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

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