  We consider the semilinear problem \[ \Delta u = \lambda_+ \left(-\log
u^+\right) 1_{\{u > 0\}} - \lambda_- \left(-\log u^- \right) 1_{\{u < 0\}}
\qquad \hbox{ in } B_1, \] where $B_1$ is the unit ball in $\mathbb{R}^n$ and
assume $\lambda_+, \lambda_- > 0$. Using a monotonicity formula argument, we
prove an optimal regularity result for solutions: $\nabla u$ 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 non-integrable unless
one already knows the optimal regularity of the solution: this puts us in a
catch-22 situation.
