A non-local one-phase free boundary problem from obstacle to cavitation

We consider a one-phase free boundary problem of the minimizer of the energy $$J_{\gamma}(u)=\frac{1}{2}\int_{(B_1^{n+1})^+}{y^{1-2s}|\nabla u(x,y)|^2dxdy}+\int_{B_1^{n}\times \{y=0\}}{u^{\gamma}dx},$$ with constants $0<s,\gamma<1$. It is an intermediate case of the fractional cavitation problem (as $\gamma=0$) and the fractional obstacle problem (as $\gamma=1$). We prove that the blow-up near every free boundary point is homogeneous of degree $\beta=\frac{2s}{2-\gamma}$, and flat free boundary is $C^{1,\theta}$ when $\gamma$ is close to 0.