Layer Profiles of Solutions to Elliptic Problems under Parameter-Dependent Boundary Conditions

Abstract

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 $R^N$, under one of the boundary conditions $u_{=}\lambda$, $\frac{\partial u}{\partial \nu }=\lambda$, $\frac{\partial u}{\partial \nu }=\lambda u$ or $\frac{\partial u}{\partial \nu }=\lambda u–uq$ 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 $\lambda \to \infty$. Our analysis is completed with the study of the same type of problems involving the $p$-Laplacian operator.