Open Access

Multiple solutions of discrete Schrödinger equations with growing potentials

Advances in Difference Equations20162016:275

https://doi.org/10.1186/s13662-016-1003-3

Received: 19 September 2016

Accepted: 17 October 2016

Published: 26 October 2016

Abstract

Under some weaker conditions than elsewhere, we obtain infinitely many homoclinic solutions for a class of discrete Schrödinger equations in infinite m dimensional lattices with nonlinearities being superlinear at infinity by using variational methods. Our result extends some existing results in the literature.

Keywords

discrete nonlinear Schrödinger equations variational methods superlinear homoclinic solutions

MSC

35Q51 35Q55 39A12 39A70

1 Introduction and main results

The discrete nonlinear Schrödinger equation is one of the most important discrete models, which plays an important role in many fields; for example, in biomolecular chains [1], nonlinear optics [2], Bose-Einstein condensates [3], and so on. In recent decades, a lot of results have been achieved in the study of homoclinic solutions for periodic discrete nonlinear Schrödinger equations; see [414], and so on. But we notice that there are only a few results of non-periodic discrete nonlinear Schrödinger equations, such as [1523]. The authors of [16, 17, 19, 22] studied the case in infinite one dimensional lattices (i.e., \(n\in Z\)), but the authors of [15, 18, 20, 21, 23] studied the case in infinite m dimensional lattices (i.e., \(n\in Z^{m}\)).

Inspired by the above results, we will study homoclinic solutions of the following non-periodic discrete nonlinear equation in infinite m dimensional lattices by more general conditions than some existing results:
$$ -\Delta u_{n}+v_{n}u_{n}-\omega u_{n}= f_{n}(u_{n}),\quad n\in Z^{m}, $$
(1.1)
where
$$ \begin{aligned} \Delta u_{n}={}&u_{(n_{1}+1,n_{2},\ldots, n_{m})}+u_{(n_{1},n_{2}+1, \ldots,n_{m})}+ \cdots +u_{(n_{1},n_{2},\ldots,n_{m}+1)}-2mu_{(n_{1},n _{2},\ldots,n_{m})} \\ &{}+u_{(n_{1}-1,n_{2},\ldots,n_{m})}+u_{(n_{1},n_{2}-1,\ldots,n_{m})}+u _{(n_{1},n_{2},\ldots,n_{m}-1)} \end{aligned} $$
is the discrete Laplace operator in m dimensional space, \(\omega \in R\), \(V=\{v_{n}\}_{n\in Z^{m}}\), and \(\{u_{n}\}_{n \in Z^{m}}\) are sequences of real numbers, and the nonlinearities \(f_{n}\) satisfy the condition:
$$ f_{n}\bigl(e^{i\omega }s\bigr)=e^{i\omega }f_{n}(s), \quad \forall \omega \in R, \forall (n,s)\in Z^{m}\times R. $$
As usual, homoclinic solutions of equation (1.1) satisfy the following boundary condition:
$$ \lim_{\vert n \vert =\vert n_{1} \vert +\vert n_{2} \vert +\cdots +\vert n_{m} \vert \rightarrow \infty }u_{n}=0. $$
(1.2)
Here we are interested in the existence of infinitely many nontrivial homoclinic solutions for (1.1) (‘u is nontrivial’ means \(u_{n} \not \equiv 0\)). The problem (1.1) comes from the study of standing waves for the discrete nonlinear Schrödinger equation
$$ i\dot{\psi }_{n}=-\Delta \psi_{n}+v_{n} \psi_{n}-f_{n}(\psi_{n}), \quad n\in Z^{m}. $$
(1.3)
By the definition of standing waves (\(\psi_{n}=u_{n}e^{-i\omega t}\) with (1.2)), we see that (1.3) becomes (1.1). Therefore, the problem of the existence of standing waves of (1.3) has been reduced to that on the existence of homoclinic solutions of (1.1).

In order to overcome the difficulties caused by the unboundedness of \(Z^{m}\) and the lack of periodic conditions, we make some suitable assumptions and get the following result.

Theorem 1.1

The problem (1.1) has infinitely many nontrivial homoclinic solutions if \(f_{n}(-s)=-f_{n}(s)\) for all \((n,s)\in Z^{m}\times R\) and the following conditions hold:
\((V_{1})\)
\(V =\{v_{n}\}_{n\in Z^{m}}\) is bounded from below and satisfies
$$ \lim_{\vert n \vert \rightarrow +\infty }v_{n}=+\infty. $$
(1.4)
\((F_{1})\)
\(f_{n}\in C(R, R)\), \(f_{n}(s)=o(s)\) as \(s\to 0\), and there exist \(a_{1}>0\) and \(\nu >2\) such that
$$ \bigl\vert f_{n}(s) \bigr\vert \leq a_{1} \bigl(1+ \vert s \vert ^{\nu -1}\bigr), \quad \forall (n,s)\in Z^{m} \times R. $$
\((F_{2})\)

\(\lim_{\vert s \vert \to +\infty }\frac{F_{n}(s)}{\vert s \vert ^{2}}=+ \infty \), \(\forall n\in Z^{m}\), where \(F_{n}(s):=\int_{0}^{s}f _{n}(t)\,dt, (n,s)\in Z^{m}\times R\).

\((F_{3})\)
There exist two positive constants b and \(\varrho >\max \{1,\nu -2\}\) such that
$$ \liminf_{\vert s \vert \to +\infty }\frac{f_{n}(s)s-2F_{n}(s)}{\vert s \vert ^{\varrho }} \geq b, \quad \forall n\in Z^{m}. $$
\((F_{4})\)
\(\frac{1}{2}f_{n}(s)s> F_{n}(s)\) if \(s\neq 0\), \(F_{n}(s)\ge 0\), \(\forall (n,s)\in Z^{m}\times R\), and
$$ \liminf_{\vert s \vert \to 0}\frac{f_{n}(s)s-2F_{n}(s)}{\vert s \vert ^{\iota }}\geq a_{2} \quad \textit{for some } a_{2}>0 \textit{ and } \iota \in [1, \nu ], \forall n\in Z^{m}. $$

To explain the rationality of the assumptions for the nonlinear terms \(f_{n}\), we give the following example. It is easy to check that the functions given in the following example satisfy our assumptions.

Example 1.1

Let
$$ F_{n}(s)=a_{n} \bigl( \vert s \vert ^{p}+(p-2)\vert s \vert ^{p-\varepsilon }\sin^{2}\bigl( \vert s \vert ^{ \varepsilon }/\varepsilon \bigr) \bigr), \quad s \in R, $$
where \(a_{n}\ge C>0\) for all \(n\in Z^{m}\), \(p>2\), and \(0<\varepsilon <p-2\). Note that
$$ f_{n}(s)s-2F_{n}(s)=(p-2)a_{n} \bigl[ (p-2- \varepsilon )\vert s \vert ^{p-\varepsilon }\sin^{2} \bigl( \vert s \vert ^{\varepsilon }/\varepsilon \bigr) + \bigl( 1+\sin \bigl( 2\vert s \vert ^{\varepsilon }/\varepsilon \bigr) \bigr) \vert s \vert ^{p} \bigr]. $$

Remark 1.1

Our result extends some results [15, 18, 20, 21, 23] in infinite m dimensional lattices.
  1. (1)

    The results [15, 18, 20, 21] are all about the positive definite case (\(\omega <\inf \sigma (-\Delta +V)\)), but the temporal frequency \(\omega \in R\) in our paper.

     
  2. (2)
    The authors of [15, 18, 20, 21] all used the conditions \((V_{1})\), \((F_{1})\), and \((F_{2})\). Besides, the authors of [15, 18] also used the following monotony condition:
    $$ \frac{f_{n}(s)}{s} \text{ is increasing for } s>0 \text{ and decreasing for } s< 0. $$
    (1.5)
    The authors [20, 21] also used the following Ambrosetti-Rabinowitz superlinear condition: there exists \(\nu >2\) such that
    $$ 0< \nu F_{n}(s)\leq f_{n}(s)s, \quad \forall s\in R \backslash \{0\}. $$
    (1.6)
    But we use local conditions \((F_{3})\) and \((F_{4})\) to replace the conditions (1.5) and (1.6). The functions of Example 1.1 satisfy our conditions \((F_{1})\)-\((F_{4})\), but they do not satisfy (1.5) and (1.6), which shows that our conditions are weaker than the above conditions.
     
  3. (3)

    The results in [23] also rely on the monotony condition (1.5).

     

In Section 2, we establish the variational framework of (1.1) and give some preliminary lemmas. In Section 3, we give the detailed proof of our main result.

2 Preliminary lemmas

Let
$$ \begin{aligned} &l^{p}\equiv l^{p}\bigl(Z^{m} \bigr) := \biggl\{ u=\{u_{n}\}: n\in Z ^{m}, u_{n} \in R, \Vert u \Vert _{l^{p}}= \biggl( \sum _{n\in Z^{m}}\vert u_{n} \vert ^{p} \biggr) ^{1/p}< \infty \biggr\} , \\ &\quad p\in [1,+\infty), \end{aligned} $$
be real sequence spaces. Clearly, the following elementary embedding relations hold:
$$ l^{p}\subset l^{q}, \quad \Vert u \Vert _{l^{q}}\leq \Vert u \Vert _{l^{p}}, \quad 1 \le p\le q\le \infty,\text{ where } \Vert u \Vert _{l^{\infty }}:= \max _{n\in Z^{m}}\vert u_{n} \vert . $$
Let \(L:=-\triangle +V\) be defined by \(Lu_{n}:=-\triangle u_{n}+v_{n} u _{n}\) for \(u\in l^{2}\). Let E be the form domain of L, i.e., \(E:=\mathcal{D}\) \((L^{1/2})\) (the domain of \(L^{1/2}\)). Under our assumptions, the operator L is an unbounded self-adjoint operator in \(l^{2}\). Since the operator − is bounded in \(l^{2}\), it is easy to see that \(E=\{u\in l^{2}: V^{1/2}u\in l^{2} \}\), where \(V^{1/2}u\) is defined by \(V^{1/2}u_{n}:= v^{1/2}_{n} u_{n}\) for \(u\in l^{2}\). We define, respectively, on E the inner product and norm by
$$ (u,v)_{E}:=(u,v)_{l^{2}}+\bigl(L^{1/2}u,L^{1/2}v \bigr)_{l^{2}}\quad \text{and} \quad \Vert u \Vert _{E}=(u,u)_{E}^{1/2}, $$
where \((u,v)_{l^{2}}\) is the inner product in \(l^{2}\). Then E is a Hilbert space.

Lemma 2.1

[21]

If (1.4) holds, then we have:
  1. (1)

    The embedding maps from E into \(l^{p}\) are compact, \(\forall p\in [2,\infty ]\).

     
  2. (2)

    The spectrum \(\sigma (L)\) is discrete and consists of simple eigenvalues accumulating to +∞.

     
By Lemma 2.1(2), we can assume that
$$ \lambda_{1}-\omega < \lambda_{2}-\omega < \cdots < \lambda_{k}-\omega < \cdots \to +\infty $$
are all eigenvalues of \(L-\omega \) and \(e_{k}\) is the associated normalized eigenfunction with the eigenvalue \(\lambda_{k}-\omega \) for each k, i.e., \((L-\omega )e_{k} = (\lambda_{k}-\omega )e _{k}\) and \(\Vert e_{k} \Vert _{l^{2}}=1\), \(k = 1, 2,\ldots\) . Moreover, \(\{e_{k}: k = 1, 2,\ldots \}\) is an orthonormal basis of \(l^{2}\). Let \(\sharp (D)\) denote the number i with \(i\in D\). Let
$$ k_{1}: = \sharp \bigl(\{i: \lambda_{i}-\omega < 0\}\bigr), \qquad k_{0}: = \sharp \bigl(\{i: \lambda_{i}-\omega =0\} \bigr), \qquad k_{2}: = k_{0}+k_{1}, $$
(2.1)
and
$$ E^{-}:=\operatorname{span}\{e_{1},\ldots,e_{k_{1}} \}, \qquad E^{0}:= \operatorname{span}\{e_{k_{1}+1},\ldots ,e_{k_{2}}\}, \qquad E^{+}:=\overline{ \operatorname{span} \{e_{k_{2}+1},\ldots \}}, $$
where the closure is taken with respect to the norm \(\Vert \cdot \Vert _{E}\). Then one has the orthogonal decomposition
$$ E=E^{-}\oplus E^{0}\oplus E^{+} $$
with respect to the inner product \((\cdot, \cdot)_{E}\). Now, we introduce, respectively, on E the following inner product and norm:
$$ (u,v):=\bigl(u^{0},v^{0}\bigr)_{l^{2}}+ \bigl(L^{\frac{1}{2}}u,L^{\frac{1}{2}}v\bigr)_{l ^{2}},\quad \Vert u \Vert =(u,u)^{\frac{1}{2}}, $$
where \(u, v\in E=E^{-}\oplus E^{0}\oplus E^{+}\) with \(u=u^{-} + u^{0} + u^{+}\) and \(v=v^{-} +v^{0} + v^{+}\). Clearly, the norms \(\Vert \cdot \Vert \) and \(\Vert \cdot \Vert _{E}\) are equivalent, and the decomposition \(E=E^{-}\oplus E ^{0}\oplus E^{+}\) is also orthogonal with respect to both inner products \((\cdot, \cdot)\) and \((\cdot, \cdot)_{l^{2}}\).
From the above arguments, we consider the functional Φ on E defined by
$$ \begin{aligned} \Phi (u)&=\frac{1}{2}\bigl((L-\omega )u,u \bigr)_{l^{2}}-\sum_{n\in Z ^{m}}F_{n}(u_{n}) \\ &=\frac{1}{2}\bigl\Vert u^{+} \bigr\Vert ^{2} - \frac{1}{2}\bigl\Vert u^{-} \bigr\Vert ^{2} -I(u), \end{aligned} $$
where \(I(u):=\sum_{n\in Z^{m}}F_{n}(u_{n})\). Under our assumptions, \(I,\Phi \in C^{1}(E,R)\), and the derivatives are given by
$$\begin{aligned} \bigl\langle \Phi '(u),v\bigr\rangle =\bigl(u^{+},v^{+} \bigr)-\bigl(u^{-},v^{-}\bigr)-\bigl\langle I'(u),v \bigr\rangle ,\qquad \bigl\langle I'(u),v\bigr\rangle =\sum_{n\in Z^{m}}f_{n}(u _{n})v_{n}, \end{aligned}$$
where \(u, v\in E=E^{-}\oplus E^{0}\oplus E^{+}\) with \(u=u^{-} + u^{0} + u^{+}\) and \(v=v^{-} +v^{0} + v^{+}\). The standard argument shows that nonzero critical points of Φ are nontrivial solutions of (1.1). We shall use the following critical point theorem to prove our main result.

Lemma 2.2

[24]

Let \(E=\overline{ \bigoplus_{j=1}^{\infty }X_{j}} \quad (\dim X_{j}<\infty, \forall j\in N)\) be a Banach space with the norm \(\Vert \cdot \Vert \), \(Y_{k}= \bigoplus_{j=1}^{k}X_{j}\), and \(Z_{k}=\overline{\bigoplus_{j=k}^{ \infty }X_{j}}\). Let the functional \(\Phi_{\lambda }=A(u)-\lambda B(u) \in C^{1}: E\rightarrow R\), \(\lambda \in [1,2]\). Assume that \(\Phi_{\lambda }\) satisfies
\((F_{1})\)

\(\Phi_{\lambda }\) maps bounded sets to bounded sets for \(\lambda \in [1,2]\), and \(\Phi_{\lambda }(-u)=\Phi_{\lambda }(u)\), \(\forall (\lambda,u)\in [1,2]\times E\).

\((F_{2})\)

\(B(u)\geq 0\), \(\forall u\in E\), \(A(u)\to \infty \) or \(B(u)\to \infty \) as \(\Vert u \Vert \to \infty \).

\((F_{3})\)
There exist \(r_{k}>\rho_{k}>0\) such that
$$\begin{aligned} \alpha_{k}(\lambda ):=\inf_{u\in Z_{k},\Vert u \Vert =\rho_{k}} \Phi_{\lambda }(u)> \beta_{k}(\lambda ):=\max_{u\in Y_{k},\Vert u \Vert =r_{k}} \Phi_{\lambda }(u), \quad \forall \lambda \in [1,2]. \end{aligned}$$
Then
$$ \alpha_{k}(\lambda )\leq \zeta_{k}(\lambda ):= \inf _{\gamma \in \Gamma_{k}}\max_{u\in B_{k}}\Phi_{\lambda }\bigl( \gamma (u)\bigr), \quad \forall \lambda \in [1,2], $$
where \(B_{k}:=\{u\in Y_{k}: \Vert u \Vert \leq r_{k}\}\) and \(\Gamma_{k}:=\{\gamma \in C(B_{k},E)\vert \gamma\textit{ is odd}, \gamma \vert _{\partial B_{k}}=id\}\). Moreover, for a.e. \(\lambda \in [1,2]\), there exists a sequence \(\{u^{km}(\lambda )\}_{m=1}^{\infty }\) such that
$$ \sup_{m}\bigl\Vert u^{km}(\lambda ) \bigr\Vert < \infty, \qquad \Phi '_{\lambda }\bigl(u^{km}( \lambda )\bigr)=0\quad and \quad \Phi_{\lambda }\bigl(u^{km}(\lambda ) \bigr)\to \zeta _{k}(\lambda )\quad \textit{as } m\to \infty. $$
Note that \(\dim E^{0}\) and \(\dim E^{-}\) are finite, we choose an orthonormal basis \(\{e_{j}\}_{j=1}^{k_{1}}\) of \(E^{-}\), an orthonormal basis \(\{e_{j}\}_{j=k_{1}+1}^{k_{2}}\) of \(E^{0}\), and an orthonormal basis \(\{e_{j}\}_{j=k_{2}+1}^{\infty }\) of \(E^{+}\), where \(k_{1}\) and \(k_{2}\) are defined in (2.1). Then \(\{e_{j}\}_{j=1}^{\infty }\) is an orthonormal basis of E. Let \(X_{j}:=Re_{j}\), then \(Y_{k}= \bigoplus_{m=1}^{k} X_{m}=\operatorname{span}\{e_{1},\ldots,e_{k}\}\) and \(Z_{k}=\overline{\bigoplus_{m=k}^{\infty }X_{m}}=\overline{ \operatorname{span}\{e_{k},\ldots \}}\) for all \(k\in N\). In order to apply Lemma 2.2 to prove our main result, we define the functionals A, B, and \(\Phi_{\lambda }\) on E by
$$ A(u)=\frac{1}{2}\bigl\Vert u^{+} \bigr\Vert ^{2}, \qquad B(u)=\frac{1}{2}\bigl\Vert u^{-} \bigr\Vert ^{2}+ \sum_{n\in Z^{m}}F_{n}(u_{n}) $$
and
$$ \Phi_{\lambda }(u)=A(u)-\lambda B(u)=\frac{1}{2}\bigl\Vert u^{+} \bigr\Vert ^{2}-\lambda \biggl( \frac{1}{2} \bigl\Vert u^{-} \bigr\Vert ^{2}+\sum _{n\in Z^{m}}F_{n}(u_{n}) \biggr), \quad \forall u\in E, \forall \lambda \in [1,2]. $$
Clearly, \(\Phi_{\lambda }\in C^{1}(E,R)\), \(\forall \lambda \in [1,2]\).

Lemma 2.3

If \((F_{4})\) holds, then \((F_{2})\) in Lemma  2.2 holds.

Proof

Obviously, \(B(u)\geq 0\) for all \(u\in E\) by \((F_{4})\) and the definition of \(B(u)\). From the Fact 1 in the Appendix, we see that there is a constant \(\epsilon > 0\) such that
$$ \sharp \bigl(\bigl\{ n\in Z^{m}: \vert u_{n} \vert \geq \epsilon \Vert u \Vert \bigr\} \bigr)\geq 1, \quad \forall u\in H\backslash \{0 \}, $$
(2.2)
for any finite-dimensional subspace \(H\subset E\). Let \(\Lambda_{u}:= \{n\in Z^{m}: \vert u_{n} \vert \geq \epsilon \Vert u \Vert \}\), \(\forall u\in H \backslash \{0\}\). Then by (2.2),
$$ \sharp (\Lambda_{u})\geq 1,\quad \forall u\in H\backslash \{0\}. $$
(2.3)
\((F_{2})\) implies that there are \(R_{1},R_{2}>0\) such that
$$ F_{n}(s)\geq R_{1}\vert s \vert ^{2},\quad \forall (n,s)\in Z^{m}\times R \text{ with } \vert s \vert \geq R_{2}. $$
(2.4)
For any \(u\in H\) with \(\Vert u \Vert \ge R_{2}/\epsilon \), we have
$$ \vert u_{n} \vert \geq R_{2},\quad \forall n\in \Lambda_{u}. $$
(2.5)
Note that \(F_{n}(s)\ge 0\) for all \((n,s)\in Z^{m}\times R\), it follows from (2.3)-(2.5) and the definitions of \(B(u)\) and \(\Lambda_{u}\) that, for any \(u\in H\) with \(\Vert u \Vert \ge R_{2}/ \epsilon \),
$$ \begin{aligned} B(u)&=\frac{1}{2}\bigl\Vert u^{-} \bigr\Vert ^{2}+\sum_{n\in Z^{m}}F_{n}(u_{n}) \\ &\ge\sum_{n\in \Lambda_{u}}F_{n}(u_{n}) \\ &\ge \sum_{n\in \Lambda_{u}}R_{1}\vert u_{n} \vert ^{2} \\ &\ge R_{1}\epsilon^{2}\Vert u \Vert ^{2}\cdot \sharp (\Lambda_{u}) \ge R_{1} \epsilon^{2} \Vert u \Vert ^{2}. \end{aligned} $$
It implies
$$\begin{aligned} B(u)\to \infty \quad \text{as } \Vert u \Vert \rightarrow \infty \text{ on } E^{-}\oplus E^{0}, \end{aligned}$$
which is due to \(E^{-}\oplus E^{0}\) being of finite dimension. It follows from the fact \(E=E^{-}\oplus E^{0} \oplus E^{+}\) and the definitions of A and B that we have
$$\begin{aligned} A(u)\to \infty \quad \text{or} \quad B(u)\to \infty \quad \text{as } \Vert u \Vert \rightarrow \infty, \forall u\in E. \end{aligned}$$
The proof is completed. □

Lemma 2.4

If the assumptions in Theorem  1.1 are satisfied, then \((F_{3})\) in Lemma  2.2 holds.

Proof

(a) Note that \((F_{1})\) implies that for any \(\varepsilon >0\) there exists \(C_{\varepsilon }\) such that
$$ \bigl\vert F_{n}(s) \bigr\vert \le \varepsilon \vert s \vert ^{2}+C_{\varepsilon }\vert s \vert ^{\nu }, \quad \forall (n,s)\in Z^{m}\times R. $$
It follows from the definition of \(\Phi_{\lambda }\) that
$$ \begin{aligned}[b] \Phi_{\lambda }(u)&\geq \frac{1}{2}\Vert u \Vert ^{2}-2\sum_{n\in Z ^{m}} F_{n}(u_{n}) \\ &\geq \frac{1}{2}\Vert u \Vert ^{2}-2\sum _{n\in Z^{m}}\bigl(\varepsilon \bigl\vert u_{n}\bigl\vert ^{2}+C_{\varepsilon } \bigr\vert u_{n} \bigr\vert ^{\nu }\bigr),\quad \forall (\lambda,u) \in [1,2]\times E^{+}. \end{aligned} $$
(2.6)
Let
$$ l_{2}(k):=\sup_{u\in Z_{k}\backslash \{0\}} \frac{\Vert u \Vert _{l^{2}}}{\Vert u \Vert },\qquad l_{\nu }(k):= \sup_{u\in Z_{k}\backslash \{0\}}\frac{\Vert u \Vert _{l^{\nu }}}{\Vert u \Vert }, \quad \forall k\in N. $$
(2.7)
Note that
$$ l_{2}(k)\to 0,\quad l_{\nu }(k)\to 0 \quad \text{as } k\to \infty, $$
(2.8)
which will be proved in the appendix. Obviously, \(Z_{k}\subset E^{+}\) for all \(k\ge k_{2}+1\) (\(k_{2}+1\) is defined above Lemma 2.3), thus it follows from (2.6)-(2.7) that for any \(k\ge k_{2}+1\) we have
$$ \Phi_{\lambda }(u)\geq \frac{1}{2}\Vert u \Vert ^{2}-2 \varepsilon l_{2}^{2}(k) \Vert u \Vert ^{2}-2C_{\varepsilon }l^{\nu }_{\nu }(k)\Vert u \Vert ^{\nu }, \quad \forall (\lambda,u)\in [1,2]\times Z_{k}. $$
(2.9)
Let
$$ \rho_{k}:= \bigl( 1-16\varepsilon l_{2}^{2}(k) \bigr) \bigl( 16C_{\varepsilon}l^{\nu }_{\nu }(k) \bigr) ^{\frac{1}{2-\nu }}. $$
(2.10)
By (2.8), there exists a large enough \(k_{3}> k_{2}+1\) such that
$$ 0< 16\varepsilon l_{2}^{2}(k)< 1, \quad \forall k> k_{3}. $$
(2.11)
By (2.8), (2.10), (2.11), and \(\nu >2\), we have
$$ \rho_{k}\to \infty \quad \text{as } k\to \infty. $$
(2.12)
By (2.9)-(2.11), we have
$$ \alpha_{k}(\lambda ):=\inf_{u\in Z_{k},\Vert u \Vert =\rho_{k}} \Phi_{\lambda }(u) \ge \rho_{k}^{2}/4>0,\quad \forall k \ge k_{3}. $$
(b) Note that \(Y_{k}\) is of finite dimension, thus (2.2) implies that for any \(k\in N\) there exists a constant \(\epsilon_{k}> 0\) such that
$$ \sharp \bigl(\bigl\{ n\in Z^{m}: \vert u_{n} \vert \geq \epsilon_{k}\Vert u \Vert \bigr\} \bigr)\geq 1, \quad \forall u\in Y_{k}\backslash \{0\}. $$
(2.13)
By \((F_{2})\), for any \(k\in N\), there exists a constant \(S_{k}>0\) such that
$$ F_{n}(s)\ge \frac{\vert s \vert ^{2}}{\epsilon_{k}^{2}},\quad \forall (n,s) \in Z^{m} \times R \text{ with } \vert s \vert \ge S_{k}. $$
(2.14)
For any \(k\in N\) and \(u\in Y_{k}\) with \(\Vert u \Vert \ge S_{k}/\epsilon _{k}\), by (2.13), (2.14), and the fact \(F_{n}(s)\ge 0\), we have
$$ \begin{aligned}[b] \Phi_{\lambda }(u)&\leq \frac{1}{2}\bigl\Vert u^{+} \bigr\Vert ^{2}-\sum_{n\in Z ^{m}} F_{n}(u_{n}) \\ &\leq \frac{1}{2}\Vert u \Vert ^{2}- \sum _{n\in \{n\in Z^{m}: \vert u_{n} \vert \geq \epsilon_{k}\Vert u \Vert \}}\frac{\vert u_{n} \vert ^{2}}{ \epsilon_{k}^{2}} \\ &\leq \frac{1}{2}\Vert u \Vert ^{2}-\frac{\epsilon_{k}^{2} \Vert u \Vert ^{2}}{\epsilon _{k}^{2}}\cdot \sharp \bigl(\bigl\{ n\in Z^{m}: \vert u_{n} \vert \geq \epsilon \Vert u \Vert \bigr\} \bigr) \\ &\leq \frac{1}{2}\Vert u \Vert ^{2}-\Vert u \Vert ^{2}=-\frac{1}{2}\Vert u \Vert ^{2}, \quad \forall \lambda \in [1,2]. \end{aligned} $$
(2.15)
Now for any \(k\in N\), if we choose
$$ r_{k}>\max \{\rho_{k},S_{k}/ \epsilon_{k}\}, $$
then (2.15) implies that
$$ \beta_{k}(\lambda ):=\max_{u\in Y_{k},\Vert u \Vert =r_{k}} \Phi_{\lambda }(u) \le -r_{k}^{2}/2< 0,\quad \forall k\in N. $$
Therefore, the proof is finished. □

3 Proof of the main result

Proof of Theorem 1.1

It is easy to check that \((F_{1})\) of Lemma 2.2 holds. Besides, \((F_{2})\) and \((F_{3})\) hold for all \(k\ge k_{3}\) by Lemmas 2.3 and 2.4. Thus Lemma 2.2 implies that for any \(k\ge k_{3}\) and a.e. \(\lambda \in [1, 2]\) there exists a sequence \(\{u_{i}^{k}(\lambda )\}_{i=1}^{\infty }\subset E\) such that
$$ \sup_{i}\bigl\Vert u_{i}^{k}(\lambda ) \bigr\Vert < \infty, \qquad \Phi '_{\lambda }\bigl(u _{i}^{k}(\lambda )\bigr)=0\quad \text{and} \quad \Phi_{\lambda }\bigl(u_{i}^{k}( \lambda )\bigr)\to \zeta_{k}(\lambda )\quad \text{as } i\to \infty, $$
(3.1)
where
$$ \zeta_{k}(\lambda ):=\inf_{\gamma \in \Gamma_{k}}\max _{u\in B_{k}} \Phi_{\lambda }\bigl(\gamma (u)\bigr),\quad \forall \lambda \in [1,2], $$
with \(B_{k}:=\{u\in Y_{k}: \Vert u \Vert \leq r_{k}\}\) and \(\Gamma_{k}:=\{\gamma \in C(B_{k},E)\vert \gamma \text{ is odd}, \gamma \vert _{\partial B_{k}}=id\}\). Furthermore, it follows from the proof of Lemma 2.4 that
$$ \zeta_{k}(\lambda )\in [\overline{\alpha }_{k},\overline{\zeta }_{k}], \quad \forall k\ge k_{3}, $$
(3.2)
where \(\overline{\zeta }_{k}:=\max_{u\in B_{k}}\Phi_{1}(u)\) and \(\overline{\alpha }_{k}:=\rho_{k}^{2}/4\to \infty \) as \(k\to \infty \) by (2.12). By (3.1), for each \(k\ge k_{3}\), there exist \(\lambda_{j} \to 1\) as \(j\to \infty \) and \(\{u_{i}^{k}(\lambda_{j})\}_{i=1}^{\infty }\subset E\) such that
$$ \begin{aligned}[b] &\sup_{i}\bigl\Vert u_{i}^{k}( \lambda_{j}) \bigr\Vert < \infty, \qquad \Phi '_{\lambda_{j}} \bigl(u _{i}^{k}(\lambda_{j})\bigr)=0\quad \text{and} \quad \Phi_{\lambda_{j}}\bigl(u _{i}^{k}( \lambda_{j})\bigr)\to \zeta_{k}(\lambda_{j}) \\ &\quad\text{as } i \to \infty. \end{aligned} $$
(3.3)

Claim 1

\(\{u_{i}^{k}(\lambda_{j})\}_{i=1}^{\infty }\) in (3.3) has a strong convergent subsequence.

Proof

Note that \(\sup_{i}\Vert u_{i}^{k}(\lambda_{j}) \Vert < \infty \) for each \(k\ge k_{3}\), without loss of generality, we may assume
$$ \begin{aligned} &\bigl(u_{i}^{k}(\lambda_{j}) \bigr)^{-} \to \bigl(u^{k}_{j} \bigr)^{-}, \qquad \bigl(u_{i}^{k}( \lambda_{j})\bigr)^{0} \to \bigl(u^{k}_{j} \bigr)^{0}\quad \text{and} \quad \bigl(u_{i} ^{k}(\lambda_{j})\bigr)^{+} \rightharpoonup \bigl(u^{k}_{j}\bigr)^{+} \\ &\quad\text{as } i\to \infty, \forall j\in N, \end{aligned} $$
(3.4)
for some \(u^{k}_{j}=(u^{k}_{j})^{-}+(u^{k}_{j})^{0}+(u^{k}_{j})^{+} \in E=E^{-} + E^{0} + E^{+}\) since \(\dim (E^{-}\oplus E^{0})<\infty \). By virtue of the Riesz representation theorem, \(\Phi '_{\lambda_{j}}: E \to E^{\ast }\) and \(I': E \to E^{\ast }\) can be viewed as \(\Phi '_{\lambda_{j}}: E \to E\) and \(I': E \to E\), respectively, where \(E^{\ast }\) is the dual space of E. Note that (3.3) implies that for each \(k\ge k_{3}\)
$$ 0=\Phi '_{\lambda_{j}}\bigl(u_{i}^{k}( \lambda_{j})\bigr)=\bigl(u_{i}^{k}( \lambda_{j})\bigr)^{+} -\lambda_{j} \bigl[ \bigl(u_{i}^{k}(\lambda_{j})\bigr)^{-} + I'\bigl(u_{i}^{k}(\lambda _{j}) \bigr) \bigr],\quad \forall i,j \in N, $$
that is,
$$ \bigl(u_{i}^{k}(\lambda_{j}) \bigr)^{+}=\lambda_{j} \bigl[ \bigl(u_{i}^{k}(\lambda_{j})\bigr)^{-} + I'\bigl(u_{i}^{k}(\lambda_{j})\bigr) \bigr],\quad \forall i,j \in N. $$
(3.5)
By the standard argument (see [25, 26]), we know \(I': E \to E^{\ast }\) is compact. Therefore, \(I': E \to E\) is also compact. It follows from (3.4) and (3.5) that the right-hand side of (3.5) converges strongly in E. Combining this with (3.4), we have
$$ \lim_{i\to \infty }u_{i}^{k}( \lambda_{j})=u_{j}^{k} \in E, \quad \forall j\in N \text{ and } k\ge k_{3}. $$
(3.6)
So Claim 1 is true. □
By (3.2), (3.3), and (3.6), we have
$$ \Phi '_{\lambda_{j}}\bigl(u_{j}^{k}\bigr)=0 \quad \text{and} \quad \Phi_{\lambda_{j}}\bigl(u_{j}^{k} \bigr)\in [\overline{\alpha }_{k},\overline{ \zeta }_{k}], \quad \forall j\in N \text{ and } k\ge k_{3}. $$
(3.7)
In fact, we can see \(\{u_{j}^{k}\}_{j=1}^{\infty }\) is bounded in E, which will be proved in the appendix. Besides, by a similar proof to Claim 1, we can also see that \(\{u_{j}^{k}\}_{j=1}^{\infty }\) possesses a strong convergent subsequence in E for all \(k\ge k_{3}\). Without loss of generality, we may assume
$$ u_{j}^{k} \to u^{k} \quad \text{as } j\to \infty,\forall k\ge k_{3}. $$
For each \(k\ge k_{3}\), by (3.7), the limit \(u^{k}\) is just a critical point of \(\Phi =\Phi_{1}\) with \(\Phi (u^{k})\in [\overline{\alpha } _{k},\overline{\zeta }_{k}]\). Since \(\overline{\alpha }_{k}\to \infty \) as \(k\to \infty \) in (3.2), we get infinitely many nontrivial critical points of Φ. Therefore, we see that problem (1.1) possesses infinitely many nontrivial homoclinic solutions. The proof of Theorem 1.1 is completed.  □

Declarations

Acknowledgements

Research supported by National Natural Science Foundation of China (No. 11401011).

We would like to thank the editors and referees for their valuable comments which have led to an improvement of the presentation of this paper.

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 Mathematical Sciences, University of Jinan

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