## Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity

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- by Adimurthi and Massimo Grossi
- Proc. Amer. Math. Soc.
**132**(2004), 1013-1019 - DOI: https://doi.org/10.1090/S0002-9939-03-07301-5
- Published electronically: November 10, 2003
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## 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 $\lim \limits _{p\rightarrow \infty }||u_{p}||_{\infty }=\sqrt e$.## References

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## Bibliographic 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
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**132**(2004), 1013-1019 - MSC (2000): Primary 35J20, 35B40
- DOI: https://doi.org/10.1090/S0002-9939-03-07301-5
- MathSciNet review: 2045416