  We present several Liouville type results for the $p$-Laplacian in $\R^N$.
Suppose that
  $h$ is a nonnegative regular function such that $$ h(x) = a|x|^\gamma\ {\rm
for}\ |x|\ {\rm large},\ a>0\ {\rm and}\ \gamma> -p. $$ We obtain the following
non -existence result:
  1) Suppose that $N>p>1$, and $u\in W^{1,p}_{loc} (\R^N)\cap {\cal C} (\R^N)$
is a nonnegative weak solution of $ - {\rm div} (|\nabla u|^{p-2 }\nabla u)
\geq h(x) u^q \;\;\mbox{in }\; \R^N $ . Suppose that $p-1< q\leq
{(N+\gamma)(p-1)\over N-p}$ then $u\equiv 0$.
  2) Let $N\leq p$. If $u\in W^{1,p}_{loc} (\R^N)\cap {\cal C} (\R^N)$ is a
weak solution bounded below of $-{\rm div} (|\nabla u|^{p-2 }\nabla u)\geq 0$
in $\R^N$ then $u$ is constant.
  3) Let $N>p$ if $u$ is bounded from below and $-{\rm div} (|\nabla u|^{p-2
}\nabla u)=0$ in $\R^N$ then $u$ is constant.
  4)If $ -\Delta_p u+h(x) u^q\leq 0, $. If $q> p-1$, then $u\equiv 0$.
