Singularly perturbed nonlinear second order elliptic equation (Q1101603)

The author studies singular perturbation problems of some semilinear second order elliptic equations with nonlinear boundary value conditions \[ \epsilon^ 2\sum_{i,j}\partial /\partial \chi_ i[a_{ij}(\chi,\epsilon)\partial u/\partial \chi_ i- h(\chi,u,\epsilon)]=0\quad (\chi \in \Omega \subset R^ n); \] \[ \beta (\chi,u,\partial u/\partial \ell,\epsilon)=0\quad (\chi \in \partial \Omega) \] where \(\epsilon\) is a small positive parameter and \(\partial u/\partial \ell\) is a directional derivative, which lies on an oblique vector \({\vec \ell}(x,\epsilon)\). He gives a construction of the asymptotic solutions and a proof of their asymptotic correctness, which is based on the principle of contraction mapping.