Consider the following obstacle problem in the whole domain $\mathbb{R}^n$
min{$\Delta u$, $u$-$\phi$}=0
with prescribed boundary value $\lim_{|x|\rightarrow\infty}u(x)=0$ and $\phi$ (can be assumed to be smooth enough) is the obstalce which is strictly concave (in an open set strictly containing coincidence set)and compactly supported in $\mathbb{R}^n$ .
Can we say the coincidence set $\{u=\phi\}$ is connected? Or weakly, can we say every component of coincidence set is nonempty ? Or is there any assumption that can guarantee the coincidence set is connected?