Extremal function for Moser-Trudinger type Inequality with Logarithmic weight

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 $\beta =0$.