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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 $β\in [0,1)$ and $w_0(x) = |\log |x||^β$ then $$ \sup_{\int_B |\grad u|^2w_0 \leq 1 , u \in H_{0,rad}^1(w_0,B)} \int_B e^{αu^{\frac{2}{1-β}}} dx < \infty,$$ if and only if $α\leq α_β= 2\left[2π(1-β) \right]^{\frac{1}{1-β}}.$ We prove the existence of an extremal function for the above inequality for the critical case when $α= α_β$ thereby generalizing the result of Carleson-Chang who proved the case when $β=0$.