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Existence results for a non-relativistic Chern-Simons model with purely mutual interaction

Aleks Jevnikar, Sang-Hyuck Moon

TL;DR

This work analyzes a two-component, purely mutual interaction mean-field system on a compact surface $M$, arising from a non-relativistic $[U(1)]^2$ Chern–Simons theory, which leads to a singular Liouville-type system with an indefinite energy functional. The authors introduce a constrained variational framework by decoupling variables via $u_1=F-G$, $u_2=F+G$ and minimizing over $G$ to obtain a reduced functional $\widetilde{J}_\rho$; they then apply Morse theory to study topology changes of sublevels, establishing existence and multiplicity results under not-in-$\Lambda$ and large $ ho$ conditions. They prove that for $M=\mathbb{S}^2$ and $\alpha_j=0$ there exists a solution, and for genus $g>0$ with $\alpha_j\ge0$ there are at least $\binom{k+g-1}{g-1}$ solutions for generic data, provided $(\rho_1,\rho_2)\notin \Lambda$ and $\rho_i$ are sufficiently large with $(\rho_1+\rho_2)/2<8(k+1)\pi$. The results extend prior work on coupled Liouville systems by accommodating positive genus and singular weights, using improved Moser–Trudinger inequalities and formal barycenter/barycentric techniques to count solutions via low-sublevel topology.

Abstract

We are concerned with a skew-symmetric singular Liouville system arising in non-relativistic Chern-Simons theory. Based on its variational structure, we establish existence and multiplicity results. Since the energy functional is indefinite, standard variational approaches do not apply directly. We overcome this difficulty by introducing a suitable constrained problem and implementing a Morse-theoretical argument

Existence results for a non-relativistic Chern-Simons model with purely mutual interaction

TL;DR

This work analyzes a two-component, purely mutual interaction mean-field system on a compact surface , arising from a non-relativistic Chern–Simons theory, which leads to a singular Liouville-type system with an indefinite energy functional. The authors introduce a constrained variational framework by decoupling variables via , and minimizing over to obtain a reduced functional ; they then apply Morse theory to study topology changes of sublevels, establishing existence and multiplicity results under not-in- and large conditions. They prove that for and there exists a solution, and for genus with there are at least solutions for generic data, provided and are sufficiently large with . The results extend prior work on coupled Liouville systems by accommodating positive genus and singular weights, using improved Moser–Trudinger inequalities and formal barycenter/barycentric techniques to count solutions via low-sublevel topology.

Abstract

We are concerned with a skew-symmetric singular Liouville system arising in non-relativistic Chern-Simons theory. Based on its variational structure, we establish existence and multiplicity results. Since the energy functional is indefinite, standard variational approaches do not apply directly. We overcome this difficulty by introducing a suitable constrained problem and implementing a Morse-theoretical argument
Paper Structure (6 sections, 15 theorems, 77 equations, 1 figure)

This paper contains 6 sections, 15 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Self-dual equations
  3. Proof of the main results
  4. A constrained problem
  5. The zero genus case
  6. The positive genus case

Key Result

Theorem 1.1

Suppose $(\rho_1,\rho_2)\notin\Lambda$ and that Then, we have: 1. If $M=\mathbb{S}^2$ and all $\alpha _j=0$, then system has a solution. 2. If $M$ has positive genus $\emph{g}>0$ and $\alpha _j\geq0$, then for a generic choice of the metric $g$ and of the functions $h_1,h_2$ it holds

Figures (1)

  • Figure 1:

Theorems & Definitions (22)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Lemma 3.5
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • ...and 12 more