Singular Trudinger--Moser inequality involving $L^{p}$ norm in bounded domain

Abstract

In this paper, we use the method of blow-up analysis and capacity estimate to derive the singular Trudinger--Moser inequality involving $N$-Finsler--Laplacian and $L^{p}$ norm, precisely, for any $p>1$, $0\leq\gamma<\gamma_{1}:= \inf\limits_{u\in W^{1, N}_{0}(\Omega)\backslash \{0\}}\frac{\int_{\Omega}F^{N}(\nabla u)dx}{\| u\|_p^N}$ and $0\leq\beta<N$, we have \begin{align} \sup_{u\in W_{0}^{1,N}(\Omega),\;\int_{\Omega}F^{N}(\nabla u)dx-\gamma\| u\|_p^N\leq1}\int_{\Omega}\frac{e^{\lambda_{N}(1-\frac{\beta}{N})\lvert u\rvert^{\frac{N}{N-1}}}}{F^{o}(x)^{\beta}}\;\mathrm{d}x<+\infty\notag, \end{align} where $\lambda_{N}=N^{\frac{N}{N-1}} \kappa_{N}^{\frac{1}{N-1}}$ and $\kappa_{N}$ is the volume of a unit Wulff ball in $\mathbb{R}^N$, moreover, extremal functions for the inequality are also obtained. When $F=\lvert\cdot\rvert$ and $p=N$, we can obtain the singular version of Tintarev type inequality by the obove inequality, namely, for any $0\leq\alpha<\alpha_{1}(\Omega):=\inf\limits_{u\in W^{1, N}_{0}(\Omega)\backslash \{0\}}\frac{\int_{\Omega}|\nabla u|^Ndx}{\| u\|_N^N}$ and $0\leq\beta<N$, it holds $$ \sup_{u\in W_{0}^{1,N}(\Omega),\;\int_{\Omega}\lvert\nabla u\rvert^{N}\;\mathrm{d}x-\alpha\|u\|_{N}^{N}\leq1}\int_{\Omega}\frac{e^{\alpha_{N}(1-\frac{\beta}{N})\lvert u\rvert^{\frac{N}{N-1}}}}{\lvert x\rvert^{\beta}}\;\mathrm{d}x<+\infty, $$ where $\alpha_{N}:=N^{\frac{N}{N-1}}\omega_{N}^{\frac{1}{N-1}}$ and $ \omega_{N}$ is the volume of unit ball in $\mathbb{R}^{N}$. Our results extend many well-known Trudinger--Moser type inequalities to more general setting.