  This paper concerns the existence of a solution for the following class of
semipositone quasilinear problems
  \begin{equation*}
  \left \{
  \begin{array}{rclcl}
  -\Delta_p u = h(x)(f(u)-a),\ &
  u > 0 & \mbox{in} & \mathbb{R}^N,
  \end{array}
  \right.
  \end{equation*} where $1<p<N$, $a>0$, $ f:[0,+\infty) \to [0,+\infty)$ is a
function with subcritical growth and $f(0)=0$, while $h:\mathbb{R}^N \to
(0,+\infty)$ is a continuous function that satisfies some technical conditions.
We prove via nonsmooth critical points theory and comparison principle, that a
solution exists for $a$ small enough. We also provide a version of Hopf's Lemma
and a Liouville-type result for the $p$-Laplacian in the whole $\mathbb{R}^N$.
