manuscripta mathematica

, Volume 94, Issue 1, pp 337–346

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

  • Flucher Martin
  • Wei Juncheng
Article

DOI: 10.1007/BF02677858

Cite this article as:
Martin, F. & Juncheng, W. Manuscripta Math (1997) 94: 337. doi:10.1007/BF02677858

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. Brézis 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.

Key words

semilinear elliptic equation critical Sobolev exponent maximum point concentration 

Mathematics Subject Classification (1991)

35J20 35B40 

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Flucher Martin
    • 1
  • Wei Juncheng
    • 2
  1. 1.Mathematisches InstitutUniversität BaselBaselSwitzerland
  2. 2.Department of MathematicsThe Chinese University of Hong KongHong Kong

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