Results on homoclinic solutions of a partial difference equation involving the mean curvature operator

1.   Introduction

In the present paper, our goal is to investigate the existence and multiplicity of nontrivial homoclinic solutions of the following second-order partial difference equation involving the mean curvature operator with a parameter $\lambda > 0$:

Here, forward difference operators are given by $\Delta_{1}u(k, l) = u(k+1, l)-u(k, l)$ and $\Delta_{2}u(k, l) = u(k, l$$+1)-u(k, l)$. For all $s\in\mathbb{R}$, let the mean curvature operator be denoted by $\phi_{c}(s) = \frac{s}{\sqrt{1+s^{2}}}$ and $\Phi_{c}(s): = \int^{s}_{0}\phi_{c}(t)dt = \sqrt{1+s^{2}}-1$. For all $(k, l)\in\mathbb{Z}^{2}$, we assume that the nonlinear term $f((k, l), u)$ is continuous in $u$ with $F\left((k, l), u\right) = \int^{u}_{0}f\left((k, l), \tau\right)d\tau$ and fulfills the following basic hypotheses:

$(F_1)$ $\lim\limits_{|u|\rightarrow 0}\frac{f((k, l), u)}{u} = 0$;

$(F_2)$ $\lim\limits_{|u|\rightarrow +\infty}\frac{F((k, l), u)}{u^{2}} = \infty$;

$(F_3)$ there exist $\alpha > 2$ and $d > 0$ such that $|f((k, l), u)|\leq d|u|^{\alpha-1}$ for all $u\in\mathbb{R}$;

$(F_4)$ there exists a constant $\beta\geq1$ such that $\beta G((k, l), u)\geq G((k, l), \nu u)$, here $u\in\mathbb{R}$, $\nu\in [0, 1]$ and $G((k, l), u) = f((k, l), u)u-2F((k, l), u)$;

$(F_5)$ $\sup\limits_{|u|\leq T}|F((k, l), u)|\in l^{1}$ for all $T > 0$;

$(F_6)$ $f((k, l), -u) = -f((k, l), u)$ for all $u\in\mathbb{R}$.

Further, we assume that the potential $b: \mathbb{Z}^{2}\rightarrow \mathbb{R}$ satisfies

$(B)$ $b(k, l)\geq b_{\ast} > 0$, $\lim\limits_{|k|+|l|\rightarrow \infty}b(k, l) = +\infty$, where $b_{\ast} = \min\{b(k, l): (k, l)\in\mathbb{Z}^{2}\}$.

Difference equations, regarded as discrete analogues of differential equations, have been used extensively in a variety of fields, including biology, economics, computer science, machine learning, artificial intelligence, and other fields over the last few decades ^[1,2,3]. It has sparked a lot of interest and attention from academics. And abundant research results on various aspects have been acquired, such as periodic solutions ^[4,5], boundary value problems ^[6,7], homoclinic solutions ^[8,9], heteroclinic solutions ^[10,11], etc. It is worthy of pointing out that the theory of difference equations developed quickly as a result of the groundbreaking work of Guo and Yu ^[12], who were the first to study difference equations using the variational method and critical point theory.

It is widely acknowledged that difference equations play an important role in mathematical modeling for real-world problem solving. As our time continues to progress, more and more factors need to be taken into account in many aspects of our lives. Consequently, partial difference equations with two or more variables appear in a wide range of domains, including fluid dynamics, mechanical engineering analysis, population growth, quantum mechanics, and image processing ^{[13,14,15,16]}. In this situation, studying partial difference equations makes sense, and numerous significant works have been published recently. Here mention a couple; Long reviewed the discrete Kirchhoff-type problems and provided multiple results on nontrivial solutions ^[17,18] and least energy sign-changing solutions ^[19]. Homoclinic solutions for partial difference equations with $p$-Laplacian were displayed in ^[20]. Very recently, the nonexistence and existence of periodic solutions for partial difference equations were presented in ^[21,22].

The mean curvature operator, involved in Eq (1.1), has broad applicability and important theoretical significance. It has been widely used to describe a variety of problems. For example, the dynamic problem of combustible gas ^[23], the capillarity problem in hydrodynamics ^[24], and the flux-limited diffusion phenomenon ^[25]. We refer the reader to ^[26] for more detail. Much like partial difference equations with $p$-Laplacian, partial difference equations involving mean curvature operators and their various modified forms have drawn a lot of attention in recent decades. For instance, given parameter $\lambda > 0$, Wang and Zhou ^[27] considered Eq (1.1) with Dirichlet boundary conditions and obtained at least three solutions.

It is evident from the above-mentioned results that partial difference equations are widely applied and thoroughly researched. However, most given results are dependent on finite-dimensional space. Meanwhile, homoclinic solutions, without periodic condition assumptions, are studied in an infinite-dimensional space, which brings us an obstacle to overcome the lack of compactness of corresponding variational functional. There is, of course, comparatively less work. Moreover, Eq (1.1) involves both the parameter $\lambda$ and the $\phi_c$-Laplacian, which both make the study more challenging and complex. Therefore, it will be interesting and significant to deal with homoclinic solutions of Eq (1.1).

Inspired by the aforementioned reasons, we shall manage to investigate the existence and multiplicity of homoclinic solutions of Eq (1.1) by critical point theory. As usual, we say $u = \{u(k, l)\}_{(k, l)\in\mathbb{Z}^{2}}\neq 0$ is a nontrivial homoclinic solution of Eq (1.1) refers to $u$ solving Eq (1.1) and satisfying

Now we are in a position to state our main results, which read as follows.

Theorem 1.1. If $(B)$ and $(F_{1})$–$(F_{4})$ are valid, then Eq (1.1) admits at least one nontrivial homoclinic solution for all $\lambda > \frac{4+b_{\ast}}{2b_{\ast}}$.

Theorem 1.2. If $(B)$ and $(F_{2})$–$(F_{6})$ are satisfied, then Eq (1.1) has infinitely many homoclinic solutions for all $\lambda > 0$.

We also display the following remarks to demonstrate that our obtained results are easier to verify and more widely applied.

Remark 1.1. In ^[28], the authors studied

and obtained some similar results to Theorems 1.1 and 1.2. Obviously, (1.2) is a very special case of Eq (1.1). And our results improve and generalize those in ^[28].

Remark 1.2. The nonlinearity $F$ can be sign-changing, which is much weaker and more applicable than the nonnegative case ($F((k, l), u)\geq 0$ for all $(k, l)\in\mathbb{Z}^{2}$ and $u\in\mathbb{R}$) in many related articles, for example, ^[29,30,31].

Remark 1.3. As known to all, either a classical Ambrosetti-Rabinowitz condition

$(AR)$ there exist constants $\mu > 2$ and $d_{0} > 0$ such that

or a generalization,

$(AR)'$ there exists $\mu > 2$ and $d_{0} > 0$ such that

plays a vital role in the application of critical point theory to ensure that any $(C)_{c}$ sequence of the corresponding energy functional is bounded. Meanwhile, $(AR)$ or $(AR)'$ indicates that there exist $r_{1}$, $r_{2} > 0$ such that

That is $(F_{2})$. However, it is not necessary for $F$ to fulfill $(AR)'$ in many practical problems. For instance, for $((k, l), u)\in\mathbb{Z}^{2}\times\mathbb{R}$, let

Then

We depict them as Figure 1 for the reader's convenience.

It can be certified that $F((k, l), u)$ does not satisfy $(AR)'$. But it does satisfy $(F_{1})$–$(F_{6})$. In this sense, our results release $(AR)'$ or $(AR)$ somehow and improve some existing results.

The structure of this paper is developed as follows: In Section 2, we introduce some basic results and establish the variational structure related to Eq (1.1). Moreover, two essential lemmas, including the compactness of the $(C)_c$ sequence and the compact embedding of spaces $X$ to space $l^{2}$, are provided. Detailed proofs of main results are presented in Section 3. Finally, we illustrate our results with two examples in Section 4 as an end.

2.   Variational structure and some auxiliary results

To prove the main results, we construct the corresponding variational functional of Eq (1.1) and provide some essential lemmas.

For $1\leq r < +\infty$, denote the set of all functions $u: \mathbb{Z}^{2}\rightarrow \mathbb{R}$ by

For any $u\in l^{r}$, we define

and let $l^{\infty}$ represent the set of all functions $u: \mathbb{Z}^{2}\rightarrow \mathbb{R}$ such that

Then $(l^r, {\|\cdot\|}_{l^r})$ is a reflexive Banach space. Moreover, for $1\leq \kappa\leq \iota \leq +\infty$, there holds

Denoted reflexive Banach space by $(X, \|\cdot\|_{X})$, where

and

Then we have

For $\lambda > 0$, define an energy functional $I_{\lambda}: X\rightarrow \mathbb{R}$, associated to Eq (1.1), as

Clearly

Then definitions of $\Phi$ and $\Psi$ guarantee that $\Phi$ and $\Psi$ are continuously Gateaux differentiable. Thus, $I_{\lambda}\in C^{1}(X, \mathbb{R})$. Moreover, for any $u, v\in X$, direct computation yields that

and

Namely,

Since $v\in X$ is arbitrary, (2.4) implies $\langle I_{\lambda}'(u), v\rangle = 0$ if and only if

which is exactly Eq (1.1). Consequently, it is sufficient to seek nonzero critical points of $I_\lambda$ (defined by (2.3)) to obtain nontrivial homoclinic solutions of Eq (1.1).

For $I_{\lambda}\in C^1(X, \mathbb{R})$, we list the definition of the ${(C)}_c$ condition and two famous theorems that are essential tools to verify our main results.

Definition 2.1. Let $I$ be a $C^{1}$-functional defined on a real reflexive Banach space $X$. A Cerami sequence ($(C)_c$ sequence for short) refers to a sequence $\{u_n\}\subset X$ that fulfills

We say that $I$ satisfies the ${(C)}_c$ condition in $X$ if any ${(C)}_c$ sequence for $I$ has a convergent subsequence for some $c\in \mathbb{R}$.

Lemma 2.1. (Mountain Pass Lemma^[32]) Let $I$ be a $C^{1}$-functional defined on a real reflexive Banach space $X$. Suppose that

$(J_{1})$ there exist $\gamma$, $\widetilde{a} > 0$ such that $I(u)\geq \widetilde{a}$ for $\|u\| = \gamma$;

$(J_{2})$ there is an $e\in X$ with $\|e\|\geq \gamma$ such that $I(e)\leq 0$.

Then $I$ possesses a $(C)_c$ sequence with $c = \inf\limits_{g\in\Gamma}\max\limits_{s\in[0, 1]}I(g(s))\geq \widetilde{a}$, where

Further, $I$ attains a critical value $c$ if the ${(C)}_c$ condition is met.

Before introducing the fountain theorem, we give some notations to help illustrate the content. Let reflexive Banach space $X$ be separable. Then for every $k\in\mathbb{N}$, there is a finite dimensional space $X_{k}\subset X$ such that $X = \overline{\bigoplus_{k\in\mathbb{N}}X_{k}}$. Set

Lemma 2.2. (Fountain theorem^[33]) Suppose that $I\in C^{1}(X, \mathbb{R})$ fulfills the $(C)_c$ condition for some $c\in \mathbb{R}$ and $I(-u) = I(u)$ for $u\in X$. For every $n\in\mathbb{N}$, if there exist $\rho_{n} > r_{n} > 0$ satisfying

$(i)$ $H_{n}: = \inf\limits_{u\in T_{n}, \|u\|_{X} = r_{n}}{I(u)}\rightarrow \infty$ as $n\rightarrow \infty$;

$(ii)$ $I_{n}: = \max\limits_{u\in S_{n}, \|u\|_{X} = \rho_{n}}{I(u)}\leq 0$.

Then I has an unbounded sequence of critical values.

Lemma 2.3. Assume that condition $(B)$ holds. Then the embedding $X\hookrightarrow l^{2}$ is compact.

Proof. Let $\{u_{n}\}\subset X$ be bounded. By the Banach-Steinhaus theorem, we obtain that $\sup\limits_{n\in\mathbb{N}}\|u_{n}\|_{X} < \infty$. Then there is a constant $M_{0} > 0$ such that

Taking a subsequence, still denoted by $\{u_{n}\}$, we have

Without loss of generality, we assume $u = 0$, in particular $u_{n}(k, l)\rightarrow 0$ as $n\rightarrow +\infty$ for all $(k, l)\in\mathbb{Z}^{2}$. In view of $(B)$, for any $\varepsilon > 0$, there is $N_0\in\mathbb{N}$ such that

Then

Additionally, the continuity of the finite sum implies that there exists $N_{1}\in \mathbb{N}$ such that

Choosing $N = \max\{N_0, N_1\}$ and jointing (2.5) with (2.6), it follows that

which means that $u_{n}\rightarrow 0$ in $l^{2}$. Therefore, the embedding $X\hookrightarrow l^{2}$ is compact. □

Lemma 2.4. Assume $(B)$ and $(F_{2})$–$(F_{4})$ hold. Then for $c\in\mathbb{R}$, $I_{\lambda}$ satisfies the $(C)_c$ condition.

Proof. Set $\{u_{n}\}$ be a $(C)_c$ sequence of $I_{\lambda}$, that is,

First, for any fixed $\lambda > 0$, we are to show that $\{u_{n}\}$ is bounded in $X$. Arguing indirectly, we assume $\|u_{n}\|_{X}\rightarrow +\infty$ as $n\rightarrow \infty$. Let $w_{n} = u_{n}/\|u_{n}\|_{X}$, then $\|w_{n}\|_{X} = 1$, which means that $\{w_{n}\}$ is bounded in $X$ and has a weakly convergence subsequence. Suppose that there is $w\in X$ such that $w_{n}\rightharpoonup w$ in $X$. In view of Lemma 2.3, we have

By $(F_{3})$, for all $(k, l)\in\mathbb{Z}^{2}$ and $u\in\mathbb{R}$, simple calculation gives that

Case 1. We show $w\neq0$ is impossible. Let $\Lambda = \{(k, l)\in\mathbb{Z}^{2}: w(k, l)\neq 0\}$. Due to $I_{\lambda}(u_{n})\rightarrow c$ as $n\rightarrow \infty$, there exists a constant $C$ such that $I_{\lambda}(u_{n})\geq C$. Moreover, $\Phi_{c}(s): = \sqrt{1+s^{2}}-1$ means that $\Phi_{c}(u)\leq\frac{1}{2}u^{2}$. Combining with (2.2), we obtain

Dividing both sides of (2.9) by $\|u_{n}\|_X^{2}$, it follows that

However, fixing $(k_{0}, l_{0})\in \Lambda$, we have

Thus, $(F_{2})$ means that

which is a contradiction of (2.10). Subsequently, $w\neq0$ is absurd.

Case 2. We verify $w = 0$ is false. Define

Recall $\|u_{n}\|_{X}\rightarrow +\infty$ as $n\rightarrow \infty$. Then, for any $M > 1$, there is an $n\in\mathbb{N}$ sufficiently large such that $\|u_{n}\|_{X}\geq 2M^{\frac{1}{2}}$. Write

Then there holds

Moreover, (2.8) and $w_{n}\rightarrow w$ in $l^{2}$ ensure that

Hence, combining (2.11) with (2.12), we obtain

Consequently, by the arbitrariness of $M$, (2.13) means

Since $I_\lambda\in C^1(X, \mathbb{R})$ and $I_{\lambda}(\nu_{n}u_{n}) = \max\limits_{\nu\in [0, 1]}{I_{\lambda}(\nu u_{n})}$, $I_{\lambda}(\nu u_{n})$ reaches its maximum at $\nu_{n}\in(0, 1)$ as $n$ is large enough. Thus,

On the other side, based on the fact that $\Phi_{c}(u)-\frac{1}{2}\phi_{c}(u)u = \sqrt{1+u^2}-1-\frac{1}{2}\frac{u^2}{\sqrt{1+u^2}}$ is an even function and increases in $[0, \infty)$, (2.15), and $(F_{4})$ indicate that

Then by (2.7), we deduce that

which is a contradiction with (2.14). Hence, $w = 0$ is also invalid. Therefore, $\|u_{n}\|_{X}\rightarrow +\infty$ as $n\rightarrow \infty$ is impossible. And $\{u_{n}\}$ is bounded.

Next, we intend to verify that the bounded sequence $\{u_{n}\}$ has a convergent subsequence. Notice that $\{u_{n}\}$ has a weakly convergent subsequence, which might still be denoted by $\{u_{n}\}$. Suppose $u_{n}\rightharpoonup u$ in $X$; by Lemma 2.3, we have

Making use of $(F_{3})$ and the Hölder inequality, we have that

Since $\{u_{n}\}$ is bounded in $X$ and $u_{n}\rightarrow u$ in $l^{2}$, (2.1), (2.2), and (2.16) ensure that

Notice the fact that $\phi_{c}(u)$ is strictly increasing for all $u\in \mathbb{R}$, we obtain

Hence,

That is,

Further, weak convergence and boundedness of the $(C)_{c}$ sequence in $X$ guarantee that

Thanks to (2.17)–(2.19), we draw a conclusion that

which just is

Consequently, the above procedure of proof manifests that $I_{\lambda}: X\rightarrow \mathbb{R}$ satisfies the $(C)_c$ condition. And this completes the proof. □

3.   Proofs of main results

In this section, we exhibit associated proofs of Theorems 1.1 and 1.2 at length.

Proof of Theorem 1.1. Owing to $(F_{1})$, we know that for any $\varepsilon > 0$, there is $\delta > 0$ such that

Let $\partial B_\gamma$ be the boundary of $B_\gamma$, where $B_{\gamma}(0)$ is an open ball with center $0$ and radius $\gamma > 0$.

Select $\gamma = \|u\|_{X} = \sqrt{b_{\ast}}\delta$. By (2.2), it follows that

For all $u\in {B_{\gamma}(0)}\setminus{\{0\}}$, (3.1) implies that

Then by the definitions of $I_{\lambda}$ and $\Phi_c$, there holds

Take $\varepsilon = \frac{b_{\ast}}{4\lambda}$ in (3.2), we find that

which manifests that $I_\lambda$ satisfies $(J_1)$ of Lemma 2.1.

Next, we verify $(J_2)$ is fulfilled. Since $b_{\ast} = \min\{b(k, l): (k, l)\in\mathbb{Z}^{2}\}$, there is $(k_{\ast}, l_{\ast})\in\mathbb{Z}^{2}$ satisfying $b(k_{\ast}, l_{\ast}) = b_{\ast}$. Define $e = \{e(k, l)\}$ by

By $(F_{2})$, there is $\xi > 0$ such that

Let $u = te\in X$, for sufficiently large $|t|$ such that $|t| > \xi$, we obtain that

Remind $\lambda > \frac{4+b_{\ast}}{2b_{\ast}}$. Then (3.3) arrives that $I_{\lambda}(te)\rightarrow -\infty$, as $|t|\rightarrow \infty$. Thus, there exists $t_{0}\in \mathbb{R}$ such that

which means that $I_\lambda$ satisfies $(J_2)$. Subsequently, Lemma 2.1 ensures that $I_\lambda$ admits a $(C)_c$ sequence with

where

Moreover, Lemma 2.4 verifies that $I_\lambda$ satisfies the $(C)_c$ condition. Using Lemma 2.1 once more, we know that $I_\lambda$ gets a critical value $c\geq\widetilde{a} > 0$. Thereby, $I_\lambda$ possesses a nontrivial critical point $u$ with $I_\lambda(u) = c$. Subsequently, Eq (1.1) has at least one nontrivial homoclinic solution in $X$. □

The proof of Theorem 1.2 is done by fountain theorem. We display the verification at length as follow.

Proof of Theorem 1.2. First, by $(F_{6})$ and Lemma 2.4, we obtain that $I_{\lambda}\in C^{1}(X, \mathbb{R})$ is an even function and satisfies the $(C)_{c}$ condition. According to Lemma 2.2, it is shown that both $(i)$ and $(ii)$ are valid. As a matter of convenience, we list some notations and basic results needed for later use. For $(i, j)\in\mathbb{Z}^{2}$, define $\eta_{(i, j)} = \{\eta_{(i, j)}(k, l)\}$ as follows:

Then $X = \overline{span\{\eta_{(i, j)}: (i, j)\in\mathbb{Z}^{2}\}}$. Denote

Choose $\vartheta_{n} = \sup\limits_{u\in T_{n}, \|u\|_{X} = 1}\|u\|_{l^{2}}$, then

where $\alpha$ is given by $(F_{3})$.

Let $r_{n} = \left(\frac{4\lambda d\vartheta_{n}^{\alpha}}{\alpha}\right)^{\frac{1}{2-\alpha}}$. Then we obtain that $r_{n} > 0$ and $\lim\limits_{n\rightarrow \infty}r_{n} = +\infty$. In view of (2.8), (3.4) with $\|u\|_{X} = r_{n}$ for $u\in T_{n}$, we have

For $\alpha > 2$, it follows that

Thus, $(i)$ of Lemma 2.2 holds.

Since $S_{n}$ is finite-dimensional, which implies that all forms of $S_{n}$ are equivalent. Hence, there is $B_{n} > 0$ such that

In view of $(F_{2})$, we can obtain that there is a $T > 0$ such that

From $(F_{5})$, it follows that there is a non-negative $\theta\in l^{1}$ satisfying

For any $u\in S_{n}$ with $\|u\|_{\infty} > T$, write

Similar to (2.2), one has

Noting that $\sup\limits_{(k, l)\in L_{u, 1}}|u(k, l)| = \sup\limits_{(k, l)\in\mathbb{Z}^{2}}|u(k, l)| = \|u\|_{\infty}$, it can be shown that

Therefore, for any fixed $\lambda > 0$, from (3.6)–(3.9), we obtain that

which means that we can choose $\rho_{n}$ with $\rho_{n} > r_{n} > 0$ large enough such that

Thus, $(ii)$ of Lemma 2.2 holds as well. Consequently, Lemma 2.2 is satisfied, and $I_{\lambda}$ has a critical point sequence $\{u_{n}\}\subset X$ satisfying $I_{\lambda}(u_{n})\rightarrow \infty$ as $n\rightarrow \infty$. This completes the proof of Theorem 1.2. □

4.   Examples

In sequence, we state two examples to illustrate the applicability of our conclusions.

Example 4.1. Given $b(k, l) = k^{2}+l^{2}+1$. For any $(k, l)\in\mathbb{Z}^{2}$, consider Eq (1.1) with

By (4.1), it follows that

And $F$ is a sign-changing function; see Figure 2. Moreover,

So $(F_{1})$ is fulfilled. If $|u|\rightarrow +\infty$, there holds

Thus, $(F_{2})$ holds.

Choose $\alpha = 5$ and $d = 2$, we obtain

Then $(F_{3})$ is true.

Further, direct computation yields that

which ensures that

Therefore, taking $\beta = 3$, (4.2) implies that $(F_{4})$ is true. In addition, it can be verified that $b(k, l)$ satisfies $(B)$.

Consequently, $f$ defined by $(4.1)$ satisfies all assumptions of Theorem $1.1$, and Eq (1.1) possesses at least one nontrivial homoclinic solution for all $\lambda > \frac{5}{2}$.

Example 4.2. For any $(k, l)\in\mathbb{Z}^{2}$, take

Consider Eq (1.1) with (4.3).

From (4.3), we know that $b(k, l)$ satisfies $(B)$, and $f((k, l), u)$ is odd in $u$ that fulfills $(F_{6})$. Moreover, direct computations yield that

and $(F_{5})$ holds. Then

which shows that $(F_{2})$ is true.

Choose $\alpha = 3$ and $d = \frac{\pi}{2}$, we obtain

Then $(F_{3})$ holds.

Further, for any $(k, l)\in\mathbb{Z}^{2}$, there holds

Namely, $G((k, l), u)\geq 0$. Take $\beta = 2$. By (4.4), we get $(F_{4})$ is met.

Then $f$, defined by (4.3), satisfies all assumptions of Theorem 1.2. As a result, Eq (1.1) admits infinitely many solutions for $\lambda > 0$.

Author contributions

Yuhua Long: Conceptualization, Funding acquisition, Visualization, Writing-original draft, Writing-review & editing; Sha Li: Conceptualization, Visualization, Writing-original draft.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by National Natural Science Foundation of China (No. 12471177).

Conflict of interest

Authors state no conflict of interest.