For a wide class of nonlinearities
f
(
u
)
f(u)
satisfying
\[
f
(
0
)
=
f
(
a
)
=
0
,
f
(
u
)
>
0
in
(
0
,
a
)
and
f
(
u
)
>
0
in
(
a
,
∞
)
,
\mbox { $f(0)=f(a)=0$, $f(u)>0$ in $(0,a)$ and $f(u)>0$ in $(a,\infty )$,}
\]
we show that any nonnegative solution of the quasilinear equation
−
Δ
p
u
=
f
(
u
)
-\Delta _p u= f(u)
over the entire
R
N
\mathbb {R}^N
must be a constant. Our results improve or complement some recently obtained Liouville type theorems. In particular, we completely answer a question left open by Du and Guo.