Existence of Solutions to a Generalized Self-dual Chern-Simons Equation on Graphs

Abstract

Let G = (V,E) be a connected finite graph and Δ be the usual graph Laplacian. In this paper, the authors consider a generalized self-dual Chern-Simons equation on the graph G

(0.1)

\mathrm{\Delta }u=-\lambda {\mathrm{e}}^{F(u)}[{\mathrm{e}}^{F(u)}-1{]}^2+4\pi \sum _{j=1}^M{\delta }_{p_j},

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Delta u = - \lambda {{\rm{e}}^{F( u )}}{[ {{{\rm{e}}^{F( u )}} - 1} ]^2} + 4\pi \sum\limits_{j = 1}^M {{\delta _{{p_j}}}},$$\end{document}

where

F(u)=\{\begin{array}{cc}\tilde{F}\left(u\right),& u\le 0,\\ 0,& u>0,\end{array}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F(u) =\begin{cases}{\tilde F\left( u \right)}, & u \le 0,\\0, & u > 0,\end{cases}$$\end{document}

\tilde{F}(u)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tilde F(u)$$\end{document}

satisfies

u=1+\tilde{F}(u)-{\mathrm{e}}^{\tilde{\mathrm{F}}(\mathrm{u})}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$u=1+\tilde F(u)-\rm{{e}}^{\tilde F(u)}$$\end{document}

, λ > 0, M is any fixed positive integer,

{\delta }_{p_j}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\delta_{p_{j}}$$\end{document}

is the Dirac delta mass at the vertex p_j, and p₁, p₂, …, p_M are arbitrarily chosen distinct vertices on the graph. They first prove that there is a critical value λ_c such that if λ ≥ λ_c, then the generalized self-dual Chern-Simons equation has a solution u_λ. Applying the existence result, they develop a new method to construct a solution of (0.1) which is monotonic with respect to λ when λ ≥ λ_c. Then they establish that there exist at least two solutions of the equation for λ > λ_c via the variational method. Furthermore, they give a fine estimate of the monotone solution, which can be applied to other related problems.

Keywords

Chern-Simons equation / Finite graph / Existence / Variational method / 35A01 / 35A15 / 35R02

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