Bubbling solutions of mixed type for a general non-Abelian Chern-Simons-Higgs system of rank 2 over a torus

Abstract

We consider a general non-Abelian Chern-Simons-Higgs system of rank 2 Delta u(i) + 1/epsilon(2) (Sigma(2)(j=1) K(ji)e(uj) - Sigma(2)(j=1)Sigma(2)(k=1) K(kj)K(ji)e(uj) e(uk)) = 4 pi Sigma(N)(l=1) m(i,l)delta(pl) (i = 1, 2) (0.1) over a flat torus, where m(1,) (l) >= 0, m(2, l) >= 0, ( m(1,l), m(2, l)) not equal (0, 0) for l = 1,..., N, delta(p) is the Dirac measure at p, K is a non-degenerate 2 x 2 matrix of the form K = (1 + a -a -b 1 + b). When a > -1, b > -1, and a + b > -1, Eqs (0.1) are expected to have three types solutions: topological, non-topological and mixed type solutions. Concerning the existence of various type solutions, there are requirements that a > 0 and b > 0, or a and b are close to 0 in the literature. It is still open for generic a and b. We partially answer this question and show that (0.1) possesses bubbling mixed type solutions provided that epsilon is small and (a, b) satisfies (1.17). (C) 2021 Elsevier Ltd. All rights reserved.