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

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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.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051, Basel, Switzerland

    Flucher Martin

  2. Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    Wei Juncheng

Authors
  1. Flucher Martin
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  2. Wei Juncheng
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Martin, F., Juncheng, W. Semilinear dirichlet problem with nearly critical exponent, asymptotic location of hot spots. Manuscripta Math 94, 337–346 (1997). https://doi.org/10.1007/BF02677858

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  • DOI: https://doi.org/10.1007/BF02677858

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