Related papers: Existence theorems for a generalized Chern-Simons …

Let $G=(V,E)$ be a connected finite graph. We study the existence of solutions for the following generalized Chern-Simons equation on $G$ \begin{equation*} \Delta u=\lambda \mathrm{e}^{u}\left(\mathrm{e}^{u}-1\right)^{5}+4 \pi…

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 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}…

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…

In this paper, we consider a system of equations arising from the $\text{U}(1)\times \text{U}(1)$ Abelian Chern-Simons model \begin{eqnarray*}\left\{\begin{aligned} \Delta u…

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 $f$…

We study a system of equations arising in the Chern-Simons model on finite graphs. Using the iteration scheme and the upper and lower solutions method, we get existence of solutions in the non-critical case. The critical case is dealt with…

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 consider the topological solutions to the skew-symmetric Chern-Simons system on lattice graphs: $$\left\{\begin{aligned} \Delta u &=\lambda\mathrm{e}^{\upsilon}(\mathrm{e}^{u}-1)+4\pi\sum\limits_{j=1}^{k_1}m_j\delta_{p_j},…

We study a class of generalized Chern-Simons equations on discrete lattice graphs. By an iterative scheme combined with an exhaustion argument, we establish the existence of topological solutions, which is also the maximal topological…

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…

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 $\rho>0$…

For $n \geq 2$, consider $\mathbb{Z}^n$ as a lattice graph. We explore a generalized Chern-Simons equation on $\mathbb{Z}^n$. Employing the method of exhaustion, we prove that there exists a global solution that also qualifies as a…

We investigate the scalar Chern-Simons equation $-\Delta u + e^u(e^u-1) = \mu$ in cases where there is no solution for a given nonnegative finite measure $\mu$. Approximating $\mu$ by a sequence of nonnegative $L^1$ functions or finite…

Let $\mathscr G:= (V,E)$ be a weighted locally finite graph whose finite measure $\mu$ has a positive lower bound. Motivated by wide interest in the current literature, in this paper we study the existence of classical solutions for a class…

Let $(V,\mu)$ be an infinite, connected, locally finite weighted graph. We study the problem of existence or non-existence of positive solutions to a semi-linear elliptic inequality \begin{equation*} \Delta u+u^{\sigma}\leq0\quad…

In this paper, we investigate the existence of multiple positive solutions to the following multi-critical Schr\"{o}dinger equation \begin{equation} \label{p} \begin{cases} -\Delta u+\lambda V(x)u=\mu…

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…

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…

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 \{…