Skip to content

Advertisement

  • Research
  • Open Access

Ground state solutions for periodic discrete nonlinear Schrödinger equations with saturable nonlinearities

Advances in Difference Equations20182018:176

https://doi.org/10.1186/s13662-018-1574-2

  • Received: 23 February 2018
  • Accepted: 20 March 2018
  • Published:

Abstract

In this paper, we study a class of periodic discrete nonlinear Schrödinger equations with asymptotically linear nonlinearities, and prove the existence of ground state solutions (i.e., nontrivial solutions with the least possible energy) by the linking theorem directly. By weakening some conditions, our conclusions extend some existing results.

Keywords

  • Discrete nonlinear Schrödinger equations
  • Ground state solutions
  • Spectral gap
  • Cerami sequence
  • Saturable nonlinearity

MSC

  • 35Q51
  • 35Q55
  • 39A12
  • 39A70

1 Introduction

The discrete nonlinear Schrödinger (DNLS) equations belong to the most important inherently discrete models, playing a crucial role in the modeling of a great variety of phenomena, ranging from solid-state and condensed-matter physics to biology ([13]). Particularly, they have been successfully applied to the modeling of localized pulse propagation in optical fibers and wave guides, to the study of energy relaxation in solids, to the behavior of amorphous material, to the modeling of self-trapping of vibrational energy in proteins or studies related to the denaturation of the DNA double strand (see [46] and the references therein).

Recently, many works in the literature have considered the existence of discrete solitons of the DNLS equations; see Refs. [712]. Results are obtained for such equations with superlinear nonlinearity [1319] and saturable nonlinearity [2023].

Assume that M is a positive integer. We consider the following DNLS equation:
$$ i\dot{\psi }_{n} = -\Delta \psi_{n} + \varepsilon_{n}\psi_{n}-f_{n}(u _{n}), \quad n=(n_{1},n_{2},\ldots,n_{M})\in \mathbb{Z} ^{M}, $$
(1.1)
where
$$ \begin{aligned} \Delta \psi_{n}&= \psi_{(n_{1}+1,n_{2},\ldots,n_{M})} + \psi_{(n_{1},n_{2}+1,\ldots,n_{M})}+\cdots+\psi_{(n_{1},n_{2},\ldots,n_{M}+1)} - 2m\psi_{(n_{1},n_{2},\ldots,n_{M})} \\ &\quad {} +\psi_{(n_{1}-1,n_{2},\ldots,n_{M})}+\psi_{(n_{1},n_{2}-1,\ldots,n_{M})}+ \cdots+\psi_{(n_{1},n_{2},\ldots,n_{M}-1)} \end{aligned} $$
is the discrete Laplacian in M spatial dimension. The given sequence \(\{\varepsilon_{n}\}\) is assumed to be real-valued and T-periodic in n, i.e., for \(n=(n_{1},n_{2},\ldots,n_{M})\in \mathbb{Z}^{M}\),
$$ \varepsilon_{(n_{1}+T_{1},n_{2},\ldots,n_{M})}= \varepsilon_{(n_{1},n_{2}+T_{2},\ldots,n_{M})}=\cdots = \varepsilon_{(n_{1},n_{2},\ldots,n_{M}+T _{M})}=\varepsilon_{(n_{1},n_{2},\ldots,n_{M})}, $$
where \(T=(T_{1},T_{2},\ldots,T_{M})\), \(T_{i}\) is a positive integer, \(i=1,2,\ldots,M\). We assume that \(f_{n}(0)=0\) for \(n\in \mathbb{Z}^{M}\) and the nonlinearity \(f_{n}(u)\) is T-periodic in n and gauge invariant in u, i.e.,
$$ f_{n}\bigl(e^{i\theta }u\bigr)=e^{i\theta }f_{n}(u), \quad \theta \in \mathbb{R}. $$
Solitons of (1.1) are spatially localized time-periodic solutions and decay to 0 at infinity, that is, \(\psi_{n}\) has the form
$$ \psi_{n} = u_{n}e^{-i\omega t} \quad \text{and}\quad \lim _{ \vert n \vert \rightarrow \infty }\psi_{n} = 0, $$
where \(\vert n \vert = \vert n_{1} \vert + \vert n_{2} \vert +\cdots+ \vert n_{M} \vert \) is the length of multi-index n, \(\{u_{n}\}\) is a real-valued sequence and \(\omega \in \mathbb{R}\) is the temporal frequency. Then (1.1) becomes
$$ -\Delta u_{n} + \varepsilon_{n}u_{n}- \omega u_{n} = f_{n}(u_{n}), \quad n\in \mathbb{Z}^{M}, $$
(1.2)
with
$$ \lim_{ \vert n \vert \rightarrow \infty }u_{n} = 0. $$
(1.3)
Naturally, if we look for solitons of (1.1), we just need to find the solutions of (1.2) satisfying (1.3).

Set \(F_{n}(s) = \int_{0}^{s}f_{n}(t) \,dt\), \(t\in \mathbb{R}\). This paper is organized as follows. In Sect. 2, we will introduce the variational framework associated with problem (1.2), Sect. 3 is devoted to the proof of the existence of ground state solutions for equation (1.2).

2 Variational framework and main results

Let
$$ l^{p}\equiv l^{p}\bigl(\mathbb{Z}^{M}\bigr)=\biggl\{ u=\{u_{n}\}_{n\in \mathbb{Z}^{M}}: \forall n\in \mathbb{Z}^{M},u_{n} \in \mathbb{R}, \Vert u \Vert _{l^{p}}=\biggl( \sum _{n\in \mathbb{Z}^{M}} \vert u_{n} \vert ^{p} \biggr)^{\frac{1}{p}}< \infty \biggr\} . $$
Then the following embedding between \(l^{p}\) spaces holds:
$$ l^{q}\subset l^{p}, \quad\quad \Vert u \Vert _{l^{p}} \leqslant \Vert u \Vert _{l^{q}}, \quad 1\leqslant q\leqslant p\leqslant \infty . $$

Let \(L=-\triangle +\varepsilon_{n}\), \(A = L - \omega \), the work space \(E = l^{2}(\mathbb{Z}^{M})\). It is well known that the spectral of L in E (denoted by \(\sigma (L)\)) has a band structure, i.e., \(\sigma (L)\) is a union of a finite number of closed intervals (see [24]), then the complement \(\mathbb{R}\setminus \sigma (L)\) consists of a finite number of open intervals called spectral gaps and two of them are semi-infinite. We fix one finite gap and denote it by \((\alpha ,\beta )\). Note that every element of E automatically satisfies (1.3).

In this paper, we impose the following assumptions on ω and \(f_{n}\):
\((V_{1})\)

\(\omega \in (\alpha ,\beta )\).

\((f_{1})\)

\(f_{n}\in C(\mathbb{R},\mathbb{R})\) and \(f_{n}\) is T-periodic in n, \(F_{n}(s)\geqslant 0\).

\((f_{2})\)

\(f_{n}(s)=o(s)\) as \(\vert s \vert \rightarrow 0\) for all \(n\in \mathbb{Z}^{M}\).

\((f_{3})\)
\(f_{n}(s) = V_{n}s+g_{n}(s)\), where \(V_{n}\) is T-periodic in n, \(0< V_{n}<\infty \), and there exists a \(u^{0} \in E_{0}^{+}\) such that
$$ \bigl\Vert u^{0} \bigr\Vert ^{2}- \Vert w \Vert ^{2}-\sum_{n\in \mathbb{Z}^{M}}V_{n} \bigl(u_{n}^{0}+w _{n}\bigr)^{2}< 0 \quad \forall w\in E^{-}; $$
(2.1)
\(sg_{n}(s)\leqslant 0\), \(f_{n}(s)g_{n}(s)<0\) for \(0< \vert s \vert \leqslant \alpha_{0}\) for some \(\alpha_{0}>0\), \(g_{n}(s)=o( \vert s \vert )\) as \(\vert s \vert \rightarrow \infty \) for all \(n\in \mathbb{Z}^{M}\).
\((f_{4})\)

\(s\mapsto f_{n}(s)/ \vert s \vert \) is increasing on \((-\infty ,0)\) and \((0,\infty )\) for all \(n\in \mathbb{Z}^{M}\).

Consider the functional Φ defined on E by
$$ \Phi (u) = \frac{1}{2}(Au,u)_{E} - \Psi (u),\quad\quad \Psi (u) = \sum_{n\in \mathbb{Z}^{M}}F_{n}(u_{n}). $$
(2.2)
Here \((\cdot ,\cdot )_{E}\) is the usual inner product in E. The corresponding norm is denoted by \(\Vert u \Vert _{E}\). Then \(\Phi , \Psi \in C^{1}(E,\mathbb{R})\) and the derivative of Φ is given by
$$ \bigl\langle \Phi '(u),v\bigr\rangle = (Au,v)_{E} - \bigl\langle \Psi '(u),v\bigr\rangle = (Au,v)_{E} - \sum_{n\in \mathbb{Z}^{M}}f_{n}(u_{n})v_{n}, \quad \forall u,v\in E. $$
(2.3)

Equation (2.3) implies that (1.2) is the corresponding Euler–Lagrange equation for Φ. Therefore, we have reduced the problem of finding a nontrivial solution of (1.2) to that of seeking a nonzero critical point of the functional Φ in E. By \((V_{1})\), we have \(\sigma (A)\subset \mathbb{R}\setminus (\alpha - \omega ,\beta -\omega )\). So, \(E = E^{+}\oplus E^{-}\) corresponds to the spectral decomposition of A with respect to the positive and negative parts of the spectrum.

Moreover, for any \(u,v\in E\), letting \(u = u^{+} + u^{-}\) with \(u^{\pm }\in E^{\pm }\) and \(v = v^{+} + v^{-}\) with \(v^{\pm }\in E ^{\pm }\), we can define an equivalent inner product \((\cdot ,\cdot )\) and the corresponding norm \(\Vert \cdot \Vert \) on E by
$$ (u,v)=\bigl(Au^{+},v^{+}\bigr)_{E}- \bigl(Au^{-},v^{-}\bigr)_{E}, \quad \Vert u \Vert =(u,u)^{ \frac{1}{2}}, $$
respectively. Therefore, Φ can be written as
$$ \Phi (u)=\frac{1}{2}\bigl( \bigl\Vert u^{+} \bigr\Vert ^{2}- \bigl\Vert u^{-} \bigr\Vert ^{2}\bigr)- \sum_{n\in \mathbb{Z}^{M}}F_{n}(u_{n}) $$
and we also have
$$ \bigl\langle \Phi '(u),u\bigr\rangle = \bigl\Vert u^{+} \bigr\Vert ^{2}- \bigl\Vert u^{-} \bigr\Vert ^{2}- \sum_{n\in \mathbb{Z}^{M}}f_{n}(u_{n})u_{n}. $$
If u is a nontrivial solution of problem (1.2), then \(u\in \mathcal{M}\), where
$$ \mathcal{M} := \bigl\{ u\in E\setminus E^{-}: \bigl\langle \Phi '(u),u\bigr\rangle = \bigl\langle \Phi '(u),v\bigr\rangle =0\text{ for all } v\in E^{-}\bigr\} . $$
The above set was first introduced in [25]. Set \(m:=\inf_{u\in \mathcal{M}}\Phi (u)\). Inspired by previous work due to Tang [26], we investigate the existence of ground state solutions of (1.2). His main idea is to find a minimizing Cerami sequence for Φ outside \(\mathcal{M}\) by using the diagonal method; see Lemma 3.10.

Now, our main result is the following:

Theorem 2.1

Suppose that \((V_{1})\), \((f_{1})\)\((f_{4})\) are satisfied. Then Eq. (1.2) has at least a nontrivial ground state solution \(\bar{u}\in E\) such that \(\Phi (\bar{u})=\inf_{\mathcal{M}}\Phi >0\). Moreover,
$$ \bigl\Vert \bar{u}^{+} \bigr\Vert ^{2}- \bigl\Vert \bar{u}^{-} \bigr\Vert ^{2}-\sum _{n\in \mathbb{Z}^{M}}V _{n}(\bar{u}_{n})^{2}< 0. $$

Remark 2.2

Set \(L^{+}=L| _{E^{+}}\), \(L^{-}=L| _{E^{-}}\). Note that if we substitute condition \((f_{3})\) by \((f'_{3})\):
\((f'_{3})\)

\(f_{n}(s) = V_{n}s+g_{n}(s)\), where \(V_{n}\) is T-periodic in n, \(\inf V_{n}>\inf \sigma (L^{+})-\omega \), \(g_{n}(s)=o( \vert s \vert )\) as \(\vert s \vert \rightarrow \infty \) for all \(n\in \mathbb{Z}^{M}\), and \(0< sf_{n}(s)< V_{n}s^{2}\) for all \(n\in \mathbb{Z}^{M}\) and \(s\neq 0\),

then the conclusions of Theorem 2.1 can also be derived because \((f'_{3})\) implies \((f_{3})\).
Here, we mention that, as a result of Theorem 2.1, the least energy value m has a minimax characterization given by
$$ m=\Phi (\bar{u})=\min_{v\in E_{0}^{+}\setminus \{0\}} \max_{u\in E^{-}\oplus \mathbb{R}^{+}v}\Phi (u), $$
where \(E_{0}^{+}\) is defined in (3.6).

Remark 2.3

In [16], A. Mai and Z. Zhou treated the discrete nonlinear Schrödinger equation with superquadratic nonlinearity, they required the condition
\((f'_{4})\)

\(s\mapsto f_{n}(s)/ \vert s \vert \) is strictly increasing on \((-\infty ,0)\) and \((0,\infty )\) for all \(n\in \mathbb{Z}\),

and obtained ground state solutions by using the generalized Nehari manifold approach developed by Szulkin and Weth [27]. In [15], they considered the following DNLS equation in M dimensional lattices:
$$ -\Delta u_{n} + \varepsilon_{n}u_{n}-\omega u_{n} = f(n,u_{n}), \quad n\in \mathbb{Z}^{M}. $$
By using S. Liu’s method in [28], they replaced \((f'_{4})\) by \((f_{4})\), and applied the generalized linking theorem of Li and Szulkin [29] to obtain the ground state solutions.

Remark 2.4

We point out that when the nonlinear term is asymptotically linear, Chen and Ma [20] proved (1.2) has at least one ground state solution by employing generalized Nehari manifold method if \((f'_{4})\) holds. In our paper, we use \((f_{4})\) instead of \((f'_{4})\). Sun and Ma [30] treated the multiplicity of solutions on problem (1.2), where the nonlinearities contains both asymptotically linear nonlinearity and superlinear nonlinearity. When they considered the asymptotically linear case, they assumed that \((V_{1})\), \((f_{1})\), \((f_{2})\) and the following two assumptions hold:
  1. (SM1)

    \(f_{n}(s)-V_{n}s=o( \vert s \vert )\) as \(\vert s \vert \rightarrow \infty \) with \(\inf V_{n}>\beta -\omega \);

     
  2. (SM2)

    \(\widetilde{F}_{n}(s)>0\) if \(s\neq 0\) and \(\liminf_{ \vert s \vert \rightarrow \infty }\widetilde{F}_{n}(s)>0\),

     
where \(\widetilde{F}_{n}(s)=\frac{1}{2}f_{n}(s)s-F_{n}(s)\). Note that \((f_{4})\) implies that \(\widetilde{F}_{n}(s)\geqslant 0\), which is weaker than (SM2), and the assumption \(\inf V_{n}>\beta -\omega \) in (SM1) is stronger than our assumption that \(V_{n}>0\). Particularly, if \((V_{1})\), \((f_{1})\), \((f_{2})\) and the following two assumptions hold:
  1. (G3)

    \(\widetilde{F}_{n}(s)>0\) if \(s\neq 0\);

     
  2. (G4)
    There is \(\zeta \in (0,\frac{\eta }{2}]\) such that
    $$ \frac{f_{n}(s)}{s}\geqslant \frac{\eta }{2}-\zeta \quad \Rightarrow\quad \widetilde{F}_{n}(s)\geqslant \zeta ,\quad \forall n\in \mathbb{Z}. $$
     
Then a necessary and sufficient condition for the existence of gap solitons of the DNLS equations was given by Chen, Ma and Wang [21] when \(V_{n}\equiv V_{0}\) (\(n\in \mathbb{Z}\), \(0< V_{0}\in \mathbb{R}\)).

In order to complete our proof, let us recall the following linking theorem taken from [29].

Proposition 2.5

Let E be a real Hilbert space with \(E = E^{+}\oplus E^{-}\) and \(E^{+}\bot E^{-}\) and let \(I\in C^{1}(E,\mathbb{R})\) be of the form
$$ I(u)=\frac{1}{2}\bigl( \bigl\Vert u^{+} \bigr\Vert ^{2}- \bigl\Vert u^{-} \bigr\Vert ^{2} \bigr)-J(u),\quad u=u^{+}+u ^{-}\in E^{+}\oplus E^{-}. $$
Suppose that the following assumptions are satisfied:
  1. (1)

    \(J\in C^{1}(E,\mathbb{R})\) is bounded from below and weakly sequentially lower semi-continuous;

     
  2. (2)

    \(J'\) is weakly sequentially continuous;

     
  3. (3)
    there exist \(r>\rho >0\) and \(e\in E^{+}\) with \(\Vert e \Vert =1\) such that
    $$ \kappa :=\inf I\bigl(S_{\rho }^{+}\bigr)>\sup I(\partial Q), $$
    where
    $$ S_{\rho }^{+} = \bigl\{ u\in E^{+}| \Vert u \Vert =\rho \bigr\} ,\quad\quad Q=\bigl\{ w+se : w\in E ^{-}, s\geqslant 0, \Vert w+se \Vert \leqslant r \bigr\} . $$
     
Then, for some \(c\in [\kappa ,\sup I(Q)]\), there exists a sequence \(\{u_{n}\}\subset E\) satisfying
$$ I(u_{n})\rightarrow c, \quad\quad \bigl\Vert I'(u_{n}) \bigr\Vert \bigl(1+ \Vert u_{n} \Vert \bigr)\rightarrow 0. $$
(2.4)

Such a sequence is called a Cerami sequence on the level c, or a \((C)_{c}\) sequence.

3 Proof of Theorem 2.1

This section is devoted to a proof of Theorem 2.1, and we begin with an important property of Ψ defined in (2.1).

Lemma 3.1

Suppose that \((f_{1})\)\((f_{3})\) are satisfied, then Ψ is nonnegative, weakly sequentially lower semi-continuous, and \(\Psi '\) is weakly sequentially continuous.

Proof

The proof is similar to the proof of Lemma 3.1 in [15], we omit it here. □

Lemma 3.2

([26])

Suppose that \(h(x,t)\) is nondecreasing in \(t\in \mathbb{R}\) and \(h(x,0)=0\) for any \(x\in \mathbb{R}^{N}\). Then
$$ \biggl(\frac{1-\theta^{2}}{2}\tau - \theta \sigma \biggr)h(x,t) \vert \tau \vert \geqslant \int_{\theta \tau +\sigma }^{\tau }h(x,s) \vert s \vert \,ds\quad \forall \theta \geqslant 0,\tau ,\sigma \in \mathbb{R}. $$
(3.1)

Lemma 3.3

Assume \((V_{1})\), \((f_{1})\)\((f_{4})\). Then
$$ \begin{aligned}[b] &\Phi (u) \geqslant \Phi (\theta u+w) + \frac{1}{2} \Vert w \Vert ^{2}+\frac{1- \theta^{2}}{2}\bigl\langle \Phi '(u),u\bigr\rangle -\theta \bigl\langle \Phi '(u),w \bigr\rangle \\ &\quad \forall \theta \geqslant 0, u\in E, w\in E^{-}. \end{aligned} $$
(3.2)

Proof

It follows from \((f_{4})\) and Lemma 3.2 that
$$ \biggl(\frac{1-\theta^{2}}{2}\tau - \theta \sigma \biggr)f_{n}(\tau ) \geqslant \int_{\theta \tau +\sigma }^{\tau }f_{n}(s) \,ds\quad \forall \theta \geqslant 0,\tau ,\sigma \in \mathbb{R}. $$
(3.3)
Then, in the same way as Lemma 2.4 in [26], we can show that (3.2) holds. □

Corollary 3.4

Assume \((V_{1})\), \((f_{1})\)\((f_{4})\). Then, for \(u\in \mathcal{M}\),
$$ \Phi (u) \geqslant \Phi (tu+w) \quad \forall t\geqslant 0, w\in E^{-}. $$
Especially, we have
$$ \begin{aligned} \Phi (u) &\geqslant \frac{t^{2}}{2}\bigl( \bigl\Vert u^{+} \bigr\Vert ^{2}+ \bigl\Vert u^{-} \bigr\Vert ^{2}\bigr)- \sum _{n\in \mathbb{Z}^{M}}F_{n}\bigl(tu_{n}^{+} \bigr) +\frac{1-t^{2}}{2}\bigl\langle \Phi '(u),u\bigr\rangle \\ & \quad {} + t^{2}\bigl\langle \Phi '(u),u^{-}\bigr\rangle \quad \forall u\in E, t \geqslant 0. \end{aligned} $$

The following lemma gives an important computation technique to obtain the linking geometry.

Lemma 3.5

Assume \((V_{1})\), \((f_{1})\)\((f_{3})\). Then
$$ \begin{aligned}[b] \tau \bigl\langle \Phi '(u), \tau u+2v\bigr\rangle &\geqslant \tau^{2} \bigl\Vert u^{+} \bigr\Vert ^{2}- \bigl\Vert \tau u^{-}+v \bigr\Vert ^{2}+ \Vert v \Vert ^{2}- \sum_{n\in \mathbb{Z}^{M}}V_{n}( \tau u_{n}+v_{n})^{2} \\ & \quad {} +\tau^{2}\sum_{n\in \mathbb{Z}^{M}}\frac{V_{n}f_{n}(u_{n})u_{n}-[f _{n}(u_{n})]^{2}}{V_{n}} \quad \forall u\in E, \tau \in \mathbb{R}, v \in E^{-}. \end{aligned} $$
(3.4)
It follows from (3.4) that
$$ \begin{aligned}[b] &\bigl\Vert u^{+} \bigr\Vert ^{2} - \bigl\Vert u^{-}+v \bigr\Vert ^{2}-\sum_{n\in \mathbb{Z}^{M}}V_{n}(u_{n}+v _{n})^{2} \\ &\quad \leqslant - \Vert v \Vert ^{2}-\sum_{n\in \mathbb{Z}^{M}} \frac{V_{n}f_{n}(u _{n})u_{n}-[f_{n}(u_{n})]^{2}}{V_{n}} \quad \forall u\in \mathcal{M}, v\in E^{-}. \end{aligned} $$
(3.5)

Proof

The proof is similar to the proof of Lemma 2.7 in [26], we omit it here. □

Lemma 3.6

Assume \((V_{1})\), \((f_{1})\)\((f_{4})\). Then
  1. (i)
    there exists \(\rho >0\) such that
    $$ m = \inf_{\mathcal{M}}\Phi \geqslant \kappa :=\inf \bigl\{ \Phi (u):u \in E ^{+}, \Vert u \Vert =\rho \bigr\} >0; $$
     
  2. (ii)

    \(\Vert u^{+} \Vert \geqslant \max \{ \Vert u^{-} \Vert ,\sqrt{2m}\}\) for all \(u\in \mathcal{M}\).

     

Proof

(i) By \((f_{2})\) and \((f_{3})\), we can see that, for any \(\varepsilon >0\), there exists a constant \(C_{\varepsilon }>0\) such that
$$ \bigl\vert f_{n}(s) \bigr\vert \leqslant \varepsilon \vert s \vert + C_{\varepsilon } \vert s \vert ^{p-1} \quad \textrm{and}\quad \bigl\vert F_{n}(s) \bigr\vert \leqslant \varepsilon s^{2} + C_{\varepsilon } \vert s \vert ^{p}, \quad p\geqslant 2. $$
For \(u\in E^{+}\), since \(\Vert \cdot \Vert \) is equivalent to the E norm on \(E^{+}\), and recalling that \(E\subset l^{p}\) for \(2\leqslant p\leqslant \infty \) with \(\Vert u \Vert _{l^{p}} \leqslant \Vert u \Vert _{E}\), we have
$$ \Phi (u)= \frac{1}{2} \Vert u \Vert ^{2}-\sum _{n\in \mathbb{Z}^{M}}F_{n}(u_{n}) =o\bigl( \Vert u \Vert ^{2}\bigr). $$
Hence we can get \(\kappa :=\inf \{\Phi (u):u\in E^{+}, \Vert u \Vert =\rho \}>0\) if \(\rho >0\) is small enough. Since for every \(u\in \mathcal{M}\) there exists \(t>0\) such that \(tu^{+}\in S_{\rho }\), and by Corollary 3.4, \(\Phi (tu^{+})=\Phi (tu-tu^{-})\leqslant \Phi (u)\), then it turns out that \(\inf_{\mathcal{M}}\Phi \geqslant \inf_{S_{\rho }} \Phi \).
(ii) For \(u\in \mathcal{M}\), we have
$$ m\leqslant \frac{1}{2}\bigl( \bigl\Vert u^{+} \bigr\Vert ^{2}- \bigl\Vert u^{-} \bigr\Vert ^{2}\bigr)- \sum_{n\in \mathbb{Z}^{M}}F_{n}(u_{n}) \leqslant \frac{1}{2}\bigl( \bigl\Vert u^{+} \bigr\Vert ^{2}- \bigl\Vert u^{-} \bigr\Vert ^{2}\bigr), $$
hence \(\Vert u^{+} \Vert \geqslant \max \{\sqrt{2m}, \Vert u^{-} \Vert \}\). □
Now, we define a set \(E_{0}^{+}\) mentioned above as follows:
$$ E_{0}^{+} = \biggl\{ u\in E^{+} \setminus \{0\}: \Vert u \Vert ^{2}- \Vert w \Vert ^{2}- \sum_{n\in \mathbb{Z}^{M}}V_{n}(u_{n}+w_{n})^{2}< 0 \ \forall w \in E^{-}\biggr\} . $$
(3.6)
It is easy to see that \((f_{3})\) guarantees that the set \(E_{0}^{+}\) is not empty.

Lemma 3.7

Suppose that \((V_{1})\), \((f_{1})\)\((f_{3})\) are satisfied. Then, for any \(e\in E_{0}^{+}\), \(\sup \Phi (E^{-}\oplus \mathbb{R^{+}}e)<\infty \) and there is \(R_{e}\) such that
$$ \Phi (u)\leqslant 0 \quad \forall u\in E^{-}\oplus \mathbb{R^{+}}e, \Vert u \Vert \geqslant R_{e}. $$

Proof

Arguing by contradiction, suppose that there exists \(\{w^{j}+s_{j}e\} \subset E^{-}\oplus \mathbb{R^{+}}e\) with \(\Vert w^{j}+s_{j}e \Vert \rightarrow \infty \), such that \(\Phi (w^{j}+s_{j}e)\geqslant 0\). Set
$$ v^{j}=\frac{w^{j}+s_{j}e}{ \Vert w^{j}+s_{j}e \Vert }=\bigl(v^{j}\bigr)^{-}+ \tau_{j}e; $$
then \(\Vert (v^{j})^{-}+\tau^{j}e \Vert = \Vert v^{j} \Vert =1\). Then
$$ 0\leqslant \frac{\Phi (w^{j}+s_{j}e)}{ \Vert w^{j}+s_{j}e \Vert ^{2}}=\frac{\tau _{j}^{2}}{2} \Vert e \Vert ^{2} -\frac{1}{2} \bigl\Vert \bigl(v^{j} \bigr)^{-} \bigr\Vert ^{2}- \sum _{n\in \mathbb{Z}^{M}}\frac{F_{n}(w_{n}^{j}+s_{j}e_{n})}{(w_{n} ^{j}+s_{j}e_{n})^{2}}\bigl(v_{n}^{j} \bigr)^{2}. $$
(3.7)
Hence \(\frac{1}{2}\leqslant \tau_{j}^{2} \Vert e \Vert ^{2}\leqslant 1\). Passing to a subsequence, we have \(v^{j}\rightharpoonup v\) in E, \(v_{n}^{j} \rightarrow v_{n}\) for all \(n\in \mathbb{Z}^{M}\) and \(\tau_{j}\rightarrow \tau >0\) as \(j\rightarrow \infty \). Since \(e\in E_{0}^{+}\), there exists a finite set \(A\subset \mathbb{Z}^{M}\) such that
$$ \tau^{2} \Vert e \Vert ^{2}- \bigl\Vert v^{-} \bigr\Vert ^{2}-\sum_{n\in A}V_{n} \bigl(\tau e_{n}+v_{n} ^{-}\bigr)^{2}< 0. $$
(3.8)
Let \(G_{n}(s)=\int_{0}^{s}g_{n}(t) \,dt\), then \(F_{n}(s)=\frac{1}{2}V _{n}s^{2}+G_{n}(s)\). From (3.7), we obtain
$$ \begin{aligned} 0 &\leqslant \frac{\tau_{j}^{2}}{2} \Vert e \Vert ^{2} -\frac{1}{2} \bigl\Vert \bigl(v^{j} \bigr)^{-} \bigr\Vert ^{2}- \sum _{n\in A}\frac{F_{n}(w_{n}^{j}+s_{j}e_{n})}{(w_{n}^{j}+s _{j}e_{n})^{2}}\bigl(v_{n}^{j} \bigr)^{2} \\ &=\frac{\tau_{j}^{2}}{2} \Vert e \Vert ^{2} -\frac{1}{2} \bigl\Vert \bigl(v^{j}\bigr)^{-} \bigr\Vert ^{2}- \frac{1}{2}\sum_{n\in A}V_{n} \bigl(v_{n}^{j}\bigr)^{2}-\sum _{n\in A}\frac{G_{n}(w _{n}^{j}+s_{j}e_{n})}{(w_{n}^{j}+s_{j}e_{n})^{2}}\bigl(v_{n}^{j} \bigr)^{2}. \end{aligned} $$
By \((f_{2})\) and \((f_{3})\), there exists some \(C_{1}>0\) such that \(\vert G_{n}(s) \vert \leqslant C_{1}s^{2}\), and \(G_{n}(s)/s^{2}\rightarrow 0\) as \(\vert s \vert \rightarrow \infty \) for all \(n\in \mathbb{Z}^{M}\). Since \((v^{j})^{-}\rightharpoonup v^{-}\) in E and \(v_{n}^{j}\rightarrow v _{n}\) as \(j\rightarrow \infty \) for all \(n\in \mathbb{Z}^{M}\), we have
$$ \sum_{n\in A} \frac{G_{n}(w_{n}^{j}+s_{j}e_{n})}{(w_{n}^{j}+s_{j}e_{n})^{2}}\bigl(v_{n} ^{j}\bigr)^{2}=o(1). $$
Therefore,
$$ 0\leqslant \tau^{2} \Vert e \Vert ^{2} - \bigl\Vert v^{-} \bigr\Vert ^{2}- \sum_{n\in A}V_{n} \bigl( \tau e_{n}+v_{n}^{-}\bigr)^{2}, $$
which contradicts with (3.8). □

Corollary 3.8

Assume \((V_{1})\), \((f_{1})\)\((f_{3})\). Let \(e\in E_{0}^{+}\) with \(\Vert e \Vert =1\). Then there is a \(r_{0}>\rho \) such that \(\sup \Phi (\partial Q)\leqslant 0\) for \(r\geqslant r_{0}\), where
$$ Q = \bigl\{ w+se : w\in E^{-}, s\geqslant 0, \Vert w+se \Vert \leqslant r \bigr\} . $$
(3.9)

Lemma 3.9

Assume \((V_{1})\), \((f_{1})\)\((f_{4})\). Then,
  1. (i)

    For any \(u\in E_{0}^{+}\), \(\mathcal{M}\cap (E^{-}\oplus \mathbb{R^{+}}u)\neq \emptyset \), that is, there exist \(t(u)>0\) and \(w(u)\in E^{-}\) such that \(t(u)u+w(u)\in \mathcal{M}\).

     
  2. (ii)
    There exists constant \(c\in [\kappa ,\sup \Phi (Q)]\) and a sequence \(\{u^{j}\}\subset E\) satisfying
    $$ \Phi \bigl(u^{j}\bigr)\rightarrow c,\quad\quad \bigl\Vert \Phi '\bigl(u^{j}\bigr) \bigr\Vert \bigl(1+ \bigl\Vert u^{j} \bigr\Vert \bigr)\rightarrow 0, $$
    where Q is defined by (3.9).
     

Proof

(i) The proof is similar to Lemma 2.12 in [26], we omit it here.

(ii) The conclusion is a directly corollary of Proposition 2.5 and Lemma 3.6(i) and Corollary 3.8. □

The following lemma takes an important part in demonstrating the existence of ground state solutions for problem (1.2).

Lemma 3.10

Assume \((V_{1})\), \((f_{1})\)\((f_{4})\). Then there exist a constant \(c_{*}\in [\kappa ,m]\) and a sequence \(\{u^{j}\}\subset E\) satisfying
$$ \Phi \bigl(u^{j}\bigr)\rightarrow c_{*}, \quad\quad \bigl\Vert \Phi '\bigl(u^{j}\bigr) \bigr\Vert \bigl(1+ \bigl\Vert u^{j} \bigr\Vert \bigr) \rightarrow 0. $$
(3.10)

Proof

First, choosing \(v^{k}\in \mathcal{M}\) such that
$$ m\leqslant \Phi \bigl(v^{k}\bigr)< m+ \frac{1}{k},\quad k\in \mathbb{N}. $$
(3.11)
By Lemma 3.6(ii), \(\Vert (v^{k})^{+} \Vert \geqslant \sqrt{2m}>0\). Since \(v^{k}\in E\), then \(\lim_{ \vert n \vert \rightarrow \infty } \vert v_{n}^{k} \vert =0\), i.e., there is an infinite subset of \(\mathbb{Z}^{M}\) where \(\vert v_{n}^{k} \vert \leqslant \alpha_{0}\), \(k\in \mathbb{N}\). Note that \((f_{3})\) asserts that \(f_{n}(s)g_{n}(s)\leqslant 0\) for all \(n\in \mathbb{Z}^{M}\) when \(\vert s \vert \geqslant 0\), then we have
$$ \sum_{n\in \mathbb{Z}^{M}}\frac{f_{n}(v_{n}^{k})g_{n}(v_{n}^{k})}{V _{n}}< 0. $$
(3.12)
Let \(e^{k}=(v^{k})^{+}/ \Vert (v^{k})^{+} \Vert \). Then \(e^{k}\in E^{+}\) and \(\Vert e^{k} \Vert =1\). According to (3.5) and (3.12),
$$ \begin{aligned} &\bigl\Vert e^{k} \bigr\Vert ^{2} - \Vert w \Vert ^{2}-\sum _{n\in \mathbb{Z}^{M}}V_{n}\bigl(e_{n}^{k}+w _{n}\bigr)^{2} \\ & \quad = \frac{ \Vert (v^{k})^{+} \Vert ^{2}}{ \Vert (v^{k})^{+} \Vert ^{2}}- \Vert w \Vert ^{2}- \sum _{n\in \mathbb{Z}^{M}}V_{n} \biggl(\frac{v_{n}^{k}}{ \Vert (v^{k})^{+} \Vert }+w _{n}-\frac{(v_{n}^{k})^{-}}{ \Vert (v^{k})^{+} \Vert }\biggr)^{2} \\ & \quad \leqslant - \biggl\Vert w-\frac{(v^{k})^{-}}{ \Vert (v^{k})^{+} \Vert } \biggr\Vert ^{2}- \frac{1}{ \Vert (v^{k})^{+} \Vert ^{2}} \sum_{n\in \mathbb{Z}^{M}}\frac{v_{n}^{k}f(n,v _{n}^{k})V_{n}-[f(n,v_{n}^{k})]^{2}}{V_{n}} \\ & \quad = - \biggl\Vert w-\frac{(v^{k})^{-}}{ \Vert (v^{k})^{+} \Vert } \biggr\Vert ^{2}+ \frac{1}{ \Vert (v ^{k})^{+} \Vert ^{2}} \sum_{n\in \mathbb{Z}^{M}}\frac{f(n,v_{n}^{k})g(n,v _{n}^{k})}{V_{n}}< 0 \quad \forall w\in E^{-}. \end{aligned} $$
This shows that \(e^{k}\in E_{0}^{+}\). By virtue of Corollary 3.8, there exists \(r_{k}>\max \{\rho , \Vert v^{k} \Vert \}\) such that \(\sup \Phi (\partial Q_{k})\leqslant 0\), where
$$ Q_{k}=\bigl\{ w+se_{k}:w\in E^{-},s\geqslant 0, \Vert w+se_{k} \Vert \leqslant r_{k} \bigr\} , \quad k\in \mathbb{N}. $$
(3.13)
Therefore, applying Lemma 3.9(ii) to the above set \(Q_{k}\), there exist a positive constant \(c_{k}\in [\kappa ,\sup \Phi (Q_{k})]\) and a sequence \(\{u^{k,j}\}_{j\in \mathbb{N}}\subset E\) satisfying
$$ \Phi \bigl(u^{k,j}\bigr)\rightarrow c_{k}, \quad\quad \bigl\Vert \Phi '\bigl(u^{k,j}\bigr) \bigr\Vert \bigl(1+ \bigl\Vert u^{k,j} \bigr\Vert \bigr)\rightarrow 0,\quad k\in \mathbb{N}, $$
(3.14)
where \(j\rightarrow \infty \). In view of Corollary 3.4, we derive
$$ \Phi \bigl(v^{k}\bigr)\geqslant \Phi \bigl(w+tv^{k}\bigr),\quad \forall t\geqslant 0,w \in E^{-}. $$
(3.15)
Since \(v^{k}\in Q_{k}\), it follows from (3.13) and (3.15) that \(\Phi (v^{k})=\sup \Phi (Q_{k})\). Hence, by (3.11) and (3.14),
$$ \Phi \bigl(u^{k,j}\bigr)\rightarrow c_{k}< m+ \frac{1}{k},\quad\quad \bigl\Vert \Phi '\bigl(u^{k,j} \bigr) \bigr\Vert \bigl(1+ \bigl\Vert u^{k,j} \bigr\Vert \bigr) \rightarrow 0,\quad k\in \mathbb{N}. $$
Now, we can choose a sequence \(\{j_{k}\}\subset \mathbb{N}\) such that
$$ \kappa -\frac{1}{k}< \Phi \bigl(u^{k,j_{k}}\bigr)\rightarrow c_{k}< m+\frac{1}{k}, \quad\quad \bigl\Vert \Phi ' \bigl(u^{k,j_{k}}\bigr) \bigr\Vert \bigl(1+ \bigl\Vert u^{k,j_{k}} \bigr\Vert \bigr)\rightarrow 0,\quad k \in \mathbb{N}. $$
Let \(u^{k}=u^{k,j_{k}}\), \(k\in \mathbb{N}\). Then, passing to a subsequence if necessary, we have
$$ \Phi \bigl(u^{j}\bigr)\rightarrow c_{*}\in [\kappa ,m], \quad\quad \bigl\Vert \Phi '\bigl(u^{j}\bigr) \bigr\Vert \bigl(1+ \bigl\Vert u^{j} \bigr\Vert \bigr)\rightarrow 0. $$
 □

Lemma 3.11

Assume \((V_{1})\), \((f_{1})\)\((f_{4})\). Then any sequence \(\{u^{j}\} \subset E\) satisfying (3.10) is bounded in E.

Proof

Arguing by contradiction, suppose there exists a sequence \(\{u^{j}\} \subset E\) such that \(\Vert u^{j} \Vert \rightarrow \infty \). Let \(v^{j}:= \Vert u ^{j} \Vert ^{-1}u^{j}\), then \(\Vert v^{j} \Vert =1\). Since \(\Vert \cdot \Vert \) is equivalent to the E norm on \(E^{+}\), there exists a constant \(C>0\) such that \(\Vert (v^{j})^{+} \Vert _{E}\leqslant C\). After passing to a subsequence, we have \(v^{j}\rightharpoonup v\) in E and \(v_{n}^{j}\rightarrow v_{n}\) as \(j\rightarrow \infty \) for all \(n\in \mathbb{Z}^{M}\).

We claim that there exist \(\delta >0\) and \(n_{j}\in \mathbb{Z}^{M}\) satisfy
$$ \bigl\vert \bigl(v_{n_{j}}^{j}\bigr)^{+} \bigr\vert \geqslant \delta . $$
Indeed, if not, then \((v^{j})^{+}\rightarrow 0\) in \(l^{\infty }\) as \(j\rightarrow \infty \). Since \(\Vert (v^{j})^{+} \Vert _{E}\) is bounded, and note the simple fact that
$$ \bigl\Vert \bigl(v^{j}\bigr)^{+} \bigr\Vert _{l^{q}}^{q}\leqslant \bigl\Vert \bigl(v^{j} \bigr)^{+} \bigr\Vert _{l^{\infty }} ^{q-2 } \bigl\Vert \bigl(v^{j}\bigr)^{+} \bigr\Vert _{l^{2}}^{2}, \quad q>2. $$
(3.16)
If \(\delta =0\), then by (3.16) we can deduce that \((v^{j})^{+} \rightarrow 0\) in all \(l^{q}\), \(q>2\). Fix \(R>[2(1+c_{*})]^{1/2}\), and \(p>2\), in view of \((f_{1})\) and \((f_{3})\), choose \(\varepsilon = \frac{1}{4(RC)^{2}}>0\), there exists \(C_{\varepsilon }>0\) such that \(\vert F_{n}(s) \vert \leqslant \varepsilon \vert s \vert ^{2}+C_{\varepsilon } \vert s \vert ^{p}\). Therefore,
$$ \limsup_{j\rightarrow \infty }\sum_{n\in \mathbb{Z}^{M}}F_{n} \bigl(R\bigl(v_{n} ^{j}\bigr)^{+}\bigr)\leqslant \Bigl[\varepsilon (RC)^{2} + R^{p} C_{\varepsilon } \lim _{j\rightarrow \infty } \bigl\Vert \bigl(v^{j}\bigr)^{+} \bigr\Vert _{l^{p}}^{p} \Bigr] = \frac{1}{4}. $$
(3.17)
Let \(t_{j} = R/ \Vert u^{j} \Vert \). By Corollary 3.4 and (3.10), (3.17),
$$ \begin{aligned} c_{*}+o(1) &=\Phi \bigl(u^{j} \bigr)\geqslant \frac{t_{j}^{2}}{2}\bigl( \bigl\Vert \bigl(u^{j} \bigr)^{+} \bigr\Vert ^{2}+ \bigl\Vert \bigl(u^{j}\bigr)^{-} \bigr\Vert ^{2}\bigr)-\sum _{n\in \mathbb{Z}^{M}}F_{n}\bigl(t_{j} \bigl(u_{n} ^{j}\bigr)^{+}\bigr) \\ &\quad {} +\frac{1-t_{j}^{2}}{2}\bigl\langle \Phi '\bigl(u^{j} \bigr),u^{j}\bigr\rangle + t_{j} ^{2}\bigl\langle \Phi '\bigl(u^{j}\bigr),\bigl(u^{j} \bigr)^{-}\bigr\rangle \\ &=\frac{R^{2}}{2}\bigl( \bigl\Vert \bigl(v^{j} \bigr)^{+} \bigr\Vert ^{2}+ \bigl\Vert \bigl(v^{j}\bigr)^{-} \bigr\Vert ^{2}\bigr)- \sum_{n\in \mathbb{Z}^{M}}F\bigl(n,R\bigl(v_{n}^{j} \bigr)^{+}\bigr) \\ &\quad {} +\biggl(\frac{1}{2}-\frac{R^{2}}{2 \Vert u^{j} \Vert ^{2}}\biggr)\bigl\langle \Phi '\bigl(u^{j}\bigr),u ^{j}\bigr\rangle + \frac{R^{2}}{ \Vert u^{j} \Vert ^{2}}\bigl\langle \Phi '\bigl(u^{j}\bigr), \bigl(u^{j}\bigr)^{-} \bigr\rangle \\ &=\frac{R^{2}}{2}-\sum_{n\in \mathbb{Z}^{M}}F_{n} \bigl(R\bigl(v_{n}^{j}\bigr)^{+}\bigr)+o(1) \geqslant \frac{R^{2}}{2}-\frac{1}{4}+o(1) >c_{*}+ \frac{3}{4}+o(1), \end{aligned} $$
which implies that \(\delta >0\), i.e., the claim holds. We derive that \(v^{+}\neq \mathbf{0}\) and so \(v\neq \mathbf{0}\).
Let \(n\in \mathbb{Z}^{M}\) be such that \(v_{n}\neq 0\), then \(\vert u_{n} ^{j} \vert = \vert v_{n}^{j} \vert \cdot \Vert u^{j} \Vert \rightarrow \infty \) as \(j\rightarrow \infty \). It is well known that \(l_{0}\) denotes the vector space of all finite sequences, i.e., sequences \(u=\{u(n)\}\) such that \(supp u=\{n \in \mathbb{Z}^{M}:u(n)\neq 0\}\) is a finite set. Obviously, \(l_{0}\) is a dense subspace of \(l^{p}\) with \(1\leqslant p<\infty \). For any \(\phi \in l_{0}\), we have
$$ \begin{aligned} \bigl\langle \Phi '\bigl(u^{j} \bigr),\phi \bigr\rangle &=\bigl(\bigl(u^{j}\bigr)^{+}- \bigl(u^{j}\bigr)^{-},\phi \bigr)- \sum _{n\in \mathbb{Z}^{M}}V_{n}u_{n}^{j} \phi_{n} - \sum_{n\in \mathbb{Z}^{M}}g_{n} \bigl(u_{n}^{j}\bigr)\phi_{n} \\ &= \bigl\Vert u^{j} \bigr\Vert \biggl[ \bigl( \bigl(v^{j}\bigr)^{+}-\bigl(v^{j} \bigr)^{-},\phi \bigr)-\sum_{n\in \mathbb{Z} ^{M}}V_{n}v_{n}^{j} \phi_{n} -\sum_{n\in \mathbb{Z}^{M}}\frac{g_{n}(u _{n}^{j})}{u_{n}^{j}}v_{n}^{j} \phi_{n} \biggr]. \end{aligned} $$
From (3.10), we derive
$$ \bigl(\bigl(v^{j}\bigr)^{+}-\bigl(v^{j} \bigr)^{-},\phi \bigr)-\sum_{n\in \mathbb{Z}^{M}}V_{n}v_{n} ^{j}\phi_{n} -\sum_{n\in \mathbb{Z}^{M}} \frac{g_{n}(u_{n}^{j})}{u_{n} ^{j}}v_{n}^{j}\phi_{n} = o(1). $$
Note that
$$ \begin{aligned} \biggl\vert \sum_{n\in \mathbb{Z}^{M}} \frac{g_{n}(u_{n}^{j})}{u_{n}^{j}}v_{n} ^{j}\phi_{n} \biggr\vert & \leqslant \sum_{n\in \mathbb{Z}^{M}}\chi_{n} \biggl\vert \frac{g _{n}(u_{n}^{j})}{u_{n}^{j}} \biggr\vert \bigl\vert v_{n}^{j}-v_{n} \bigr\vert \vert \phi_{n} \vert + \sum _{n\in \mathbb{Z}^{M}} \biggl\vert \frac{g_{n}(u_{n}^{j})}{u_{n}^{j}} \biggr\vert \vert v_{n} \vert \vert \phi_{n} \vert \\ &\leqslant C\sum_{n\in supp \phi } \bigl\vert v_{n}^{j}-v_{n} \bigr\vert \vert \phi_{n} \vert + \sum_{\{n\in \mathbb{Z}^{M}:v_{n}\neq 0\}} \biggl\vert \frac{g_{n}(u_{n}^{j})}{u _{n}^{j}} \biggr\vert \vert v_{n} \vert \vert \phi_{n} \vert =o(1). \end{aligned} $$
Therefore,
$$ \bigl(v^{+}-v^{-},\phi \bigr)-\sum _{n\in \mathbb{Z}^{M}}V_{n}v_{n}\phi_{n}=0, $$
i.e.,
$$ \bigl((L-\omega )v,\phi \bigr)_{E} = \sum _{n\in \mathbb{Z}^{M}}V_{n}v_{n}\phi _{n}. $$
This gives a contradiction since it is well known that the operator \(L - V\) has no eigenvalue in E, where the operator V is defined as follows:
$$ V : E\rightarrow E,\quad\quad (Vu)_{n} = V_{n}u_{n}. $$
Thus, \(\{u^{j}\}\) is bounded and so the lemma is proved. □

Proof of Theorem 2.1

Lemmas 3.10 and 3.11 imply that there exists a bounded \((C)_{c_{*}}\) sequence \(\{u^{j}\}\subset E\). A standard argument shows that \(u^{j}\rightharpoonup u\neq \mathbf{0}\in E\) as \(j\rightarrow \infty \) after passing to a subsequence, and \(\Phi '(u)=0\). This shows that \(u\in \mathcal{M}\) and so \(\Phi (u)\geqslant m\). Note that \((f_{2})\) and \((f_{4})\) imply that \(\frac{1}{2}f_{n}(u_{n}^{j})u_{n} ^{j}-F_{n}(u_{n}^{j})\geqslant 0\), it follows from (3.10) and Fatou’s lemma that
$$\begin{aligned} m&\geqslant c_{*} =\lim_{j\rightarrow \infty } \biggl[\Phi \bigl(u^{j}\bigr)- \frac{1}{2}\bigl\langle \Phi '\bigl(u^{j}\bigr),u^{j}\bigr\rangle \biggr] \\ &=\lim_{j\rightarrow \infty }\sum_{n\in \mathbb{Z}^{M}}\biggl[ \frac{1}{2}f _{n}\bigl(u_{n}^{j} \bigr)u_{n}^{j}-F_{n}\bigl(u_{n}^{j} \bigr)\biggr] \\ &\geqslant \sum_{n\in \mathbb{Z}^{M}}\biggl[\frac{1}{2}f_{n}(u_{n})u_{n}-F _{n}(u_{n})\biggr] \\ &=\Phi (u)-\frac{1}{2}\bigl\langle \Phi '(u),u\bigr\rangle = \Phi (u). \end{aligned}$$
Hence, we derive that \(m=\Phi (u)=\inf_{\mathcal{M}}\Phi \). □

Remark 3.12

It seems possible to generalize the results of Theorem 2.1 to the following DNLS equation:
$$ Lu_{n}-\omega u_{n} = \sigma f_{n}(u_{n}), \quad n\in \mathbb{Z}^{M}, $$
(*)
where L is a Jacobi operator [24] given by
$$ \begin{aligned} Lu_{n}&= a_{1(n_{1},n_{2},\ldots,n_{M})}u_{(n_{1}+1,n_{2},\ldots,n_{M})} + a _{1(n_{1}-1,n_{2},\ldots,n_{M})}u_{(n_{1}-1,n_{2}+1,\ldots,n_{M})} \\ & \quad {} +a_{2(n_{1},n_{2},\ldots,n_{M})}u_{(n_{1},n_{2}+1,\ldots,n_{M})}+a_{2(n _{1},n_{2}-1,\ldots,n_{M})}u_{(n_{1},n_{2}-1,\ldots,n_{M})} \\ & \quad {} +\cdots+a_{M(n_{1},n_{2},\ldots,n_{M})}u_{(n_{1},n_{2},\ldots,n_{M}+1)} + a _{M(n_{1},n_{2},\ldots,n_{M}-1)}u_{(n_{1},n_{2},\ldots,n_{M}-1)} \\ & \quad {} +b_{(n_{1},n_{2},\ldots,n_{M})}u_{(n_{1},n_{2},\ldots,n_{M})} , \end{aligned} $$
where \(\{a_{kn}\}\) (\(k=1,2,\ldots,M\)) and \(\{b_{n}\}\) are real-valued T-periodic sequences, \(\sigma =\pm 1\). Indeed, (1.2) is a special case of equation (*) when \(a_{kn}\equiv -1\) (\(k=1,2,\ldots,M\)) and \(b_{n}=2M+\varepsilon_{n}\), \(\sigma =1\).

Declarations

Acknowledgements

We would like to thank the referee for his/her valuable comments and helpful suggestions, which have led to an improvement of the presentation of this paper. This work is supported by the Specialized Fund for the Doctoral Program of Higher Education and the National Natural Science Foundation of China (Grant No. 11571187).

Authors’ contributions

All authors contributed equally to the manuscript. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

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.

Authors’ Affiliations

(1)
School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou, China
(2)
School of Mathematical Sciences and LPMC, Nankai University, Tianjin, China

References

  1. Christodoulides, D.N., Lederer, F., Silberberg, Y.: Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature 424, 817–823 (2003) View ArticleGoogle Scholar
  2. Kopidakis, G., Aubry, S., Tsironis, G.P.: Targeted energy transfer through discrete breathers in nonlinear systems. Phys. Rev. Lett. 87, 165501 (2001) View ArticleGoogle Scholar
  3. Livi, R., Franzosi, R., Oppo, G.-L.: Self-localization of Bose–Einstein condensates in optical lattices via boundary dissipation. Phys. Rev. Lett. 97, 060401 (2006) View ArticleGoogle Scholar
  4. Kevreides, P.G., Rasmussen, K., Bishop, A.R.: The discrete nonlinear Schrödinger equation: a survey of recent results. Int. J. Mod. Phys. B 15, 2883–2900 (2001) Google Scholar
  5. Kevshar, Y.S., Agrawal, G.P.: Optical Solitons: From Fibers to Photonic Crystals. Academic Press, San Diego (2003) Google Scholar
  6. Weinstein, M.: Excitation thresholds for nonlinear localized modes on lattices. Nonlinearity 12, 673–691 (1999) MathSciNetView ArticleMATHGoogle Scholar
  7. Arioli, G., Gazzola, F.: Periodic motions of an infinite lattice of particles with nearest neighbor interaction. Nonlinear Anal. 26, 1103–1114 (1996) MathSciNetView ArticleMATHGoogle Scholar
  8. Aubry, S.: Breathers in nonlinear lattices: existence, linear stability and quantization. Physica D 103, 201–250 (1997) MathSciNetMATHGoogle Scholar
  9. Aubry, S., Kopidakis, G., Kadelburg, V.: Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems. Discrete Contin. Dyn. Syst., Ser. B 1, 271–298 (2001) MathSciNetView ArticleMATHGoogle Scholar
  10. Cuevas, J., Kevrekidis, P.G., Frantzeskakis, D.J., Malomed, B.A.: Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity. Physica D 238, 67–76 (2009) MathSciNetMATHGoogle Scholar
  11. James, G.: Centre manifold reduction for quasilinear discrete systems. J. Nonlinear Sci. 13, 27–63 (2003) MathSciNetView ArticleMATHGoogle Scholar
  12. Mackay, R.S., Aubry, S.: Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity 7, 1623–1643 (1994) MathSciNetView ArticleMATHGoogle Scholar
  13. Chen, G.W., Ma, S.W.: Discrete nonlinear Schrödinger equations with superlinear nonlinearities. Appl. Math. Comput. 218, 5496–5507 (2012) MathSciNetMATHGoogle Scholar
  14. Ma, D., Zhou, Z.: Existence and multiplicity results of homoclinic solutions for the DNLS equations with unbounded potentials. Abstr. Appl. Anal. 2012, 703596 (2012) MathSciNetMATHGoogle Scholar
  15. Mai, A., Zhou, Z.: Discrete solitons for periodic discrete nonlinear Schrödinger equations. Appl. Math. Comput. 222, 34–41 (2013) MathSciNetMATHGoogle Scholar
  16. Mai, A., Zhou, Z.: Ground state solutions for the periodic discrete nonlinear Schrödinger equations with superlinear nonlinearities. Abstr. Appl. Anal. 2013, 317139 (2013) View ArticleMATHGoogle Scholar
  17. Pankov, A.: Gap solitons in periodic discrete nonlinear Schrödinger equations. Nonlinearity 19(1), 27–40 (2006) MathSciNetView ArticleMATHGoogle Scholar
  18. Pankov, A.: Gap solitons in periodic discrete nonlinear Schrödinger equations. II. A generalized Nehari manifold approach. Discrete Contin. Dyn. Syst., Ser. A 19, 419–430 (2007) View ArticleMATHGoogle Scholar
  19. Yang, Z., Chen, W., Ding, Y.: Solutions for discrete periodic Schrödinger equations with spectrum 0. Acta Appl. Math. 110, 1475–1488 (2010) MathSciNetView ArticleMATHGoogle Scholar
  20. Chen, G.W., Ma, S.W.: Ground state and geometrically distinct solitons of discrete nonlinear Schrödinger equations with saturable nonlinearities. Stud. Appl. Math. 131, 389–413 (2013) MathSciNetView ArticleMATHGoogle Scholar
  21. Chen, G.W., Ma, S.W., Wang, Z.Q.: Standing waves for discrete Schrödinger equations in infinite lattices with saturable nonlinearities. J. Differ. Equ. 261, 3493–3518 (2016) View ArticleMATHGoogle Scholar
  22. Shi, H.: Gap solitons in periodic discrete nonlinear Schrödinger equations with nonlinearity. Acta Appl. Math. 109, 1065–1075 (2010) MathSciNetView ArticleMATHGoogle Scholar
  23. Zhou, Z., Yu, J., Chen, Y.: On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity. Nonlinearity 23, 1727–1740 (2010) MathSciNetView ArticleMATHGoogle Scholar
  24. Teschl, G.: Jacobi Operators and Completely Ingretable Nonlinear Lattices. Am. Math. Soc., Providence (2000) MATHGoogle Scholar
  25. Pankov, A.: Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math. 73(1), 259–287 (2005) MathSciNetView ArticleMATHGoogle Scholar
  26. Tang, X.H.: Non-Nehari-manifold method for asymptotically linear Schrödinger equation. J. Aust. Math. Soc. 98, 104–116 (2015) MathSciNetView ArticleMATHGoogle Scholar
  27. Szulkin, A., Weth, T.: Ground state solutions for some indefinite variational problems. J. Funct. Anal. 257, 3802–3822 (2009) MathSciNetView ArticleMATHGoogle Scholar
  28. Liu, S.B.: On superlinear Schrödinger equations with periodic potential. Calc. Var. 45, 1–9 (2002) View ArticleMATHGoogle Scholar
  29. Li, G., Szulkin, A.: An asymptotically periodic Schrödinger equation with indefinite linear part. Commun. Contemp. Math. 4, 763–776 (2002) MathSciNetView ArticleMATHGoogle Scholar
  30. Sun, J.J., Ma, S.W.: Multiple solutions for discrete periodic nonlinear Schrödinger equations. J. Math. Phys. 56, 1413–1442 (2015) View ArticleGoogle Scholar

Copyright

© The Author(s) 2018

Advertisement