We consider the following semilinear elliptic equation involving the fractional Laplacian \(\begin{aligned} (-\triangle )^su=-u^{-p} \hbox { in } B_1, \end{aligned}\) where \(p>1\) , \(s\in (0,1)\) , \((-\triangle )^s\) is the s-Laplacian and \(B_1=B_1(0)\) is the unit ball in \(\mathbb {R}^N\) . We first establish an optimal Hölder regularity estimate for solutions by using blow-up analysis and Liouville-type theorems. Subsequently, we give a convergence result for sequences of solutions with uniform Hölder continuity. These results are also used to show that the Hausdorff dimension of the rupture set \(\{u=0\}\) satisfies: \(\dim _{\mathcal {H}} \{u=0\} \le N-2 \hbox { if } \frac{p+1}{2p}<s<1;\) \(\dim _{\mathcal {H}} \{u=0\} \le N-1 \hbox { if } 0<s\le \frac{p+1}{2p}\) . In particular, the latter one is a new phenomenon arising from the fractional Laplacian.
