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Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity

Authors:	Adimurthi Massimo Grossi

Institution:	T.I.F.R. Centre, P.O. Box 1234, Bangalore 560012, India ; Università di Roma ``La Sapienza", P.le Aldo Moro, 2, 00185 Roma, Italy

Abstract:	In this paper we give asymptotic estimates of the least energy solution of the functional $\begin{displaymath}J(u) =\int_\Omega \vert\nabla u\vert^2 \quad\hbox{constrained on the manifold }\int _\Omega \vert u\vert^{p+1}=1\end{displaymath}$ as 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 $\lim\limits_{p\rightarrow\infty}\vert\vert u_{p}\vert\vert _{\infty}=\sqrt e$ .

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