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.
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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 [4–14], and so on. But we notice that there are only a few results of non-periodic discrete nonlinear Schrödinger equations, such as [15–23]. 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:
where
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:
As usual, homoclinic solutions of equation (1.1) satisfy the following boundary condition:
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
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
where \(a_{n}\ge C>0\) for all \(n\in Z^{m}\), \(p>2\), and \(0<\varepsilon <p-2\). Note that
Remark 1.1
Our result extends some results [15, 18, 20, 21, 23] in infinite m dimensional lattices.
-
(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)
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)
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
be real sequence spaces. Clearly, the following elementary embedding relations hold:
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
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)
The embedding maps from E into \(l^{p}\) are compact, \(\forall p\in [2,\infty ]\).
-
(2)
The spectrum \(\sigma (L)\) is discrete and consists of simple eigenvalues accumulating to +∞.
By Lemma 2.1(2), we can assume that
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
and
where the closure is taken with respect to the norm \(\Vert \cdot \Vert _{E}\). Then one has the orthogonal decomposition
with respect to the inner product \((\cdot, \cdot)_{E}\). Now, we introduce, respectively, on E the following inner product and norm:
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
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
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
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
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
and
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
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),
\((F_{2})\) implies that there are \(R_{1},R_{2}>0\) such that
For any \(u\in H\) with \(\Vert u \Vert \ge R_{2}/\epsilon \), we have
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 \(\Vert u \Vert \ge R_{2}/ \epsilon \),
It implies
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
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
It follows from the definition of \(\Phi_{\lambda }\) that
Let
Note that
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
Let
By (2.8), there exists a large enough \(k_{3}> k_{2}+1\) such that
By (2.8), (2.10), (2.11), and \(\nu >2\), we have
(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
By \((F_{2})\), for any \(k\in N\), there exists a constant \(S_{k}>0\) such that
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
Now for any \(k\in N\), if we choose
then (2.15) implies that
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
where
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
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
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
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 \(k\ge k_{3}\)
that is,
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
So Claim 1 is true. □
By (3.2), (3.3), and (3.6), we have
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
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. □
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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.
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Appendix
Appendix
Fact 1
The result (2.2) holds.
Proof
If not, for any \(j\in N\), there exists \(u^{j} \in H\backslash \{0\}\) such that
Let \(v^{j}:=\frac{u^{j}}{\Vert u^{j} \Vert }\in H\), then \(\Vert v^{j} \Vert =1\) and
Note that since H is finite dimensional, passing to a subsequence if necessary, we may assume \(v^{j}\to v\) in E for some \(v \in H\). Evidently, \(\Vert v \Vert =1\). Note that any two norms on H are equivalent, thus by Lemma 2.1(1), we have
The fact that \(\Vert v \Vert =1\) implies \(\Vert v \Vert _{l^{\infty }}= \max_{n\in Z^{m}}\vert v_{n} \vert >0\). By the definition of the norm \(\Vert \cdot \Vert _{l^{\infty }}\), there exists a constant \(\delta_{0}> 0\) such that
For any \(j\in N\), let
Set \(\Lambda_{0}:=\{n\in Z^{m}: \vert v_{n} \vert \geq \delta_{0}\}\). Then for j large enough, by (A.1) and (A.3), we have
It follows from the definitions of \(\Lambda_{j}\) and \(\Lambda_{0}\) that for j large enough we have
This is in contradiction to (A.2). Therefore, (2.2) holds. □
Fact 2
The result (2.8) holds.
Proof
It is clear that \(0< l_{\nu }(k+1)\le l_{\nu }(k)\), so that \(l_{\nu }(k)\to l\ge 0\) as \(k\to \infty \). For every \(k\ge 0\), there exists \(u^{k}\in Z_{k}\) such that \(\Vert u^{k} \Vert =1\) and \(\Vert u^{k} \Vert _{l^{ \nu }}>l_{\nu }(k)/2\). By the definition of \(Z_{k}\), \(u^{k}\rightharpoonup 0\) in E, then \(u^{k} \rightarrow 0\) in \(l^{\nu }\) dues to Lemma 2.1(1). Therefore, we have \(l=0\), that is, \(l_{\nu }(k)\to 0\). Similarly, \(l_{2}(k)\to 0\). Therefore, (2.8) holds. □
Fact 3
\(\{u_{j}^{k}\}_{j=1}^{\infty }\) is bounded in E.
Proof
Note that \((F_{3})\) implies that there exists a constant \(L_{0}>0\) such that
For notational simplicity, we will set
throughout this paragraph. Note that \((F_{4})\) implies \(\frac{1}{2}f _{n}(s)s-F_{n}(s)\ge 0\) for all \((n,s)\in Z^{m}\times R\), it follows from (3.7), (A.4), and the definition of \(\Phi_{\lambda }\) that
where \(\Pi_{j}:=\{n\in Z^{m}: \vert u_{j,n} \vert \ge L_{0}\}\). It follows from (3.7) that
for some \(D_{1}>0\). Note that \((F_{4})\) implies that there exists a constant \(L_{1}\in (0,L_{0})\) such that
Similar to (A.5), by (A.7), \((F_{1})\), and the fact \(\frac{1}{2}f_{n}(s)s-F _{n}(s)>0\) if \(s\neq 0\) (see \((F_{4})\)), we get
for some \(D_{2},D_{3}>0\). It follows from (3.7) that
for some \(D_{4} >0\). For any \(j\in N\), let \(\chi_{j}: Z^{m}\to R\) be the indicator of \(\Pi_{j}\), that is,
Then by (A.6) and the definitions of \(\Pi_{j}\) and \(\chi_{j}\), we have
It follows from the equivalence of any two norms on the finite-dimensional space \(E^{0}\oplus E^{-}\) and Hölder’s inequality that
for some \(c_{1},c_{2} > 0\), where \(\varrho ':=\frac{\varrho }{\varrho -1}\). Consequently, we get
By the equivalence of norms \(\Vert \cdot \Vert _{l^{2}}\) and \(\Vert \cdot \Vert \) on \(E^{0}\oplus E^{-}\), we know there exists \(c_{3} > 0\) such that
Therefore,
By the definition of \(\Phi_{\lambda_{j}}\), we have
Note that \((F_{1})\) implies that for any \(\varepsilon >0\) there exists \(C_{\varepsilon }\) such that
Therefore, by (3.7), (A.8)-(A.12), and the Sobolev embedding theorem we have
for some \(c_{4},c_{5},c_{6}> 0\), where \(\iota \le \nu \) is defined in \((F_{4})\). If \(\nu -\varrho \ge 0\), by (A.6) and Lemma 2.1(1), we have
for some \(c_{7}> 0\). If \(\nu -\varrho <0\), by (A.6) and the definition of \(\Pi_{j}\), we have
Note that \(\nu -\varrho <2\) (see \((F_{3})\)) and \(\varepsilon >0\) is arbitrary, thus it follows from (A.13)-(A.15) that \(\{u_{j}\}\) is bounded in E. □
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Jia, L., Chen, G. Multiple solutions of discrete Schrödinger equations with growing potentials. Adv Differ Equ 2016, 275 (2016). https://doi.org/10.1186/s13662-016-1003-3
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DOI: https://doi.org/10.1186/s13662-016-1003-3