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

Abstract

We consider the unique positive solution to the equation
Δ_u_ = ur in Ω, where r > 1 and Ω is a smooth bounded domain of ℝ_N_, under one of the boundary conditions u = λ, ∂u/∂ν = λ, ∂u/∂ν = λ_u_ or ∂u/∂ν = λ_u_ – uq on ∂Ω, q > 1. The main interest is determining the exact layer behavior of this solution near ∂Ω in terms of the parameter  λ as λ → ∞. Our analysis is completed with the study of the same type of problems involving the p-Laplacian operator.