Multiple blowing-up and concentrating solutions for Liouville-type equations with singular sources under mixed boundary conditions

In this article, we mainly construct multiple blowing-up and concentrating solutions for a class of Liouville-type equations under mixed boundary conditions: {-Delta v = epsilon(2)e(v) - 4 pi Sigma(N)(i-1) alpha(i)delta(pi), in Omega, epsilon(1 - t)partial derivative v/partial derivative v + tb(x)v = 0, on partial derivative Omega, for epsilon small, where , a"broken vertical bar is a bounded, smooth domain in , I" := {p (1), ..., p (N) } aS, a"broken vertical bar is the set of singular sources, delta (p) denotes the Dirac mass at p, nu denotes unit outward normal vector to a,a"broken vertical bar and b(x) > 0 is a smooth function on a,a"broken vertical bar.