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The critical Sobolev exponent in two dimensions

Published online by Cambridge University Press:  14 November 2011

Bryce McLeod  and
Kevin McLeod
Bryce McLeod
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles, Oxford OX1 3LB, U.K.
Kevin McLeod
Affiliation:
Department of Mathematics, University of Indiana, Bloomington, Indiana, U.S.A.
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Synopsis

The object of the paper is to investigate solutions of equations of the form

with

and in particular to look at the asymptotic behaviour of these solutions as γ ↑∞. It is found that, if tγ is the first zero of ϑ, then

while tγ is bounded below if p < 2. This answers questions raised by Brezis concerning solutions of −Δu = eup in R2

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 109 , Issue 1-2 , 1988 , pp. 1 - 15
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

1Brezis, H. and Nirenberg, L.. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36 (1983), 437–477.CrossRefGoogle Scholar
2Gidas, B., Ni, W. M., and Nirenberg, L.. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), 209–243.CrossRefGoogle Scholar
3Atkinson, F. V. and Peletier, L. A.. Ground states and Dirichlet problems for –Δu =f(u) in R2. Arch. Rational Mech. Anal. 96 (1986), 147–165.CrossRefGoogle Scholar