A free boundary problem involving limiting Sobolev exponents (Q1081107)

We present existence and nonexistence results for the variational inequality: \[ 0\neq u\geq \psi ^ +,\quad \int _{\Omega}\nabla u\nabla (v-u)\geq \int _{\Omega}u^{2^ *-1}(v-u),\quad v\in H^ 1_ 0(\Omega),\quad v\geq \psi \quad a.e.\quad in\quad \Omega \] where \(\Omega\) is a smooth bounded domain in \({\mathbb{R}}^ N\), \(\psi \in H^{2,s}(\Omega)\), \(s>N\), \(\psi <0\) on \(\partial \Omega\), and \(2^ *=2N/(N-2)\). In particular, we show there are no solutions if \(\psi\) is too large, while there are two distinct solutions \(0<\underline u\leq \bar u\) if \(\psi\) is positive somewhere and sufficiently small.