Geodesic
Nonexistence of Maxima for Perturbations of Some Inequalities with Critical Growth
Canadian mathematical bulletin, Tome 39 (1996) no. 2, pp. 227-237

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-1996-029-1
Mots-clés : 49J45, 28A20, 26A46, 30A10, nonexistence of extremals, lack of upper semicontinuity, nonlinear functional, convergence in measure, Moser-Trudinger inequality, Chang-Marshall inequality, Dirichlet space, Dirichlet integral, optimization problems 
Pruss, Alexander R. Nonexistence of Maxima for Perturbations of Some Inequalities with Critical Growth. Canadian mathematical bulletin, Tome 39 (1996) no. 2, pp. 227-237. doi: 10.4153/CMB-1996-029-1
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