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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Two dimensional elliptic equation
with critical nonlinear growth


Authors: Takayoshi Ogawa and Takashi Suzuki
Journal: Trans. Amer. Math. Soc. 350 (1998), 4897-4918
MSC (1991): Primary 35J60, 35P30, 35J20
DOI: https://doi.org/10.1090/S0002-9947-98-02269-7
MathSciNet review: 1641254
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the asymptotic behavior of solutions to a semilinear elliptic equation associated with the critical nonlinear growth in two dimensions.

\begin{equation*}\left\{ \begin{array}{cc} -\Delta u= \lambda ue^{u^2}, \ u>0 & \ \text{in} \ \ \Omega , \\ u=0 & \ \text{on} \ \ \partial \Omega , \end{array} \right. \tag{1.1} \end{equation*}

where $\Omega$ is a unit disk in $\mathbb{R}^2$ and $\lambda$ denotes a positive parameter. We show that for a radially symmetric solution of (1.1) satisfies

\begin{equation*}\int _{D}\left\vert\nabla u\right\vert^{2}dx\rightarrow 4\pi,\quad\lambda \searrow 0. \end{equation*}

Moreover, by using the Pohozaev identity to the rescaled equation, we show that for any finite energy radially symmetric solutions to (1.1), there is a rescaled asymptotics such as

\begin{equation*}u_m^2(\gamma _m x)-u_m^2 (\gamma _m)\to 2\log\frac{2}{1+|x|^2} \quad\text{as }\lambda _m\searrow 0 \end{equation*}

locally uniformly in $x\in\mathbb R^2$. We also show some extensions of the above results for general two dimensional domains.


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Additional Information

Takayoshi Ogawa
Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106
Address at time of publication: Graduate School of Mathematics, Kyushu University 36, Fukuoka, 812-8581, Japan
Email: ogawa@math.kyushu-u.ac.jp

Takashi Suzuki
Affiliation: Department of Mathematics, Osaka University, Toyonaka, Osaka 560, Japan
Email: takashi@math.sci.osaka-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-98-02269-7
Received by editor(s): January 29, 1996
Additional Notes: The first author is on long-term leave from the Graduate School of Polymathematics, Nagoya University, Nagoya 464-01 Japan.
Article copyright: © Copyright 1998 American Mathematical Society