Asymptotic behavior of least energy solutions for a 2D nonlinear Neumann problem with large exponent

Abstract

In this paper, we consider the following elliptic problem with the nonlinear Neumann boundary condition:(Ep){−Δu+u=0onΩ,u>0onΩ,∂u∂ν=upon∂Ω, where Ω is a smooth bounded domain in R2, ν is the outer unit normal vector to ∂Ω, and p>1 is any positive number.We study the asymptotic behavior of least energy solutions to (Ep) when the nonlinear exponent p gets large. Following the arguments of X. Ren and J.C. Wei [13,14], we show that the least energy solutions remain bounded uniformly in p, and it develops one peak on the boundary, the location of which is controlled by the Green function associated to the linear problem.