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ISSN 1088-6850(e) ISSN 0002-9947(p)

Two dimensional elliptic equation with critical nonlinear growth

Author(s): Takayoshi Ogawa; Takashi Suzuki
Journal: Trans. Amer. Math. Soc. 350 (1998), 4897-4918.
MSC (1991): Primary 35J60, 35P30, 35J20
MathSciNet review: 1641254
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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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