  We deal with existence and non-existence of non-negative entire solutions
that blow-up at infinity for a quasilinear problem depending on a non-negative
real parameter. Our main objectives in this paper are to provide far more
general conditions for existence and non-existence of solutions. To this end,
we explore an associated $\mu$-parameter convective ground state problem, sub
and super solutions method combined and an approximation arguments to show
existence of solutions. To show the result of non-existence of solutions, we
follow an idea due to Mitidieri-Pohozaev.
