Existence and multiplicity of nontrivial solutions to discrete elliptic Dirichlet problems

1.   Introduction

Let $\mathbb{N}$ and $\mathbb{Z}$ be the sets of natural numbers and integers, respectively. For integers $a$, $b$, define $\mathbb{Z}(a, b): = \{a, a+1, ..., b\}$ with $a\leq b$. Given the integers $T_1$, $T_2 \geq 2$, we write $\Omega: = \mathbb{Z}(1, T_{1})\times\mathbb{Z}(1, T_{2})$. Consider the existence and multiplicity of nontrivial solutions to the following discrete elliptic problem:

with Dirichlet boundary conditions

where $\Delta_{1}$, $\Delta_{2}$ are the forward difference operators defined by $\Delta_{1}u(i, j) = u(i+1, j)-u(i, j)$, $\Delta_{2}u(i, j) = u(i, j+1)-u(i, j)$, and $\Delta^2u(i, j) = \Delta(\Delta u(i, j))$. Here, $f((i, j), u)$ is continuously differentiable with respect to $u$ and $f((i, j), 0) = 0$.

Advances in modern computing devices have made it increasingly convenient to determine the behavior of complex systems through simulations, contributing greatly to the increasing interest in discrete problems. As a result ^{[1,2,3,4,5,6,7]}, difference equations have been widely investigated and numerous results have been obtained ^{[8,9,10,11,12,13,14,15,16]}. With the development of science and technology in modern society and the progress of mathematical research, the study of difference equations has gradually shifted to the study of partial difference equations. For example, ^[17,18,19] deal with discrete Kirchhoff-type problems, whereas ^[20,21,22] present several results on multiple solutions to partial difference equations.

Equation (1.1) is a partial difference equation involving bivariate sequences with two independent integer variables over $\Omega$. It is elementary, but illustrative of many problems that are of interest in various branches of science, such as chemical reactions, population dynamics with spatial migration, and even the computation and analysis of finite difference equations ^[23,24]. Therefore, Eq (1.1) has attracted considerable attention. For example, ^[24] shows that Eq (1.1) possesses at least two nontrivial solutions, while Zhang ^[25] studied Eq (1.1) using the extremum principle. Moreover, Eq (1.1) is regarded as a discrete analog of the partial differential equation

which has been extensively studied. Consequently, investigating problem (1.1)–(1.2) is of practical significance.

Morse theory is a very powerful tool for studying the existence of multiple solutions to both differential and difference equations having a variational structure ^{[26,27,28,29,30]}. Very recently, Long ^[18,19,20] studied partial difference equations via Morse theory and obtained rich results on the existence and multiplicity of nontrivial solutions. This encourages us to consider the existence and multiplicity of nontrivial solutions for problem (1.1)–(1.2) using Morse theory.

The remainder of this paper is organized as follows. In Section 2, the variational structure and the corresponding functional are established according to (1.1)–(1.2). Moreover, we recall some related definitions and propositions that are beneficial to our results. Section 3 displays our main results and the corresponding proofs. Finally, two examples and numerical simulations are provided to illustrate our main results in Section 4.

2.   Variational structure and some auxiliary results

Let $E$ be a $T_1T_2$-dimensional Euclidean space equipped with the usual inner product $(\cdot, \cdot)$ and norm $|\cdot|$. Let

Define the inner product $\langle \cdot, \cdot\rangle$ on $S$ as

and let the induced norm be

Then, $S$ is a Hilbert space and is isomorphic to $E$. Here and hereafter, we take $u \in S$ as an extension of $u \in E$ when necessary.

Consider the functional $I: S\rightarrow \mathbb{R}$ in the following form:

where $F((i, j), u) = \int_0^uf((i, j), \tau)d\tau$. Note that $f((i, j), u)$ is continuously differentiable with respect to $u$. It is clear that $I\in C^2(S, \mathbb{R})$ and solutions of the problem (1.1)–(1.2) are precisely the critical points of $I(u)$. Moreover, for any $u, v\in S$, using the Dirichlet boundary conditions gives

Let $\Xi$ be the discrete Laplacian, which is defined by $\Xi u(i, j) = \Delta_1^2u(i-1, j)+\Delta_2^2u(i, j-1)$. According to the conclusion of ^[31], we know that $-\Xi$ is invertible and the distinct Dirichlet eigenvalues of $-\Xi$ on $[1, T_1]\times [1, T_2]$ can be denoted by $0 < \lambda_1 < \lambda_2\leq \lambda_3 \leq \cdots \leq \lambda_{T_1T_2}$. Let $\phi_k = (\phi_k(1), \phi_k(2), \cdots, \phi_k(T_1T_2))^{tr}$, $k\in[1, T_1T_2]$, where each $\phi_k$ is an eigenvector corresponding to the eigenvalue $\lambda_k$. Let $W^{-} = {\mbox{span}}\{\phi_{1}, ..., \phi_{k-1}\}$, $W^{0} = {\mbox{span}}\{\phi_{k}\}$, $W^{+} = (W^{-}\oplus W^{0})^{\bot}$. Then, $S$ can be expressed as

For later use, we define another norm as $\|u\|_{2} = \left(\sum\limits^{T_{1}}\limits_{i = 1}\sum\limits^{T_{2}} \limits_{j = 1}|u(i, j)|^{2}\right)^{\frac{1}{2}}$. Then, for any $u\in S$, we have that

Next, we recall some preliminaries with respect to Morse theory.

We say that the functional $I$ satisfies the Cerami condition ($(C)$ for short) if any sequence $\{u_{n}\}\subseteq S$ satisfying $I(u_{n})\rightarrow c$, $(1+\|u_{n}\|)\|I'(u_{n})\|\rightarrow 0$ as $n\rightarrow \infty$ has a convergent subsequence. Note that if $(C)$ is satisfied, then the deformation condition ($(D)$ for short) is satisfied ^[32].

Definition 2.1. ^[28,33] Let $u_{0}$ be an isolated critical group of $I$ with $I(u_{0}) = c\in\mathbb{R}$, and $U$ be a neighborhood of $u_{0}$. The group

is called the $q$-th critical group of $I$ at $u_{0}$. Let $\kappa = \{u\in S|I'(u) = 0\}$. For all $a\in \mathbb{R}$, each critical point of $I$ is greater than $a$ and $I\in C^{2}(S, \mathbb{R})$ satisfies $(D)$. Then, the group

is called the $q$-th critical group of $I$ at infinity.

To calculate the critical group at infinity, we need the following auxiliary proposition.

Proposition 2.1. ^[34]Suppose that $S$ is a Hilbert space, $\{I_{t}\in C^{2}(S, \mathbb{R})|t\in [0, 1]\}$. $I'_{t}$ and $\partial_{t}I_{t}$ are locally continuous. If $I_0$ and $I_1$ satisfy $(C)$, and there exist $a\in\mathbb{R}$ and $\delta > 0$ such that

then

In particular, if there is some $R > 0$ such that

and

then Eq (2.3) is satisfied.

The following three propositions are important in obtaining some nonzero critical points.

Proposition 2.2. ^[26]Let $S$ be a real Hilbert space, $I\in C^{2}(S, \mathbb{R})$. Suppose that $u_{0}$ is the isolated critical point of $I$ with limited Morse index $\mu(u_{0})$ and null dimension $\nu(u_{0})$. $I''(u_{0})$ is a Fredholm operator. Moreover, if $u_{0}$ is the local minimizer of $I$, then

Proposition 2.3. ^[34]Assume that $I\in C^{2}(S, \mathbb{R})$ with $S = S^{+}\oplus S^{-}$ and 0 is the isolated critical point of $I$. If $I$ has a local linking structure at 0 with $k = \dim S^{-} < \infty$, then

Proposition 2.4. ^[35]Let $I\in C^{2}(S, \mathbb{R})$ satisfy $(D)$. Then,

$(J_{1})$ if $C_{q}(I, \infty)\ncong 0$ holds for some $q$, then $I$ possesses a critical point $u$ such that $C_{q}(I, u)\ncong 0$;

$(J_{2})$ if 0 is the isolated critical point of $I$ and there exists some $q$ such that $C_{q}(I, \infty)\ncong C_{q}(I, 0)$, then $I$ has a nonzero critical point.

In our proofs, we also require the following Mountain Pass Lemma.

Proposition 2.5. ^[33]Let $S$ be a real Banach space and $I\in C^{1}(S, \mathbb{R})$ satisfy the Palais–Smale condition ($(PS)$ for short). Further, if $I(0) = 0$ and

$(J_{3})$ there exist constants $\rho$, $a > 0$ such that$I|_{\partial B_{\rho}}\geq a$,

$(J_{4})$ there is some $e\in S\backslash B_{\rho}$ such that $I(e)\leq 0$.

Then, $I$ possesses a critical value $c\geq a$ given by

where

3.   Main results and proofs

In this section, we state our main results and provide detailed proofs. First, the following assumptions are required:

${\bf(f_{1})}$ There exists $k\geq2$ such that

${\bf(f_{2})}$ There exists a subsequence $\{u_{n, k}^{(1)}\}\subseteq {\mbox{span}}\{\phi_k\}$ such that $\frac{\|u_{n, k}^{(1)}\;\|}{\|u_{n}\|}\rightarrow1$ as $\|u_{n}\|\rightarrow \infty$. Then, there exist $\delta_{1}$, $N_{1} > 0$ such that

${\bf(f_{3})}$ There exists a subsequence $\{u_{n, k+1}^{(1)}\}\subseteq {\mbox{span}}\{\phi_{k+1}\}$ such that $\frac{\|u_{n, k+1}^{(1)}\;\;\|}{\|u_{n}\|}\rightarrow1$ as $\|u_{n}\|\rightarrow \infty$. Then, there exist $\delta_{2}$, $N_{2} > 0$ such that

We are now in a position to state our main results.

Theorem 3.1. Let ${\bf{(f_1)}}$, ${\bf{(f_2)}}$, ${\bf{(f_3)}}$ hold. Moreover, for all $(i, j)\in\Omega$,

$({\bf{V_1}}) \quad F''((i, j), 0) < \lambda_{1}$,

$({\bf{V_2}}) \quad F''((i, j), u) > 0$ for all $u\in\mathbb{R}$

are satisfied. Then, problem (1.1)–(1.2) possesses at least three nontrivial solutions, among which one is positive and one is negative.

Consider the following sign condition:

$({\bf{F_{0}}^{\pm}})$ There exists $\delta > 0$ such that

Then, we can state the following theorem.

Theorem 3.2. Suppose ${\bf{(f_1)}}$, ${\bf{(f_2)}}$, ${\bf{(f_3)}}$ are satisfied. For all $(i, j)\in\Omega$, let

$({\bf{V_3}}) \quad f'((i, j), 0) = \lambda_{m}$,

$({\bf{V_4}})$ there exists $u_0\neq 0$ such that $f((i, j), u_0) = 0$.

Then, problem (1.1)–(1.2) possesses at least four nontrivial solutions if one of the following conditions is met:

${\bf(i)} \; {\bf{(F_{0}}^{+})}$ with $2\leq m\neq k$,     ${\bf(ii)} \quad {\bf{(F_{0}}^{-})}$ with $2 < m\neq k+1$.

Theorem 3.3. Assume ${\bf{(f_1)}}$, ${\bf{(f_2)}}$, ${\bf{(f_3)}}$, and ${\bf{(V_3)}}$ are true. Further, if $k = 1$ and either

${\bf(iii)} \quad {\bf{(F_{0}}^{+})}$ with $m\neq1$,     ${\bf(iv)} \quad {\bf{(F_{0}}^{-})}$ with $m\neq2$,

then problem (1.1)–(1.2) admits at least two nontrivial solutions.

To prove our results, $(C)$ is necessary. First, we present a detailed proof to show that $I$ satisfies $(C)$.

Lemma 3.1. Let ${\bf{(f_1)}}$, ${\bf{(f_2)}}$, and ${\bf{(f_3)}}$ hold. Then, $I$ satisfies $(C)$.

Proof. Suppose that $\{u_{n}\}\subseteq S$ and there exists a constant $c$ such that

Because $S$ is a $T_1T_2$-dimensional space, it suffices to show that $\{u_{n}\}$ is bounded. Otherwise, we can assume that

Denote ${\overline{u}}_{n} = \frac{u_{n}}{\|u_{n}\|}$. Then, $\|{\overline{u}}_{n}\| = 1$, which means that $\{\overline{u}_{n}\}$ has some subsequences. Without loss of generality, we set the subsequence to be the sequence. Moreover, there exists $\overline{u}\in S$ with $\|\overline{u}\| = 1$ such that

For any $\varphi\in S$, we obtain

From $\bf({f_1})$, there exist $b\geq\lambda_{k+1}$ and $N > 0$ such that

Set $b_{1} = \frac{b}{\|u_{n}\|}$ such that, for $n > N$, we have

If $n\leq N$, then because $f((i, j), \cdot)$ is continuous in $\cdot$, we have

Therefore, Eqs (3.1) and (3.2) ensure that $\{\frac{f((i, j), u_{n})}{\|u_{n}\|}\}$ is bounded. Consequently, $\{\frac{f((i, j), u_{n})}{\|u_{n}\|}\}$ has a convergent subsequence. We still denote this by $\{\frac{f((i, j), u_{n})}{\|u_{n}\|}\}$. Using ${\bf{(f_1)}}$ once more, we can assume that there exists some $p$ satisfying $\lambda_{k}\leq p\leq\lambda_{k+1}$ such that

Hence,

which means that $\overline{u}$ is the nontrivial solution of

with boundary conditions

Together with the maximum principle and unique continuation property, this implies that $p\equiv\lambda_{k}$ or $p\equiv\lambda_{k+1}$ for $\lambda_{k}\leq p\leq\lambda_{k+1}$.

If $p\equiv\lambda_{k}$, then $\overline{u}\in W^{0}$ and

In fact, if $\overline{u}\notin W^{0}$, then $p\overline{u} = 0$, which leads to $\|\overline{u}\| = 0$. In view of $\|\overline{u}\| = 1$, this is a contradiction. Therefore, as $n\rightarrow \infty$, we have that

Obviously, Eq (3.3) is inconsistent with $({\bf{f_2}})$.

If $p\equiv\lambda_{k+1}$, then $\overline{u}\in {\mbox{span}}\{\phi_{k+1}\}$ and

Thus,

as $n\rightarrow \infty$, which contradicts ${\bf{(f_3)}}$. Therefore, $\{u_{n}\}$ is bounded.

To calculate critical groups at infinity, we have the following lemma.

Lemma 3.2. Let $\mu_{\infty} = \dim (W^{0}\oplus W^{-})$. If ${\bf{(f_1)}}$, ${\bf{(f_2)}}$, and ${\bf{(f_3)}}$ are satisfied, then $C_{q}(I, \infty)\cong\delta_{q, \mu_{\infty}}\mathbb{Z}$.

Proof. First, for $t\in[0, 1]$, let $I_{t}: S\rightarrow \mathbb{R}$ be given as

In the following, we prove that Eq (2.4) in Proposition 2.1 is true. Otherwise, there exists $\{u_{n}\}\subseteq S$, $t_{n}\in[0, 1]$ such that

Denote $\overline{u}_{n} = \frac{u_{n}}{\|u_{n}\|}$. Then, $\|\overline{u}_{n}\| = 1$ and there exists $\overline{u}\in S$ satisfying $\|\overline{u}\| = 1$ such that $\overline{u}_{n}\rightarrow \overline{u}$ as $n\rightarrow \infty$. From Lemma 3.1, we know that

Suppose that $t_{n}\rightarrow t_0$, whereby is easy to show that $\overline{u}$ is a solution subject to

where $\xi(t_0) = \frac{1-t_0}{2}\lambda_{k}+\frac{1-t_0}{2}\lambda_{k+1}+t_{0}p$ and $\lambda_{k}\leq\xi(t_0)\leq\lambda_{k+1}$. By the maximum principle and unique continuation property, we find that

Furthermore, $t_0 = 1$ means that $t_{n}\rightarrow1$ as $n\rightarrow \infty$. Therefore,

Note that ${\bf{(f_2)}}$ and ${\bf{(f_3)}}$ are valid, and so

or

Considering $\lambda_{k}\leq\lambda_{k+1}$, we have

which implies that

Therefore,

or

which guarantees that Eq (2.4) is satisfied.

It is easy to show that Eq (2.5) is satisfied and $I_0$, $I_1$ satisfy $(C)$. In fact, because

we have that

which is impossible for $\|\overline{u}\| = 1$. Consequently, $I_0$ satisfies $(C)$. Moreover,

is continuous. According to Proposition 2.1, we have that

Note that $u = 0$ is a unique nondegenerate critical point of $I_0$ with $\mu_{0} = \dim (W^{0}\oplus W^{-})$. Thus,

Further, we have that $C_{q}(I, \infty)\cong\delta_{q, \mu_{\infty}}\mathbb{Z}$.

To gain some mountain pass-type critical points by applying the cut-technique, we verify the following compactness conditions.

Lemma 3.3. Let

such that

Then, the functional $I^{+}:S\rightarrow \mathbb{R}$ defined by

satisfies $(PS)$, where $F^{+}((i, j), u) = \int^{u}_{0}f^{+}((i, j), \tau)d\tau$.

Proof. Let $\{u_{n}\}\subseteq S$ be a $(PS)_c$ sequence, that is,

Similar to Lemma 3.1, assume that $\{u_{n}\}$ is unbounded. Then, we have

Denote $v_{n} = \frac{u_{n}}{\|u_{n}\|}$, so that $\|v_{n}\| = 1$. As $S$ is a $T_1T_2$-dimensional Hilbert space, there exists $v\in S$ satisfying $\|v\| = 1$ such that $v_{n}\rightarrow v$ as $n\rightarrow \infty$. Hence, for any $\varphi\in S$, it holds that

For any $(i, j)\in\Omega$, write $v^{+}(i, j) = \max\{v(i, j), 0\}$. Then, there exists $\alpha$ satisfying $\lambda_{k}\leq\alpha\leq\lambda_{k+1}$ such that

Together with Eq (3.6), this yields

which implies that $v$ is the nontrivial solution of

with boundary conditions

Denote $v(i_{0}, j_{0}): = \min\{v(i, j)|(i, j)\in\Omega\}$. We aim to show that $v(i_{0}, j_{0}) > 0$. Otherwise, $\alpha v^{+}(i, j) = 0$ and

which means that $\Delta_{1}v(i_{0}-1, j_{0}) = \Delta_{2}v(i_{0}, j_{0}-1) = 0$. Moreover, $v(i_{0}-1, j_{0}) = v(i_{0}, j_{0}) = v(i_{0}, j_{0}-1)$. Therefore, $v\equiv 0$ for all $(i, j)\in\Omega$. Additionally, $v$ is the nontrivial solution of Eqs (3.7)–(3.8), which implies that $v(i_{0}, j_{0}) > 0$, and so $v(i, j) > 0$. Recall that $\lambda_{k}\leq\alpha\leq\lambda_{k+1}$, we so we have that $\{u_{n}\}$ is bounded.

In the same manner as Lemma 3.3, we can state the following.

Lemma 3.4. If

where

then the functional $I^{-}:S\rightarrow \mathbb{R}$ defined by

satisfies $(PS)$, where $F^{-}((i, j), u) = \int^{u}_{0}f^{-}((i, j), \tau)d\tau$.

Lemma 3.5. If all conditions of Theorem 3.1 are fulfilled, then $I^{+}$ has a critical point $u^{+} > 0$ such that $C_{q}(I^{+}, u^{+})\cong\delta_{q, 1}\mathbb{Z}$ and $I^{-}$ has a critical point $u^{-} < 0$ such that $C_{q}(I^{-}, u^{-})\cong\delta_{q, 1}\mathbb{Z}$.

Proof. We prove the case of $I^{+}$; the proof of $I^{-}$ is similar and is omitted for brevity. With the aid of Proposition 2.5, we need only prove that $I^{+}$ satisfies ${\bf{(J_{3}})}$, ${\bf{(J_{4}})}$. According to $({\bf{V_1}})$, there exist $\rho$, $\rho_{1} > 0$ such that $F''((i, j), 0) < \rho_{1} < \lambda_{1}$ and

Then, for any $(i, j)\in\Omega$ and $u\in S$ with $\|u\|\leq\sqrt{\lambda_{1}}\rho_{1}$, we have

with $U = \{(i, j)\in\Omega|u(i, j)\geq0\}$, which ensures ${\bf{(J_{3}})}$ is valid.

Using $({\bf{f_1}})$, there exist $\gamma > \lambda_{k-1}(\geq\lambda_{1})$, $b_2\in\mathbb{R}$ such that

For $t > \rho$, choose $e\in {\mbox {span}} \{\phi_{1}\}$. Then, we have

Therefore, ${\bf{(J_{4}})}$ is satisfied.

By the Mountain Pass Lemma, $I^{+}$ possesses a critical point $u^{+}\neq 0$. Moreover, there exists a sequence $\{u_{n}^{+}\}$ such that $I'^{+}(u_{n}^{+})\rightarrow I'^{+}(u^{+})$ as $n\rightarrow \infty$. For any $\varphi\in S$, we have

Letting $n\rightarrow \infty$,

which leads to

with $\lambda_{k}\leq\alpha\leq\lambda_{k+1}$ and $u^{+}(i, j): = \max\{u(i, j), 0\}$. Similar to the proof of Lemma 3.3, $u^{+}$ is positive.

We now calculate $C_{q}(I^{+}, u^{+})$. Recalling Eq (2.1), we find that

and there exists some $v_{0}\neq 0$ such that

which implies that $v_{0}$ is a solution of

Combining this with $({\bf{V_2}})$, it follows that

admits an eigenvalue $\lambda = 1$ such that $\dim\ker(I''(u^{+})) = 1$. Hence, we conclude that $C_{q}(I, u^{+})\cong C_{q}(I^{+}, u^{+})\cong\delta_{q, 1}\mathbb{Z}$. This completes the proof.

It is time for us to give the detailed proof of Theorem 3.1 using Proposition 2.4.

Proof of Theorem 3.1 Given $I'(0) = 0$ and $({\bf{V_1}})$, then

which implies that 0 is the local minimizer of $I$. From Proposition 2.2, we have

Furthermore, Lemma 3.2 ensures that

Then, according to Proposition 2.4, there exists some critical point $u_{1}\neq 0$ of $I$ such that

Consequently, we conclude that $u^{+}$, $u^{-}$ and $u_{1}$ are nontrivial critical points of $I$ with $u^{+} > 0$ and $u^{-} < 0$. The proof of Theorem 3.1 is achieved.

Before verifying Theorem 3.2 using Proposition 2.3, we need the following lemma about local linking.

Lemma 3.6. Let ${\bf{(V_3)}}$ and ${\bf{(F_{0}}^{+})}$ (or ${\bf{(F_{0}}^{-})}$) hold. Then, $I$ has a local linking at 0 with respect to

where $S^{-} = \mathit{\mbox{span}}\{\phi_{1}, \phi_{2}, ..., \phi_{m}\}$ (or $S^{-} = \mathit{\mbox{span}}\{\phi_{1}, \phi_{2}, ..., \phi_{m-1}\}$).

Proof. Suppose that ${\bf{(F_{0}}^{+})}$ is satisfied. Then, there exists $\delta > 0$ such that $|u(i, j)|\leq\delta$, $\|u\|\leq\delta\sqrt{\lambda_{T_1T_2}}$, and

For $u\in S^{-}$ with $0 < \|u\|\leq\delta\sqrt{\lambda_{T_1T_2}}$, we have

For $u\in S^{+}$ with $0 < \|u\| < \delta\sqrt{\lambda_{T_1T_2}}$, we obtain

Obviously, $I(0) = 0$. Combining this with Eqs (3.12) and (3.13), it is clear that $I$ has a local linking at 0.

Proof of Theorem 3.2 Denote $\mu_{0} = \dim{\mbox{span}}\{\phi_{1}, ..., \phi_{m-1}\}$, $\nu_{0} = \dim{\mbox{span}}\{\phi_{m}\}$. Let $({\bf{V_3}})$ be true. Then, 0 is degenerate. Taking account of Proposition 2.3, we find that

In contrast, Lemma 3.2 gives $C_{q}(I, \infty)\cong\delta_{q, \mu_{\infty}}\mathbb{Z}$. Therefore, Proposition 2.4 guarantees the existence of $u^{*}$ such that

Moreover, $m\neq k$ implies that $\mu_{0}+\nu_{0}\neq\mu_{\infty}$. Thus, $u^{*}\neq0$.

From $({\bf{V_4}})$, there exists $u_0\neq 0$ such that $f((i, j), u_0) = 0$. Without loss of generality, we can assume that $u_{0} > 0$. In the sequence, we intend to obtain the local minimizer of $I$. Define

and let

where $\widetilde{F}((i, j), u) = \int_0^u\widetilde{f}((i, j), \tau)d\tau$. Then, $\widetilde{I}$ is coercive and continuous. Therefore, there exists a minimizer $\widetilde{u}_{0}$ of $\widetilde{I}$. From the maximum principle, we deduce that $\widetilde{u}_{0} = 0$ or $0 < \widetilde{u}_{0}(i, j) < u_{0}$ for all $(i, j)\in\Omega$. Moreover, $({\bf{V_3}})$ means that 0 is not a minimizer. Consequently, $\widetilde{u}_{0}\neq0$ is a local minimizer of $I$ and $C_{q}(I, \widetilde{u}_{0})\cong\delta_{q, 0}\mathbb{Z}$, $q\in\mathbb{Z}$.

Denote $\widehat{F}((i, j), u) = \int_0^u\widehat{f}((i, j), \tau)d\tau$, where

The corresponding functional is then given by

If $v$ is a nonzero critical point of $\widehat{I}$, then $\widetilde{u}_{0}+v$ is a critical point of $I$ with

Define

and construct the corresponding functional as

with $\widehat{F}^{+}((i, j), u) = \int_0^u\widehat{f}^{+}((i, j), \tau)d\tau$. Then, $\widetilde{u}_{0}$ is a local minimizer of $I$ for $\widehat{I}^{+}$ satisfying $(PS)$, which leads to $v = 0$ being a local minimizer of $\widehat{I}^{+}$. Thus, ${\bf{(J_{3}})}$ is fulfilled. Applying ${\bf{(f_1)}}$ yields

which ensures that ${\bf{(J_{4}})}$ is satisfied. By Lemma 2.5, $\widehat{I}^{+}$ possesses a critical point $v^{+} > 0$. Furthermore, $\widehat{I}$ possesses a critical point $v^{+}$ with $C_{q}(\widehat{I}, v^{+})\cong\delta_{q, 1}\mathbb{Z}$, $q\in\mathbb{Z}$. As a result, $u^{+} = \widetilde{u}_{0}+v^{+}$ is a critical point of $I$ satisfying $C_{q}(I, u^{+})\cong\delta_{q, 1}\mathbb{Z}$, $q\in\mathbb{Z}$. Similarly, $u^{-} < \widetilde{u}_{0}$ is a critical point of $I$ satisfying $C_{q}(I, u^{-})\cong\delta_{q, 1}\mathbb{Z}$. Therefore, $u^{*}$, $\widetilde{u}_{0}$, $u^{\pm}$ are four nontrivial solutions of $I$ and $\widetilde{u}_{0}$, $u^{+}$ are positive.

For the case $u_{0} < 0$, repeating the above steps shows that $I$ has four nontrivial solutions, among which there are two negative solutions. Therefore, $I$ admits four nontrivial solutions and the proof is finished.

Proof of Theorem 3.3 Similar to the proof of Theorem 3.2, we find that $C_{q}(I, 0)\cong\delta_{q, \mu_{0}+\nu_{0}}\mathbb{Z}$, $q\in\mathbb{Z}$. Because $k = 1$, $C_{q}(I, \infty)\cong\delta_{q, 1}\mathbb{Z}$ and there exists some critical point $\overline{u}^{*}$ such that $C_{1}(I, \overline{u}^{*})\ncong0$. Moreover, $u^{*}$ is the critical point of $I$ satisfying $C_{q}(I, \overline{u}^{*})\cong\delta_{q, 1}\mathbb{Z}$. From $m\neq1$, we conclude that $\overline{u}^{*}\neq0$. If $\kappa = \{0, \overline{u}^{*}\}$, the Morse equality implies that

Of course, Eq (3.16) is impossible. Therefore, problem (1.1)–(1.2) possesses at least two nontrivial solutions.

4.   Examples

Finally, we present two examples to verify the feasibility of our results.

Example 4.1. Take $T_{1} = 3$, $T_{2} = 2$, and consider

with the boundary value conditions of Eq (1.2).

Because $f((i, j), u) = \frac{(\frac{\lambda_{1}}{2}-\frac{\lambda_{k}+\lambda_{k+1}}{2})u(i, j)}{1+[u(i, j)]^{2}}+\frac{\lambda_{k}+\lambda_{k+1}}{2}u(i, j)$, it follows that

It is not difficult to verify that $f((i, j), 0) = 0$, $F''((i, j), 0) = \frac{\lambda_{1}}{2} < \lambda_{1}$, and

which means that $({\bf{f_1}})$, $({\bf{V_1}})$ and $({\bf{V_2}})$ are satisfied. If $\frac{\|u_{n, k}^{(1)}\;\|}{\|u_{n}\|}\rightarrow1$ as $\|u_{n}\|\rightarrow \infty$, then we obtain

Therefore, ${\bf{(f_2)}}$ is satisfied. Similarly, ${\bf{(f_3)}}$ is valid. Therefore, Theorem 3.1 guarantees that problem (4.1)–(1.2) admits at least three nontrivial solutions, of which one is positive and one is negative.

Example 4.2. Take $T_{1} = 3$, $T_{2} = 2$, and consider

with the boundary value conditions of Eq (1.2).

Denote

Then, $f((i, j), 0) = 0$ and there exists $u = \pm\sqrt{\frac{4\lambda_{m}}{\lambda_{k}+\lambda_{k+1}}}\neq0$ such that $f((i, j), u) = 0$, which means that $({\bf{V_4}})$ is satisfied.

A direct computation yields

and

and so $f'((i, j), 0) = \lambda_{m}$ and $({\bf{V_3}})$ is valid.

As $\|u_{n}\|\rightarrow \infty$, if $\frac{\|u_{n, k}^{(1)}\;\|}{\|u_{n}\|}\rightarrow1$, then

Therefore, $({\bf{f_2}})$ is satisfied. Similar to $({\bf{f_2}})$, we can show that $({\bf{f_3}})$ is satisfied.

In the following, we verify $({\bf{f_{1}}})$ and $({\bf{F_{0}}^{+}})$. If we write

then $A$ is positive-definite and the eigenvalues of $A$ are

Let $m = 2$, $k = 3$. Then, $5-\sqrt{2}\leq\liminf\limits_{|u|\rightarrow \infty}\frac{f((i, j), u)}{u} = \limsup\limits_{|u|\rightarrow \infty}\frac{f((i, j), u)}{u} = 4\leq3+\sqrt{2}$, which means that $({\bf{f_{1}}})$ is satisfied. Further, there exists $\delta > 0$ such that, when $|u(i, j)|\leq\delta$, the following holds for any $(i, j)\in\mathbb{Z}(1, 3)\times\mathbb{Z}(1, 2)$:

In fact, for any $(i, j)\in\Omega$, we can choose $\delta = 1 > 0$, and then $0 < [u(i, j)]^{2}\leq1$ for $0 < |u(i, j)|\leq1$. This means that $\ln(2-u^{2})\geq0$ and $2F((i, j), u)-3u^{2}\geq0$. Thus, $({\bf{F_{0}}^{+}})$ holds and all conditions of Theorem 3.2 are satisfied. Consequently, Theorem 3.2 ensures that (4.2)–(1.2) admits at least four nontrivial solutions.

More clearly, using Matlab, we find that problem (4.2)–(1.2) has 36 nontrivial solutions. Some examples of these solutions are as follows: $(0.5061, 0.6548, 0.5061, 0.5061, 0.6548, 0.5061)$, $(-0.5061, -0.6548, -0.5061, -0.5061, -0.6548, -0.5061)$, $(-1.9858, 2.1015, 0.6084, 1.9858, -2.1015, -0.6084)$, and $(1.2137, -4.7492, 1.2137, 13.9612, -1.5032, 13.9612)$.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11971126). The authors wish to thank the handling editor and the anonymous referees for their valuable comments and suggestions.

Conflict of interest

All authors declare no conflicts of interest in this paper.