Abstract
We study the nonlinear boundary value problem
in
,
on
, where
is a bounded domain in
with smooth boundary,
,
are
positive real numbers,
,
,
,
are
continuous functions on
,
and
are
weight functions in generalized Lebesgue spaces
and
, respectively, such that
in an open set
with
and
on
.
We prove, under appropriate conditions that for any
there
exists
sufficiently small such that for any
the above nonhomogeneous quasilinear problem
has a nontrivial positive weak solution. The proof relies on some
variational arguments based on Ekeland's variational principle.