arXiv subjects

Shoudong Man

Publications and source records attributed to Shoudong Man.

6 recordsLinked to original sources

arXiv2025

Gradient estimates for the weighted porous medium equation on graphs

In this paper, we study the gradient estimates for the positive solutions of the weighted porous medium equation $$Δu^{m}=δ(x)u_{t}+ψu^{m}$$ on graphs for $m>1$, which is a nonlinear version of the heat equation. Moreover, as applications, we derive a Harnack inequality and the estimates of the porous medium kernel on graphs. The obtained results extend the results of Y. Lin, S. Liu and Y. Yang for the heat equation [8, 9].

math.DG↗

arXiv2023

Existence results for some nonlinear elliptic systems on graphs

In this paper, several nonlinear elliptic systems are investigated on graphs. One type of the sobolev embedding theorem and a new version of the strong maximum principle are established. Then, by using the variational method, the existence of different types of solutions to some elliptic systems is confirmed. Such problems extend the existence results on closed Riemann surface to graphs and extend the existence results for one single equation on graphs [A. Grigor'yan, Y. Lin, Y. Yang, J. Differential Equations, 2016] to nonlinear elliptic systems on graphs. Such problems can also be viewed as one type of discrete version of the elliptic systems on Euclidean space and Riemannian manifold.

math.AP↗

arXiv2019

On a class of nonlinear Schrödinger equation on finite graphs

Suppose that $G=(V, E)$ is a finite graph with the vertex set $V$ and the edge set $E$. Let $Δ$ be the usual graph Laplacian. Consider the following nonlinear Schr$\ddot{o}$dinger type equation of the form $$ \left \{ \begin{array}{lcr} -Δu-αu=f(x,u),\\ u\in W^{1,2}(V),\\ \end{array} \right. $$ on graph $G$, where $f(x,u):V\times\mathbb{R}\rightarrow\mathbb{R}$ is a nonlinear function and $α$ is a parameter. Firstly, we prove the Trudinger-Moser inequality on graph $G$, and under the assumption that $G$ satisfies the curvature-dimension type inequality $CD(m, ξ)$, we prove an integral inequality on $G$. Then by using the two inequalities, we prove that there exists a positive solution to the nonlinear Schr$\ddot{o}$dinger type equation if $α<\frac{2λ^{2}}{m(λ-ξ)}$, where $λ$ is the eigenvalue of the graph Laplacian. Our work provides remarkable improvements to the previous results.

math.DG↗

arXiv2019

On a class of quasilinear elliptic equation with indefinite weights on graphs

Suppose that $G=(V, E)$ is a connected locally finite graph with the vertex set $V$ and the edge set $E$. Let $Ω\subset V$ be a bounded domain. Consider the following quasilinear elliptic equation on graph $G$ $$ \left \{ \begin{array}{lcr} -Δ_{p}u= λK(x)|u|^{p-2}u+f(x,u), \ \ x\inΩ^{\circ}, u=0, \ \ x\in\partial Ω, \\ \end{array} \right. $$ where $Ω^{\circ}$ and $\partial Ω$ denote the interior and the boundary of $Ω$ respectively, $Δ_{p}$ is the discrete $p$-Laplacian, $K(x)$ is a given function which may change sign, $λ$ is the eigenvalue parameter and $f(x,u)$ has exponential growth. We prove the existence and monotonicity of the principal eigenvalue of the corresponding eigenvalue problem. Furthermore, we also obtain the existence of a positive solution by using variational methods.

math.DG↗

arXiv2014

Hypergraphs with Spectral Radius at most $(r-1)!\sqrt[r]{2+\sqrt{5}}$

In our previous paper, we classified all $r$-uniform hypergraphs with spectral radius at most $(r-1)!\sqrt[r]{4}$, which directly generalizes Smith's theorem for the graph case $r=2$. It is nature to ask the structures of the hypergraphs with spectral radius slightly beyond $(r-1)!\sqrt[r]{4}$. For $r=2$, the graphs with spectral radius at most $\sqrt{2+\sqrt{5}}$ are classified by [{\em Brouwer-Neumaier, Linear Algebra Appl., 1989}]. Here we consider the $r$-uniform hypergraphs $H$ with spectral radius at most $(r-1)!\sqrt[r]{2+\sqrt{5}}$. We show that $H$ must have a quipus-structure, which is similar to the graphs with spectral radius at most $\frac{3}{2}\sqrt{2}$ [{\em Woo-Neumaier, Graphs Combin., 2007}].

math.SP↗

arXiv2014

Connected Hypergraphs with Small Spectral Radius

In 1970 Smith classified all connected graphs with the spectral radius at most $2$. Here the spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Recently, the definition of spectral radius has been extended to $r$-uniform hypergraphs. In this paper, we generalize the Smith's theorem to $r$-uniform hypergraphs. We show that the smallest limit point of the spectral radii of connected $r$-uniform hypergraphs is $ρ_r=(r-1)!\sqrt[r]{4}$. We discovered a novel method for computing the spectral radius of hypergraphs, and classified all connected $r$-uniform hypergraphs with spectral radius at most $ρ_r$.

math.CO↗