Nonexistence of Maxima for Perturbations of Some Inequalities with Critical Growth

Pruss, Alexander R.

Canadian mathematical bulletin, Tome 39 (1996) no. 2, pp. 227-237

Résumé

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.

Détail

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