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 $mathbb{R}^N$ $(Ngeq 2)$, $p>1$, $g=|x|^{frac{2p}{N}beta}(dx_1^2+cdots+dx_N^2)$ be a conical metric on $mathbb{B}$, and $lambda_p(mathbb{B})=infleft{int_mathbb{B}| abla u|^Ndx: uin W_0^{1,N}(mathbb{B}),,int_mathbb{B}|u|^pdx=1right}$. We prove that for any $betageq 0$ 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 $uin W_0^{1,N}(mathbb{B})$ with $int_mathbb{B}| abla u|^Ndx-alpha(int_mathbb{B}|u|^p|x|^{pbeta}dx)^{N/p}leq 1$, there holds $$int_mathbb{B}e^{alpha_N(1+frac{p}{N}beta)|u|^{frac{N}{N-1}}}|x|^{pbeta}dxleq C,$$ where $|x|^{pbeta}dx=dv_g$, $alpha_N=Nomega_{N-1}^{1/(N-1)}$, $omega_{N-1}$ 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 $p=N=2$, $betageq 0$ and $alpha=0$ was studied by de Figueiredo-do O-dos Santos cite{F-do-dos}.