$C^{\infty}$ regularity of the Alt-Phillips Functional for negative powers

In this paper, we study the regularity of the free boundary for minimizers of the Alt-Phillips functional with negative powers $$\mathcal{E}_{\gamma}(u)=\int_{\Omega}\frac{1}{2}|\nabla u|^2+\frac{1}{\gamma}u^{-\gamma}\chi_{\{u>0\}}dx,\quad\gamma\in(0,2).$$ We proved that the free boundaries are $C^{\infty}$ at regular points. A key technical tool is the linearized operator for the PDE satisfied by the partial derivatives of a solution to the Alt-Phillips Euler-Lagrange equation in the negative power case. For this operator we establish a comparison principle, which may have further applications to the Alt-Phillips problem with negative powers.