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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity


Authors: Adimurthi and Massimo Grossi
Journal: Proc. Amer. Math. Soc. 132 (2004), 1013-1019
MSC (2000): Primary 35J20, 35B40
Published electronically: November 10, 2003
MathSciNet review: 2045416
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Abstract: In this paper we give asymptotic estimates of the least energy solution $u_p$ 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 $p$ 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 $\lim\limits_{p\rightarrow\infty}\vert\vert u_{p}\vert\vert _{\infty}=\sqrt e$.


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Additional Information

Adimurthi
Affiliation: T.I.F.R. Centre, P.O. Box 1234, Bangalore 560012, India
Email: aditi@math.tifrbng.res.in

Massimo Grossi
Affiliation: Università di Roma “La Sapienza", P.le Aldo Moro, 2, 00185 Roma, Italy
Email: grossi@mat.uniroma1.it

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07301-5
PII: S 0002-9939(03)07301-5
Received by editor(s): September 7, 2002
Published electronically: November 10, 2003
Additional Notes: Supported by M.U.R.S.T., project “Variational methods and nonlinear differential equations"
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2003 American Mathematical Society