We study the nonlinear boundary value problem
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.