Existence of Strictly Positive Solutions for a Kirchhoff-Type Equation with the Dirichlet Boundary on Locally Finite Graphs

Abstract

In this paper, we investigate the existence of multiple solutions for a Kirchhoff-type equation with Dirichlet boundary conditions defined on locally finite graphs. Our study extends some previous results on nonlinear Laplacian equations to the more complex Kirchhoff equation which incorporates a nonlocal term. By employing an abstract three critical points theorem that is based on Morse theory, we provide sufficient conditions that guarantee the existence of at least three distinct solutions, including two strictly positive solutions. We also present an example to verify our results.

1. Introduction and Main Results

In order to describe the vibration of the strings which is affected by the length of the strings, the following Kirchhoff equation

$$\begin{array}{c}\hfill \rho {\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}-\left({\displaystyle \frac{P_0}{h}}+{\displaystyle \frac{E}{2L}}{\int }_0^L{\left|{\displaystyle \frac{\partial u}{\partial x}}\right|}^{2}dx\right){\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}=0,\end{array}$$

(1)

is presented by Kirchhoff who proposed a mathematical model describing the transverse vibrations of elastic strings [1] in 1883, where physical parameters include the following: h is the area of the cross section, $\rho$ is the mass density of the string, L is the length of the string, E is the Young modulus of the material, and $P_0$ is the initial tension. This equation marked the first introduction of a nonlocal term (the integral of the gradient squared), distinguishing it from classical linear wave equations. In 1978, Lions [2] revolutionized the analysis of Kirchhoff-type equations by applying functional analytic techniques to a stationary version of the model, laying the groundwork for the applications of modern functional methods to Kirchhoff-type equations. The stationary Kirchhoff equation with Dirichlet boundary conditions and nonlinear perturbations is now widely studied in the form:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-\left(a+b{\int }_{\mathsf{\Omega }}{|\nabla u|}^{2}dx\right)\Delta u=f(x,u),x\in \mathsf{\Omega }\hfill \\ u=0,x\in \partial \mathsf{\Omega },\hfill \end{array}\right.\end{array}$$

(2)

where $a,b>0$, $\mathsf{\Omega }\subset {\mathbb{R}}^N(N\ge 1)$ is a bounded domain and $f:\mathsf{\Omega }\times \mathbb{R}\to \mathbb{R}$ is a nonlinear term, and numerous studies have explored the existence, multiplicity, and qualitative properties of solutions for such equations with contributions, for example, [3,4,5,6,7,8,9] and references therein. Alves, Correa, and Ma [3] established the existence of positive solutions to a generalized Kirchhoff-type equations with Dirichlet boundary conditions by employing the mountain pass theorem. Perera and Zhang [4] investigated nontrivial solutions of Equation (2) using the Yang index and critical groups. Subsequently, Shuai [5] demonstrated that the Kirchhoff-type problem (2) admits at least one least energy sign-changing solution through a combination of constraint variational methods and the quantitative deformation lemma. Building on this work, Tang and Cheng [6] refined the results by introducing novel analytical techniques and the Non-Nehari manifold method, proving that the energy of least energy sign-changing solutions strictly exceeds twice that of least energy solutions. In a related study, Tang and Chen [7] developed new approaches to establish the existence of a Nehari–Pohozaev-type ground state solution and a least energy solution for (2) in the case where $\mathsf{\Omega }={\mathbb{R}}^N$, under weaker assumptions. Meanwhile, Han and Yao [8] explored sign-changing ground state solutions for a class of p-Laplacian Kirchhoff-type problems, leveraging tools such as the quantitative deformation lemma, degree theory, the Non-Nehari manifold method, and refined mathematical techniques. Additionally, Feng et al. [9] classified the Kirchhoff equation with Sobolev critical exponent into four distinct cases, proving the existence and multiplicity of normalized solutions under appropriate conditions.

In the past two decades, the study of partial differential equations (PDEs) on graphs has emerged as a vibrant field, bridging discrete mathematics and continuous analysis. The study of PDEs on graphs transcends theoretical curiosity, serving as a cornerstone in bridging discrete mathematics and applied sciences, with a significant role in both practical and theoretical realms.

In practical applications, partial differential equations (PDEs) on graphs have emerged as fundamental tools across multiple disciplines. The foundational work by Chung and Berenstein [10] established key theoretical insights into weighted Laplacians and harmonic functions on graphs, providing critical methodologies for solving inverse problems related to network connectivity and link conductivity analysis. Elmoataz et al. [11] made significant advances by introducing a novel family of graph p-Laplacian operators incorporating gradient terms. Their work demonstrated how these operators could be unified within a comprehensive framework to address various inverse problems in image processing, 3D point cloud analysis, and machine learning applications. Further developments were achieved by Ennaji et al. [12], who provided new perspectives on tug-of-war games and their associated PDEs on graphs. Their research revealed that translating these games to graph structures generates diverse nonlocal elliptic and parabolic PDEs, leading to innovative algorithms for interpolation problems with practical applications in cultural heritage preservation and medical imaging. In the domain of neural networks, Gao et al. [13] made a notable contribution by developing a Graph Learning Neural Network (GLNN) that synergistically combines data-driven and task-oriented graph optimization. Their approach demonstrated superior performance compared to existing state-of-the-art methods in semi-supervised classification tasks on standard social and citation network datasets. Most recently, Martinet and Bungert [14] introduced a groundbreaking mesh-free shape optimization framework that represents shapes as neural network level sets and employs graph Laplacians for PDE approximation. Their work particularly highlighted the method’s remarkable versatility through successful applications to three distinct shape optimization challenges.

Theoretically, there are some new challenges and interesting problems for the partial differential equations defined on graphs, which are caused mainly by the particular definition of gradient on graph. A pioneer work was given by [10], where Chung–Berenstein firstly introduced the definition of the weighted Laplacian operator and gradient on finite graphs. In 2016, Grigor’yan-Lin-Yang [15] established the variational frameworks of the Yamabe equations, p-Laplacian equations, and poly-Laplacian equations defined on finite graphs or local finite graphs (with the Dirichlet boundary condition), and they then found that the equations have at least one nontrivial solution via the mountain pass theorem. After that, a series of works on the existence and multiplicity of solutions for different types of partial differential equations on finite graphs or locally finite graphs appeared, for example, Kazdan–Warner equations [16,17,18,19], heat equations [20,21,22], nonlinear Schrödinger equations [23], p-Laplacian equations [24,25], $(p,q)$-Laplacian systems [26], poly-Laplacian equations [27,28], biharmonic equations [29], and Kirchhoff equations [30,31,32].

Next, we review some basic definitions and conclusions developed on locally finite graphs. More details can be seen in [15,33]. A graph $G=(V,E)$ consists of a vertex set V and an edge set E, where each edge connects two vertices. A graph is categorized as locally finite when, for every $x\in V$, there are only finitely many $y\in V$ such that $xy\in E$, where $xy$ signifies the edge connecting x and y. A graph is deemed connected if any two vertices x and y can be joined via a finite sequence of edges. For adjacent vertices $x,y\in V$ with $xy\in E$, we assume that the edge weight satisfies ${\omega }_{xy}={\omega }_{yx}>0$. For any $x\in V$, the finite positive measure $\mu :V\to {\mathbb{R}}^{+}$ is defined as $\mu \left(x\right)={\sum }_{y\sim x}{\omega }_{xy}$, with $y\sim x$ indicating that y is linked to x. The distance $d(x,y)$ of two vertices $x,y\in V$ is determined by the minimum number of edges that connect x and y. We call that $\mathsf{\Omega }\subset V$ is a bounded domain within V, if the distance $d(x,y)$ is uniformly bounded from above for any $x,y\in \mathsf{\Omega }$. We denote the boundary of $\mathsf{\Omega }$ by $\partial \mathsf{\Omega }$ which is defined as

$$\partial \mathsf{\Omega }:=\{y\in V,y\notin \mathsf{\Omega }:\exists x\in \mathsf{\Omega }\mathrm{such}\mathrm{that}xy\in E\}$$

and the interior of $\mathsf{\Omega }$ by ${\mathsf{\Omega }}^{\circ }$. It is easy to see then that $\mathsf{\Omega }={\mathsf{\Omega }}^{\circ }$. For any function $u:\mathsf{\Omega }\to \mathbb{R}$, define

$$\begin{array}{c}\hfill \Delta u\left(x\right)={\displaystyle \frac{1}{\mu \left(x\right)}}\sum _{y\sim x}{w}_{xy}(u\left(y\right)-u\left(x\right)).\end{array}$$

(3)

The corresponding gradient bilinear form $\mathsf{\Gamma }(u,v)$ for the two functions u and v at $x\in \mathsf{\Omega }$ is defined as

$$\begin{array}{c}\hfill \mathsf{\Gamma }(u,v)\left(x\right)={\displaystyle \frac{1}{2\mu \left(x\right)}}\sum _{y\sim x}{w}_{xy}(u\left(y\right)-u\left(x\right))(v\left(y\right)-v\left(x\right)):=\nabla u·\nabla v.\end{array}$$

(4)

Specifically, we set $\mathsf{\Gamma }\left(u\right)=\mathsf{\Gamma }(u,u)$ and denote the length of its gradient by

$$\begin{array}{c}\hfill |\nabla u|\left(x\right)=\sqrt{\mathsf{\Gamma }\left(u\right)\left(x\right)}={\left({\displaystyle \frac{1}{2\mu \left(x\right)}}\sum _{y\sim x}{w}_{xy}{(u\left(y\right)-u\left(x\right))}^2\right)}^{{\displaystyle \frac{1}{2}}}.\end{array}$$

(5)

In order to compare with the Euclidean setting, for any function $u:\mathsf{\Omega }\to \mathbb{R}$, the integral of u over $\mathsf{\Omega }$ is defined as

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}u\left(x\right)d\mu =\sum _{x\in \mathsf{\Omega }}\mu \left(x\right)u\left(x\right).\end{array}$$

(6)

From the distributional perspective, $\Delta u$ admits the following representation: for every u belonging to ${\mathcal{C}}_c\left(\mathsf{\Omega }\right)$,

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}\Delta u\varphi d\mu =-{\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}\mathsf{\Gamma }(u,\varphi )d\mu ,\end{array}$$

(7)

where

$$\begin{array}{c}\hfill {\mathcal{C}}_c\left(\mathsf{\Omega }\right)=\left\{{u:\mathsf{\Omega }\to \mathbb{R}|suppu\subset \mathsf{\Omega },u|}_{\partial \mathsf{\Omega }}=0\right\}.\end{array}$$

Recently, in [34], Liu considered the following nonlinear Laplacian equation with the Dirichlet boundary condition on locally finite graphs $G=(V,E)$:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\Delta u=f\left(u\right),\hfill & \mathrm{in}{\mathsf{\Omega }}^{\circ },\hfill \\ u=0,\hfill & \mathrm{on}\partial \mathsf{\Omega },\hfill \end{array}\right.\end{array}$$

where $f:\mathbb{R}\to \mathbb{R}$. With the help of a three-solutions theorem in [35], he proved the equation has at least three solutions, of which one is trivial and the others are strictly positive.

Inspired by [34], in this paper, our aim is to extend the result in [34] to Kirchhoff-type equations by incorporating the nonlocal term $\left(a+b{\left({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^{2}d\mu \right)}^k\right)$, which introduces additional mathematical complexity due to its higher-order dependence on the gradient. To be precise, we employ an abstract three critical points theorem in [35], which is essentially based on Morse theory, to study the existence of three solutions for the following Kirchhoff-type equation on a locally finite graph $G=(V,E)$:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}-(a+b({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^{2}d\mu {)}^k)\Delta u=f\left(u\right),\hfill & \mathrm{in}{\mathsf{\Omega }}^{\circ },\hfill \\ u=0,\hfill & \mathrm{on}\partial \mathsf{\Omega },\hfill \end{array}\right.\end{array}$$

(8)

where $a,b>0$ are constants, $k\ge 1$ is an integer, $f:\mathbb{R}\to \mathbb{R}$ is a nonlinear function, and $\mathsf{\Omega }$ is a connected bounded domain with non-empty interior ${\mathsf{\Omega }}^{\circ }$ and boundary $\partial \mathsf{\Omega }$. To the best of our knowledge, there are three works on the Kirchhoff-type equations on finite graphs or locally finite graphs, see [30,31,32]. In [30], Yu, Xie, and Zhang investigated the existence and multiplicity of solutions for a class of $(p,q)$-Kirchhoff system with combined nonlinearities on graphs by using the well-known mountain-pass theorem and Ekeland’s variational principle. In [31], Pan and Ji were concerned with the existence and convergence of the least energy sign-changing solutions for nonlinear Kirchhoff equations on locally finite graphs by using the constrained variational method. In [32], Ou and Zhang investigated the existence of least energy sign-changing solutions and ground state solutions for a class of Kirchhoff-type equations with a general power law, logarithmic nonlinearity, and a Dirichlet boundary value on a locally finite graph and found that the sign-changing least energy is larger than twice of the ground state energy. We observe that the nonlinear term f is required to be a combination of convex and concave components in [30], while it is assumed to satisfy a super-quartic growth condition in [31,32]. Notably, none of these works addressed the existence of positive solutions. Based on this observation, we investigate the sub-quartic case (corresponding to condition $\left(f_2\right)$ below with $k=1$) and establish the existence of strictly positive solutions.

Our main result is presented as follows.

Theorem 1. 

Suppose $G=(V,E)$ is a locally finite graph and $\mathsf{\Omega }\subset V$ is a connected bounded domain with ${\mathsf{\Omega }}^{\circ }\ne \varnothing$ and $\partial \mathsf{\Omega }\ne \varnothing$. Let $0<{\lambda }_1\left(\mathsf{\Omega }\right)<{\lambda }_2\left(\mathsf{\Omega }\right)<\dots <{\lambda }_l\left(\mathsf{\Omega }\right)$ denote the distinct eigenvalues of $-\Delta$ under the Dirichet boundary condition on Ω, and the function $f:\mathbb{R}\to \mathbb{R}$ meets the subsequent conditions:

($f_1$) $f\left(s\right)$ is continuously differentiable, $f\left(s\right)\ge 0$ for every s belonging to $[0,+\infty )$ and $f\left(0\right)=0$;

($f_2$) there are constants $0<c_1<{\displaystyle \frac{b{\lambda }_1{\left(\mathsf{\Omega }\right)}^{k+1}}{2(k+1)}}$ and $c_2>0$ such that

$$\begin{array}{c}\hfill 0\le F\left(s\right)={\int }_0^{s}f\left(t\right)dt\le c_1{s}^{2(k+1)}+c_2,\end{array}$$

for all $s\in [0,+\infty )$;

($f_3$) $f^{\prime }\left(0\right)>0$, $f^{\prime }\left(0\right)\notin \sigma (-\Delta )=\{{\lambda }_1\left(\mathsf{\Omega }\right),{\lambda }_2\left(\mathsf{\Omega }\right),\dots ,{\lambda }_l\left(\mathsf{\Omega }\right)\}$, and one of the following assumptions holds:

(i) there is an $m\in \mathbb{N}$ such that $a{\lambda }_{2m-1}\left(\mathsf{\Omega }\right)<f^{\prime }\left(0\right)<a{\lambda }_{2m}\left(\mathsf{\Omega }\right)$;

(ii) $f^{\prime }\left(0\right)>a{\lambda }_l\left(\mathsf{\Omega }\right)$ if l is odd.

Then Equation (8) has at least three distinct solutions, one trivial and the others strictly positive.

Example 1. 

Examples meeting the requirements of Theorem 1 are attainable. For instance, let $a=1$, $b={\displaystyle \frac{\pi +1}{16}}$, $k=1$ and

$$\begin{array}{c}\hfill f\left(s\right)=s(arctan{s}^2+{\displaystyle \frac{1}{2}}).\end{array}$$

(9)

Then $f\left(s\right)\ge 0$ for all s within $[0,+\infty )$, $f^{\prime }\left(0\right)={\displaystyle \frac{1}{2}}$ and

$$F\left(s\right)={\displaystyle \frac{1}{2}}{s}^{2}arctan{s}^2-{\displaystyle \frac{1}{4}}ln(1+s^4)+{\displaystyle \frac{1}{4}}{s}^2.$$

It can be readily confirmed that f fulfills conditions $\left(f_1\right)$–$\left(f_3\right)$ if we select $c_1={\displaystyle \frac{\pi +1}{4}}$ and a suitable graph such that its eigenvalues satisfy

$${\left({\displaystyle \frac{2(k+1)c_1}{b}}\right)}^{{\displaystyle \frac{1}{k+1}}}={\displaystyle \frac{1}{4}}<{\lambda }_1\left(\mathsf{\Omega }\right)<{\displaystyle \frac{1}{2}}=f^{\prime }\left(0\right)<{\lambda }_2\left(\mathsf{\Omega }\right).$$

Remark 1. 

By a comparative analysis between our Theorem 1 and Theorem 1.2 in [34], it is easy to see that our condition $\left(f_2\right)$ is allowed to possess a higher power law growth, and $\left(f_3\right)$ is also slightly different from that in [34], which are essentially affected by the nonlocal Kirchhoff term.

The subsequent structure of the paper is as below. Within the second section, we review some conclusions developed on the locally finite graphs and two abstract critical points theorems used in our proofs. We devote Section 3 to finalizing the proof of Theorem 1 by developing and applying several lemmas.

2. Preliminaries

Let $W_0^{1,2}\left(\mathsf{\Omega }\right)$ denote the completion of ${\mathcal{C}}_c\left(\mathsf{\Omega }\right)$ with respect to the subsequent norm

$$\begin{array}{c}\hfill \parallel u\parallel ={\left({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u\left(x\right)|}^{2}d\mu \right)}^{{\displaystyle \frac{1}{2}}}.\end{array}$$

(10)

It is easy to obtain that $W_0^{1,2}\left(\mathsf{\Omega }\right)$ is a space of finite dimension due to the reason that $\mathsf{\Omega }$ only contains finite vertexes, and $W_0^{1,2}\left(\mathsf{\Omega }\right)$ is a Hilbert Space with the inner product

$$\begin{array}{c}\hfill {(u,v)}_{W_0^{1,2}\left(\mathsf{\Omega }\right)}={\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}\mathsf{\Gamma }(u,v)d\mu ={\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}\nabla u\nabla vd\mu .\end{array}$$

The following embedding theorem obviously holds.

Lemma 1 

([15]). Let $G=(V,E)$ be a locally finite graph and Ω be a connected bounded domain. Then $W_0^{1,2}\left(\mathsf{\Omega }\right)$ is pre-compact. That is, for any bounded sequence $\left\{u_n\right\}\subset W_0^{1,2}\left(\mathsf{\Omega }\right)$, there is some $u_0\in W_0^{1,2}\left(\mathsf{\Omega }\right)$ such that, up to a subsequence, $u_n\to u_0$ as $k\to +\infty$.

Let ${\lambda }_1\left(\mathsf{\Omega }\right)>0$ denote the first eigenvalue of $-\Delta$ with the Dirichlet boundary condition, which is expressed by

$$\begin{array}{c}\hfill {\lambda }_1\left(\mathsf{\Omega }\right)=\underset{{u\neg \equiv 0,u|}_{\partial \mathsf{\Omega }}=0,u\in W_0^{1,2}\left(\mathsf{\Omega }\right)}{inf}{\displaystyle \frac{{\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^{2}d\mu }{{\int }_{\mathsf{\Omega }}{u}^{2}d\mu }}.\end{array}$$

(11)

Then the first characteristic subspace can be written as $E_{{\lambda }_1\left(\mathsf{\Omega }\right)}=\{u\in W_0^{1,2}\left(\mathsf{\Omega }\right)|-\Delta u=$${\lambda }_1\left(\mathsf{\Omega }\right)u\}$ with dimension $m_1$. The second eigenvalue of the operator $-\Delta$ is defined as

$$\begin{array}{c}\hfill {\lambda }_2\left(\mathsf{\Omega }\right)=\underset{{u\not\equiv 0,u|}_{\partial \mathsf{\Omega }}=0,u\in E_{{\lambda }_1^{\perp }\left(\mathsf{\Omega }\right)}}{inf}{\displaystyle \frac{{\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^{2}d\mu }{{\int }_{\mathsf{\Omega }}{u}^{2}d\mu }},\end{array}$$

where $E_{{\lambda }_1^{\perp }\left(\mathsf{\Omega }\right)}$ representing the orthogonal complement space of $E_{{\lambda }_1\left(\mathsf{\Omega }\right)}$ within the Sobolev space $W_0^{1,2}\left(\mathsf{\Omega }\right)$, is denoted as

$$\begin{array}{c}\hfill E_{{\lambda }_1^{\perp }\left(\mathsf{\Omega }\right)}=\left\{u\in W_0^{1,2}\left(\mathsf{\Omega }\right)|\forall v\in E_{{\lambda }_1\left(\mathsf{\Omega }\right)},{(u,v)}_{W_0^{1,2}\left(\mathsf{\Omega }\right)}=0\right\}.\end{array}$$

Then, the corresponding second characteristic subspace $E_{{\lambda }_2\left(\mathsf{\Omega }\right)}$ with dimension $m_2$ is attainable. By virtue of $W_0^{1,2}\left(\mathsf{\Omega }\right)$ being a finite-dimensional linear space, we obtain all eigenvalues of the operator $-\Delta$ and the associated characteristic subspaces showcased as follows:

$$\begin{array}{c}\hfill 0<{\lambda }_1\left(\mathsf{\Omega }\right)<{\lambda }_2\left(\mathsf{\Omega }\right)<\dots <{\lambda }_l\left(\mathsf{\Omega }\right),\end{array}$$

and

$$\begin{array}{c}\hfill W_0^{1,2}\left(\mathsf{\Omega }\right)=E_{{\lambda }_1\left(\mathsf{\Omega }\right)}\oplus E_{{\lambda }_2\left(\mathsf{\Omega }\right)}\oplus \dots \oplus E_{{\lambda }_l\left(\mathsf{\Omega }\right)}.\end{array}$$

It is easy to obtain that $m_1+m_2+\dots +m_l=dim\left(W_0^{1,2}\left(\mathsf{\Omega }\right)\right)$, where $m_i$ is the multiplicity of ${\lambda }_i\left(\mathsf{\Omega }\right)$, $i=1,\dots ,l$. (see [34]).

By (11), it is easy to obtain the following conclusions: for any $u\in W_0^{1,2}\left(\mathsf{\Omega }\right)$,

$$\begin{array}{c}\hfill {\parallel u\parallel }_{\infty }^2\le {\int }_{\mathsf{\Omega }}{\left|u\right|}^{2}d\mu \le {\displaystyle \frac{1}{{\lambda }_1\left(\mathsf{\Omega }\right)}}{\int }_{\mathsf{\Omega }}{|\nabla u|}^{2}d\mu ,\end{array}$$

(12)

where ${\parallel u\parallel }_{\infty }={max}_{x\in \mathsf{\Omega }}\left|u\left(x\right)\right|$, and for any given $p\ge 2$,

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p}d\mu \le {\parallel u\parallel }_{\infty }^{p-2}{\int }_{\mathsf{\Omega }}{\left|u\right|}^{2}d\mu \le {\left({\int }_{\mathsf{\Omega }}{\left|u\right|}^{2}d\mu \right)}^{{\displaystyle \frac{p-2}{2}}}{\int }_{\mathsf{\Omega }}{\left|u\right|}^{2}d\mu ={\left({\int }_{\mathsf{\Omega }}{\left|u\right|}^{2}d\mu \right)}^{{\displaystyle \frac{p}{2}}}\le {\displaystyle \frac{1}{{\lambda }_1{\left(\mathsf{\Omega }\right)}^{\frac{p}{2}}}}{\left({\int }_{\mathsf{\Omega }}{|\nabla u|}^{2}d\mu \right)}^{{\displaystyle \frac{p}{2}}}.\end{array}$$

(13)

Definition 1 

([36]). Let X be a Banach space and $I:X\to \mathbb{R}$ be a $C^1$-functional. The functional I satisfies the Palais–Smale condition (abbreviated as the (PS)-condition) if every sequence $\left\{u_n\right\}\subset X$ such that $I\left(u_n\right)$ is bounded, and $I^{\prime }\left(u_n\right)\to 0$ in $X^{*}$ (the dual space of X) admits a strongly convergent subsequence in X.

We will use the following critical point theorems to prove our main results.

Lemma 2 

([35]). Consider H as a Hilbert space. Let J be a function $\in C^2(H,R)$ that satisfies the $\left(PS\right)$-condition, is bounded below, and has $p_0$ as a non-degenerate, non-minimum critical point of J with finite index $j\ge 1$. Then J is shown to have at least three mutually distinct critical points.

Lemma 3 

([36]). Let X be a Banach space. Assume that $J\in C^1(X,\mathbb{R})$ is bounded from below (above) and conforms to the (PS)-condition. Then c, given by $c=\underset{u\in X}{inf}J\left(u\right)(c=\underset{u\in X}{sup}J\left(u\right))$ stands as a critical value of J.

3. Proofs

Define the energy functional $J:W_0^{1,2}\left(\mathsf{\Omega }\right)\to \mathbb{R}$ by

$$\begin{array}{c}\hfill J\left(u\right)={\displaystyle \frac{a}{2}}{\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^{2}d\mu +{\displaystyle \frac{b}{2(k+1)}}{\left({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^{2}d\mu \right)}^{k+1}-{\int }_{\mathsf{\Omega }}F\left(u\right)d\mu .\end{array}$$

(14)

Under the assumption $\left(f_1\right)$, for any $\varphi \in {\mathcal{C}}_c\left(\mathsf{\Omega }\right)$, by a standard calculation, we can obtain that the Fréchet derivative of $J\left(u\right)$ is the following

$$J^{\prime }\left(u\right)\left(\varphi \right)\begin{array}{c}=a{\int }_{\mathsf{\Omega }}\nabla u·\nabla \varphi d\mu +b{\left({\int }_{\mathsf{\Omega }}{|\nabla u|}^{2}d\mu \right)}^k{\int }_{\mathsf{\Omega }}\nabla u·\nabla \varphi d\mu -{\int }_{\mathsf{\Omega }}f\left(u\right)\varphi d\mu \hfill \\ ={\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{(a+b\parallel u\parallel }^{2k})\mathsf{\Gamma }(u,\varphi )d\mu -{\int }_{\mathsf{\Omega }}f\left(u\right)\varphi d\mu ,\forall \varphi \in {\mathcal{C}}_c\left(\mathsf{\Omega }\right).\hfill \end{array}$$

(15)

Definition 2. 

If $u\in W_0^{1,2}\left(\mathsf{\Omega }\right)$ satisfies

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}\left(a+{b\parallel u\parallel }^{2k}\right)\mathsf{\Gamma }(u,\varphi )d\mu ={\int }_{\mathsf{\Omega }}f\left(u\right)\varphi d\mu \end{array}$$

(16)

for any $\varphi \in {\mathcal{C}}_c\left(\mathsf{\Omega }\right)$, then u is known as a weak solution of Equation (8).

Lemma 4. 

If $\left(f_1\right)$ (or $\left(f_1^{\prime }\right)$) holds and $u\in W_0^{1,2}\left(\mathsf{\Omega }\right)$ is a nontrivial weak solution of Equation (1), then it is a strictly positive weak solution of Equation (8).

Proof. 

Let

$$\begin{array}{c}\hfill u^{+}=max\{u,0\},u^{-}=min\{u,0\}\end{array}$$

and

$$\begin{array}{c}\hfill \tilde{f}\left(s\right)=\left\{\begin{array}{cc}0,\hfill & s<0,\hfill \\ f\left(s\right),\hfill & s\ge 0.\hfill \end{array}\right.\end{array}$$

Suppose that $u\in W_0^{1,2}\left(\mathsf{\Omega }\right)$ is the weak solution of the following equation:

$$\begin{array}{c}\hfill -\left(a+b({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^k\right)\Delta u=\tilde{f}\left(u\right).\end{array}$$

(17)

By Definition 2 and taking $u^{-}$ as the test function, we obtain

$$\begin{array}{c}\hfill -(a+b({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^k){\int }_{\mathsf{\Omega }}\Delta u·u^{-}d\mu ={\int }_{\mathsf{\Omega }}\tilde{f}\left(u\right)·u^{-}d\mu .\end{array}$$

Noting that $u=u^{+}+u^{-}$ and $u^{+}{u}^{-}=0$, it follows from (7) that

$$0\begin{array}{c}\le (a+b({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^k){\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u^{-}|}^{2}d\mu \hfill \\ =(a+b({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^k){\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}\mathsf{\Gamma }(u^{-},u^{-})d\mu \hfill \\ =-(a+b({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^k){\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}\Delta u^{-}{u}^{-}d\mu \hfill \\ =-(a+b({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^k){\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}\Delta u{u}^{-}d\mu +(a+b({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^k){\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}\Delta u^{+}{u}^{-}d\mu \hfill \\ ={\int }_{\mathsf{\Omega }}\tilde{f}\left(u\right)u^{-}d\mu +(a+b({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^k){\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}\Delta u^{+}{u}^{-}d\mu \hfill \\ =\sum _{x\in \mathsf{\Omega }}\tilde{f}\left(u\right)u^{-}\left(x\right)\mu \left(x\right)+(a+b({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^k)\sum _{x\in \mathsf{\Omega }\cup \partial \mathsf{\Omega }}{\displaystyle \frac{1}{\mu \left(x\right)}}\sum _{y\sim x}{w}_{xy}\left(u^{+}\left(y\right)-u^{+}\left(x\right)\right)u^{-}\left(x\right)\mu \left(x\right)\hfill \\ =\sum _{x\in \mathsf{\Omega }}\tilde{f}\left(u\right)u^{-}\left(x\right)\mu \left(x\right)+(a+b({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^k)\sum _{x\in \mathsf{\Omega }\cup \partial \mathsf{\Omega }}\sum _{y\sim x}{w}_{xy}\left(u^{+}\left(y\right)-u^{+}\left(x\right)\right)u^{-}\left(x\right)\hfill \\ =\sum _{x\in \mathsf{\Omega }}\tilde{f}\left(u\right)u^{-}\left(x\right)\mu \left(x\right)+(a+b({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^k)\sum _{x\in \mathsf{\Omega }\cup \partial \mathsf{\Omega }}\sum _{y\sim x}{w}_{xy}{u}^{-}\left(x\right)u^{+}\left(y\right)\hfill \\ =\sum _{x\in \mathsf{\Omega }\setminus {\mathsf{\Omega }}^{\prime }}f\left(u\left(x\right)\right)u^{-}\left(x\right)\mu \left(x\right)+(a+b({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^k)\sum _{x\in \mathsf{\Omega }\cup \partial \mathsf{\Omega }}\sum _{y\sim x}{w}_{xy}{u}^{-}\left(x\right)u^{+}\left(y\right)\hfill \\ \le 0.\hfill \end{array}$$

where ${\mathsf{\Omega }}^{\prime }=\{x\in \mathsf{\Omega }|u\left(x\right)<0\}$, which shows that $\left(a+(b{\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^k\right){\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u^{-}|}^2$$d\mu =0$. It follows that $|\nabla u^{-}{|}^2\equiv 0$ for all $x\in \mathsf{\Omega }\cup \partial \mathsf{\Omega }$, so $u^{-}\left(x\right)\equiv 0$. Thus, we obtain $u\left(x\right)\ge 0$ for all $x\in \mathsf{\Omega }$. Furthermore, we claim that $u\left(x\right)>0$ for all $x\in \mathsf{\Omega }$. To prove this, by contradiction, suppose there is some $x_0$ such that $u\left(x_0\right)={min}_{x\in \mathsf{\Omega }}u\left(x\right)=0$. Then, inserting it into Equation (17), we have $-\left(a+b({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^k\right)\Delta u\left(x_0\right)=\tilde{f}\left(u\left(x_0\right)\right)=f\left(0\right)=0$ by $\left(f_1\right)$. It follows that $\Delta u\left(x_0\right)=0$. Thus, $u\left(y\right)=u\left(x_0\right)=0$ for all $y\sim x_0$ with the help of the fact $u\left(x_0\right)={min}_{x\in \mathsf{\Omega }}u\left(x\right)$. Then $u\left(x\right)=0$ for all $x\in \mathsf{\Omega }$ since $\mathsf{\Omega }$ is a connected bounded domain, which will contradict the fact that u is a nontrivial weak solution. Hence u is a strictly positive weak solution of Equation (17). Then $\tilde{f}\left(u\left(x\right)\right)=f\left(u\left(x\right)\right)$. Therefore, if $u\in W_0^{1,2}\left(\mathsf{\Omega }\right)$ is a nontrivial weak solution of Equation (17), then it is strictly positive weak solution of Equation (8). In other words, the problem seeking the strictly positive weak solution of Equation (8) can be reduced to that seeking the nontrivial weak solution of Equation (17). The proof is completed. □

Furthermore, together with the following Lemma 5, u is also the strictly positive point-wise solution of Equation (8).

Lemma 5. 

If $u\in W_0^{1,2}\left(\mathsf{\Omega }\right)$ is a weak solution of Equation (8), then u is also a point-wise solution of Equation (8).

Proof. 

Since $u\in W_0^{1,2}\left(\mathsf{\Omega }\right)$ is a weak solution of Equation (8), then by (7) and (16), we have

$$\begin{array}{c}\hfill -(a+{b\parallel u\parallel }_{W_0^{1,2}\left(\mathsf{\Omega }\right)}^{2k}){\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}\Delta u\varphi d\mu ={\int }_{\mathsf{\Omega }}f\left(u\right)\varphi d\mu \end{array}$$

(18)

for any $\varphi \in {\mathcal{C}}_c\left(\mathsf{\Omega }\right)$. For any fixed $x_0\in \mathsf{\Omega }$, take a test function $\varphi :\mathsf{\Omega }\to \mathbb{R}$ in (18) as

$$\begin{array}{c}\hfill \varphi \left(x\right)=\left\{\begin{array}{cc}1,\hfill & x=x_0,\hfill \\ 0,\hfill & x\ne x_0.\hfill \end{array}\right.\end{array}$$

Thus, we have

$$\begin{array}{c}\hfill -(a+b({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^k)\Delta u\left(x_0\right)=f\left(u\left(x_0\right)\right).\end{array}$$

Since $x_0$ is arbitrary, this completes the proof. □

By Lemma 4 and its proof, without loss of generality, in the sequel we assume that

$$f\left(s\right)\equiv 0,foralls<0$$

(19)

in Equation (8). Next, we verify that the variational functional J satisfies all assumptions in Lemmas 2 and 3, and then complete the proofs of Theorem 1 by a series of lemmas.

Lemma 6. 

Assume that f satisfies the condition $\left(f_2\right)$. Then $J\left(u\right)$ is coercive and bounded from below for all $u\in W_0^{1,2}\left(\mathsf{\Omega }\right)$.

Proof. 

By $\left(f_2\right)$ and (19), it is easy to obtain that

$$\begin{array}{c}\hfill 0\le F\left(s\right)\le c_1{\left|s\right|}^{2(k+1)}+c_2,foralls\in \mathbb{R},\end{array}$$

(20)

where the constants $c_1,c_2$ are defined as in $\left(f_2\right)$.

Then inserting (20) into (14), together with (13), we obtain

$$J\left(u\right)\begin{array}{c}\ge {\displaystyle \frac{a}{2}}{\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^{2}d\mu +{\displaystyle \frac{b}{2(k+1)}}{\left({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^{2}d\mu \right)}^{k+1}-{\int }_{\mathsf{\Omega }}{c}_1{\left|u\right|}^{2(k+1)}d\mu -c_2{\int }_{\mathsf{\Omega }}d\mu \hfill \\ \ge {\displaystyle \frac{b}{2(k+1)}}{\left({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^{2}d\mu \right)}^{k+1}-{\displaystyle \frac{c_1}{{\lambda }_1{\left(\mathsf{\Omega }\right)}^{k+1}}}{\left({\int }_{\mathsf{\Omega }\cup \partial \mathsf{\Omega }}{|\nabla u|}^{2}d\mu \right)}^{k+1}-c_2\mu \left(\mathsf{\Omega }\right)\hfill \\ =\left({\displaystyle \frac{b}{2(k+1)}}-{\displaystyle \frac{c_1}{{\lambda }_1{\left(\mathsf{\Omega }\right)}^{k+1}}}\right){\parallel u\parallel }_{W_0^{1,2}\left(\mathsf{\Omega }\right)}^{k+1}-c_2\mu \left(\mathsf{\Omega }\right),\hfill \end{array}$$

(21)

where $\mu \left(\mathsf{\Omega }\right)={\int }_{\mathsf{\Omega }}d\mu ={\sum }_{x\in \mathsf{\Omega }}\mu \left(x\right)$ represents the volume of $\mathsf{\Omega }$. Since $c_1<{\displaystyle \frac{b{\lambda }_1{\left(\mathsf{\Omega }\right)}^{k+1}}{2(k+1)}}$, then (21) shows that $J\left(u\right)$ is coercive and then bounded from below on $W_0^{1,2}\left(\mathsf{\Omega }\right)$. □

Lemma 7. 

Assume that f fulfills conditions $\left(f_1\right)$ (or $\left(f_1^{\prime }\right)$) and $\left(f_2\right)$. Then the $\left(PS\right)$-condition holds for $J\left(u\right)$.

Proof. 

Assume that there is a sequence $\left\{u_n\right\}$ where $J\left(u_n\right)$ is bounded and $J^{\prime }\left(u_n\right)\to 0$ as $n\to +\infty$. Then due to the coercivity of J, it follows that $\left\{u_n\right\}$ is bounded within $W_0^{1,2}\left(\mathsf{\Omega }\right)$. By Lemma 1, upon considering a subsequence, $u_n$ converges to a function $u_0$ in $W_0^{1,2}\left(\mathsf{\Omega }\right)$. Consequently, the $\left(PS\right)$-condition is established. □

Lemma 8. 

Assume that $\left(f_1\right)$–$\left(f_3\right)$ hold. Then $u=0$ serves as a non-degenerate, non-minimum critical point for $J\left(u\right)$, with a finite index $j\ge 1$.

Proof. 

By (15) and $\left(f_1\right)$, it is easy to see that $J^{\prime }\left(0\right)\left(\varphi \right)=0$ for any $\varphi \in C_c\left(\mathsf{\Omega }\right)$, which shows that $u=0$ is critical point of $J\left(u\right)$ and then it is a weak solution of (1). For any $\varphi ,\psi \in C_c\left(\mathsf{\Omega }\right)$, the second order Fréchet derivative of $J\left(u\right)$ can be calculated as:

$$\begin{array}{cc}\hspace{1em}& ⟨J^{″}\left(u\right)\varphi ,\psi ⟩\hfill \\ =& {\displaystyle \frac{d^{2}J(u+t\varphi +s\psi )}{dtds}}{|}_{t=0,s=0}\hfill \\ =& (a+b({\int }_{\mathsf{\Omega }}{|\nabla u|}^{2}d\mu {)}^k){\int }_{\mathsf{\Omega }}\nabla \varphi ·\nabla \psi d\mu \hfill \\ \hspace{1em}& +2bk({\int }_{\mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^{k-1}{\int }_{\mathsf{\Omega }}\nabla u·\nabla \varphi d\mu {\int }_{\mathsf{\Omega }}\nabla u·\nabla \psi d\mu -{\int }_{\mathsf{\Omega }}{f}^{\prime }\left(u\right)\varphi \psi d\mu \hfill \\ =& {\left((a+b({\int }_{\mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^k)\varphi +2bk({\int }_{\mathsf{\Omega }}{|\nabla u|}^2{d\mu )}^{k-1}{\int }_{\mathsf{\Omega }}\nabla u·\nabla \varphi d\mu u-{(-\Delta )}^{-1}{f}^{\prime }\left(u\right)\varphi ,\psi \right)}_{W_0^{1,2}\left(\mathsf{\Omega }\right)}.\hfill \end{array}$$

Especially, when $u=0$, we have

$$\begin{array}{c}\hfill ⟨J^{″}\left(0\right)\varphi ,\psi ⟩={\left(a\varphi -{(-\Delta )}^{-1}{f}^{\prime }\left(0\right)\varphi ,\psi \right)}_{W_0^{1,2}\left(\mathsf{\Omega }\right)}={\left((a·id-{(-\Delta )}^{-1}{f}^{\prime }\left(0\right))\varphi ,\psi \right)}_{W_0^{1,2}\left(\mathsf{\Omega }\right)}.\end{array}$$

Hence,

$$\begin{array}{c}\hfill J^{″}\left(0\right)=a·id-{(-\Delta )}^{-1}{f}^{\prime }\left(0\right).\end{array}$$

(22)

Note that $f^{\prime }\left(0\right)>0$ and $0<{\lambda }_1\left(\mathsf{\Omega }\right)<{\lambda }_2\left(\mathsf{\Omega }\right)<\dots <{\lambda }_l\left(\mathsf{\Omega }\right)$ are all distinct eigenvalues of $-\Delta$ with the Dirichlet boundary condition. Then all eigenvalues of $J^{″}\left(0\right)$ are $a-{\displaystyle \frac{f^{\prime }\left(0\right)}{{\lambda }_1\left(\mathsf{\Omega }\right)}}<a-{\displaystyle \frac{f^{\prime }\left(0\right)}{{\lambda }_2\left(\mathsf{\Omega }\right)}}<\dots <a-{\displaystyle \frac{f^{\prime }\left(0\right)}{{\lambda }_l\left(\mathsf{\Omega }\right)}}$. Thus, by condition $\left(f_3\right)$, we can get that det $J^{″}\left(0\right)={\mathsf{\Pi }}_{i=1}^l(a-{\displaystyle \frac{f^{\prime }\left(0\right)}{{\lambda }_i\left(\mathsf{\Omega }\right)}})<0$. Then combining the fact that $J^{\prime }\left(0\right)=0$, $u=0$ is a non-degenerate non-minimum critical point of $J\left(u\right)$. Obviously, the condition $\left(f_3\right)$ also implies that the number of all negative eigenvalue of $J^{″}\left(0\right)$ is at least one, that is, the Morse index ind $\left(J^{″}\left(0\right)\right)\ge 1$. □

Proof of Theorem 1 

Lemmas 6 and 7 imply the functional J satisfies all conditions in Lemma 3. Hence, the functional J has a global minimum $c={inf}_{u\in W_0^{1,2}\left(\mathsf{\Omega }\right)}J\left(u\right)$, which is a critical point of J. Since $u=0$ is a non-minimum critical point of $J\left(u\right)$ and $J\left(0\right)=0$, then $c<0$. Hence, together with Lemma 2, we conclude that J has at least three distinct critical points, two of them are nontrivial and one of them is the global minimum point. The proof is completed. □

4. Conclusions

By employing a three-critical-point theorem derived from Morse theory, we establish the existence of two strictly positive solutions for nonlocal Kirchhoff equations on a locally finite graph with Dirichlet boundary conditions. A key challenge arises from the nonlocal Kirchhoff term when verifying that $u=0$ is a non-degenerate critical point of $J\left(u\right)$ with finite Morse index $j\ge 1$ and is not a local minimum. Our work distinguishes itself from previous studies, such as those in [30,31,32], by focusing on nonlinear terms f satisfying a sub-quartic growth condition, as opposed to convex–concave or super-quartic nonlinearities. For future research, an interesting challenging direction would be to investigate Kirchhoff equations with a potential term $h\left(x\right)u$ (where $h:V\to \mathbb{R})$ on the entire locally finite graph $G=(V,E)$ without Dirichlet boundary constraints. This direction presents significant difficulties due to the infinite-dimensional nature of the vertex set V, which leads to a lack of compactness in Sobolev embeddings. Additionally, the spectral properties of the operator $-\Delta u+h\left(x\right)u$ remain unclear, further complicating the analysis.