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 u_p of the functional J ( u ) = ∫ Ω | ∇ u | 2 constrained on the manifold  ∫ Ω | u | p + 1 = 1 \begin{equation*} J(u) =\int _\Omega |\nabla u|^2 \quad \text {constrained on the manifold }\int _\Omega |u|^{p+1}=1\end{equation*} as p p goes to infinity. Here Ω \Omega is a smooth bounded domain of R 2 \mathbb {R}^2 . 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 \lim \limits _{p\rightarrow \infty }||u_{p}||_{\infty }=\sqrt e .