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 \begin{equation*} J(u) =\int _\Omega |\nabla u|^2 \quad \text {constrained on the manifold }\int _\Omega |u|^{p+1}=1\end{equation*} as $p$ goes to infinity. Here $\Omega$ is a smooth bounded domain of $\mathbb {R}^2$. Among other results we give a positive answer to a question raised by Chen, Ni, and Zhou (2000) by showing that $\lim \limits _{p\rightarrow \infty }||u_{p}||_{\infty }=\sqrt e$.