Abstract
We investigate the existence and nonexistence of nonnegative radial solutions to exterior problems of the form
Δ
H
m
u
(
q
)
+
λ
ψ
(
q
)
K
(
r
(
q
)
)
f
(
r
2
−
Q
(
q
)
,
u
(
q
)
)
=
0
{\Delta }_{{{\mathbb{H}}}^{m}}u\left(q)+\lambda \psi \left(q)K\left(r\left(q))f\left({r}^{2-Q}\left(q),u\left(q))=0
in
B
1
c
{B}_{1}^{c}
, under the Dirichlet boundary conditions
u
=
0
u=0
on
∂
B
1
\partial {B}_{1}
and
lim
r
(
q
)
→
∞
u
(
q
)
=
0
{\mathrm{lim}}_{r\left(q)\to \infty }u\left(q)=0
. Here,
λ
≥
0
\lambda \ge 0
is a parameter,
Δ
H
m
{\Delta }_{{{\mathbb{H}}}^{m}}
is the Kohn Laplacian on the Heisenberg group
H
m
=
R
2
m
+
1
{{\mathbb{H}}}^{m}={{\mathbb{R}}}^{2m+1}
,
m
>
1
m\gt 1
,
Q
=
2
m
+
2
Q=2m+2
,
B
1
{B}_{1}
is the unit ball in
H
m
{{\mathbb{H}}}^{m}
,
B
1
c
{B}_{1}^{c}
is the complement of
B
1
{B}_{1}
, and
ψ
(
q
)
=
∣
z
∣
2
r
2
(
q
)
\psi \left(q)=\frac{| z{| }^{2}}{{r}^{2}\left(q)}
. Namely, under certain conditions on
K
K
and
f
f
, we show that there exists a critical parameter
λ
∗
∈
(
0
,
∞
]
{\lambda }^{\ast }\in \left(0,\infty ]
in the following sense. If
0
≤
λ
<
λ
∗
0\le \lambda \lt {\lambda }^{\ast }
, the above problem admits a unique nonnegative radial solution
u
λ
{u}_{\lambda }
; if
λ
∗
<
∞
{\lambda }^{\ast }\lt \infty
and
λ
≥
λ
∗
\lambda \ge {\lambda }^{\ast }
, the problem admits no nonnegative radial solution. When
0
≤
λ
<
λ
∗
0\le \lambda \lt {\lambda }^{\ast }
, a numerical algorithm that converges to
u
λ
{u}_{\lambda }
is provided and the continuity of
u
λ
{u}_{\lambda }
with respect to
λ
\lambda
, as well as the behavior of
u
λ
{u}_{\lambda }
as
λ
→
λ
∗
−
\lambda \to {{\lambda }^{\ast }}^{-}
, are studied. Moreover, sufficient conditions on the the behavior of
f
(
t
,
s
)
f\left(t,s)
as
s
→
∞
s\to \infty
are obtained, for which
λ
∗
=
∞
{\lambda }^{\ast }=\infty
or
λ
∗
<
∞
{\lambda }^{\ast }\lt \infty
. Our approach is based on partial ordering methods and fixed point theory in cones.