Abstract
We investigate the existence and nonexistence of nonnegative radial solutions to exterior problems of the form
in
, under the Dirichlet boundary conditions
on
and
. Here,
is a parameter,
is the Kohn Laplacian on the Heisenberg group
,
,
,
is the unit ball in
,
is the complement of
, and
. Namely, under certain conditions on
and
, we show that there exists a critical parameter
in the following sense. If
, the above problem admits a unique nonnegative radial solution
; if
and
, the problem admits no nonnegative radial solution. When
, a numerical algorithm that converges to
is provided and the continuity of
with respect to
, as well as the behavior of
as
, are studied. Moreover, sufficient conditions on the the behavior of
as
are obtained, for which
or
. Our approach is based on partial ordering methods and fixed point theory in cones.