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 $$ Δu=λ\mathrm{e}^u(\mathrm{e}^{bu}-1)+4π\sum\limits_{j=1}^{N}δ_{p_j} $$ 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.

Theorems & Definitions (21)

• Theorem 1.1
• Remark 1
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5