The Alt-Phillips functional for negative powers

We develop the free boundary regularity for nonnegative minimizers of the Alt-Phillips functional for negative power potentials $${\int }_{\mathrm{\Omega }}(\frac{1}{2}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2+u^{\gamma }{\chi }_{\{u>0\}})dx,\gamma \in (-2,0),$$ and establish a $\mathrm{\Gamma }$-convergence result of the rescaled energies to the perimeter functional as $\gamma \to -2$.