Abstract
We consider the elliptic boundary blow-up problem
where is a smooth
bounded domain of ,
,
are positive continuous functions supported in disjoint subdomains
,
of
, respectively,
and
is a parameter. We show that there exists
such that no positive
solutions exist when ,
while a minimal positive solution exists for every
. Under the additional
hypotheses that and
intersect along a
smooth -dimensional
manifold
and ,
have a convenient
decay near ,
we show that a second positive solution exists for every
if
. Our
proofs are mainly based on continuation methods.