On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions

Abstract

Moser-Trudinger inequality was generalised by Calanchi-Ruf to the following version: If β ∈ [0, 1) and w 0 (x) = | log |x|| β(n−1) or log e |x| β(n−1)2010 Mathematics Subject Classification. Primary: 35B38, 35J20, 47N20, 26D10; Secondary: 46E35.

Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations

“…Nguyen proved the existence of a maximizer for this inequality when β is sufficiently small. The question of the attainability of the inequality (3) has been also considered by P. Roy in [39] for the case N = 2, and in [40] for higher dimensions.…”

“…Using the fact that the function s −→ Φ(s) = e s −1 s is increasing on [0, +∞[, from (40) it follows that…”

N- Laplacian problems with critical double exponential nonlinearities

“…N −1 and ω N −1 is the surface area of the unit ball in R N . Recently, the influence of weights on limiting inequalities of Trudinger-Moser type has been studied, for example, see [3,9,4,5,14,15]. If ω ∈ L 1 (Ω) is a non-negative function, we introduce the weighted Sobolev space…”