Remarks on singular trudinger-moser type inequalities
Xiaobao Zhu · Communications on Pure & Applied Analysis · 2019
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( abla u)^n dx)^{1/n} $ and $ (\int_{\mathbb{R}^n}(F( abla 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{document}$ \begin{align*} \sup\limits_{u\in W^{1, n}_0(\Omega), \int_\Omega F( abla u)^n dx\leq1}\int_\Omega \frac{e^{\lambda|u|^{\frac{n}{n-1}}}}{F^0(x)^{\beta}}dx0 $, $ \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.