Homoclinic solutions of discrete $p$-Laplacian equations containing both advance and retardation

1.   Introduction

Consider the following $2m$th-order nonlinear $p$-Laplacian difference equation containing both advance and retardation

Here $p > 1$ is a real number, $\phi_p(s) = |s|^{p-2}s$ for all $s \in \mathbb{R}$, $\Delta$ is the forward difference operator defined by $\Delta u_k = u_{k+1}-u_k$, $\Delta^j u_k = \Delta(\Delta^{j-1} u_k)$ for $j\ge 2$, $\{r_n\}$ and $\{\omega_n\}$ are real positive $T$-periodic sequences for a positive integer $T$, $f\in C(\mathbb{Z}\times \mathbb{R}^{3}, \mathbb{R})$ with $f$ being $T$-periodic in the first variable.

Special cases of (1.1) are produced, for example, when we look for standing waves of the discrete nonlinear Schrödinger (DNLS) equation,

Assume that the nonlinearity is gauge invariant, i.e.,

Since solitons are spatially localized time-periodic solutions and decay to zero at infinity, $\psi_n$ has the form

where $\{u_n\}$ is a real valued sequence and $\omega \in \mathbb{R}$ is the temporal frequency. Then we arrive at the nonlinear equation

Clearly, (1.2) is a special form of (1.1) with $m = 1$ and $p = 2$ but without advance or retardation.

We assume that $f(n, 0, 0, 0) = 0$ for each $n \in \mathbb{Z}$, then $\{u_n\} = \{0\}$ is a solution of (1.1), which is called the trivial solution. As usual, we say that a solution $u = \{u_n\}$ of (1.1) is homoclinic (to 0) if $\lim_{\left|n\right|\rightarrow \infty}u_n = 0.$ In addition, if $\{u_n\}\neq\{0\}$, then $u$ is called a nontrivial homoclinic solution.

Critical point theory was introduced into discrete systems by Guo-Yu ^[1] in 2003 to study the existence of periodic and subharmonic solutions. It has been proved to be a powerful tool for studying the existence of homoclinic solutions for discrete nonlinear systems ^[2]. Among them, the theory of difference equations has been widely used to examine discrete models appearing in many fields ^[3,4]. In recent years, the existence of homoclinic and heteroclinic solutions and boundary value problems for various difference equations have been investigated by many researchers ^{[5,6,7,8,9,10,11,12,13,14]}. For example, some researchers have studied the following nonlinear difference equation with a coercive weight function

where $\lambda$ is a positive real parameter, $a, b:\mathbb{Z}\rightarrow (0, +\infty)$. By means of critical point theory, Iannizzotto and Tersian ^[6] have proved the existence of at least two nontrivial homoclinic solutions when $\lambda$ is big enough of (1.3). Moreover, infinitely many homoclinic solutions were obtained in ^[12] by employing Nehari manifold methods, and in ^[11] by applying the fountain theorem.

In particular, difference equations containing both advance and retardation have important background and applications in the field of cybernetics and biological mathematics ^[15,16]. Thus they have received considerably attention. For some recent works, we refer readers to ^[7,10,17,18] and references therein. For instances, by using the mountain pass theorem and periodic approximations, Shi et al. ^[10] studied the existence of a nontrivial homoclinic orbit of

where $M$ is a given nonnegative integer. Kong ^[7] employed the critical point theory to study the existence of at least three homoclinic solutions for the following $p$-Laplacian difference equation with both advance and retardation

$k\in \mathbb{Z}$, where $\lambda$ is a positive real parameter, $a, b:\mathbb{Z}\rightarrow (0, +\infty)$. Unlike the problem we studied, in this article, the author requires that $b(k)$ is unbounded.

Inspired by the above interesting research, we shall attempt to establish the new sufficient conditions on the existence of nontrivial homoclinic solutions for more general nonlinear terms of (1.1), see remarks 1 and 2 for details. To wit, we have

Theorem 1.1 Assume that there exists a function $F\in C^1(\mathbb{Z}\times \mathbb{R}^2, \mathbb{R})$ having the following properties with $p > 2$.

$(T_1)$ For $n\in \mathbb{Z}, v_1, v_2, v_3\in \mathbb{R}$, $F(n+T, v_1, v_2) = F(n, v_1, v_2)$ and

where we denote by

$(T_2)$

$(T_3)$ $\partial_i F(n, v_1, v_2) = o(|(v_1, v_2)|) \ \ \ {as}\ \ \ (v_1, v_2)\rightarrow (0, 0) \ \ {for \; all}\ \ n\in \mathbb{Z}, \ \ i = 2, 3;$

$(T_4)$ There exists a real sequence $\{a_n\}$ such that

$(T_5)$ $\partial_2 F(n, v_1, v_2)v_1+\partial_3 F(n, v_1, v_2)v_2 - pF(n, v_1, v_2) > 0$ for all $(n, v_1, v_2)\in \mathbb{Z} \times \mathbb{R}^2 \setminus \{(0, 0)\}$.

If $p a_n > \bar{r} 2^{mp}$ for each $n \in \mathbb{Z}$, then (1.1) has at least a nontrivial solution $u$ in $l^2$, where $\bar{r} = \max_{n\in \mathbb{Z}}\{r_n\}$.

Theorem 1.2 Assume that there exists $F\in C^1(\mathbb{Z}\times \mathbb{R}^2, \mathbb{R})$ satisfying $(T_1), (T_2), (T_3)$ and the following properties with $1 < p\leq2$.

$(T_6)$ There exists a real sequence $\{b_n\}$ such that

$(T_7)$ $\partial_2 F(n, v_1, v_2)v_1+\partial_3 F(n, v_1, v_2)v_2 - 2F(n, v_1, v_2) > 0$ for all $(n, v_1, v_2)\in \mathbb{Z} \times \mathbb{R}^2 \setminus \{(0, 0)\};$

$(T_8)$ $\partial_2 F(n, v_1, v_2)v_1+\partial_3 F(n, v_1, v_2)v_2 - 2F(n, v_1, v_2)\rightarrow +\infty$ as $|v_1|+|v_2|\rightarrow \infty$.

If $2b_n > \omega_n$ for each $n \in \mathbb{Z}$, then (1.1) has at least a nontrivial solution $u$ in $l^2$.

Remark 1. If a solution $\{u_n\}$ of (1.1) is in $l^2$, then $\lim_{|n|\to\infty}u_n = 0$ and $\{u_n\}$ is a homoclinic solution. The condition $(T_4)$ implies that the nonlinearity $F$ can be mixed super $p$-linear with asymptotically p-linear at $\infty$ and $(T_6)$ implies that the nonlinear term $F$ can be mixed superquadratic linear with asymptotically quadratic linear at $\infty$. In some references, the nonlinear $f$ is assumed to be either only superlinear or only asymptotically linear at $\infty$, which plays an important role in establishing the existence of nontrivial homoclinic solutions.

Remark 2. If $m = 1$, $r_n\equiv1$, and $f(n, u_{n+1}, u_n, u_{n-1}) = g(n, u_n)$, then Theorem 1.1 reduces to Theorem 2.2 in ^[9] when $\phi$-Laplacian is $p$-Laplacian. Moreover, our sufficient conditions are based on the limit superior and limit inferior, which are more applicable.

This rest of the paper is organized as follows. In Section 2, we establish the variational framework associated with (1.1) and cite the Mountain Pass Lemma. Section 3 and Section 4 are devoted to the proofs of Theorem 1.1 and Theorem 1.2, respectively. The paper concludes with an example to illustrate the applicability of the main results.

2.   The variational structure

We first establish the corresponding variational framework for (1.1).

Let $S$ be the set of all two-sided sequences, that is,

Then $S$ is a vector space with $au+bv = \{au_n+bv_n\}$ for $u, v\in S, a, b\in \mathbb{R}$. For any fixed positive integer $k$, we define the subspace $E_k$ of $S$ as

Obviously, $E_k$ is isomorphic to $\mathbb{R}^{2kT}$ and we identify $u = (u_1, u_2, \cdots, u_{2kT})^\ast\in E_k$, where * denotes the transpose of a vector. $E_k$ can be equipped with the inner product $(\cdot, \cdot)_k$ and norm $\|\cdot\|_k$ defined respectively by

and

In $E_k$, we also define the equivalent norms $\|\cdot\|_{k\infty}$ by

and $\|\cdot\|_{kp}$ by

By Hölder inequality and Jensen inequality, we have

where

For $p\geq 1$, let

For simplicity, the inner product and norm in $l^2$ are denoted by $(\cdot, \cdot)$ and $\|\cdot\|$, respectively.

Consider the functional $J_k$ in $E_k$ defined by

whose Fréchet derivative is given by

for $u, v\in E_k$.

Equation (2.3) implies that (1.1) is the corresponding Euler-Lagrange equation for $J_k$. It is easy to see that the critical points of $J_k$ in $E_k$ are exactly $2kT$-periodic solutions of the difference equation (1.1).

Let $P$ be the $2kT\times 2kT$ matrix corresponding to the quadratic form $\sum^{2kT}_{k = 1}(\Delta{u_k})^2$ with $u_{2kT+1} = u_1$ for $k\in \mathbb{Z}$, that is,

By matrix theory, the eigenvalues of $P$ are

It follows that $\lambda_0 = 0, \lambda_1 > 0, \lambda_2 > 0, \cdots, \lambda_{2kT-1} > 0$. Moreover, $\lambda_{\max} = \max\{\lambda_1, \lambda_2, \cdots, \lambda_{2kT-1}\} = 4$.

For the readers' convenience, we now cite the Mountain Pass Lemma. Let $H$ be a Hilbert space and $C^1(H, \mathbb{R})$ denote the set of functionals that are Fréchet differentiable and their Fréchet derivatives are continuous on $H$, $B_r$ be the open ball in $H$ with radius $r$ and center 0, and $\partial B_r$ denote its boundary.

Definition 2.1 Let $J\in C^1(H, \mathbb{R})$. A sequence $\{x_j\}\subset H$ is called a Cerami sequence ($(C)$ sequence for short) for $J$ if $J(x_j)\rightarrow c$ for some $c\in \mathbb{R}$ and $(1+\|x_j\|)J^\prime(x_j)\rightarrow0$ as $j\rightarrow \infty$. We say $J$ satisfies the Cerami condition ($(C)$ condition for short) if any $(C)$ sequence for $J$ possesses a convergent subsequence.

Lemma 2.1 (Mountain Pass Lemma ^[19]) If $J\in C^1(H, \mathbb{R})$ and satisfies the following conditions: there exist $e\in H \backslash \{0\}$ and $r\in(0, \|e\|)$ such that $\max\{J(0), J(e)\} < \inf_{u\in \partial B_r}J(u)$. Then there exists a $(C)$ sequence $\{u_n\}$ for the mountain pass level $c$ which is defined by

where

Finally, by similar arguments as those in ^[18], we can obtain the following result.

Lemma 2.2 For $u\in E_{k}$, we have

By Lemma 2.2 and (2.1), for $u\in E_k$,

3.   Proof of Theorem 1.1

In order to prove Theorem 1.1, we need some preparation. Denote $\omega_* = \min_{n\in \mathbb{Z}}\{\omega_n\}.$

Lemma 3.1 Under the assumptions of Theorem 1.1, the functional $J_k$ satisfies the $(C)$ condition.

$Proof.$ Let $\{u^{(j)}\}\subset E_k$ be a $(C)$ sequence for $J_k$. We need to show that $\{u^{(j)}\}$ has a convergent subsequence. Since $E_k$ is finite dimensional, it suffices to show that $\|u^{(j)}\|_k$ is bounded. By assumption, $J_k(u^{(j)})\rightarrow c$ for some $c\in \mathbb{R}$ and $(1+\|u^{(j)}\|_k)J_k^\prime(u^{(j)})\rightarrow 0$ as $j\rightarrow \infty$. Then there exists $M > 0$ such that $|J_k(u^{(j)})|\leq M$ and $\|(1+\|u^{(j)}\|_k)J_k^\prime(u^{(j)})\| \leq M$ for $j\in \mathbb{N}$. So we have $\|u^{(j)}\|_k\|J_k^\prime(u^{(j)})\|\leq\|(1+\|u^{(j)}\|_k)J_k^\prime(u^{(j)})\| \leq M$ for $j\in \mathbb{N}$. Then by (2.2), (2.3) and $(T_5)$, we have

Choose $\delta > 0$ such that

This and (3.1) imply that $|u_n^{(j)}|\leq \delta$ for $n\in \mathbb{Z}$, that is,

Since $E_k$ is finite dimensional, $\|\cdot\|_k$ and $\|\cdot\|_{k\infty}$ are equivalent. Then (3.2) implies that $\{\|u^{(j)}\|_k\}$ is bounded. The proof is completed.

Lemma 3.2 Under the assumptions of Theorem 1.1, there exists $n_0\in \mathbb{N}$ such that $J_k$ has at least a nonzero critical point $u^{(k)}$ in $E_k$ for each $k\geq n_0$.

$Proof.$ We first show that $J_k$ satisfies conditions of Lemma 2.1. From $(T_2)$, there exists $r > 0$ such that

Then, for $u\in E_k$ with $\|u\|_k \leq r$,

Taking $a = \frac{1}{4}\omega_*r^2$ gives $J_k|_{\partial B_r}\geq a > 0$.

Since $a_n > \frac{\bar{r}}{p} 2^{mp}$ for each $n \in \mathbb{Z}$, there exists $\varepsilon\in (0, 1)$ such that

For a given $e = \{e_n\}\in l^2$ with $\sum_{n = -\infty}^{\infty}|e_{n}|^p = 1$. Let $n_0$ be large enough such that

For $k\geq n_0$, define $e^{(k)}\in E_k$ by

By $(T_4)$, there exists $\mu_0 > r$, such that

Then, for $\mu \geq \mu_0$,

Noticing that $p > 2$ and $\frac{\bar{r}}{p} 2^{mp} +2(\varepsilon-a_n)(1-\varepsilon) < 0$, there exists $\mu' > \mu_0$ such that

It can easily be seen that $J_k(0) = 0$. Then we have $r\in (0, \|\mu'e^{(k)}\|_k)$ and

Now that we have verified all assumptions of Lemma 2.1, we know $J_k$ possesses a $(C)$ sequence $\{u_j^{(k)}\}$ for the mountain pass level $c_k\geq a$ with

where

According to Lemma 3.1, $\{u_j^{(k)}\}$ has a convergent subsequence $\{u_{j_m}^{(k)}\}$ such that $u_{j_m}^{(k)}\rightarrow u^{(k)}$ as $j_m\rightarrow \infty$ for some $u^{(k)} \in E_k$. Since $J_k \in C^1(E_k, \mathbb{R})$, we have

as $j_m\rightarrow \infty$. By the uniqueness of the limit, we obtain that $u^{(k)}$ is a critical point of $J_k$ corresponding to $c_k$. Moreover, $u^{(k)}$ is nonzero as $c_k\geq a > 0$.

Lemma 3.3 There exist constants $\alpha, \beta, N > 0$ such that

hold for every critical point $u^{(k)}$ of $J_k$ in $E_k$ with $k \geq n_0$ obtained in Lemma 3.2.

$Proof.$ For $k\geq n_0$, we define $h_k\in \Gamma_k$ as $h_k(s) = s\mu_0e^{(k)}$ for $s\in[0, 1]$. Similarly to the derivation of ^[18], we can find

Obviously, $M_0 > 0$ is independent of $k$.

Since $u^{(k)}$ is a critical point of $J_k$, by (2.2), (2.3) and (3.3), we have

Choose $\beta > 0$ such that

This combined with (3.4) implies that $|u_n^{(k)}|\leq \beta$ for each $n\in \mathbb{Z}$, that is,

From (2.3), we have

By $(T_3)$, there exists $\alpha > 0$ such that

which together with (3.5) produces

Arguing by a contradiction, we have

In view of $(T_5)$ and (3.4), we have

Let $N = \sqrt{\frac{p M_0}{\omega_*\left(\frac{p}{2}-1\right)}}$. Then we have

The proof is complete.

Now, we are ready to prove Theorem 1.1.

According to Lemma 3.2, there exists $n_0\in \mathbb{N}$ such that for every $k > n_0$, $J_k$ has a critical point $u^{(k)} = \{u_n^{(k)}\}\in E_k$. Moreover, there exists $n_k\in \mathbb{Z}$ such that

Note that

By the periodicity of $\{\omega_n\}$ and $f(n, u_{n+1}, u_n, u_{n-1})$, we see that $\{u_{n+T}^{(k)}\}$ is also a solution of (3.8). Without loss of generality, we may assume that $0\leq n_k\leq T-1$ in (3.7). Moreover, passing to a subsequence of $\{u^{(k)}\}$ if necessary, we can also assume that $n_k = n^\ast$ for $k\geq n_0$ and some integer $n^\ast$ between $0$ and $T-1$. It follows from (3.7) that we can choose a subsequence, still denoted by $\{u^{(k)}\}$, such that

Then $u = \{u_n\}$ is a nonzero sequence as (3.7) implies $|u_{n^\ast}|\geq \alpha$. It remains to show that $u = \{u_n\}\in l^2$ and it is a solution of (1.1).

Let

Since $F(n, u_1, u_2)$ is continuously differentiable in the second and third variables and $T$-periodic in $n$, for $n\in \mathbb{Z}, \alpha \leq |u_1|+|u_2| \leq 2\beta$, let

It is clear that $d_1, d_2 > 0$. Thus, for $n\in B_k$,

This combined with (3.4), (3.5) and (3.6) gives us

It follows that

Given $\varrho\in \mathbb{N}$, for $k > \max\{\varrho, n_0\}$, it follows from (3.9) that

It is clear that $\sum_{n = -\varrho}^{\varrho}u_n^2\leq \frac{2d_1M_0}{d_2\omega_*}$ as $k\rightarrow \infty$ and hence $u = \{u_n\}\in l^2$ by the arbitrariness of $\varrho$.

Now, for each $n\in \mathbb{Z}$, letting $k\rightarrow \infty$ in (3.8) gives us

that is, $u = \{u_n\}$ satisfies (1.1).

Consequently, we infer that $u = \{u_n\}$ is a nontrivial solution of (1.1). This completes the proof of Theorem 1.1.

4.   Proof of Theorem 1.2

The proof of Theorem 1.2 is quite similar to that of Theorem 1.1. But some of the arguments are different. As a result, we provide some details below.

Lemma 4.1 Under the assumptions of Theorem 1.2, the functional $J_k$ satisfies the $(C)$ condition.

$Proof.$ Let $\{u^{(j)}\}\subset E_k$ be a $(C)$ sequence for $J_k$. As in the proof of Lemma 3.1, there exists $M > 0$ such that $|J_k(u^{(j)})|\leq M$ and $\|u^{(j)}\|_k\|J_k^\prime(u^{(j)})\| \leq M$ for $j\in \mathbb{N}$. Then by (2.2), (2.3) and $1 < p\leq2$, we have

From $(T_8)$, there exists $\delta > 0$ such that

Then (4.1) and $(T_7)$ imply that $|u_n^{(j)}|\leq \delta$ for $n\in \mathbb{Z}$, that is,

Since $E_k$ is finite dimensional, $\|\cdot\|_k$ and $\|\cdot\|_{k\infty}$ are equivalent. Then (4.2) tells us that $\{\|u^{(j)}\|_k\}$ is bounded and hence $\{u^{(j)}\}$ has a convergent subsequence. This completes the proof.

Lemma 4.2 Under the assumptions of Theorem 1.2, there exists $n_0\in \mathbb{N}$ such that $J_k$ has at least a nonzero critical point $u^{(k)}$ in $E_k$ for each $k\geq n_0$.

$Proof.$ Proceeding as in the proof of Lemma 3.2, there exist $r > 0$ and $a > 0$ such that $J_k|_{\partial B_r}\geq a > 0$. Since $2b_n > \omega_n$, there exists $d > 0$ such that

Let $\varepsilon\in (0, 1)$ satisfy

There exists $e = \{e_n\}\in l^2$ with $\sum_{n = -\infty}^{\infty}| e_{n}|^2 = 1$ such that $\sum_{n = -\infty}^{\infty}|\Delta^m e_{n}|^2 < \varepsilon$. Let $n_0$ be large enough such that

For $k\geq n_0$, define $e^{(k)}\in E_k$ by

By $(T_6)$, there exists $\mu_0 > \max\{r, 1\}$ such that

Then, for $\mu \geq \mu_0$,

Thus

The remaining arguments are the same as those in the proof of Lemma 3.2.

Lemma 4.3 There exist $\alpha', \beta' > 0$ such that

holds for every critical point $u^{(k)}$ of $J_k$ in $E_k$ with $k \geq n_0$ obtained in Lemma 4.2.

$Proof.$ we can find $M_1 > 0$ (independent of $k$) such that $J_k(u^{(k)}) \leq M_1$ for $k\geq n_0$. Since $u^{(k)}$ is a critical point of $J_k$, by (2.2) and (2.3), we have

From $(T_8)$, there exists $\beta' > 0$ such that

This and (4.3) together imply that $|u_n^{(k)}|\leq \beta'$ for each $n\in \mathbb{Z}$, that is,

Then similar argumengts as those in the proof of Lemma 3.3 yield

Then Theorem 1.2 can be proved in the same manner as that for Theorem 1.1 and hence we omit the details.

5.   Example

In this section, we give an example to illustrate Theorem 1.1.

Example 5.1. Consider the difference equation (1.1), where

where $\theta > p > 2$, $T$ is a given positive integer. Take

Then

It is easy to see that all the assumptions of Theorem 1.1 are satisfied. Consequently, equation (1.1) has at least a nontrivial solution $u$ in $l^2$.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11971126) and the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT$_{-}$16R16).

Conflict of interest

The authors declare there is no conflicts of interest.