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

TL;DR

This paper establishes a Trudinger-Moser inequality for functions on the unit ball with a conical metric, extending previous results and proving the existence of extremal functions under certain conditions.

Contribution

It generalizes the Trudinger-Moser inequality to conical metrics in the unit ball for a range of parameters, including the existence of extremal functions.

Findings

Proved inequality for radially symmetric functions with conical metrics.

Identified conditions on parameters for the inequality to hold.

Established existence of extremal functions.

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 $\mathbb{R}^N$ $(N\geq 2)$, $p>1$, $g=|x|^{\frac{2p}{N}\beta}(dx_1^2+\cdots+dx_N^2)$ 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}.