  We prove a sharp H\"older continuity estimates of rupture sets for sequences
of solutions of the following nonlinear problem with negative exponent $$
\Delta u= \frac{1}{u^p}, \ p>1, \ \mbox{in} \ \Omega .$$ As a consequence, we
prove the existence of rupture solutions with isolated ruptures in a bounded
convex domain in $\R^2$.
