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 $$ \sup_{\int_B |\grad u|^2w_0 \leq 1 , u \in H_{0,rad}^1(w_0,B)} \int_B e^{αu^{\frac{2}{1-β}}} dx < \infty,$$ if and only if $\alpha \le {\alpha }_{\beta }=2{\left[2\pi (1-\beta )\right]}^{\frac{1}{1-\beta }}.$ 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 $\beta =0$.

To appear in "Nonlinear Analysis- TMA"