Remarks on singular trudinger-moser type inequalities

Let $ \Omega\subset\mathbb{R}^n $ be a bounded domain. Let $ F: \mathbb{R}^n\rightarrow[0, +\infty) $ be a convex function of class $ C^2(\mathbb{R}^n\setminus\{0\}) $, which is even and positively homogeneous of degree $ 1 $. For such a function $ F $, there exist two positive constants $ a_1\leq a_2 $ such that $ a_1|\xi|\leq F(\xi)\leq a_2|\xi|\; (\forall\xi\in\mathbb{R}^n) $. Therefore, $ (\int_\Omega F(\nabla u)^n dx)^{1/n} $ and $ (\int_{\mathbb{R}^n}(F(\nabla u)^n+\tau |u|^n)dx)^{1/n} $ $ (\tau>0) $ are equivalent with the standard norms on $ W^{1, n}_0(\Omega) $ and $ W^{1, n}(\mathbb{R}^n) $ respectively. In this paper, we prove that

$ \begin{align*} \sup\limits_{u\in W^{1, n}_0(\Omega), \int_\Omega F(\nabla u)^n dx\leq1}\int_\Omega \frac{e^{\lambda|u|^{\frac{n}{n-1}}}}{F^0(x)^{\beta}}dx<+\infty \Leftrightarrow\frac{\lambda}{\lambda_n}+\frac{\beta}{n}\leq1 \end{align*} $

and

$ \begin{align*} \sup\limits_{u\in W^{1, n}(\mathbb{R}^n), \int_{\mathbb{R}^n}(F(\nabla u)^n+\tau |u|^n)dx\leq1}\int_{\mathbb{R}^n}\frac{e^{\lambda|u|^{\frac{n}{n-1}}}-\sum_{k = 0}^{n-2}\frac{\lambda^k|u|^{\frac{nk}{n-1}}}{k!}}{F^0(x)^{\beta}}dx<\infty\nonumber\ \\ \Leftrightarrow\frac{\lambda}{\lambda_n}+\frac{\beta}{n}\leq1, \end{align*} $

where $ F^0 $ is the polar function of $ F $, $ \lambda>0 $, $ \beta\in[0, n) $, $ \tau>0 $, $ \lambda_n = n^{\frac{n}{n-1}}\kappa_n^{\frac{1}{n-1}} $ and $ \kappa_n $ is the volume of the unit Wulff ball. Extremal functions for above two supremums are also considered.