A Trudinger-Moser inequality for conical metric in the unit ball

In this note, we prove a Trudinger-Moser inequality for conical metric in the unit ball. Precisely, let $\mathbb{B}$ be the unit ball in $\mathbb{R}^N$ $(N\geq 2)$, $p>1$, $g=|x|^{\frac{2p}{N}\beta}(dx_1^2+\cdots+dx_N^2)$ be a conical metric on $\mathbb{B}$, and $\lambda_p(\mathbb{B})=\inf\left\{\int_\mathbb{B}|\nabla u|^Ndx: u\in W_0^{1,N}(\mathbb{B}),\,\int_\mathbb{B}|u|^pdx=1\right\}$. We prove that for any $\beta\geq 0$ and $\alpha<(1+\frac{p}{N}\beta)^{N-1+\frac{N}{p}}\lambda_p(\mathbb{B})$, there exists a constant $C$ such that for all radially symmetric functions $u\in W_0^{1,N}(\mathbb{B})$ with $\int_\mathbb{B}|\nabla u|^Ndx-\alpha(\int_\mathbb{B}|u|^p|x|^{p\beta}dx)^{N/p}\leq 1$, there holds $$\int_\mathbb{B}e^{\alpha_N(1+\frac{p}{N}\beta)|u|^{\frac{N}{N-1}}}|x|^{p\beta}dx\leq C,$$ where $|x|^{p\beta}dx=dv_g$, $\alpha_N=N\omega_{N-1}^{1/(N-1)}$, $\omega_{N-1}$ is the area of the unit sphere in $\mathbb{R}^N$; moreover, extremal functions for such inequalities exist. The case $p=N$, $-1<\beta<0$ and $\alpha=0$ was considered by Adimurthi-Sandeep \cite{A-S}, while the case $p=N=2$, $\beta\geq 0$ and $\alpha=0$ was studied by de Figueiredo-do \'O-dos Santos \cite{F-do-dos}.