Existence theorems for a general 2 × 2 non-Abelian Chern–Simons–Higgs system over a torus

Abstract

In this paper we study a general 2×2 non-Abelian Chern–Simons–Higgs system of the formΔui+1ε2(∑j=12Kjieuj−∑j=12∑k=12KkjKjieujeuk)=4π∑j=1Niδpij(x),i=1,2 over a flat 2-torus T2, where ε>0, δp is the Dirac measure at p, Ni∈N (i=1,2), K is a non-degenerate 2×2 matrix of the form K=(1+a−a−b1+b), which may cover the physically interesting case when K is a Cartan matrix (of a rank 2 semisimple Lie algebra). Concerning the existence results of this type system over T2, usually in the literature there is a requirement that a,b>0. However, it is an open problem so far for the solvability about such system with a,b<0, which naturally appears in several Chern–Simons–Higgs models with some specific gauge groups. We partially solve this problem by showing that there exists a constant ε0>0 such that this system admits a solution over the torus if 0<ε<ε0 provided |a|,|b| are suitably small. Furthermore, if ab≥0 in addition, with suitable condition on a,b,N1,N2, this system admits a mountain-pass solution. Our argument is based on a perturbation approach and the mountain-pass lemma.