A dichotomy result for locally compact sober dcpos

The second author proved in [7] that each cartesian closed category
of pointed domains and Scott-continuous functions is contained in either
the category of Lawson-compact domains or that of L-domains, and this
result eventually led to a classification of continuous domains with respect
to cartesian closedness, as laid out in [8].
In this paper, we generalise this result to the category LcS of pointed
locally compact sober dcpos and Scott-continuous functions, and show
that any cartesian closed full subcategory of LcS is contained in either
the category of stably compact dcpos or that of L-dcpos. (Note that
for domains Lawson-compactness and stable compactness are equivalent.)
As we will show, this entails that any candidate for solving the Jung-Tix
problem in LcS must be stably compact.
To prove our dichotomy result, we first show that any dcpo with a core-
compact function space must be meet-continuous; then we prove that a
function space in LcS is meet-continuous only if either its input dcpo is
coherent or its output dcpo has complete principal ideals.