- Research
- Open Access
- Published:
Homoclinic orbits for damped vibration systems with asymptotically quadratic or subquadratic potentials
Advances in Difference Equations volume 2016, Article number: 78 (2016)
Abstract
In this paper, we study a class of damped vibration systems,
where \(W(t,u)\) is of indefinite sign. By using a critical point theorem of Ding, we establish a new criterion to guarantee that the above system has infinitely many nontrivial homoclinic orbits under the assumption that \(W(t,u)\) is asymptotically quadratic or subquadratic as \(|u|\rightarrow\infty\). Recent results in the literature are generalized and significantly improved.
1 Introduction
In this paper, we consider the following damped vibration system:
where \(u\in\mathbb{R}^{N}\), B is an antisymmetric \(N\times N\) constant matrix, \(L\in C(\mathbb{R},\mathbb{R}^{N\times N})\) is a symmetric matrix-valued function, and \(W\in C^{1}(\mathbb{R}\times \mathbb{R}^{N},\mathbb{R})\). As usual, we say that a solution u of system (1.1) is homoclinic to zero if \(u\in C^{2}(\mathbb {R},\mathbb{R}^{N})\), \(u(t)\rightarrow0\), and \(\dot{u}(t)\rightarrow 0\) as \(|t|\rightarrow\infty\). In addition, if \(u(t)\not\equiv0\), then \(u(t)\) is called a nontrivial homoclinic solution.
Homoclinic orbits have been found in various models of continuous dynamical systems and play an important role in the study of the behavior of dynamical systems; see [1]. Thus, the study of homoclinic orbits has become one of the most important directions in the research of dynamical systems.
When \(B=0\), system (1.1) reduces to the following second order Hamiltonian system:
As a special case of dynamical systems, Hamiltonian systems play an important role in practical problems concerning relativistic mechanics, gas dynamics, nuclear physics, fluid mechanics. With the aids of the variational methods, the existence and multiplicity of homoclinic orbits for (1.2) have been extensively investigated in many recent papers; see [2–24].
Compared with the case where \(B=0\), the case where \(B\neq0\) is more difficult. The study of homoclinic orbits for system (1.1) has attracted a lot of attention by many researchers; see [25–33]. This work is mainly based on variational methods. Some of the authors considered the superquadratic case [27, 29, 30, 32, 33]; the authors of [25, 28, 31] considered the subquadratic case; for the asymptotically quadratic case, except for [26], few researchers have investigated this case. More precisely, in [25], Chen studied system (1.1) under the assumption that \(W(t,u)\) is subquadratic as \(|u|\rightarrow\infty \). In detail, he obtained the following result.
Theorem 1.1
([25])
Assume that the following conditions hold.
- (L1):
-
There exists a constant \(\upsilon>1\) such that
$$\operatorname{meas}\bigl\{ t\in\mathbb{R}:|t|^{-\upsilon}L(t)< TI_{N} \bigr\} ,\quad \forall T>0, $$where \(I_{N}\) denotes the \(N\times N\) identity matrix.
- (L2):
-
There exists a constant \(\beta\geq0\) such that
$$l(t):=\inf_{|u|=1}\bigl(L(t)u,u\bigr)\geq-\beta, \quad t\in \mathbb{R}. $$ - (H1):
-
\(W(t,u)\geq0\), \(\forall(t,u)\in\mathbb{R}\times \mathbb{R}^{N}\), and there exist constants \(\kappa\in(0,2)\) and \(R_{0}>0\) such that
$$\bigl(\nabla W(t,u),u\bigr)\leq\kappa W(t,u),\quad \forall t\in\mathbb{R} \textit{ and } |u|\geq R_{0} $$and
$$\bigl(\nabla W(t,u),u\bigr)\leq2 W(t,u), \quad \forall t\in\mathbb{R} \textit{ and } |u|\leq R_{0}. $$ - (H2):
-
There exists \(a>0\) such that
$$W(t,u)\leq a|u|, \quad \forall t\in\mathbb{R} \textit{ and } |u|\leq R_{0}. $$ - (H3):
-
\(\liminf_{|u|\rightarrow\infty}\frac {W(t,u)}{|u|}\geq b\) uniformly in \(t\in\mathbb{R}\), where \(b>0\) is a constant.
- (H4):
-
\(\lim_{|u|\rightarrow0}\frac {W(t,u)}{|u|^{2}}=+\infty\) uniformly in \(t\in\mathbb{R}\).
- (H5):
-
\(W(t,-u)=W(t,u)\), \(\forall(t,u)\in\mathbb{R}\times \mathbb{R}^{N}\).
Then system (1.1) has infinitely many nontrivial homoclinic orbits.
In [26], \(W(t,u)\) being asymptotically quadratic as \(|u|\rightarrow\infty\), by using the variant fountain theorem, Chen obtained the following result.
Theorem 1.2
([26])
Assume that (L1), (L2), and (H5) hold. Moreover, we assume that the following conditions are satisfied:
- (H6):
-
\(\widetilde{W}(t,u)=W(t,u)-(1/2)(\nabla W(t,u),u)\rightarrow+\infty\) as \(|u|\rightarrow\infty\) uniformly in \(t\in\mathbb{R}\).
- (H7):
-
There are constants \(\tau\in(1,2)\) and \(a_{1},a_{2},a_{2}>0\) such that
$$a_{3}|u|^{\tau}\leq W(t,u)\leq a_{1}|u|,\quad \forall t\in\mathbb{R} \textit{ and } |u|\leq a_{2}. $$ - (H8):
-
\(\limsup_{|u|\rightarrow0}\frac{|\nabla W(t,u)|^{\tau/(\tau-1)}}{\widetilde{W}(t,u)}=P(t)\) uniformly in t, \(|P(t)|<\infty\).
- (H9):
-
\(W(t,u)\geq(1/2)(\nabla W(t,u),u)\geq0\), \(\forall(t,u)\in\mathbb{R}\times\mathbb{R}^{N}\).
- (H10):
-
\(\lim_{|u|\rightarrow\infty}\frac {W(t,u)}{|u|^{2}}=f(t)\) uniformly in t, where \(\inf_{t\in\mathbb {R}}f(t)\leq\sup_{t\in\mathbb{R}}f(t)<+\infty\).
Then system (1.1) has infinitely many nontrivial homoclinic orbits.
Motivated by the above facts, in this paper, our aim is to generalize some results in [25, 26]. Moreover, our approach is different from [25, 26].
We will use the following conditions:
- (W1):
-
\(\nabla W(t,u)=S(t)u+\nabla G(t,u)\), where \(S:\mathbb {R}\rightarrow\mathbb{R}^{N\times N}\) is bounded symmetric \(N\times N\) matrix-valued function.
- (W2):
-
\(W(t,0)=G(t,0)=0\) and there exist \(\bar{a},\bar {b}>0\) and \(0<\nu<1\) such that
$$\bigl\vert \nabla G(t,u)\bigr\vert \leq\bar{a}+\bar{b}\vert u\vert ^{\nu}, \quad \forall(t,u)\in \mathbb{R}\times\mathbb{R}^{N}. $$ - (W3):
-
\(\widetilde{W}(n,u)\geq0\), \(\forall(t,u)\in \mathbb{R}\times\mathbb{R}^{N}\).
Now, we state our main result.
Theorem 1.3
Assume that (L1), (L2), (H4)-(H6), (W1)-(W3) hold. Then system (1.1) has infinitely many nontrivial homoclinic orbits.
Remark 1.1
When \(S\not\equiv0\), Theorem 1.3 generalizes Theorem 1.2. First of all, we remove conditions (H8) and \(W(t,u)\geq0\) for any \((t,u)\in\mathbb {R}\times\mathbb{R}^{N}\). Second, condition (H4) is weaker than (H7). In fact, condition (H7) implies (H4). Third, it is clear that condition (W3) is weaker than (H9). Furthermore, there are many functions satisfying our Theorem 1.3 and not satisfying Theorem 1.2. For example, let \(W(t,u)=g(t)|u|^{2}+|u|\) for all \((t,u)\in\mathbb {R}\times\mathbb{R}^{N}\), where \(g:\mathbb{R}\rightarrow\mathbb{R}\) is bounded continuous function such that \(g\not\equiv0\). It is easy to check that \(W(t,u)\) satisfies all conditions of Theorem 1.3, but it does not satisfy condition (H8) of Theorem 1.2.
Remark 1.2
When \(S\equiv0\), Theorem 1.3 generalizes Theorem 1.1. First, conditions (H1) and (H3) imply conditions (H6) and (W3). Second, conditions (H1) and (H2) imply (W2). Third, we remove conditions (H3) and \(W(t,u)\geq 0\) for any \((t,u)\in\mathbb{R}\times\mathbb{R}^{N}\). Furthermore, there are many functions \(W(t,u)\) satisfying our Theorem 1.3 and not satisfying Theorem 1.1. For example, set
for all \((t,u)\in\mathbb{R}\times\mathbb{R}^{N}\). It is easy to check that \(W(t,u)\) satisfies all conditions of Theorem 1.3, but it does not satisfy condition (H1) of Theorem 1.1. Set
for all \((t,u)\in\mathbb{R}\times\mathbb{R}^{N}\). It is easy to check that \(W(t,u)\) satisfies all conditions of Theorem 1.3, but it does not satisfy condition (H3) of Theorem 1.1.
The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proof of Theorem 1.3.
2 Preliminaries
In this section, we give the variational setting for (1.1) and some related preliminary lemmas. Let \(X:= H^{1}(\mathbb{R},\mathbb{R}^{N})\) be a Hilbert space with the inner product and the norm given, respectively, by
We define an operator \(K:X\rightarrow X\) by
Since B is an antisymmetric \(N\times N\) constant matrix, K is self-adjoint on X. Moreover, we denote by J the self-adjoint extension of the operator \(-\frac{d^{2}}{dt^{2}}+L(t)+K\) with the domain \(D(J)\subset L^{2}(\mathbb{R},\mathbb{R}^{N})\).
Let \(E:=D(|J|^{\frac{1}{2}})\) be the domain of \(|J|^{\frac{1}{2}}\), which is a Hilbert space equipped with the inner product and norm given by
for \(u,v\in E\), where \((\cdot,\cdot)_{2}\) denotes the inner product in \(L^{2}(\mathbb{R},\mathbb{R}^{N})\). Let \(\|\cdot\|_{p}\) denote the usual norm on \(L^{p}(\mathbb{R},\mathbb{R}^{N})\) (\(p\in[1,\infty]\)).
Lemma 2.1
([29])
Assume that L satisfies (L1) and (L2). Then E is compactly embedded in \(L^{p}(\mathbb{R},\mathbb{R}^{N})\) for any \(1\leq p\leq\infty\).
By Lemma 2.1, the spectrum \(\sigma(J)\) consists of eigenvalues numbered by \(\eta_{1}\leq\eta_{2}\leq\cdots\leq\eta _{k}\leq\cdots\rightarrow\infty\) (counted in their multiplicities) and a corresponding system of eigenfunctions \(\{e_{k}\}\) (\(Je_{k}=\eta _{k}e_{k}\)) which forms an orthogonal basis in \(L^{2}(\mathbb{R},\mathbb{R}^{N})\).
Set
and
Thus, we have the orthogonal decomposition
with respect to the inner product \(\langle\cdot,\cdot\rangle_{E}\). Now we introduce on E the following inner product:
and the norm
where \(u,v\in E\) with \(u=u^{-}+u^{0}+u^{+}\) and \(v=v^{-}+v^{0}+v^{+}\). It is easy to verify that \(\|\cdot\|_{E}\) and \(\|\cdot\|\) are equivalent; see [8]. Evidently, the aforementioned decomposition is also orthogonal with respect to both inner products \((\cdot,\cdot)_{2}\) and \(\langle\cdot,\cdot\rangle\).
Define the functional Ψ on E by
It follows from the assumptions that Ψ is defined on E and belongs to \(C^{1}(E,\mathbb{R})\), and one can easily check that
for any \(u,v\in E\) with \(u=u^{-}+u^{0}+u^{+}\) and \(v=v^{-}+v^{0}+v^{+}\). Furthermore, it is routine to verify that any critical point of Ψ in E is a solution of system (1.1) with \(u(\pm\infty )=0=\dot{u}(\pm\infty)\) (see [20, 21]). In view of Lemma 2.1, there exists \(D_{p}>0\) such that
where \(p\in[1,+\infty]\).
Define \(E_{j}=\mathbb{R}e_{j}\),
Lemma 2.2
Under assumptions (L1) and (L2), for \(\varsigma\in[1,+\infty]\),
Proof
It is clear that \(0<\beta_{k+1}(\varsigma)\leq \beta_{k}(\varsigma)\), so that \(\beta_{k}(\varsigma)\rightarrow\tilde {\beta}(\varsigma)\), \(k\rightarrow\infty\). For every \(k\geq0\), there exists \(u_{k}\in Z_{k}\) such that \(\|u_{k}\|=1\) and \(\|u_{k}\|_{\varsigma}>\frac{\beta_{k}}{2}\). For any \(v\in X\), let \(v=\sum_{i=1}^{\infty}\bar {b}_{i}e_{i}\), by the Cauchy-Schwartz inequality, one has
which implies that \(u_{k}\rightharpoonup0\). It follows from Lemma 2.1 that \(u_{k}\rightarrow0\) in \(L^{q}(\mathbb{R},\mathbb {R}^{N})\). Thus we have proved that \(\tilde{\beta}(\varsigma)=0\). □
By Lemma 2.2, we can choose a positive integer \(k_{0}\geq n^{+}+1\) such that
where \(m_{0}=\sup_{t\in\mathbb{R}}[\sup_{x\in\mathbb{R}^{N},|x|=1}(S(t)x,x)]\).
In order to prove our main result, we need the following lemma.
Lemma 2.3
([8])
Let E be an infinite dimensional Banach space and \(\Phi\in C^{1}(E,\mathbb{R})\) be even, satisfy the \((PS)\) condition, and \(\Psi(0)=0\). If \(X=Y\oplus Z\), where Y is finite dimensional, and Ψ satisfies
- (S1):
-
Ψ is bounded from below on Z;
- (S2):
-
for each finite dimensional subspace \(\widetilde {E}\subset E\), there are positive constants \(\rho=\rho(\widetilde {E})\) and \(\alpha=\alpha(\widetilde{E})\) such that \(\Psi|_{B_{\rho}\cap\widetilde{E}}\leq0\) and \(\Psi|_{\partial B_{\rho}\cap \widetilde{E}}\leq-\alpha\), where \(B_{\rho}=\{x\in E:\|x\|\leq\rho\}\).
Then Ψ possesses infinitely many nontrivial critical points.
Remark 2.1
As shown in [34], a deformation lemma can be proved with condition \((C)\) replacing \((PS)\) condition, and it turns out that Lemma 2.3 holds true under condition \((C)\). We say that Ψ satisfies condition \((C)\), i.e. for any \(\{u_{n}\}\subset E\), \(\{u_{n}\}\) has a convergent subsequence if \(\Psi(u_{n})\) is bounded and \((1+\|u_{n}\|)\|\Psi^{\prime}(u_{n})\| \rightarrow0\) as \(n \rightarrow\infty\).
3 Proof of Theorem 1.3
Set \(Y=Y_{k_{0}}\), \(Z=Z_{k_{0}}\).
Lemma 3.1
Suppose that (W1) and (W2) are satisfied. Then Ψ is bounded from below on Z.
Proof
By virtue of (W1), (W2), (2.3), (2.5), and (2.8), we have
as \(\|u\|\rightarrow\infty\) and \(u\in Z_{k_{0}}\). The proof is completed. □
Lemma 3.2
Assume that (H4) holds. Then for each finite dimensional subspace \(\widetilde{E}\subset E\), there are positive constants \(\rho=\rho(\widetilde{E})\) and \(\alpha=\alpha(\widetilde{E})\) such that \(\Psi|_{B_{\rho}\cap \widetilde{E}}\leq0\) and \(\Psi|_{\partial B_{\rho}\cap\widetilde {E}}\leq-\alpha\), where \(B_{\rho}=\{x\in E:\|x\|\leq\rho\}\).
Proof
Let \(\widetilde{E}\subset E\) be any finite dimensional subspace. Then there exists \(M_{0}>0\) such that
By virtue of (H4), for \(M_{0}\) given above, there exists a constant \(\sigma>0\),
In view of (2.8), for any \(u\in\widetilde{E}\) with \(\| u\|\leq\frac{\sigma}{D_{\infty}}\), we have
for any \(u=u^{-}+u^{0}+u^{+}\in\widetilde{E}\) with \(\|u\|\leq\frac{\sigma }{D_{\infty}}\). Then there exist \(\rho=\rho(\widetilde{E})>0\) and \(\alpha=\alpha(\widetilde{E})>0\) such that
The proof is completed. □
Lemma 3.3
Under the assumptions of Theorem 1.3, Ψ satisfies condition \((C)\).
Proof
Let \(\{u_{n}\}\subset E\) is a \((C)\) sequence of Ψ, that is, \(\{\Psi(u_{n})\}\) is bounded and
then there exists a constant \(M_{1}>0\) such that
for every \(n\in\mathbb{N}\). We choose \(k\geq n^{+}+1\) large enough such that
where \(m_{0}=\sup_{t\in\mathbb{R}}[\sup_{x\in\mathbb {R}^{N},|x|=1}(S(t)x,x)]\). We now prove that \(\{u_{n}\}\) is bounded in E. In fact, if not, we may assume by contradiction that \(\|u_{n}\| \rightarrow\infty\) as \(n\rightarrow\infty\). Let \(u_{n}=\tilde {w}_{n}+\tilde{v}_{n}\), \(z_{n}=\frac{u_{n}}{\|u_{n}\|}\), then \(\|z_{n}\|=1\), \(z_{n}=w_{n}+v_{n}\in E\), where \(w_{n}=\frac{\tilde{w}_{n}}{\|u_{n}\|}\), \(v_{n}=\frac{\tilde{v}_{n}}{\|u_{n}\|}\), \(\tilde{w}_{n}\in Y_{k}\), \(\tilde {v}_{n}\in Z_{k}\). After passing to a subsequence, we have \(z_{n}\rightharpoonup z\), \(w_{n}\rightarrow w\), and \(\gamma=\lim_{n\rightarrow\infty}\|v_{n}\|\) exists.
Case 1. \(\gamma=0\). Since \(\dim Y_{k}<\infty\), we obtain \(\| w_{n}\|\rightarrow\|w\|=1\). It follows from (3.7) that
By virtue of (H6), for any \(\eta>0\), there exists \(M_{2}>0\) such that
For any \(\varepsilon>0\), define \(\Lambda_{\varepsilon}:=\{t\in\mathbb {R}:|w(t)|\geq\varepsilon\}\) and \(\Lambda_{n\varepsilon}:=\{t\in \mathbb{R}:|v_{n}(t)|\geq\frac{\varepsilon}{2}\}\). First, we claim that there exists \(\varepsilon_{0}>0\) such that
Otherwise, for any positive integer m, there exists \(w_{m}\in Y_{k}\) with \(\|w_{m}\|=1\) such that
Passing to a subsequence if necessary, we may assume \(w_{m}\rightarrow w_{0}\) in E for some \(w_{0}\in Y_{k}\) since \(\dim Y_{k}<\infty\). Evidently, \(\|w_{0}\|=1\). By the equivalence of the norms on the finite dimensional space \(Y_{k}\), we have
Thus there exists \(\varepsilon_{1}>0\) such that
In fact, if not, then, for all positive integers m, we have
It implies that
Hence \(w_{0}=0\), which contradicts that \(\|w_{0}\|_{0}=1\). Thus, (3.13) holds.
Now set
and \(\Upsilon_{m}^{c}=\mathbb{R}\setminus\Upsilon_{m}=\{t\in\mathbb {R}:|w_{m}(t)|\geq\frac{1}{m}\}\). By virtue of (3.11) and (3.13), we obtain
for all positive integers m. Let m be large enough such that \(\frac {1}{2}\varepsilon_{1}-\frac{1}{m}>\frac{1}{4}\varepsilon_{1}\). Thus, for m large enough,
which is a contradiction to (3.12). Thus, there exists \(\varepsilon>0\) such that \(\operatorname{meas}(\Lambda_{\varepsilon})\geq \varepsilon\).
In view of (2.5), we obtain
Then we have \(\operatorname{meas}(\Lambda_{\varepsilon}\setminus\Lambda _{n\varepsilon})\rightarrow\operatorname{meas}(\Lambda_{\varepsilon})\) as \(n\rightarrow\infty\). Therefore, there exists \(N_{0}>0\) such that \(|z_{n}(t)|\geq\frac{\varepsilon}{3}\), \(\forall t\in\Lambda _{\varepsilon}\setminus\Lambda_{n\varepsilon}\) and \(n\geq N_{0}\), then we have \(|u_{n}(t)|\geq\frac{\varepsilon}{3}\|u_{n}\|\), \(\forall t\in \Lambda_{\varepsilon}\setminus\Lambda_{n\varepsilon}\) and \(n\geq N_{0}\). By (W3), (3.9), and (3.10), there exists \(N_{1}>0\) such that
which gives a contradiction due to the arbitrariness of η.
Case 2. \(\gamma>0\). In view of (W1), (W2), (2.5), (3.7), (3.8), and Hölder’s inequality, we get
Dividing by \(\|u_{n}\|^{2}\) on both sides of (3.15), we obtain
which gives a contradiction.
Thus, \(\{u_{n}\}\) is bounded. Next, we show that \(\{u_{n}\}\) has a convergent subsequence. In view of the boundedness of \(\{u_{n}\}\), without loss of generality, we may assume that
It follows from (2.4) that
It is clear that
By (2.5), Lemma 2.1, (W1), (W2), and Hölder’s inequality, we have
as \(n\rightarrow\infty\). Therefore, by (3.17)-(3.19), we get \(\|u_{m}^{+}-u^{+}\|\rightarrow0\) as \(n\rightarrow \infty\). Consequently, \(u_{n}\rightarrow u\). The proof is completed. □
Proof of Theorem 1.3
Obviously, \(\Psi\in C^{1}(X,\mathbb{R})\) is even and \(\Phi(0)=0\). It follows from Lemmas 3.1-3.3 that all conditions of Lemma 2.3 are satisfied. By Lemma 2.3, we see that Ψ possesses infinitely many nontrivial critical points, that is, system (1.1) possesses infinitely many nontrivial homoclinic orbits. □
References
Poincaré, H: Les Méthodes Nouvelles de la Mécanique Céleste. Gauthier-Villars, Paris (1897-1899)
Alves, CO, Carriao, PC, Miyagaki, OH: Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation. Appl. Math. Lett. 16, 639-642 (2003)
Caldiroli, P, Montecchiari, P: Homoclinic orbits for second order Hamiltonian systems with potential changing sign. Commun. Appl. Nonlinear Anal. 1(2), 97-129 (1994)
Carriao, PC, Miyagaki, OH: Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems. J. Math. Anal. Appl. 291, 203-213 (2004)
Chen, H, He, Z: Infinitely many homoclinic solutions for a class of second-order Hamiltonian systems. Adv. Differ. Equ. 2014, 161 (2014)
Coti Zelati, V, Rabinowitz, PH: Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Am. Math. Soc. 4, 693-727 (1991)
Ding, Y, Lee, C: Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems. Nonlinear Anal. 71, 1395-1413 (2009)
Ding, Y: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 25, 1095-1113 (1995)
Izydorek, M, Janczewska, J: Homoclinic solutions for a class of the second order Hamiltonian systems. J. Differ. Equ. 219, 375-389 (2005)
Korman, P, Lazer, AC: Homoclinic orbits for a class of symmetric Hamiltonian systems. Electron. J. Differ. Equ. 1994, 1 (1994)
Lv, X, Lu, S, Jiang, J: Homoclinic solutions for a class of second-order Hamiltonian systems. Nonlinear Anal., Real World Appl. 13, 176-185 (2012)
Omana, W, Willem, M: Homoclinic orbits for a class of Hamiltonian systems. Differ. Integral Equ. 5, 1115-1120 (1992)
Paturel, E: Multiple homoclinic orbits for a class of Hamiltonian systems. Calc. Var. Partial Differ. Equ. 12, 117-143 (2001)
Rabinowitz, PH: Homoclinic orbits for a class of Hamiltonian systems. Proc. R. Soc. Edinb. A 114, 33-38 (1990)
Rabinowitz, PH, Tanaka, K: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206, 473-499 (1991)
Salvatore, A: Homoclinic orbits for a special class of nonautonomous Hamiltonian systems. Nonlinear Anal. 30, 4849-4857 (1997)
Sun, J, Chen, H, Nieto, JJ: Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl. 373, 20-29 (2011)
Tang, X, Lin, X: Homoclinic solutions for a class of second-order Hamiltonian systems. J. Math. Anal. Appl. 354, 539-549 (2009)
Tang, X, Lin, X: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Proc. R. Soc. Edinb. A 141, 1103-1119 (2011)
Wang, L, Tang, C: Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition. Discrete Contin. Dyn. Syst., Ser. B 15, 255-271 (2011)
Wang, J, Xu, J, Zhang, F: Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials. Commun. Pure Appl. Anal. 10, 269-286 (2011)
Yang, J, Zhang, F: Infinitely many homoclinic orbits for the second-order Hamiltonian systems with super-quadratic potentials. Nonlinear Anal., Real World Appl. 10, 1417-1423 (2009)
Zhang, Q, Liu, C: Infinitely many homoclinic solutions for second order Hamiltonian systems. Nonlinear Anal. TMA 72, 894-903 (2010)
Zou, W, Li, S: Infinitely many homoclinic orbits for the second-order Hamiltonian systems. Appl. Math. Lett. 16, 1283-1287 (2003)
Chen, G: Non-periodic damped vibration systems with sublinear terms at infinity: infinitely many homoclinic orbits. Nonlinear Anal. 92, 168-176 (2013)
Chen, G: Nonperiodic damped vibration systems with asymptotically quadratic terms at infinity: infinitely many homoclinic orbits. Abstr. Appl. Anal. 2013, Article ID 937128 (2013)
Chen, H, He, Z: New existence and multiplicity of homoclinic solutions for second order non-autonomous systems. Electron. J. Qual. Theory Differ. Equ. 2014, 22 (2014)
Chen, P, Tang, X: Infinitely many homoclinic solutions for a class of damped vibration problems. Math. Methods Appl. Sci. 37, 2297-2307 (2014)
Sun, J, Nieto, JJ, Otero-Novoa, M: On homoclinic orbits for a class of damped vibration systems. Adv. Differ. Equ. 2012, 102 (2012)
Yuan, R, Zhang, Z: Homoclinic solutions for a class of second order non-autonomous systems. Electron. J. Differ. Equ. 2009, 128 (2009)
Zhang, Z, Yuan, R: Homoclinic solutions of some second order non-autonomous systems. Nonlinear Anal. 71, 5790-5798 (2009)
Wu, X, Zhang, W: Existence and multiplicity of homoclinic solutions for a class of damped vibration problems. Nonlinear Anal. 74, 4392-4398 (2011)
Zhu, W: Existence of homoclinic solutions for a class of second order systems. Nonlinear Anal. 75, 2455-2463 (2012)
Bartolo, P, Benci, V, Fortunato, D: Abstract critical point theorems and applications to some nonlinear problems with ‘strong’ resonance at infinity. Nonlinear Anal. 7, 981-1012 (1983)
Acknowledgements
The first author was supported by the Doctor Priming Fund Project of University of South China (2014XQD13) and National Natural Science Foundation of China (Grants Nos. 11526111 and 11501284).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of 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
Chen, H., He, Z., Ouyang, Z. et al. Homoclinic orbits for damped vibration systems with asymptotically quadratic or subquadratic potentials. Adv Differ Equ 2016, 78 (2016). https://doi.org/10.1186/s13662-016-0805-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-016-0805-7
MSC
- 34C37
- 35A15
- 37J45
Keywords
- damped vibration systems
- homoclinic orbits
- variational methods
- asymptotically quadratic theorem
- subquadratic