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

Abstract

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 ${\mathbb{R}}^N$ $(N\ge 2)$, $p>1$, $g=|x{|}^{\frac{2p}{N}\beta }(d{x}_1^2+\cdots +d{x}_N^2)$ be a conical metric on $\mathbb{B}$, and ${\lambda }_p(\mathbb{B})=\inf \left\{{\int }_{\mathbb{B}}|\nabla u{|}^{N}dx:u\in W_0^{1,N}(\mathbb{B}),{\int }_{\mathbb{B}}|u{|}^{p}dx=1\right\}$. We prove that for any $\beta \ge 0$ 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 ${\int }_{\mathbb{B}}|\nabla u{|}^{N}dx-\alpha ({\int }_{\mathbb{B}}|u{|}^p|x{|}^{p\beta }dx{)}^{N/p}\le 1$, there holds $${\int }_{\mathbb{B}}{e}^{{\alpha }_N(1+\frac{p}{N}\beta )|u{|}^{\frac{N}{N-1}}}|x{|}^{p\beta }dx\le C,$$ where $|x{|}^{p\beta }dx=d{v}_g$, ${\alpha }_N=N{\omega }_{N-1}^{1/(N-1)}$, ${\omega }_{N-1}$ 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 $p=N=2$, $\beta \ge 0$ and $\alpha =0$ was studied by de Figueiredo-do Ó-dos Santos \cite{F-do-dos}.

12 pages