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}.
Categories: math.AP