Asymptotic behavior of two semi-linear elliptic free boundary problems (Q1084266)

Given a bounded open set \(\Omega \subseteq R^ n\) with \(C^{2+\alpha}\) boundary and a monotone increasing function f(t) with \(f(0)=0\), this paper treats two related exterior free boundary problems: Problem A: Given \(\lambda >0\), determine \(u\in C_ 0^{2+\alpha_ 1}(R^ n- \Omega)\) satisfying \[ \Delta u=\lambda f(u)\quad in\quad R^ n-{\bar \Omega};\quad u=1\quad on\quad \partial \Omega. \] Problem B: Given \(c>0\), determine \(v\in C_ 0^{2+\alpha_ 1}(R^ n-\Omega)\) satisfying \[ \Delta v=f(v)\quad in\quad R^ n-{\bar \Omega};\quad v=c\quad on\quad \partial \Omega. \] In both problems, the free boundary is the boundary of the support of the sought function.