Affiliation:
1. Department of Mathematics , The University of British Columbia , Vancouver , BC, V6T 1Z2 , Canada
2. Departamento de Matemática , Universidade Federal de Juiz de Fora , Juiz de Fora , Brazil
Abstract
Abstract
We consider the Hardy–Schrödinger operator
L
γ
:=
-
Δ
𝔹
n
-
γ
V
2
{L_{\gamma}:=-\Delta_{\mathbb{B}^{n}}-\gamma{V_{2}}}
on the Poincaré ball model of the hyperbolic space
𝔹
n
{\mathbb{B}^{n}}
(
n
≥
3
{n\geq 3}
). Here
V
2
{V_{2}}
is a radially symmetric potential, which behaves like the Hardy potential around its singularity at 0, i.e.,
V
2
(
r
)
∼
1
r
2
{V_{2}(r)\sim\frac{1}{r^{2}}}
.
As in the Euclidean setting,
L
γ
{L_{\gamma}}
is positive definite whenever
γ
<
(
n
-
2
)
2
4
{\gamma<\frac{(n-2)^{2}}{4}}
, in which case we exhibit explicit solutions for the critical equation
L
γ
u
=
V
2
*
(
s
)
u
2
*
(
s
)
-
1
{L_{\gamma}u=V_{2^{*}(s)}u^{2^{*}(s)-1}}
in
𝔹
n
,
{\mathbb{B}^{n},}
where
0
≤
s
<
2
{0\leq s<2}
,
2
*
(
s
)
=
2
(
n
-
s
)
n
-
2
{2^{*}(s)=\frac{2(n-s)}{n-2}}
, and
V
2
*
(
s
)
{V_{2^{*}(s)}}
is a weight that behaves like
1
r
s
{\frac{1}{r^{s}}}
around 0. In dimensions
n
≥
5
{n\geq 5}
, the equation
L
γ
u
-
λ
u
=
V
2
*
(
s
)
u
2
*
(
s
)
-
1
{L_{\gamma}u-\lambda u=V_{2^{*}(s)}u^{2^{*}(s)-1}}
in a domain Ω of
𝔹
n
{\mathbb{B}^{n}}
away from the boundary but containing 0 has a ground state solution, whenever
0
<
γ
≤
n
(
n
-
4
)
4
{0<\gamma\leq\frac{n(n-4)}{4}}
, and
λ
>
n
-
2
n
-
4
(
n
(
n
-
4
)
4
-
γ
)
{\lambda>\frac{n-2}{n-4}(\frac{n(n-4)}{4}-\gamma)}
. On the other hand,
in dimensions 3 and 4, the existence of solutions depends on whether the domain has a positive “hyperbolic mass”
a notion that we introduce and analyze therein.
Funder
Natural Sciences and Engineering Research Council of Canada
Subject
General Mathematics,Statistical and Nonlinear Physics
Cited by
3 articles.
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