Related papers: Existence results for Kazdan-Warner type equations…

The wave equation on network is defined by $\partial_{tt}u=\Delta_{G}u+g(u)$, where $u\in\mathbb{R}^{n}$ and the graph Laplacian $\Delta_{G}$ is an operator on functions on $n$ vertices. We suppose that $g:\mathbb{R}^{n}\rightarrow…

Let $G=(V,E)$ be a locally finite graph, whose measure $\mu(x)$ have positive lower bound, and $\Delta$ be the usual graph Laplacian. Applying the mountain-pass theorem due to Ambrosetti-Rabinowitz, we establish existence results for some…

We prove that for all $k \ge 3$ and any integers $\Delta, n$ with $n \ge 2^\Delta,$ there exists a $k$-graph on $n$ vertices with maximum degree at most $\Delta$ such that $r(H)\geq\tw_{k-1}(c_k \Delta) \cdot n$ for some constant $c_k > 0$,…

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph $G=(V,E)$, which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear…

We study the following $1$-Yamabe equation on a connected finite graph $$\Delta_1u+g\mathrm{Sgn}(u)=h|u|^{\alpha-1}\mathrm{Sgn}(u),$$ where $\Delta_1$ is the discrete $1$-Laplacian, $\alpha>1$ and $g, h>0$ are known. We show that the above…

We investigate the Allen-Cahn system \begin{equation*} \Delta u-W_u(u)=0,\quad u:\mathbb{R}^2\rightarrow\mathbb{R}^2, \end{equation*} where $W\in C^2(\mathbb{R}^2,[0,+\infty))$ is a potential with three global minima. We establish the…

In this paper, we study the following Kazdan-Warner equation with sign-changing prescribed function $h$ \begin{align*} -\Delta u=8\pi\left(\frac{he^{u}}{\int_{\Sigma}he^{u}}-1\right) \end{align*} on a closed Riemann surface whose area is…

This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in…

Suppose that $G=(V, E)$ is a finite graph with the vertex set $V$ and the edge set $E$. Let $\Delta$ be the usual graph Laplacian. Consider the following nonlinear Schr$\ddot{o}$dinger type equation of the form $$ \left \{…

Let $ G=(V,E) $ be a connected finite graph and $ \Delta $ the usual graph Laplacian. In this paper, we consider a generalized self-dual Chern-Simons equation on the graph $G$ \begin{eqnarray}\label{one1}…

Given a finite simple undirected graph $G$, let $T_1(G)$ denote the subset of vertices of $G$ such that every vertex of $T_1(G)$ belongs to at least one subgraph isomorphic to a graph obtained by connecting a single vertex to two vertices…

Denote by $G=(V,E)$ a finite graph. We study a generalized Chern-Simons equation $$ \Delta u=\lambda \mathrm{e}^u(\mathrm{e}^{bu}-1)+4\pi\sum\limits_{j=1}^{N}\delta_{p_j} $$ on $G$, where $\lambda$ and $b$ are positive constants; $N$ is a…

The purpose of this paper is to develop a "calculus" on graphs that allows graph theory to have new connections to analysis. For example, our framework gives rise to many new partial differential equations on graphs, most notably a new…

Let $G$ be a connected graph and $\mathcal{P}(G)$ a graph parameter. We say that $\mathcal{P}(G)$ is feasible if $\mathcal{P}(G)$ satisfies the following properties: (I) $\mathcal{P}(G)\leq \mathcal{P}(G_{uv})$, if $G_{uv}=G[u\to v]$ for…

We investigate the Kazdan-Warner equation on a network. In this case, the differential equation is defined on each edge, while appropriate transition conditions of Kirchhoff type are prescribed at the vertices. We show that the…

Inspired by works of Cast\'eras (Pacific J. Math., 2015), Li-Zhu (Calc. Var., 2019) and Sun-Zhu (Calc. Var., 2020), we propose a heat flow for the mean field equation on a connected finite graph $G=(V,E)$. Namely $$…

The $\delta$-complement $G_\delta$ of a graph $G$, introduced in 2022 by Pai et al., is a variant of the graph complement, where two vertices are adjacent in $G_\delta$ if and only if they are of the same degree but not adjacent in $G$ or…

This paper establishes an existence theory for discrete second-order boundary value problems on non-uniform time grids using the upper and lower solution method. We consider difference equations of the form $u^{\Delta\Delta}(t_{i-1}) +…

Let $G=(V,E)$ be a connected infinite and locally finite weighted graph, $\Delta_p$ be the $p$-th discrete graph Laplacian. In this paper, we consider the $p$-th Yamabe type equation $$-\Delta_pu+h|u|^{p-2}u=gu^{\alpha-1}$$ on $G$, where…

We investigate the Allen-Cahn system \begin{equation*} \Delta u-W_u(u)=0,\quad u:\mathbb{R}^2\rightarrow\mathbb{R}^2, \end{equation*} where $W\in C^2(\mathbb{R}^2,[0,+\infty))$ is a potential with three global minima. We establish the…