Abstract
Abstract
We consider positive and spatially decaying solutions to the following Gross–Pitaevskii equation with a harmonic potential:
−
Δ
u
+
|
x
|
2
u
=
ω
u
+
|
u
|
p
−
2
u
in
R
d
,
where
d
⩾
3
, p > 2 and ω > 0. For
p
=
2
d
d
−
2
(energy-critical case) there exists a ground state u
ω
if and only if
ω
∈
(
ω
∗
,
d
)
, where
ω
∗
=
1
for d = 3 and
ω
∗
=
0
for
d
⩾
4
. We give a precise description on asymptotic behaviours of u
ω
as
ω
→
ω
∗
up to the leading order term for different values of
d
⩾
3
. When
p
>
2
d
d
−
2
(energy-supercritical case) there exists a singular solution
u
∞
for some
ω
∈
(
0
,
d
)
. We compute the Morse index of
u
∞
in the class of radial functions and show that the Morse index of
u
∞
is infinite in the oscillatory case, is equal to 1 or 2 in the monotone case for p not large enough and is equal to 1 in the monotone case for p sufficiently large.
Funder
National Natural Science Foundation of China
NSERC
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics