Desigualdades críticas e subcríticas de Trudinger-Moser e Adams em RN

Abstract

In this dissertation, we present a detailed analysis of the best constants for the Trudinger-Moser and Adams inequalities through a new approach to the critical and subcritical inequalities. Initially, we introduce Trudinger-Moser-type inequalities for a class of Sobolev spaces modeled on weighted Lebesgue spaces. These inequalities are subsequently extended to fractional dimensions. Furthermore, we establish asymptotic bounds for the subcritical supremum, which are used to remarkably demonstrate the equivalence between the critical and subcritical Trudinger-Moser and Adams inequalities. Based on this equivalence, we provide a proof of the existence of extremal functions for both the subcritical and critical extremal problems. As a consequence, we are able to explicitly calculate the value of the critical supremum in some special cases.