Semilinear dirichlet problem with nearly critical exponent, asymptotic location of hot spots |
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Authors: | Flucher Martin Wei Juncheng |
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Affiliation: | 1. Mathematisches Institut, Universit?t Basel, Rheinsprung 21, CH-4051, Basel, Switzerland 2. Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
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Abstract: | We study asymptotic properties of the positive solutions of 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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Keywords: | semilinear elliptic equation critical Sobolev exponent maximum point concentration |
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