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

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 $p>1$, $0\le \gamma <{\gamma }_1:=\underset{u\in W_0^{1,N}(\mathrm{\Omega })\mathrm{\backslash }\{0\}}{\inf }\frac{{\int }_{\mathrm{\Omega }}{F}^N(\mathrm{\nabla }u)dx}{\mathrm{\parallel }u{\mathrm{\parallel }}_p^N}$ and $0\le \beta <N$, we have \begin{align} \sup_{u\in W_{0}^{1,N}(Ω),;\int_ΩF^{N}(\nabla u)dx-γ| u|p^N\leq1}\int_Ω\frac{e^{λ{N}(1-\fracβ{N})\lvert u\rvert^{\frac{N}{N-1}}}}{F^{o}(x)^β};$\mathrm{d}$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 $F=\mid \cdot \mid$ and $p=N$, we can obtain the singular version of Tintarev type inequality by the obove inequality, namely, for any $0\le \alpha <{\alpha }_1(\mathrm{\Omega }):=\underset{u\in W_0^{1,N}(\mathrm{\Omega })\mathrm{\backslash }\{0\}}{\inf }\frac{{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{N}dx}{\mathrm{\parallel }u{\mathrm{\parallel }}_N^N}$ and $0\le \beta <N$, it holds $$\underset{u\in W_0^{1,N}(\mathrm{\Omega }),{\int }_{\mathrm{\Omega }}\mid \mathrm{\nabla }u{\mid }^N\mathrm{d}x-\alpha \mathrm{\parallel }u{\mathrm{\parallel }}_N^N\le 1}{\sup }{\int }_{\mathrm{\Omega }}\frac{e^{{\alpha }_N(1-\frac{\beta }{N})\mid u{\mid }^{\frac{N}{N-1}}}}{\mid x{\mid }^{\beta }}\mathrm{d}x<+\mathrm{\infty },$$ 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.