Existence of nontrivial solutions for a poly-Laplacian system involving concave-convex nonlinearities on locally finite graphs

1.   Introduction

The research on the existence of nontrivial solutions of elliptic partial differential equations and systems involving concave-convex nonlinearities in Euclidean space have attracted some attentions. In ^[1], Ambrosetti et al. studied the second order Laplacian equation involving concave-convex nonlinearities with a constant coefficient. With the help of the sub- and supersolutions, as well as variational arguments, they obtained some existence and multiplicity results of solutions. In ^[2], Brown and Wu also studied the second order Laplacian equation involving concave-convex nonlinearities with weight functions. With the help of the fibering method and the Nehari manifold which was introduced by Pohozaev in ^[3], they obtained that the equation has at least two nontrivial solutions. Moreover, in ^[4], Brown and Wu studied a potential operator equation. By using methods similar to those in ^[2], they obtained that the equation has at least two nontrivial solutions when the functionals related to potential operators satisfy some appropriate conditions. In ^[5], Chen et al. studied a class of second order Kirchhoff equations involving concave-convex nonlinearities and parameters. Their result was that the equation has multiple positive solutions basd on the fibering method and the Nehari manifold. In ^[6], Chen et al. studied the nonhomogeneous $p$-Kirchhoff equation involving concave-convex nonlinearities with weight functions and a perturbation. Their result was the existence of two nontrivial solutions of the equation basd on the mountain pass theorem and Ekeland's variational principle. In ^[7], Wu investigated the following second order Laplacian elliptic system:

where $\Omega \subseteq \mathbb{R}^N$ is a bounded domain and $1 < \gamma < 2 < \alpha+\beta < \frac{2N}{N-2}$. With the help of the fibering method and the Nehari manifold, he obtained that when the parameters $\lambda$ and $\mu$ belong to the appropriate range, the system has two nontrivial nonnegative solutions. In ^[8], Echarghaoui and Sersif investigated a class of second order semilinear elliptic systems involving critical Sobolev growth and concave nonlinearities. By using the fountain theorem, they obtained that the system has two completely different but infinitely many radial solution sets. Moreover, the energy functional of one solution set is positive, and the energy functional of the other solution set is negative. In ^[9], Bozhkova and Mitidieri considered the $(p, q)$-Laplacian elliptic system with the Diriclet boundary condition. Their results were on the existence and multiplicity of solutions of the system basd on the fibering method and the Nehari manifold. Moreover, by using the Pokhozhaev identity, they obtained the nonexistence result of solutions. In ^[10], Liu and Ou investigated the following $(p, q)$-Laplacian elliptic system:

where $\Omega \subseteq \mathbb{R}^{N}$ is a bounded domain, $1 < p, q < N$, $\alpha, \beta, \gamma, \eta > 0$, $1 < \alpha+\beta < \min\{p, q\}$ and $\max\{p, q\} < \gamma+\eta < \min\{p^*, q^*\}$. Their result was on the existence of two nontrivial solutions basd on the fibering method and the Nehari manifold. We refer the reader to ^{[11,12,13,14]} for more results about the elliptic systems involving concave-convex nonlinearities.

Moreover, in recent years, the research about equations on graphs have also attracted some attentions. For example, see ^[15,16,17]. In ^[15], Grigor'yan et al. considered the second order Laplacian equation with the nonlinear term satisfying the superquadratic condition and some additional assumptions on finite graphs and locally finite graphs. With the help of the mountain pass theorem, the conclusion they reached was on the existence of a nontrivial solution for the equation. Furthermore, they also investigated the $p$-Laplacian equation and poly-Laplacian equation on finite graphs and locally finite graphs and obtained some similar results. In ^[16], Han and Shao studied the $p$-Laplacian equation with the Dirichlet boundary condition on the locally finite graph. With the help of the mountain pass theorem and the Nehari manifold, their result was the existence of a positive solution and a ground-state solution for the equation. In addition, in ^[17], Han et al. studied a nonlinear biharmonic equation with a parameter $\lambda$ and the Dirichlet boundary condition on the locally finite graph. With the help of the mountain pass theorem and the method of the Nehari manifold, they obtained that when the parameter $\lambda$ is small enough, the equation has a ground-state solution. Moreover, when $\lambda \rightarrow +\infty$, the ground-state solutions converge.

Next, we recall some basic knowledge and notations of locally finite graphs, which were taken from ^[15,16,17]. Suppose that $G = (V, E)$ is a graph, where $V$ is a vertex set and $E$ is a edge set. $xy$ denotes the edge connecting $x$ with $y$. Assume that for any $x \in V$, there are only finite edges $xy \in E$; then, $(V, E)$ is called a locally finite graph. Moreover, assume that both the vertex set $V$ and edge set $E$ are finite sets; then, $(V, E)$ is called a finite graph. $\omega_{xy} > 0$ is defined as the weight of the edge $xy\in E$, and it is assumed that $\omega_{xy} = \omega_{yx}$. Furthermore, for any $x\in V$, the degree is defined as $\mbox{deg}(x) = \sum_{y\thicksim x}\omega_{xy}$, where $y\thicksim x$ denotes that $y \in V$ and $xy\in E$. $d(x, y)$ is the distance of two vertices $x$ and $y$, which is the minimal number of edges that connect $x$ with $y$. Let $\Omega\subset V$. Assume that for any $x, y \in \Omega$, there exists a positive constant $c$ such that $d(x, y) \le c$; then, $\Omega$ is a bounded domain in $V$. The definition of the boundary of $\Omega$ is as follows:

Assume that $\mu:V\to \mathbb{R}^+$ is a finite measure and it is assumed that $\mu(x)\geq \mu_0 > 0$. For any function $u:V\to\mathbb{R}$, one denotes

Let $C(V) = \{u|u:V\to \mathbb{R}\}$. Define the Laplacian operator $\Delta: C(V)\to C(V)$ by

and define the associate gradient $\Gamma(u_1, u_2)$ as

Denote $\Gamma(u): = \Gamma(u, u)$. The definition of the length of the gradient is as follows:

and the definition of the length of the m-order gradient is as follows:

For any given $s > 1$, define the $s$-Laplacian operator $\Delta_{s}:C(V)\rightarrow C(V)$ by

Let $C_c(\Omega): = \{u:V \to \mathbb{R}\big| supp\mbox{ }u\subset \Omega \mbox{ and }\forall x \in V \backslash\Omega, u(x) = 0\}$. For any function $\phi \in C_c(\Omega)$, the following equality holds:

For any $1\leq r < +\infty$, assume that the completion space of $C_c(\Omega)$ is $L^r(\Omega)$ under the norm

Moreover, the completion space of $C_c(\Omega)$ is $W_0^{m, s}(\Omega)$ under the norm

where $m$ is a positive integer and $s > 1$. For any $u\in W_0^{m, s}(\Omega)$, we also define the following norm:

$W_0^{m, s}(\Omega)$ is of finite dimension. See ^[15,16] for more details.

In this paper, our work was mainly inspired by ^{[4,6,7,10,15]}. We shall employ the mountain pass theorem and Ekeland's variational principle as in ^[6] to investigate the multiplicity of solutions for a class of poly-Laplacian systems on graphs, which can be seen as a discrete version of (1.1) on graphs in some sense, and we also obtain that a poly-Laplacian equation on a locally finite graph has a ground-state solution basd on an abstract result in ^[4], which was essentially obtained by using the fibering method and the Nehari manifold as in ^[2,4,7,10]. To be specific, we discuss the following poly-Laplacian system on a locally finite graph $G = (V, E)$:

where $\Omega\cup \partial\Omega\subset V$ is a bounded domain, $m_i, i = 1, 2$ denotes positive integers, $p, q, \gamma_1, \gamma_2 > 1$, $\lambda_1, \lambda_2, \alpha, \beta > 0$, $\max\{\gamma_1, \gamma_2\} < \min\{p, q\} \le \max\{p, q\} < \alpha+\beta$, $h_1(x), h_2(x), c(x):\Omega \to \mathbb{R}^+$ and the definitions of ${\text{£}}_{m_i, s}$ ($i = 1, 2, s = p, q)$ is expressed as follows for any function $\phi: \Omega \cup \partial\Omega \to \mathbb {R}$:

When $m = 1$, ${\text{£}}_{m, p}u = -\Delta_p u$, and when $p = 2$, ${\text{£}}_{m, p}u = (-\Delta^m)u$, which is called a poly-Laplacian operator of $u$. More details can be seen in ^[15] for the definition of ${\text{£}}_{m, p}$. Obviously, system (1.9) with $m_1 = m_2 = 1$, $p = q = 2$ and $\gamma_1 = \gamma_2 = \gamma$ is a generalization of (1.1) from the Euclidean setting to a locally finite graph.

In this paper, when $(u, v)$ is a solution of system (1.9) with either $(u, v) = (u, 0)$ or $(u, v) = (0, v)$, $(u, v)$ is called a semi-trivial solution of system (1.9). Moreover, when $(u, v)$ is a solution of system (1.9) and $(u, v)\not = (0, 0)$, $(u, v)$ is called a nontrivial solution of system (1.9).

Denote

where $C_0 = \max_{x\in \Omega}c(x)$ and $C_{m_1, p}(\Omega)$ and $C_{m_2, q}(\Omega)$ are embedding constants given in Lemma 2.1 below. Especially, we present concrete values of $C_{m_1, p}(\Omega)$ and $C_{m_2, q}(\Omega)$ if $m_1 = m_2 = 1$, $p, q\geq2$ and for each $x \in \Omega$, there is at least one $y \in \partial \Omega$ satisfying that $y \thicksim x$ (see Lemma 2.2 below).

Our main results are as follows. We suppose that $\lambda_1$ and $\lambda_2$ satisfy the following inequalities:

Theorem 1.1. Suppose that $G = (V, E)$ is a locally finite graph, $\Omega\not = \emptyset$ and $\partial \Omega\not = \emptyset$. If $(\lambda_1, \lambda_2)$ satisfies (1.11), system (1.9) admits one nontrivial solution of positive energy and one nontrivial solution of negative energy.

Remark 1.1. There exist $\lambda_1$ and $\lambda_2$ satisfying (1.11). For example, let $m_1 = 2$, $m_2 = 3$, $\gamma_1 = 2$, $\gamma_2 = 3$, $p = 4$, $q = 5$, $\alpha = 2$, $\beta = 4$ and

When $\lambda_1 = \frac{1}{5}C_{2, 4}^{-4}(\Omega)$ and $\lambda_2 = \frac{1}{6}C_{3, 5}^{-5}(\Omega)$, we can obtain that

Evidently,

Moreover,

Hence, (1.11) holds for $\lambda_1 = \frac{1}{5}C_{2, 4}^{-4}(\Omega)$ and $\lambda_2 = \frac{1}{6}C_{3, 5}^{-5}(\Omega)$.

Theorem 1.2. Suppose that $G = (V, E)$ is a locally finite graph, $\Omega\not = \emptyset$ and $\partial \Omega\not = \emptyset$. For each $\lambda_1 > 0$, assume that $(u, 0)$ is a semi-trivial solution of system (1.9). Then

where $H_1 = \max_{x\in \Omega} h_1(x)$. Similarly, for each $\lambda_2 > 0$, assume that $(0, v)$ is a semi-trivial solution of system (1.9). Then

where $H_2 = \max_{x\in \Omega} h_2(x)$.

Moreover, we also investigate the existence of a ground-state solution for the following poly-Laplacian equation on $G = (V, E)$ by applying Theorem 3.3 in ^[4]:

where $\Omega\cup \partial\Omega\subset V$ is a bounded domain, $m$ is a positive integer, $p, \gamma > 1$, $\lambda > 0$, $\gamma < p < \alpha$ and $h(x), c(x):\Omega \to \mathbb{R}^+$. Denote

where $H_0 = \max_{x \in \Omega} h(x)$ and $C_0 = \max_{x \in \Omega} c(x)$. We obtain the following result.

Theorem 1.3. Suppose that $G = (V, E)$ is a locally finite graph, $\Omega\not = \emptyset$ and $\partial \Omega\not = \emptyset$. If $\lambda \in (0, \lambda_0)$, then (1.12) admits one nontrivial solution of positive energy and one nontrivial solution of negative energy. Furthermore, if $\lambda \in (0, \lambda_{\star\star})$, the negative energy solution is the ground-state solution of (1.12).

Remark 1.2. Similar to the arguments of Theorem 1.1, applying the mountain pass theorem and Ekeland's variational principle, we can also get one nontrivial solution of positive energy and one nontrivial solution of negative energy for (1.12). We do not know whether these two solutions are different from those two solutions in Theorem 1.3 which were obtained, essentially, by using the fibering method and the Nehari manifold.

2.   Preliminaries

Define the space $W = W_0^{m_1, p}(\Omega)\times W_0^{m_2, q}(\Omega)$ with the norm

Then, $W$ is a finite dimensional Banach space. The energy functional $\psi:\; W \to \mathbb{R}$ of system (1.9) is defined as follows:

Then, $\psi(u, v) \in C^1(W, \mathbb{R})$. Moreover,

Definition 2.1. $(u, v)\in W$ is called a weak solution of system (1.9) whenever, for all $(\phi, \varphi)\in W$, the following equalities are true:

Evidently, $(u, v)\in W$ is a weak solution of system (1.9) if and only if $(u, v)$ is a critical point of $\psi$.

Proposition 2.1. Assume that $(u, v)\in W$ is a weak solution of system (1.9). Then, $(u, v)\in W$ is also a point-wise solution of (1.9).

Proof. We define two functions $\phi, \varphi:V\to \mathbb{R}$ as follows for any fixed $y\in V$:

Hence, by (2.3) and (2.4), the following holds:

By the arbitrariness of $y$, we come to the conclusion.

Lemma 2.1. (^[15]). Assume that $G = (V, E)$ is a locally finite graph: $\Omega\cup \partial\Omega\subset V$ is a bounded domain with $\Omega\not = \emptyset$. For any given $m$ and $s$ with $m \in \mathbb{N}^+$ and $s > 1$, $W_0^{m, s}(\Omega)$ is embedded in $L^r(\Omega)$ for each $1 \le r \le+\infty$. Especially, there exists a positive constant $C_{m, s}(\Omega)$ such that

where

and $|\Omega| = \sum_{x \in \Omega} \mu(x)$. Furthermore, $W_0^{m, s}(\Omega)$ is pre-compact.

Moreover, if for each $x \in \Omega$, there is at least one $y \in \partial \Omega$ satisfying that $y \thicksim x$, we can present a specific value of $C_{m, s}(\Omega)$ with $m = 1$ and $s\geq 2$. The details are as follows.

Lemma 2.2. Suppose that $G = (V, E)$ is a locally finite graph, $\Omega\cup \partial\Omega\subset V$ is a bounded domain with $\Omega\not = \emptyset$ and $\partial\Omega\not = \emptyset$, and for each $x \in \Omega$, there is at least one $y \in \partial \Omega$ satisfying that $y \thicksim x$. Then, for any $s\geq2$ and $1 \le r < +\infty$,

where

$\hat{\mu}_{\min} = \min_{x\in \Omega}\mu(x)$, $\mu_{\max} = \max_{x\in \Omega\cup\partial \Omega}\mu(x)$ and $w_{\min} = \min_{x\in \Omega\cup\partial \Omega}w_{xy}$.

Proof. The following holds:

Hence,

Moreover, by (2.8), we have

It follows from (2.9) that for any $1 \le r < +\infty$, the following holds:

Assume that $B$ is a real Banach space and $f \in C^{1}(B, \mathbb{R})$. We say that $f$ satisfies the Palais-Smale condition if any Palais-Smale sequence $\{u_n\} \subseteq B$ has a convergent subsequence, where $\{u_n\}$ is called the Palais-Smale sequence if for all $n \in \mathbb{N}$, there exists a positive constant $c$ such that $|f(u_n)|\le c$ and $f'(u_n)\rightarrow 0$ as $n\rightarrow \infty$.

Lemma 2.3. (^[18]) Assume that $B$ is a real Banach space and $f \in C^{1}(B, \mathbb{R})$, where $f$ satisfies the Palais-Smale condition and $f(0) = 0$. Moreover, if $f$ satisfies the following conditions:

(i) there exist two constants $r$ and $m$ with $r, m\in \mathbb{R}^+$ such that $f_{\partial B_{r}(0)}\ge m$, where $B_r = \{x\in B:\|x\|_B < r\}$;

(ii) there is $x\in B\backslash \bar B_{r} (0)$ satisfying that $f(x)\le 0$,

then $f$ admits a critical value $m_*\ge m$ and

where

Lemma 2.4. (^[19]) Assume that $(B, \rho)$ is a complete metric space and $f:B\rightarrow \mathbb{R}$, which is lower-semicontinuous and bounded from below. Moreover, there exist $\delta > 0$ and $x \in B$ such that

On that occasion, there exists $y \in B$ such that

Furthermore, for all $w\in B$, the following holds:

3.   Proofs for Theorems 1.1 and 1.2

Lemma 3.1. For each $(\lambda_1, \lambda_2)$ satisfying (1.11), there exists a positive constant $r_{(\lambda_1, \lambda_2)}$ such that $\psi(u, v) > 0$ whenever $||(u, v)||_W = r_{(\lambda_1, \lambda_2)}$.

Proof. Note that

Then, by using Lemma 2.1 and Young's inequality, for all $u\in W_0^{m_1, p}(\Omega)$, the following holds:

Similarly, for all $v\in W_0^{m_2, q}(\Omega)$, the following holds:

Furthermore, for all $(u, v)\in W$, Lemma 2.1 and Young's inequality imply that

Thus, (2.1) and (3.2)–(3.4) imply that when $(\lambda_1, \lambda_2) \in (0, C_{m_1, p}^{-p}(\Omega))\times (0, C_{m_2, q}^{-q}(\Omega))$, for any $(u, v) \in W$ with $\|(u, v)\|_W \le 1$, the following holds:

Note that

Define

To find $r_{(\lambda_1, \lambda_2)}$ satisfying that $\psi(u, v) > 0$ whenever $\|(u, v)\|_W = r_{(\lambda_1, \lambda_2)}$, it suffices to prove that there is $t_{(\lambda_1, \lambda_2)}^{\star}\in (0, 1]$ satisfying that $f(t_{(\lambda_1, \lambda_2)}^{\star}) > 0$. In fact, by (3.6), we have

Let $f'(t_{(\lambda_1, \lambda_2)}^{\star}) = 0$. We obtain that

Since $\lambda_1$ and $\lambda_2$ satisfy (1.11), we have that $0 < t_{(\lambda_1, \lambda_2)}^{\star} \le 1.$ Moreover,

Hence, by (1.11), we have

Let $r_{(\lambda_1, \lambda_2)} = t_{(\lambda_1, \lambda_2)}^{\star}$. Hence, we have come to the conclusion.

Lemma 3.2. For each $(\lambda_1, \lambda_2)$ satisfying (1.11), there exists $(u_{(\lambda_1, \lambda_2)}, v_{(\lambda_1, \lambda_2)})\in W$ with $\|(u_{(\lambda_1, \lambda_2)}, v_{(\lambda_1, \lambda_2)})\|_W > r_{(\lambda_1, \lambda_2)}$ such that $\psi(u_{(\lambda_1, \lambda_2)}, v_{(\lambda_1, \lambda_2)}) < 0$.

Proof. For any given $(u, v)\in W$ with $\int_{\Omega} c(x)|u|^{\alpha}|v|^{\beta}d\mu\not = 0$ and any $z\in \mathbb{R}^+$, we have

Note that $\alpha+\beta > \max\{p, q\}$. So, there exists $z_{(\lambda_1, \lambda_2)}$ large enough such that $\|(z_{(\lambda_1, \lambda_2)}u, z_{(\lambda_1, \lambda_2)}v)\|_W > r_{(\lambda_1, \lambda_2)}$ and $\psi(z_{(\lambda_1, \lambda_2)} u, z_{(\lambda_1, \lambda_2)}v) < 0$. Let $u_{(\lambda_1, \lambda_2)} = z_{(\lambda_1, \lambda_2)} u$ and $v_{(\lambda_1, \lambda_2)} = z_{(\lambda_1, \lambda_2)} v$. Then, the proof is finished.

Lemma 3.3. For each $(\lambda_1, \lambda_2)$ satisfying (1.11), $\psi$ satisfies the Palais-Smale condition.

Proof. For any Palais-Smale sequence $(u_k, v_k)\subseteq W$, there exists a constant $c$ with $c > 0$ such that

Hence, the following holds:

We claim that ${\|(u_k, v_k)\|}_W$ is bounded. In fact, if

then it follows from (3.8) that

which contradicts (3.9). If

and $\|v_k\|_{W_0^{m_2, q}(\Omega)}$ is bounded, then (3.8) implies that there exists a constant $c_1$ with $c_1 > 0$ such that

which contradicts (3.10). Similarly, if

and $\|u_k\|_{W_0^{m_1, p}(\Omega)}$ is bounded, we can also obtain a contradiction. Hence, $\|u_k\|_{W_0^{m_1, p}(\Omega)}$ and $\|v_k\|_{W_0^{m_2, q}(\Omega)}$ are bounded. Then, there exist subsequences $\{u_{k_n}\} \subset \{u_k\}$ and $\{v_{k_n}\} \subset \{v_k\}$ such that $u_{k_n}\rightharpoonup u_0$ and $v_{k_n}\rightharpoonup v_0$ for some $u_0\in W_0^{m_1, p}(\Omega)$ and $v_0\in W_0^{m_2, q}(\Omega)$ as $n\to \infty$. Moreover, Lemma 2.1 implies that

The proof is complete.

Proof of Theorem 1.1. Applying Lemmas 3.1–3.3 and Lemma 2.3, we obtain that for each $(\lambda_1, \lambda_2)$ satisfying (1.11), system (1.9) admits one nontrivial solution $(u_0, v_0)$ which has positive energy. Next, similar to the proof of Theorem 3.3 in ^[20] and Theorem 1.3 in ^[21], we prove that system (1.9) admits one nontrivial solution of negative energy. Note that $\gamma_1 < p$ and $\gamma_2 < q$. Then, it follows from (3.7) that there exists $z$ small enough such that

So,

where $r_{(\lambda_1, \lambda_2)}$ is given in Lemma 3.1 and $\bar B_{r_{(\lambda_1, \lambda_2)}} = \{(u, v)\in W\big|\|(u, v)\|_W\le r_{(\lambda_1, \lambda_2)}\}$. Moreover, by Lemma 3.1, for each $(\lambda_1, \lambda_2)$ satisfying (1.11), the following holds:

Set

By the definition of an infimum, we obtain that there is a point $(u_n, v_n)\in \bar B_{r_{(\lambda_1, \lambda_2)}}$ satisfying

Since $\psi(u, v) \in C^1(W, \mathbb{R})$, $\psi(u, v)$ is lower semicontinuous. Hence, using Lemma 2.4, we get

Moreover, (3.11) and (3.12) imply that

thus, $(u_n, v_n)\in B_{r_{(\lambda_1, \lambda_2)}}$. Set $M_n:W \rightarrow \mathbb{R}$ as

Then, $(u_n, v_n)\in B_{r_{(\lambda_1, \lambda_2)}}$ is the minimum point of $M_n$ on $\bar B_{r_{(\lambda_1, \lambda_2)}}$. Hence, for some $(u, v)\in W$ satisfying that $\|(u, v)\|_W = 1$, assume that $t > 0$ is small enough such that $(u_n+tu, v_n+tv)\in \bar B_{r_{(\lambda_1, \lambda_2)}}$. Then

By (3.13) and the definition of $M_n$, the following holds:

Similarly, if $t < 0$ and $|t|$ is small enough, then

Therefore,

Thus, (3.12) and (3.14) imply that

Hence, by using Lemma 3.3, we know that there exists a subsequence $\{(u_{n_k}, v_{n_k})\} \subset \{(u_n, v_n)\}$ satisfying that $(u_{n_k}, v_{n_k})\rightarrow (u^{\star}_0, v^{\star}_0)\in \bar B_{r_{(\lambda_1, \lambda_2)}}$ as $k \rightarrow \infty$, and that

Hence, system (1.9) admits a nontrivial solution $(u^{\star}_0, v^{\star}_0)$ which has negative energy.

Proof of Theorem 1.2. For each $\lambda_1 > 0$, assume that $(u, 0)$ is a semi-trivial solution of system (1.9). Then, we have

Hence,

Similarly, for each $\lambda_2 > 0$, if $(0, v)$ is a semi-trivial solution of system (1.9), we can also obtain

4.   Proofs for Theorem 1.3

In this section, we discuss the existence of a ground-state solution for (1.12) by using Lemma 2.5 and Theorem 3.3 in ^[4]. In ^[4], Brown and Wu researched the following operator equation basd on the fibering maps and the Nehari manifold:

where $S$ is a reflexive Banach space, $A, B, C: S \to S^*$ are homogeneous operators of degree $p-1, \alpha-1$ and $\gamma-1$ with $1 < \gamma < p < \alpha$. The energy functional of (4.1) is

the fibering map is

for all $t > 0$, and the Nehari manifold is

Define

Then, $\mathcal{N}$ can be divided into the following three parts:

In ^[4], Brown and Wu obtained the following results.

Lemma 4.1. (^[4], Lemma 2.5) For any $u \in S$, when $\langle B(u), u\rangle > 0$ and $\langle C(u), u\rangle > 0$, there exist $t_u^+$ and $t_u^-$ with $0 < t_u^+ < t_u^-$ such that $G_u(t)$ is increasing on the interval $(t_u^+, t_u^-)$ and decreasing on the interval $(0, t_u^+)$ and interval $(t_u^-, +\infty)$.

Lemma 4.2. (^[4], Theorem 3.3) If the following conditions hold:

$(H_1)$ $u \to \langle A(u), u\rangle$ is a weakly lower semicontinuous function on $S$ and there is a continuous function $\kappa: [0, +\infty) \to [0, +\infty)$ with $\kappa(s) > 0$ on $(0, +\infty)$ and $\lim_{s\rightarrow \infty}\kappa(s) = \infty$ such that for all $u \in S$, $\langle A(u), u\rangle \ge \kappa(\|u\|)\|u\|$;

$(H_2)$ there exist $u_i \in S$, $i = 1, 2$ such that

$(H_3)$ $B, C$ are strongly continuous;

$(H_4)$ there exist two positive constants $d_1, d_2$ with

such that

then (4.1) admits at least two nontrivial solutions $u_0^+$ and $u_0^-$, where

and $\mathcal{N}^0 = \emptyset$.

In the locally finite graph $G = (V, E)$ setting, let $S = W_0^{m, p}(\Omega)$ and

Similar to the arguments in ^[4], $(H_1)$–$(H_3)$ hold with $A, B$ and $C$ respectively defined by (4.6)–(4.8). Note that

Lemma 2.1 implies that the following holds:

and

Let

Then (4.4) and (4.5) hold. Moreover, note that

Then, it is easy to see that $(H_4)$ holds if $\lambda \in (0, \lambda_0)$. Thus, by Lemma 4.2, (1.12) admits at least two nontrivial solutions $u^+_0\in \mathcal{N}^+$ and $u^-_0\in\mathcal{N}^-$, and one of them must be a ground-state solution. Next, we discuss which is the ground-state solution. Note that the energy functional of (1.12) is

and for each $u\in W_0^{m, p}(\Omega)\backslash\{0\}$, the corresponding fibering map is

We can obtain that $G_u(t)$ has positive values if $\lambda \in (0, \lambda_{\star})$ where $\lambda_{\star}$ is defined by (1.13). In fact, we define

By (4.9), we have

with

Furthermore,

It follows that

Note that

Then, for all $u \in W_0^{m, p}(\Omega)\backslash\{0\}$, if $\lambda \in (0, \lambda_{\star})$, it follows that

Moreover, for all $u \in \mathcal{N}^-$, the following holds:

and

Then $G_u(1)$ is a local maximum value of $G_u(t)$ on $(0, +\infty)$. It is easy to see that for any $u \in W_0^{m, p}(\Omega)\backslash\{0\}$, we have

Then, by Lemma 4.1, for all $u \in \mathcal{N}^-$, there exist $t_u^+$ and $t_u^-$ with $0 < t_u^+ < t_u^-$ such that $G_u(t)$ is increasing on the interval $(t_u^+, t_u^-)$ and decreasing on the interval $(0, t_u^+)$ and interval $(t_u^-, +\infty)$, together with $G_u(0) = 0$ and (4.17), which implies that both the local maximum point $t_u = 1$ and $t_{0u}$ belong to the interval $(t_u^+, +\infty)$. Thus, for each $u \in \mathcal{N}^-$, we have

Similarly, for each $u \in \mathcal{N}^+$, we know that $G_u(1)$ is a local minimum value of $G_u(t)$ on $(0, +\infty)$, which is located at $(0, t_u^-)$ by Lemma 4.1. Hence, for each $u \in \mathcal{N}^+$, we have

Therefore, we conclude that (1.12) admits a nontrivial ground-state solution $u^+_0\in \mathcal{N}^+$ if $\lambda \in (0, \lambda_{\star\star})$ where $\lambda_{\star\star} = \min\{\lambda_0, \lambda_{\star}\}$.

5.   Some results on the finite graph

Assume that $G = (V, E)$ is a finite graph. Using similar arguments as for Theorems 1.1 and 1.2, we can obtain similar results for the following poly-Laplacian system on finite graph $G$:

where $m_i, i = 1, 2$ are positive integers, $p, q, \gamma_1, \gamma_2 > 1$, $\lambda_1, \lambda_2, \alpha, \beta > 0$, $\max\{\gamma_1, \gamma_2\} < \min\{p, q\} \le \max\{p, q\} < \alpha+\beta$, $a, b, h_1, h_2, c:V \to \mathbb{R}^+$. Moreover, similar to the arguments in Theorem 1.3, we can also obtain a similar result for the following equation:

where $m$ is a positive integer, $p, \gamma > 1$, $\lambda, \alpha > 0$, $\gamma < p < \alpha$, $a, h, c:V \to \mathbb{R}^+$. For any given $m$ and $s$ with $m\in \mathbb{N}^+$ and $s > 1$, the definition of $W^{m, s}(V)$ is similar to that of $W^{m, s}(\Omega)$, which changed the region from $\Omega$ to $V$; the norm is defined as follows:

Similarly, for any given $1\le r < +\infty$, the definition of $L^r(V)$ is also similar to that of $L^r(\Omega)$, and the norm is defined as follows:

For system (5.1), we work in the space of $W(V) = W^{m_1, p}(V)\times W^{m_2, q}(V)$, and for (5.2), we work in the space of $W^{m, p}(V)$. Both $W(V)$ and $W^{m, p}(V)$ are of finite dimension. See ^[15] for more details.

Denote

where $C_0(V) = \max_{x \in V}c(x)$ and $C_p(V)$ and $C_q(V)$ are embedding constants from $W^{m_1, p}(V)$ and $W^{m_2, q}(V)$ into $L^p(V)$ and $L^q(V)$, respectively, which have been obtained in ^[22] with

Next, we state the results similar to Theorems 1.1–1.3. Suppose that $\lambda_1$ and $\lambda_2$ satisfy the following inequalities:

Theorem 5.1. Assume that $G = (V, E)$ is a finite graph. If $(\lambda_1, \lambda_2)$ satisfies (5.3), then system (5.1) admits at least one nontrivial solution of positive energy and one nontrivial solution of negative energy.

Theorem 5.2. Assume that $G = (V, E)$ is a finite graph. For each $\lambda_1 > 0$, suppose that $(u, 0)$ is a semi-trivial solution of system (5.1). Then

where $H_1(V) = \max_{x\in V} h_1(x)$. Similarly, for each $\lambda_2 > 0$, suppose that $(0, v)$ is a semi-trivial solution of system (5.1). It follows that

where $H_2(V) = \max_{x\in V} h_2(x)$.

Denote

where $H_0(V) = \max_{x \in V} h(x)$ and $C_0(V) = \max_{x \in V}c(x)$.

Theorem 5.3. Assume that $G = (V, E)$ is a finite graph. If $\lambda \in (0, \lambda_0(V))$, then (5.2) admits at least one nontrivial solution of positive energy and one nontrivial solution of negative energy. Furthermore, if $\lambda \in (0, \lambda_{\star\star}(V))$, the negative energy solution is the ground-state solution of (5.2).

Use of AI tools declaration

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

Acknowledgments

This project was supported by Yunnan Fundamental Research Projects (grant no.: 202301AT070465) and Xingdian Talent Support Program for Young Talents of Yunnan Province. The authors are grateful to the reviewers for their careful review and valuable comments, which have helped us to correct the proofs of Lemmas 2.2 and 3.1, as well as improve the writing of the manuscript.

Conflict of interest

The authors state no conflict of interest.