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*}](http://www.ams.org/tran/1998-350-12/S0002-9947-98-02269-7/gif-abstract/img1.gif)
where is a unit disk in and denotes a positive parameter. We show that for a radially symmetric solution of (1.1) satisfies ![begin{equation*}int _{D}leftvertnabla urightvert^{2}dxrightarrow 4pi,quadlambda searrow 0. end{equation*}](http://www.ams.org/tran/1998-350-12/S0002-9947-98-02269-7/gif-abstract/img5.gif)
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 2logfrac{2}{1+|x|^2} quadtext{as }lambda _msearrow 0 end{equation*}](http://www.ams.org/tran/1998-350-12/S0002-9947-98-02269-7/gif-abstract/img6.gif)
locally uniformly in . We also show some extensions of the above results for general two dimensional domains. |