Sharp regularity for singular obstacle problems

We obtain sharp local $C^{1,\alpha}$ regularity of solutions for singular obstacle problems, Euler-Lagrange equation of which is given by $$\Delta_p u=\gamma(u-\varphi)^{\gamma-1}\,\text{ in }\,\{u>\varphi\},$$ for $0<\gamma<1$ and $p\ge2$. At the free boundary $\partial\{u>\varphi\}$, we prove optimal $C^{1,\tau}$ regularity of solutions, with $\tau$ given explicitly in terms of $p$, $\gamma$ and smoothness of $\varphi$, which is new even in the linear setting.