We study qualitative positivity properties of quasilinear equations of the
form
where
is a
domain in
,
,
is a symmetric and locally uniformly positive definite matrix,
is a real potential in a certain local Morrey space (depending on
),
and
Our assumptions on the coefficients of the operator for
are the minimal (in the Morrey scale) that ensure the validity of the local
Harnack inequality and hence the Hölder continuity of the solutions. For
some of the results of the paper we need slightly stronger assumptions when
.
We prove an Allegretto–Piepenbrink-type theorem for the operator
,
and extend criticality theory to our setting. Moreover, we establish a
Liouville-type theorem and obtain some perturbation results. Also, in the case
, we
examine the behaviour of a positive solution near a nonremovable isolated singularity
and characterize the existence of the positive minimal Green function for the operator
in
.
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