Layer profiles of solutions to elliptic problems under parameter-dependent boundary conditions (Q606471)

Summary: We consider the unique positive solution to the equation \(\Delta u = u^{r}\) in \(\Omega \), where \(r > 1\) and \(\Omega \) is a smooth bounded domain of \(\mathbb R^N\), under one of the boundary conditions \(u = \lambda\), \(\frac {\partial u}{\partial \nu} = \lambda\), \(\frac{\partial u}{\partial \nu} = \lambda u\) or \(\frac {\partial u}{\partial \nu } = \lambda u - u^{q}\) on \(\partial \Omega \), \(q > 1\). The main interest is determining the exact layer behavior of this solution near \(\partial \Omega \) in terms of the parameter \(\lambda \) as \(\lambda \rightarrow \infty \). Our analysis is completed with the study of the same type of problems involving the \(p\)-Laplacian operator.