Related papers: Existence results for Tzitz\'eica equation via top…

Consider a finite connected graph denoted as $G=(V, E)$. This study explores a generalized Chern-Simons Higgs model, characterized by the equation: $$ \Delta u = \lambda e^u (e^u - 1)^{2p+1} + f,$$ where $\Delta$ denotes the graph…

We are concerned with the following class of equations with exponential nonlinearities: $$ \Delta u+h_1e^u-h_2e^{-2u}=0 \qquad \mbox{in } B_1\subset\mathbb{R}^2, $$ which is related to the Tzitz\'eica equation. Here $h_1, h_2$ are two…

Let $(V,E)$ be a finite connected graph. We are concerned about the Chern-Simons Higgs model $$\Delta u=\lambda e^u(e^u-1)+f, \quad\quad\quad\quad\quad\quad{(0.1)}$$ where $\Delta$ is the graph Laplacian, $\lambda$ is a real number and $f$…

In this paper, motivated by the work of Huang-Lin-Yau (Commun. Math. Phys. 2020), Sun-Wang (Adv. Math. 2022) and Li-Sun-Yang (Calc. Var. Partial Differential Equations 2024), we investigate the existence of Kazdan-Warner type equations on a…

Let $G=(V, E)$ be a finite connected graph, where $V$ denotes the set of vertices and $E$ denotes the set of edges. We revisit the following Chern-Simons Higgs model, \begin{equation*} \Delta u=\lambda…

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}) +…

We consider the following logarithmic Schr\"{o}dinger equation $$ -\Delta u+h(x)u=u\log u^{2} $$ on a locally finite graph $G=(V,E)$, where $\Delta$ is a discrete Laplacian operator on the graph, $h$ is the potential function. Different…

We study semilinear elliptic equations on finite graphs with fully general exponential nonlinearities, thereby extending classical equations such as the Kazdan-Warner and Chern-Simons equations. A key contribution of this work is the…

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…

In this paper, we prove two existence results of solutions to mean field equations $$\Delta u+e^u=\rho\delta_0$$ and $$\Delta u=\lambda e^u(e^u-1)+4 \pi \sum_{j=1}^{M}{\delta_{p_j}}$$ on an arbitrary connected finite graph, where $\rho>0$…

We deal with a planar differential system of the form \begin{equation*} \begin{cases} \, u' = h(t,v), \\ \, v' = - \lambda a(t) g(u), \end{cases} \end{equation*} where $h$ is $T$-periodic in the first variable and strictly increasing in the…

Let $V$ be a locally finite, connected and weighted graph. We study non-existence results of non-trivial, non-negative solutions of the system $$ \begin{cases} u_{t t}-\Delta u \geq h_1|v|^p & \text { in } V \times(0, \infty), v_{t…

Let $G=(V,E)$ be a finite graph. We consider the existence of solutions to a generalized Chern-Simons-Higgs equation $$ \Delta u=-\lambda e^{g(u)}\left( e^{g(u)}-1\right)^2+4\pi\sum\limits_{j=1}^{N}\delta_{p_j} $$ on $G$, where $\lambda$ is…

Let $G=(V,E)$ be a finite or locally finite connected weighted graph, $\Delta$ be the usual graph Laplacian. Using heat kernel estimate, we prove the existence and nonexistence of global solutions for the following semilinear heat equation…

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…

We consider a class of equations with exponential non-linearities on a compact surface which arises as the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. We prove an existence result via degree theory.…

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}…

Let $G=(V,E)$ be a connected finite graph. We study the Bogomol'nyi equation \begin{equation*} \Delta u= \mathrm{e}^{u}-1 +4 \pi \sum_{s=1}^{k} n_s \delta_{z_{s}} \quad \text { on } \quad G, \end{equation*} where $z_1, z_2,\dots, z_k$ are…

In this paper, we study the semilinear elliptic equation of the form \begin{eqnarray*} -\Delta u+a(x)|u|^{p-2}u-b(x)|u|^{q-2}u=0 \end{eqnarray*} on lattice graphs $\mathbb{Z}^{N}$, where $N\geq 2$ and $2\leq p<q<+\infty$. By the…

Let $G=(V,E)$ be a finite graph and $\Delta$ be the usual graph Laplacian. Using the calculus of variations and a method of upper and lower solutions, we give various conditions such that the Kazdan-Warner equation $\Delta u=c-he^u$ has a…