论文标题
加权图上的超线性椭圆形不平等
Superlinear elliptic inequalities on weighted graphs
论文作者
论文摘要
令$（v，μ）$为无限，连接的，局部有限的加权图。 We study the problem of existence or non-existence of positive solutions to a semi-linear elliptic inequality \begin{equation*} Δu+u^σ\leq0\quad \text{in}\,\,V, \end{equation*} where $Δ$ is the standard graph Laplacian on $V$ and $σ>0$. For $σ\in(0,1]$, the inequality admits no nontrivial positive solution. For $σ>1$, assuming condition \textbf{($p_0$)} on $(V,μ)$, we obtain a sharp condition for nonexistence of positive solutions in terms of the volume growth of the graph, that is \begin{equation*} μ(o,n)\lesssim n^{\ frac {2σ} {σ-1}}（\ ln n）^{\ frac {1} {σ-1}}}} \ end {equation*} 对于v $中的某些$ o \，以及所有足够大的$ n $。对于任何$ \ varepsilon> 0 $，我们可以在同质树$ \ mathbb t_n $上与$μ（o，n）\大约n^{\ frac {\ frac {2σ} {2σ} {σ}} {σ}}}}（\ ln n n）^{\ frac {\ frac {\ frac {1} {1} {1} {1} {1} {1} {1} {1} {\ frac {v n N^{\ frac {1} {1} {1}+n n n N^{\ frac+ $（\ Mathbb t_n，μ）$以说明清晰度 $ \ frac {2σ} {σ-1} $和$ \ frac {1} {σ-1} $
Let $(V,μ)$ 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*} Δu+u^σ\leq0\quad \text{in}\,\,V, \end{equation*} where $Δ$ is the standard graph Laplacian on $V$ and $σ>0$. For $σ\in(0,1]$, the inequality admits no nontrivial positive solution. For $σ>1$, assuming condition \textbf{($p_0$)} on $(V,μ)$, we obtain a sharp condition for nonexistence of positive solutions in terms of the volume growth of the graph, that is \begin{equation*} μ(o,n)\lesssim n^{\frac{2σ}{σ-1}}(\ln n)^{\frac{1}{σ-1}} \end{equation*} for some $o\in V$ and all large enough $n$. For any $\varepsilon>0$, we can construct an example on a homogeneous tree $\mathbb T_N$ with $μ(o,n)\approx n^{\frac{2σ}{σ-1}}(\ln n)^{\frac{1}{σ-1}+\varepsilon}$, and a solution to the inequality on $(\mathbb T_N,μ)$ to illustrate the sharpness of $\frac{2σ}{σ-1}$ and $\frac{1}{σ-1}$.
