A Trudinger–Moser inequality for a conical metric in the unit ball
Tóm tắt
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).
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