Existence of bubbling solutions for the Liouville system in a torus

Abstract

We consider the following Liouville system on a parallelogram Omega in R-2: Delta u(i) + Sigma(n)(j=1) a(ij)rho(j) (h(j)e(uj)/integral(Omega)h(j)e(u)j - 1/vertical bar Omega vertical bar) = 0, i is an element of I = {1,..., n}, (0.1) where h(i) (x) is an element of C-3(Omega), h(i) (x) > 0, ui is doubly periodic on partial derivative Omega (i is an element of I), and A = (a(ij)) nxn is a non-negative constant matrix. We prove that if q is a non-degenerate critical point of Sigma n i= 1. * i log hi (x) and A satisfies certain conditions stated in Theorem 1.1, (0.1) has a sequence of fully bubbling solutions which blow up at p, as. = (.1,...,.n).. * = (. * 1,...,. * n), where. * satisfies 8p Sigma n i= 1. * i = Sigma n i= 1 Sigma nj = 1 ai j. * i. * j and Sigma n i= 1 ai j. * i. * j > 6p for j is an element of I.