论文标题
有限图上的广义Chern-Simons方程的存在定理
Existence theorems for a generalized Chern-Simons equation on finite graphs
论文作者
论文摘要
用$ g =（v，e）$表示有限图。我们研究了一个广义的Chern-Simons方程$$Δu=λ\ Mathrm {e}^u（\ Mathrm {E}^{bu}^{bu} -1）+4π\ sum \ sum \ limits_ { $ n $是一个积极的整数； $ p_1，p_2，\ cdot \ cdot \ cdot，p_n $是$ v $的不同顶点，$δ__{p_j} $是$ p_j $的dirac delta质量。我们证明存在一个临界值$λ_c$，因此，如果$λ\geqλ_c$，如果$λ<λ_c$，该方程没有解决方案。我们还证明，如果$λ>λ_c$该方程至少具有两个解决方案，其中包括用于相应功能和山间通用类型解决方案的局部最小化器。
Denote by $G=(V,E)$ a finite graph. We study a generalized Chern-Simons equation $$ Δu=λ\mathrm{e}^u(\mathrm{e}^{bu}-1)+4π\sum\limits_{j=1}^{N}δ_{p_j} $$ on $G$, where $λ$ and $b$ are positive constants; $N$ is a positive integer; $p_1, p_2, \cdot\cdot\cdot, p_N$ are distinct vertices of $V$ and $δ_{p_j}$ is the Dirac delta mass at $p_j$. We prove that there exists a critical value $λ_c$ such that the equation has a solution if $λ\geq λ_c$ and the equation has no solution if $λ<λ_c$. We also prove that if $λ>λ_c$ the equation has at least two solutions which include a local minimizer for the corresponding functional and a mountain-pass type solution.
