Asymptotic Symmetry and Local Behavior of Semilinear Elliptic Equations with Critical Sobolev Growth
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چکیده
in a punctured ball, B,(O) \ (0) c R“, n 2 3, with an isolated singularity at the origin. The model equation (1.1) arises in many physical contexts but its greatest interest in recent years lies in its relation to the Yamabe problem. From this geometric point of view, we think of u as defining the conformally flat metric El, = u4/(n-2)6i, . Equation (1.1) then says that the metric has constant scalar curvature. The recent work of Schoen and Yau [8],[9],[10] on conformally flat manifolds and the Yamabe problem has highlighted the importance of studying the distribution solutions of (1.1) and characterizing the singular set of u. A solution u of (1.1) with an isolated singularity is the simplest example of a singular distribution solution. This deceptively simple looking problem is analytically very difficult and requires the development of a new technique which we may call an asymptotic symmetry method. It is a “measure theoretic” variation of the Alexandrov reflection technique as developed by Gidas, Ni and Nirenberg [4], [5 ] . Loosely speaking, the heuristic idea of the asymptotic symmetry technique may be described as follows. After an inversion, the function u becomes defined in the complement of B,, is strictly positive on JB,, and in some sense “goes to zero” at infinity. If we could extend u to B, as a super solution of our problem,
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تاریخ انتشار 2006