Extremal function for Moser-Trudinger type Inequality with Logarithmic weight

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 $w_0(x)=\mathrm{\mid }\log \mathrm{\mid }x\mathrm{\mid }{\mathrm{\mid }}^{\beta }$ then $$\underset{{\int }_B\mathrm{\mid }\text{grad}u{\mathrm{\mid }}^2{w}_0\le 1,u\in H_{0,rad}^1(w_0,B)}{\sup }{\int }_B{e}^{\alpha u^{\frac{2}{1-\beta }}}dx<\mathrm{\infty },$$ 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 $\beta =0$.