Asymptotic estimates for the least energy solution of a planar semi-linear Neumann problem

In this work we study the asymptotic behavior of the L∞L∞ norm of the least energy solution upup of the following semi-linear Neumman problem{Δu=u,u>0in Ω,∂u∂ν=upon ∂Ω, where Ω is a smooth bounded domain in R2R2. Our main result shows that the L∞L∞ norm of upup remains bounded, and bounded away from zero as p goes to infinity, more precisely, we prove thatlimp→∞⁡‖u‖L∞(∂Ω)=e.