\infty}$ and $v\in \mathscr{N}_{A_{\infty}}$. In view of Lemma 4.1 and Fatou's lemma, we can conclude that

However, according to Lemma 4.2 we know that $c_{0} < c_{A_{\infty}}$. Evidently, this is a contradiction. Therefore, $\{\epsilon_{n}y_{n}\}$ is bounded in $\mathbb{R}$, and passing to a subsequence, we can assume that $\epsilon_{n}y_{n}\rightarrow x_{0}$. According to the above argument, for $\forall \psi\in E$, we get

Obviously, we can see that $v$ is a ground state homoclinic orbit of the following system

and $v\in \mathscr{N}_{A(x_{0})}$. Following to the proof of (4.8), we can get tht $c_{A(x_{0})}\leq c_{0}$, then, using Lemma 3.9, it follows that $A(x_0)\geq A(0)$; together with condition $(A)$, we can obtain that $A(x_0) = A(0)$. Hence, we show that $\lim\limits_{n\rightarrow \infty}\epsilon_{n}y_{n} = x_{0}$ and $x_{0}\in \mathscr{A}$. The proof is completed.

According to Lemma 4.7, we see that $v$ is a ground state homoclinic orbit of system (4.7), then, $I_{0}(v) = c_{0}$ and $I'_{0}(v) = 0$. Using Lemma 4.1 and Fatou's lemma, we directly obtain

Hence, we have

Lemma 4.8. We have the convergence conclusion: $v_{n}\to v$ in $E$ as $n\to\infty$.

Proof. Let $\eta:[0, +\infty)\rightarrow [0, 1]$ be a smooth function satisfying that $\eta(s) = 1$ if $s\leq 1$, and $\eta(s) = 0$ if $s\geq 2$. Define $\tilde{v}_{n}(t) = \eta(2|t|/n)v(t)$, then, for $q\in [2, +\infty)$, one has

Setting $\theta_{n} = v_{n}-\tilde{v}_{n}$, it is not difficult to verify that along a subsequence

and

uniformly in $\varphi\in E$ with $\|\varphi\|\leq 1$. Using the fact that $A(\epsilon_{n}t+\epsilon_{n}y_{n})\rightarrow A_{0}$ as $n\rightarrow \infty$ uniformly on any bounded set of $t$, and combining the decay of $v$ and (4.11) we can easily check the following result

Consequently, using (4.10), (4.11), (4.12) and (4.14) we infer that

which implies that $\widehat{I}_{\epsilon_{n}}(\theta_{n})\rightarrow 0$. Similarly, we also obtain

which implies that $\widehat{I}'_{\epsilon_{n}}(\theta_{n})\rightarrow 0$. Therefore

from which together with $(g_{3})$, we can infer that

Notice that $\{\|v_{n}\|_{\infty}\}$ is bounded, thus,

As a consequence, we obtain

that is, $\|\theta_{n}\|\rightarrow 0$, which together with (4.11) leads to $v_{n}\rightarrow v$ in $E$ as $n\rightarrow \infty$.

Lemma 4.9. We have that $v_{n}(t)\to0$ uniformly in $n\in \mathbb{N}$ as $t\to\infty$. Moreover, there exist $c, C > 0$ such that for all $t\in\mathbb{R}$, it holds that

Proof. Firstly, we observe that if $z$ is a homoclinic orbit of system (1.1), then it satisfies the following relation

Computing directly, we obtain

with

Setting

Applying Kato's inequality and (4.15), and using the real positivity of $(\mathscr{J}L)^{2}$, we can find some $\rho > 0$ such that

Hence, using (2.1), (4.6), (4.15) and (4.16) we conclude that there exists $\kappa > 0$ such that

Then by the sub-solution estimate ^[38], there is a $\widehat{c}_{0}$ independent of $t$; we have the following estimate

Now we claim that $v_{n}(t)\to0$ uniformly in $n\in \mathbb{N}$ as $t\to\infty$. Indeed, if it is not true, then using (4.17) we can find that there exist $c_{0} > 0$ and $t_{n}\in \mathbb{R}$ with $|t_{n}|\rightarrow \infty$ such that

this is because $v_n$ satisfies $\widehat{I}'_{\epsilon_{n}}(v_{n}) = 0$, then, the above processes still hold for $v_{n}$. From Lemma 4.8, it follows that $v_n\to v$ in $E$. Therefore, we get

which yields a contradiction. So, the claim holds.

Note that $g(s) = o(1)$ and $g'_{s}(s)s = o(1)$ as $s\rightarrow 0$; then, we can find suitable constants $0 < \delta < \frac{\rho}{2}$ and $R > 0$ such that

Combining the above relation and (4.16), we get

Let $\Lambda(t)$ be a fundamental solution of the following equation

From the uniform boundedness, we may choose $\Lambda(t)$ such that $|v_{n}(t)|\leq\delta\Lambda(t)$ holds on $|t| = R$ for all $n\in\mathbb{N}$. Let $u_{n} = |v_{n}|-\delta\Lambda$; thus, we obtain

The maximum principle yields that $u_{n}(t)\leq 0$ for $|t|\geq R$, i.e., $|v_{n}(t)|\leq \delta \Lambda(t)$ for $|t|\geq R$. As we know that there exists $c_{1} > 0$ such that

Therefore, there are constants $C, c > 0$; we obtain

We complete the proof.

Lemma 4.10. There exists $\nu > 0$ such that $\|v_{n}\|_{\infty}\geq \nu$ for all $n\in\mathbb{N}$.

Proof. According to Lemma 4.6, we can see that there exist $r > 0$ and $\delta > 0$ such that

Suppose by contradiction that $\|v_{n}\|_{\infty}\to0$ as $n\to\infty$, then, it holds that

which is absurd. This ends the proof.

Finally, based on the above facts, next we give the completed proof of Theorem 1.1.

Proof of Theorem 1.1 (completed). Suppose that $q_{n}$ is a global maximum point of $|v_{n}(t)|$ for each $n\in \mathbb{N}$, then,

Since $v_{n}(t) = z_{n}(t+y_{n})$, we can see that $p_{n} = q_{n}+y_{n}$ is a maximum point of $|z_{n}(t)|$. Lemma 4.10 shows that there exists $\nu > 0$ such that

then we know that $\{q_{n}\}$ is bounded. So, we conclude from Lemma 4.7 that

Consequently, it follows that

Furthermore, from Lemma 4.7 and Lemma 4.8, it is easy to see that $z_{n}(t+p_n)$ converges to a ground state homoclinic orbit $v$ of the following limit system

From Lemma 4.9 and the boundedness of $\{q_{n}\}$, we derive that

for some $\tilde{c}, \widetilde{C} > 0$ and all $t\in\mathbb{R}$.

Finally, we observe that Lemma 4.2 shows that, there is $\epsilon_{0} > 0$; system (1.1) has a ground state homoclinic orbit $z_{\epsilon}$ for each $\epsilon \in(0, \epsilon_{0})$. So, the conclusion (a) of Theorem 1.1 holds. Moreover, according to the above discussions, we directly obtain the following conclusions:

(b) let $t_{\epsilon}$ be the maximum point of $|z_{\epsilon}(t)|$, then,

and $z_{\epsilon}(t+t_{\epsilon})\to v$ in $E$, where $v$ is a ground state homoclinic orbit of the limit system

(c) there are two positive constants $\tilde{c}$, $\widetilde{C}$ such that

We have finished the proof of all conclusions of Theorem 1.1.

Use of AI tools declaration

The authors declare that no artificial intelligence tools were used in the creation of this article.

Acknowledgments

The research of Tianfang Wang was supported by the High Level Research Achievement Project-General Project of Natural Science of Baotou Teachers' College (BSYKJ2022-ZY09), the Youth Innovative Talent Project of Baotou City (2022). The research of Wen Zhang was supported by the Natural Science Foundation of Hunan Province (2022JJ30200), the Key project of Scientific Research Project of Department of Education of Hunan Province (22A0461), and Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province.

Conflict of interest

The authors declare that they have no competing interests.