Uniformly Elliptic Liouville Type Equations Part II: Pointwise Estimates and Location of Blow up Points

We refine the analysis, initiated in [3], [4] of the blow up phenomenon for the following two dimensional uniformly elliptic Liouville type problem in divergence form: {-div(A del u) = mu Ke(u)/integral(Omega)Ke(u) in Omega, u = 0 on partial derivative Omega. We provide a partial generalization of a result of Y.Y. Li [18] to the case A inverted iota I. To this end, in the same spirit of [2], we obtain a sharp pointwise estimate for simple blow up sequences. Moreover, we prove that if {p(1), ... , p(N)} is the blow up set corresponding to a given simple blow up sequence, then, (del detA)(p(j)) = 0, for all j = 1, ... , N. This characterization of the blow up set yields an improvement of the a priori estimates already established in [3].