Two dimensional elliptic equation with critical nonlinear growth期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Two dimensional elliptic equation with critical nonlinear growth

Authors:	Takayoshi Ogawa Takashi Suzuki

Affiliation:	Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106 ; Department of Mathematics, Osaka University, Toyonaka, Osaka 560, Japan

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 is a unit disk in $mathbb{R}^2$ and denotes a positive parameter. We show that for a radially symmetric solution of (1.1) satisfies $begin{equation}int _{D}leftvertnabla urightvert^{2}dxrightarrow 4pi,quadlambda searrow 0. end{equation}$ 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 $begin{equation}u_m^2(gamma _m x)-u_m^2 (gamma _m)to 2logfrac{2}{1+\|x\|^2} quadtext{as }lambda _msearrow 0 end{equation}$ locally uniformly in . We also show some extensions of the above results for general two dimensional domains.

Keywords:

	点击此处可从《Transactions of the American Mathematical Society》浏览原始摘要信息
	点击此处可从《Transactions of the American Mathematical Society》下载全文