A Trudinger–Moser inequality for a conical metric in the unit ball

  • Yunyan YangEmail author
  • Xiaobao Zhu


In this note, we prove a Trudinger–Moser inequality for a conical metric in the unit ball. Precisely, let \({\mathbb {B}}\) be the unit ball in \({\mathbb {R}}^N\) \((N\ge 2)\), \(p>1\), \(g=|x|^{\frac{2p}{N}\beta }(dx_1^2+\cdots +dx_N^2)\) be a conical metric on \({\mathbb {B}}\), and \(\lambda _p({\mathbb {B}})=\inf \left\{ \intop _{\mathbb {B}}|\nabla u|^Ndx: u\in W_0^{1,N}({\mathbb {B}}),\intop _{\mathbb {B}}|u|^pdx=1\right\} \). 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 \(\intop _{\mathbb {B}}|\nabla u|^Ndx-\alpha (\intop _{\mathbb {B}}|u|^p|x|^{p\beta }dx)^{N/p}\le 1\), there holds
$$\begin{aligned} \intop _{\mathbb {B}}e^{\alpha _N(1+\frac{p}{N}\beta )|u|^{\frac{N}{N-1}}}|x|^{p\beta }dx\le C, \end{aligned}$$
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 (Nonlinear Differ Equ Appl 13:585–603, 2007), while the case \(p=N=2\), \(\beta \ge 0\), and \(\alpha =0\), was studied by de Figueiredo (Proc Am Math Soc 144:3369–3380, 2016).


Trudinger–Moser inequality Blow-up analysis Conical metric 

Mathematics Subject Classification

35J15 46E35 


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This work is partly supported by the National Science Foundation of China (Grant Nos. 11471014, 11401575 and 11761131002).


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Authors and Affiliations

  1. 1.School of MathematicsRenmin University of ChinaBeijingPeople’s Republic of China

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