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 $$\int_\Omega \left(|\nabla u|^2 + u^{-\gamma} \chi_{\{u>0\}}\right) \, dx, \quad \quad \gamma \in (0,2),$$ 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 $n \le 4$.