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

• Yunyan Yang
• Xiaobao Zhu
Article

Abstract

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).

Keywords

Trudinger–Moser inequality Blow-up analysis Conical metric

35J15 46E35

Notes

Acknowledgements

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

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