Semilinear dirichlet problem with nearly critical exponent, asymptotic location of hot spots

Abstract

We study asymptotic properties of the positive solutions of $$\begin{array}{*{20}c} {\Delta u + u^{p - 1} = 0 in \Omega ,} \\ { u = 0 on \partial \Omega } \\ \end{array} $$ as the exponent tends to the critical Sobolev exponent. Brezis and Peletier conjectured that in every dimensionn ≥ 3 the maximum points of these solutions accumulate at a critical point of the Robin function. This has been confirmed by Rey and Han independently. A similar result in two dimensions has been obtained by Ren and Wei. In this paper we restrict our attention to solutions obtained as extremals of a suitable variational problem related to the best Sobolev constant. Our main result says that the maximum points of these solutions accumulate at a minimum point of the Robin function. This additional information is not accessible by the methods of Rey or Han. We present a variational approach that covers all dimensionsn ≥ 2 in a unified way.