Existence theorems for a generalized Chern-Simons equation on finite graphs

Abstract

Denote by $G=(V,E)$ a finite graph. We study a generalized Chern-Simons equation $$\Delta u=\lambda {\mathrm{e}}^u({\mathrm{e}}^{bu}-1)+4\pi \sum _{j=1}^N{\delta }_{p_j}$$ on $G$, where $\lambda$ 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 ${\delta }_{p_j}$ is the Dirac delta mass at $p_j$. We prove that there exists a critical value ${\lambda }_c$ such that the equation has a solution if $\lambda \ge {\lambda }_c$ and the equation has no solution if $\lambda <{\lambda }_c$. We also prove that if $\lambda >{\lambda }_c$ the equation has at least two solutions which include a local minimizer for the corresponding functional and a mountain-pass type solution.

15 pages