Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity

Abstract

In this paper we give asymptotic estimates of the least energy solution u_p of the functional J(u) =∫_Ω|∇u|² constrained on the manifold ∫_Ω|u|^p+1= 1 as p goes to infinity. Here Ω is a smooth bounded domain of R². Among other results we give a positive answer to a question raised by Chen, Ni, and Zhou (2000) by showing that lim_p→∞||u_p||_∞=√e.