Non-Abelian Chern–Simons–Higgs system with indefinite functional

• Published in: Nonlinear differential equations and applications 2023-05, Vol.30 (3), Article 36
• Main Authors: Huang, Hsin-Yuan, Lee, Youngae, Moon, Sang-hyuck
• Format: Article
• Language: English
• Subjects: Analysis; Mathematics; Mathematics and Statistics
• ISSN: 1021-9722; 1420-9004
• DOI: 10.1007/s00030-022-00837-5

In this paper, we are concerned with the general non-Abelian Chern–Simons–Higgs models of rank two. The corresponding self-dual equations can be reduced to a nonlinear elliptic system, and the form is determined by a non-degenerate matrix K . One of the major questions is how the matrix K affects the structure of the solutions to the self-dual equations. There have been some existence results of the solutions to the self-dual equations when det ( K ) > 0 . However, the solvability for the case det ( K ) < 0 is not fully understood in spite of its physical importance. In contrast to det ( K ) > 0 , one major difficulty for the case det ( K ) < 0 is that the energy functional associated with the elliptic system is usually indefinite. The direct variational method thus fails. We overcome this obstacle and obtain a partially positive answer for the solvability when det ( K ) < 0 by controlling the indefinite functional with a suitable constraint.