Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents

We investigate the rigidity of global minimizers $u\ge 0$ of the Alt-Phillips functional involving negative power potentials $${\int }_{\mathrm{\Omega }}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2+u^{-\gamma }{\chi }_{\{u>0\}})dx,\gamma \in (0,2),$$ when the exponent $\gamma$ is close to the extremes of the admissible values. In particular we show that global minimizers in ${\mathbb{R}}^n$ are one-dimensional if $\gamma$ is close to 2 and $n\le 7$, or if $\gamma$ is close to $0$ and $n\le 4$.