Abstract
We study the existence of homoclinic solutions for the following second-order self-adjoint discrete Hamiltonian system: \(\triangle[p(n)\triangle u(n-1)]-L(n)u(n)+\nabla W(n, u(n))=0\), where \(p(n)\), \(L(n)\), and \(W(n, x)\) are N-periodic in n, and \(\nabla W(n, x)\) is asymptotically linear in x as \(|x|\to\infty\).
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1 Introduction
Discrete Hamiltonian systems can be applied in many areas, such as physics, chemistry, and so on. For more discussions on discrete Hamiltonian systems, we refer the reader to [1, 2]. In this paper, we consider the second-order self-adjoint discrete Hamiltonian system
where \(n\in{\mathbb{Z}}\), \(u\in{\mathbb{R}}^{\mathcal{N}}\), \(\triangle u(n)=u(n+1)-u(n)\) is the forward difference, \(p, L: {\mathbb{Z}} \rightarrow{\mathbb{R}}^{\mathcal {N}\times\mathcal{N}}\) and \(W: {\mathbb{Z}}\times{\mathbb{R}}^{\mathcal{N}}\rightarrow {\mathbb{R}}\).
As usual, we say that a solution \(u(n)\) of system (1.1) is homoclinic (to 0) if \(u(n)\rightarrow0\) as \(n\to\pm\infty\). In addition, if \(u(n)\not\equiv0\) then \(u(n)\) is called a nontrivial homoclinic solution.
In recent years, several authors studied homoclinic orbits for system (1.1) or its special forms via critical point theory. For example, see [3–18]. We emphasize that in all these papers the nonlinear term was assumed to be superlinear or sublinear at infinity. To the best of our knowledge, the existence of homoclinics for asymptotically linear discrete Hamiltonian systems has not been previously studied.
In this paper, we assume that \(p(n)\) and \(L(n)\) are N-periodic \(\mathcal{N}\times\mathcal{N}\) real symmetric matrices. Let \(\mathcal{A}\) is an operator defined as follows:
Then it is easy to check that \(\mathcal{A}\) is a bounded self-adjoint operator in \(l^{2}(\mathbb {Z}, {\mathbb {R}}^{\mathcal{N}})\), where \(l^{2}(\mathbb {Z}, {\mathbb {R}}^{\mathcal{N}})\) is defined in Section 2. By the Floquet theorem, it is easy to verify that \(\mathcal{A}\) has only continuous spectrum \(\sigma(\mathcal{A})\), which is a union of bounded closed intervals.
When \(p(n)\) and \(L(n)\) are positive definite, \(\sigma(\mathcal {A})\subset(0, +\infty)\). In this case, the mountain pass theorem of Ambrosetti and Rabinowitz is a very useful tool for finding critical points of the energy functionals associated to (1.1). However, when \(p(n)\) or \(L(n)\) is not positive definite, 0 is a saddle point rather than a local minimum of the functional associated to (1.1), which is strongly indefinite. This case is difficult because the mountain-pass reduction of the definite case is not available, and it is not known if the Palais-Smale sequences are bounded. We choose this case as the object of the present paper.
To state our results, we first introduce the following assumptions:
-
(PL)
\(p(n)\) and \(L(n)\) are N-periodic \(\mathcal{N}\times\mathcal {N}\) real symmetric matrices, and
$$ \sup\bigl[\sigma(\mathcal{A})\cap(-\infty, 0)\bigr]:=\underline{ \Lambda }< 0<\bar{\Lambda}:=\inf\bigl[\sigma(\mathcal{A})\cap(0, \infty)\bigr]; $$(1.2) -
(W1)
\(W(n, x)\) is continuously differentiable in x for every \(n\in{ \mathbb {Z}}\), \(W(n, 0)=0\), \(W(n, x)\ge0\), and \(W(n, x)\) is N-periodic in n;
-
(W2)
\(\nabla W(n, x)=o(|x|)\) as \(|x|\to0\) uniformly for \(n\in \mathbb {Z}\);
-
(W3)
\(W(n, x)=\frac{1}{2}M(n)x\cdot x+ W_{\infty}(n, x)\), where \(M(n)\) is an N-periodic \(\mathcal{N}\times\mathcal{N}\) real symmetric matrix, \(\inf_{n\in \mathbb {Z}, |x|=1}M(n)x\cdot x> \bar{\Lambda}\), \(\nabla W_{\infty}(n, x)=o(|x|)\) as \(|x|\to\infty\), uniformly for \(n\in \mathbb {Z}\);
-
(W4)
\(\widetilde{W}(n, x):=\frac{1}{2}\nabla W(n, x)\cdot x-W(n, x)\ge0\), \(\forall (n, x)\in \mathbb {Z}\times \mathbb {R}^{\mathcal{N}}\), and there exists a \(\delta_{0}\in(0, \Lambda_{0})\) with \(\Lambda_{0}=\min\{-\underline{\Lambda}, \bar{\Lambda }\}\) such that
$$\frac{|\nabla W(n, x)|}{|x|}\ge\Lambda_{0}-\delta_{0} \quad\Rightarrow\quad \widetilde{W}(n, x)\ge\delta_{0}. $$
Now, we are ready to state the main result of this paper.
Theorem 1.1
Assume that p, L, and W satisfy (PL), (W1), (W2), (W3), and (W4). Then system (1.1) possesses a nontrivial homoclinic solution.
Remark 1.2
The following functions satisfy (W1)-(W4):
where \(a(n)\) and \(\alpha(n, s)\) are N-periodic positive function in n, \(\alpha(n, s)\) is non-decreasing for \(s\in[0, \infty)\), \(\alpha(n, s)\rightarrow0\) as \(s\to0\) and \(\alpha(n, s)\rightarrow b(n)\) as \(s\to\infty\) with \(\inf_{\mathbb {Z}} b>\bar{\Lambda}\), uniformly in \(n\in \mathbb {Z}\).
2 Proof of theorem
Let
As usual, for \(1\le q<\infty\), set
and
and their norms are defined by
respectively. In particular, \(l^{2}(\mathbb {Z}, {\mathbb {R}}^{\mathcal{N}})\) is a Hilbert space with the following inner product:
Let \(\{\mathcal{E}(\lambda): -a_{0} \le\lambda\le b_{0}\}\) and \(|\mathcal{A}|\) be the spectral family and the absolute value of \(\mathcal{A}\), respectively, and \(|\mathcal {A}|^{1/2}\) be the square root of \(|\mathcal{A}|\). Set \(\mathcal{U}=\mathrm{id}-\mathcal{E}(0)-\mathcal{E}(0-)\). Then \(\mathcal {U}\) commutes with \(\mathcal{A}\), \(|\mathcal{A}|\) and \(|\mathcal{A}|^{1/2}\), and \(\mathcal{A} = \mathcal{U}|\mathcal{A}|\) is the polar decomposition of \(\mathcal{A}\) (see [19, Theorem 4.3.3]).
As in [20], let \(E=l^{2}(\mathbb {Z}, {\mathbb {R}}^{\mathcal{N}})\) and
For any \(u\in E\), it is easy to see that
and
Let
Then E is a Hilbert space with the above inner product, and the corresponding norm is
By virtue of (2.1)-(2.4), one has the decomposition \(E=E^{-}\oplus E^{+}\) orthogonal with respect to both \((\cdot, \cdot)_{l^{2}}\) and \((\cdot, \cdot)\). Moreover,
and
Let X be a real Hilbert space with \(X=X^{-}\oplus X^{+}\) and \(X^{-}\bot X^{+}\). For a functional \(\varphi \in C^{1}(X, \mathbb {R})\), φ is said to be weakly sequentially lower semi-continuous if for any \(u_{k}\rightharpoonup u\) in X one has \(\varphi(u)\le\liminf_{n\to\infty}\varphi(u_{k})\), and \(\varphi'\) is said to be weakly sequentially continuous if \(\lim_{k\to\infty}\langle\varphi'(u_{k}), v\rangle= \langle\varphi'(u), v\rangle\) for each \(v\in X\).
Lemma 2.1
([21, Theorem 2.1])
Let X be a real Hilbert space with \(X=X^{-}\oplus X^{+}\) and \(X^{-}\bot X^{+}\), and let \(\varphi\in C^{1}(X, \mathbb {R})\) of the form
Suppose that the following assumptions are satisfied:
-
(LS1)
\(\psi\in C^{1}(X, \mathbb {R})\) is bounded from below and weakly sequentially lower semi-continuous;
-
(LS2)
\(\psi'\) is weakly sequentially continuous;
-
(LS3)
there exist \(r>\rho>0\) and \(e\in X^{+}\) with \(\|e\|=1\) such that
$$\kappa:=\inf\varphi\bigl(S^{+}_{\rho}\bigr) > \sup\varphi( \partial Q), $$where
$$S^{+}_{\rho}= \bigl\{ u\in X^{+} : \|u\|=\rho \bigr\} ,\qquad Q= \bigl\{ se+v : v\in X^{-}, s\ge0, \|se+v\|\le r \bigr\} . $$
Then for some \(c\ge\kappa\), there exists a sequence \(\{u_{n}\} \subset X\) satisfying
Such a sequence is called a Cerami sequence on the level c, or a \((C)_{c}\) sequence.
Now we define a functional Φ on E by
For any \(u\in E\), there exists an \(n_{0}\in \mathbb {N}\) such that \(|u(n)|\le1\) for \(|n|\ge n_{0}\). Hence, under assumptions (PL), (W1), and (W2), the functional Φ is of class \(C^{1}(E, \mathbb {R})\). Moreover,
By virtue of (2.1), (2.2), (2.3), and (2.4), one has
and
Furthermore, the critical points of Φ in E are solutions of system (1.1) with \(u(\pm\infty)=0\); see [6, 10].
Let
Then, by standard arguments, we can prove the following two lemmas.
Lemma 2.2
Suppose that (PL), (W1), and (W2) are satisfied. Then Ψ is nonnegative, weakly sequentially lower semi-continuous, and \(\Psi'\) is weakly sequentially continuous.
Lemma 2.3
Suppose that (PL), (W1), and (W2) are satisfied. Then there is a \(\rho>0\) such that \(\kappa:=\inf\Phi(S_{\rho}^{+})>0\), where \(S_{\rho }^{+}=\partial B_{\rho}\cap E^{+}\).
Let \(m_{0}:=\inf_{n\in \mathbb {Z}, |x|=1}M(n)x\cdot x\). Then (W3) implies that \(m_{0}> \bar{\Lambda}\). Since \(\sigma(\mathcal{A})\) is a union of closed intervals, we can choose \(e\in[\mathcal {E}(m_{1})-\mathcal{E}(\bar{\Lambda})]E\subseteq E^{+}\), where \(\bar{\Lambda}< m_{1}<m_{0}\). Thus,
Lemma 2.4
Suppose that (PL), (W1), (W2), and (W3) are satisfied. Then there is a \(r_{0}>0\) such that \(\sup\Phi(\partial Q)\le0\), where
Proof
Obviously, \(\Phi(w)\le0\) for \(w\in E^{-}\). It is sufficient to show that \(\Phi(w+te)\le0\) for \(t\ge0\), \(w\in E^{-}\) and \(\|w+te\|\ge r\) for large \(r>0\). Arguing indirectly, assume that for some sequence \(\{w_{k}+t_{k}e\}\subset E^{-}\oplus \mathbb {R}^{+} e\) with \(\|w_{k}+t_{k}e\| \rightarrow\infty\), \(\Phi(w_{k}+t_{k}e)\ge0\) for all \(k\in \mathbb {N}\). Set \(v_{k}=(w_{k}+t_{k}e)/\|w_{k}+t_{k}e\|=v_{k}^{-}+s_{k}e\), then \(\| v_{k}^{-}+s_{k}e\|=1\). Passing to a subsequence, we may assume that \(v_{k}\rightharpoonup v\) in E, then \(v_{k}(n)\rightarrow v(n)\) for all \(n\in \mathbb {Z}\), \(v_{k}^{-}\rightharpoonup v^{-}\) in E, \(s_{k}\rightarrow s\), and
Clearly, (2.15) yields \(s>0\). By virtue of (2.13), there exists a finite set \(\Pi\subset \mathbb {Z}\) such that
From (W3) and (2.15), one has
Clearly, \(|W_{\infty}(n, x)|\le c_{0}|x|^{2}\) for some \(c_{0}>0\) and \(W_{\infty}(n, x)/|x|^{2} \rightarrow0\) as \(|x|\to\infty\). Since \(v_{k} \rightharpoonup v\) in E, then \(v_{k}(n) \rightarrow v(n)\) for \(n\in\Pi\). Hence, one has
Hence
a contradiction to (2.16). □
Lemma 2.5
Suppose that (PL), (W1), (W2) and (W3) are satisfied. Then there exist a constant \(c>0\) and a sequence \(\{u_{k}\}\subset E\) satisfying
Proof
Lemma 2.5 is a direct corollary of Lemmas 2.1, 2.2, 2.3, and 2.4. □
Lemma 2.6
Suppose that (PL), (W1), (W2), (W3), and (W4) are satisfied. Then any sequence \(\{u_{k}\}\subset E\) satisfying (2.17) is bounded in E.
Proof
In view of (2.17), there exists a constant \(C_{0}>0\) such that
To prove the boundedness of \(\{u_{k}\}\), arguing by contradiction, suppose that \(\|u_{k}\| \to\infty\). Let \(v_{k}=u_{k}/\|u_{k}\|\). Then \(\|v_{k}\|=1\). Passing to a subsequence, we may assume that \(v_{k}\rightharpoonup v\) in E, then \(v_{k}(n)\rightarrow v(n)\) for all \(n\in \mathbb {Z}\). Let
Then by using \(\Lambda_{0}\|v_{k}\|_{2}^{2}\le\|v_{k}\|^{2}\), one has
If \(\delta:=\limsup_{k\to\infty}\|v_{k}\|_{\infty}=0\), then it follows from (W3), (W4), and (2.18) that
From (2.10), (2.11), (2.19), and (2.20), one gets
a contradiction. Thus \(\delta>0\).
Going if necessary to a subsequence, we may assume the existence of \(n_{k}\in \mathbb {Z}\) such that
Choose integers \(i_{k}\) and \(m_{k}\) with \(0\le m_{k}\le N-1\) such that \(n_{k}=i_{k}N+m_{k}\). Let \(\tilde{v}_{k}(n) =v_{k}(n+i_{k}N)\), then
Now we define \(\tilde{u}_{k}(n)=u_{k}(n+i_{k}N)\). Since \(p(n)\), \(L(n)\), and \(W(n, x)\) are N-periodic in n, then \(\tilde{u}_{k}/\|u_{k}\|=\tilde{v}_{k}\) and \(\|\tilde{u}_{k}\|=\|u_{k}\|\). Passing to a subsequence, we have \(\tilde{v}_{k}\rightharpoonup\tilde {v}\) in E, then \(\tilde{v}_{k}(n)\rightarrow \tilde{v}(n)\) for all \(n\in \mathbb {Z}\). Obviously, (2.22) implies that \(\tilde{v}(n)\ne0\) for some \(n\in\{0, 1, \ldots, N-1\}\). Let
For any \(\phi\in E_{0}\), there exists an \(n_{0}\in \mathbb {N}\) such that \(\phi (n)=0\) for all \(|n|>n_{0}\). Setting \(\phi_{k}(n)=\phi(n-i_{k}N)\), then it follows from (W3) and (2.9) that
Note that
Hence, it follows from (2.17) and (2.23) that
which yields
This shows that \(\tilde{v}\) is an eigenfunction of the operator ℬ, where
But ℬ has only continuous spectrum in E. This contradiction shows that \(\{u_{n}\}\) is bounded. □
Proof of Theorem 1.1
In view of Lemmas 2.5 and 2.6, there exists a bounded sequence \(\{u_{k}\}\subset E\) satisfying (2.17). Thus there exists a constant \(C_{3}>0\) such that
Hence, by (W1) and (W2), there exists a constant \(C_{4}>0\) such that
If \(\delta:=\limsup_{k\to\infty}\|u_{k}\|_{\infty}=0\), then
From (2.10), (2.11), (2.17), (2.26), (2.27), and (2.28), one has
This contradiction shows that \(\delta>0\).
Going if necessary to a subsequence, we may assume the existence of \(n_{k}\in \mathbb {Z}\) such that
Choose integers \(i_{k}\) and \(m_{k}\) with \(0\le m_{k}\le N-1\) such that \(n_{k}=i_{k}N+m_{k}\). Let \(v_{k}(n)=u_{k}(n+i_{k}N)\), then
Since \(p(n)\), \(L(n)\), and \(W(n, x)\) are N-periodic in n, we have \(\| v_{k}\|=\|u_{k}\|\) and
Passing to a subsequence, we have \(v_{k}\rightharpoonup v\) in E, \(v_{k}(n)\rightarrow v(n)\) for all \(n\in \mathbb {Z}\). Obviously, (2.29) implies that \(v\ne0\). It is easy to show that \(\Phi'(v)=0\). □
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Acknowledgements
This work is partially supported by the NNSF (No: 11471278) of China and Scientific Research Fund of Hunan Provincial Education Department (13A093).
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Wang, X. Homoclinic orbits for asymptotically linear discrete Hamiltonian systems. Adv Differ Equ 2015, 52 (2015). https://doi.org/10.1186/s13662-015-0390-1
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DOI: https://doi.org/10.1186/s13662-015-0390-1