Optimal regularity for a two-phase obstacle-like problem with logarithmic singularity

We consider the semilinear problem Delta u = lambda(+) (-log u(+)) 1({u>0}) - lambda(-) (-log u(-))1({u< 0}) in B-1, where B1 is the unit ball in R-n and assume lambda(+), lambda(-) > 0: Using a monotonicity formula argument, we prove an optimal regularity result for solutions: del u is a log-Lipschitz function. This problem introduces two main difficulties. The first is the lack of invariance in the scaling and blow-up of the problem. The other (more serious) issue is a term in the Weiss energy which is potentially nonintegrable unless one already knows the optimal regularity of the solution: this puts us in a catch-22 situation.