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Nonexistence of Maxima for Perturbations of Some Inequalities with Critical Growth

Published online by Cambridge University Press:  20 November 2018

Alexander R. Pruss
Alexander R. Pruss*
Affiliation:
University of British Columbia, Vancouver, B.C. V6T 1Z2 e-mail:pruss@math.ubc.ca
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Abstract

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We study the question of nonexistence of extremal functions for perturbations of some sharp inequalities such as those of Moser-Trudinger (1971) and Chang- Marshall (1985). We shall show that for each critically sharp (in a sense that will be precisely defined) inequality of the form

where is a collection of measurable functions on a finite measure space (I, μ) and O a nonnegative continuous function on [0, ∞), we have a continuous Ψ on [0, ∞) with 0 ≤ Ψ ≤ Φ, but with

not being attained even if the supremum in (1) is attained. We then apply our results to the Moser-Trudinger and Chang-Marshall inequalities. Our result is to be contrasted with the fact shown by Matheson and Pruss (1994) that if Ψ(t) = o(Φ(t) as t —> ∞ then the supremum in (2) is attained. In the present paper, we also give a converse to that fact.

Keywords

49J4528A2026A4630A10nonexistence of extremalslack of upper semicontinuitynonlinear functionalconvergence in measureMoser-Trudinger inequalityChang-Marshall inequalityDirichlet spaceDirichlet integraloptimization problems
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 2 , 01 June 1996 , pp. 227 - 237
Copyright
Copyright © Canadian Mathematical Society 1996

References

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