  In this note, we study Liouville theorems for the stable and finite Morse
index weak solutions of the quasilinear elliptic equation $-\Delta_p u= f(x)
F(u) $ in $\mathbb{R}^n$ where $p\ge 2$, $0\le f\in C(\mathbb{R}^n)$ and $F\in
C^1(\mathbb{R})$. We refer to $f(x)$ as {\it weight} and to $F(u)$ as {\it
nonlinearity}. The remarkable fact is that if the weight function is bounded
from below by a strict positive constant that is $0<C\le f$ then it does not
have much impact on the stable solutions, however, a nonnegative weight that is
$0\le f$ will push certain critical dimensions. This analytical observation has
potential to be applied in various models to push certain well-known critical
dimensions.
  For a general nonlinearity $F\in C^1(\mathbb{R})$ and $f(x)=|x|^\alpha$, we
prove Liouville theorems in dimensions $n\le \frac{4(p+\alpha)}{p-1}+p$, for
bounded radial stable solutions. For specific nonlinearities $F(u)=e^u$, $u^q$
where $q>p-1$ and $-u^{q}$ where $q<0$, known as the Gelfand, the Lane-Emden
and the negative exponent nonlinearities, respectively, we prove Liouville
theorems for both radial finite Morse index (not necessarily bounded) and
stable (not necessarily radial nor bounded) solutions.
