Discontinuous free boundary problems and vortex theory (Q1315154)

We consider an open, bounded domain \(\Omega\) of \(\mathbb{R}^ n\) with smooth boundary \(\partial \Omega\), and let \(p \in C (\mathbb{R}^ +,\mathbb{R}^ +)\) be given. We assume that \(p\) is bounded and nondecreasing, a hypothesis that covers several important applications. However, more general situations can be handled with suitable changes. Given a smooth \(q_ 0>0\), we look for an open subset \(A\) of \(\Omega\) and a function \(v \in C^ 2(\Omega \backslash \partial A) \cap C^ 1 (\Omega)\), with \(v>0\) in \(A\), satisfying: \[ -\Delta v = p(v) \text{ in } A,\quad -\Delta v=0 \text{ on } \Omega \backslash \bar A, \] \[ v=0 \text{ on } \partial A\quad \text{ and }\quad v= -q_ 0 \text{ on } \partial \Omega. \]