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 $p>1$, $0\le {\hat{I}}^3<{\hat{I}}_1^3:=\underset{u\in W_0^{1,N}(\hat{I}\text{c◯})\backslash \{0\}}{\inf }\frac{{\int }_{\hat{I}}\text{c◯}{F}^N(\nabla u)dx}{\Vert u{\Vert }_p^N}$ and $0\le {\hat{I}}^2<N$, 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 $\hat{I}{»}_N=N^{\frac{N}{N-1}}\hat{I}{º}_N^{\frac{1}{N-1}}$ and $\hat{I}{º}_N$ is the volume of a unit Wulff ball in ${\mathbb{R}}^N$, moreover, extremal functions for the inequality are also obtained. When $F=|\cdot |$ and $p=N$, we can obtain the singular version of Tintarev type inequality by the obove inequality, namely, for any $0\le \hat{I}\pm <\hat{I}{\pm }_1(\hat{I}\text{c◯}):=\underset{u\in W_0^{1,N}(\hat{I}\text{c◯})\backslash \{0\}}{\inf }\frac{{\int }_{\hat{I}}\text{c◯}|\nabla u{|}^{N}dx}{\Vert u{\Vert }_N^N}$ and $0\le {\hat{I}}^2<N$, it holds $$\underset{u\in W_0^{1,N}(\hat{I}\text{c◯}),{\int }_{\hat{I}}\text{c◯}|\nabla u{|}^N\mathrm{d}x-\hat{I}\pm \Vert u{\Vert }_N^N\le 1}{\sup }{\int }_{\hat{I}}\text{c◯}\frac{e^{\hat{I}{\pm }_N(1-\frac{\hat{I}}{²}N)|u{|}^{\frac{N}{N-1}}}}{|x{|}^2}\mathrm{d}x<+\infty ,$$ where $\hat{I}{\pm }_N:=N^{\frac{N}{N-1}}\ddot{I}{‰}_N^{\frac{1}{N-1}}$ and $\ddot{I}{‰}_N$ is the volume of unit ball in ${\mathbb{R}}^N$. Our results extend many well-known Trudinger--Moser type inequalities to more general setting.