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∈[0,1)β∈[0,1) and w0(x)=|log|x||βw0(x)=|log|x||β then sup∫B∣∇u∣2w0≤1,u∈H0,rad1(w0,B)∫Beαu21−βdx<∞, if and only if α≤αβ=2[2π(1−β)]11−β. We prove the existence of an extremal function for the above inequality for the critical case when α=αβα=αβ thereby generalizing the result of Carleson–Chang who proved the case when β=0β=0.