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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 | References | Similar Articles | Additional Information

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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  • 1. Adimurthi and M. Struwe, Global compactness properties of semilinear elliptic equations with critical exponential growth, Journal of Functional Analysis 175 (2000), 125-167. MR 2001g:35063
  • 2. H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of $ -\Delta u = V(x) e^u $ in two dimensions, Comm. Partial Differential Equations 16 (1991), 1223-1253. MR 92m:35084
  • 3. W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615-622. MR 93e:35009
  • 4. G. Chen, W. M. Ni, and J. Zhou, Algorithms and visualization for solutions of nonlinear elliptic equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 10 (2000), 1565-1612.
  • 5. K. El Medhi and M. Grossi, Asymptotic estimates and qualitative properties of an elliptic problem in dimension two, preprint.
  • 6. M. Flucher and J. Wei, Semilinear Dirichlet problem with nearly critical exponent, asymptotic location of hot spots, Manuscripta Math. 94 (1997), 337-346. MR 99b:35066
  • 7. D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Second edition, Grundlehren der mathematischen Wissenschaften, Band 224, Springer-Verlag, Berlin, 1983. MR 86c:35035
  • 8. X. Ren and J. Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc. 343 (1994), 749-763. MR 94h:35074
  • 9. X. Ren and J. Wei, Single-point condensation and least-energy solutions, Proc. Amer. Math. Soc. 124 (1996), 111-120. MR 96d:35037
  • 10. R. Schoen, On the number of constant scalar curvature metrics in a conformal class, ``Differential geometry: A symposium in honor of Manfredo Do Carmo" (H. B. Lawson and K. Tenenblat, eds.), Longman Sci. Tech., Harlow, 1991, pp. 311-320. MR 94e:53035

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Additional Information

Affiliation: T.I.F.R. Centre, P.O. Box 1234, Bangalore 560012, India

Massimo Grossi
Affiliation: Università di Roma “La Sapienza", P.le Aldo Moro, 2, 00185 Roma, Italy

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

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