# Extremal function for Moser-Trudinger type Inequality with Logarithmic
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Research paper by **Prosenjit Roy**

Indexed on: **15 Feb '16**Published on: **15 Feb '16**Published in: **Mathematics - Analysis of PDEs**

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#### Abstract

On the space of weighted radial Sobolev space, the following generalization
of Moser-Trudinger type inequality was established by Calanchi and Ruf in
dimension 2 : If $\beta \in [0,1)$ and $w_0(x) = |\log |x||^\beta $ then $$
\sup_{\int_B |\grad u|^2w_0 \leq 1 , u \in H_{0,rad}^1(w_0,B)} \int_B e^{\alpha
u^{\frac{2}{1-\beta}}} dx < \infty,$$ if and only if $\alpha \leq \alpha_\beta
= 2\left[2\pi (1-\beta) \right]^{\frac{1}{1-\beta}}.$ We prove the existence of
an extremal function for the above inequality for the critical case when
$\alpha = \alpha_\beta$ thereby generalizing the result of Carleson-Chang who
proved the case when $\beta =0$.