Singular limits in Liouville-type equations

We consider the boundary value problem Delta u + epsilon(2) k( x) e(u) = 0 in a bounded, smooth domain Omega R-2 with homogeneous Dirichlet boundary conditions. Here epsilon > 0, k( x) is a non-negative, not identically zero function. We. find conditions under which there exists a solution u(epsilon) which blows up at exactly m points as epsilon -> 0 and satisfies epsilon(2) integral Omega ke u epsilon -> 8m pi. In particular, we. and that if k epsilon C2((Omega) over bar), inf(Omega) k > 0 and is not simply connected then such a solution exists for any given m >= 1.