A Liouville type theorem for a variational problem with free boundary in three dimensions

We consider the minimum problem for the functional EΩ(u)=∫Ω(|Du|2+λ2χ{u>0})EΩ(u)=∫Ω(|Du|2+λ2χ{u>0}) in three dimensional space, where λ>0λ>0 is a constant. We will establish a Liouville type theorem for this variational problem: if u∈C(R3)u∈C(R3) is a nonnegative and nonzero global minimizer, then u(x)=λ((x−x0)⋅ν)+u(x)=λ((x−x0)⋅ν)+ for some point x0x0 and some unit vector νν.