Singular points for a free boundary problem

We will analyse the singular points, that is $X^j$ such that $u\notin C^{1,1}(B_r(X^j))$ for any $r>0$, of solutions to $$ \Delta u=-\chi_{\{u>0\}} \textrm{ in }B_1. $$ In $\mathbb{R}3$ we will show that the singular set consists of isolated points and a part that is locally contained in a one dimensional $C1$ manifold.

We will also classify the singular points in $\mathbb{R}3$ and show that there are only three kinds of such point and give explicit assymptotics for the solution at such points.