Extremal function for Moser–Trudinger type inequality with logarithmic weight

Abstract

On the space of weighted radial Sobolev space, the following generalization of MoserTrudinger type inequality was established by Calanchi and Ruf in dimension 2 : If β ∈ [0, 1) and w0(x) = | log |x||

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

“…This follows from P. L. Lions concentration compactness lemma (Lemma 2.1). The method of the proof of our main result (Theorem 1.3) follows similar idea as it is done in [6] and [28]. In Lemma 2.1, we show that a maximizing sequence can loose compactness only if it concentrates at the point x = 0.…”

“…The two dimensional version of the above theorem was first established in [5]. For n = 2 and w 0 = | log |x|| β(n−1) , the issue of existence of extremal function is addressed in [28]. Unfortunately, there is a mistake in [28].…”

“…For n = 2 and w 0 = | log |x|| β(n−1) , the issue of existence of extremal function is addressed in [28]. Unfortunately, there is a mistake in [28]. The main theorem is true for β in some small positive neighborhood of 0, after a slight modification.…”

Anisotropic Moser–Trudinger-Type Inequality with Logarithmic Weight

“…Note that when β = 0, by the Pólya-Szegö principle, (2) recovers the classical Moser-Trudinger inequality (1). Furthermore, Roy [20] proved the existence of an extremal function for inequality (2).…”

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.…”