| Summary: | We prove a Liouville-type theorem for stable solution of the singular
quasilinear elliptic equations
$$\displaylines{
-\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=f(x)|u|^{q-1}u, \quad
\text{in } \mathbb{R}^N, \cr
-\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=f(x)e^u, \quad
\text{in } \mathbb{R}^N
}$$
where $2\le p<N, -\infty<a<(N-p)/p$ and the function f(x) is continuous
and nonnegative in $\mathbb{R}^N\setminus\{0\}$ such that
$f(x)\ge c_0|x|^{b}$ as $|x|\ge R_0$, with $b>-p(1+a)$ and $c_0>0$.
The results hold for $1\le p-1<q=q_c(p,N,a,b)$ in the first equation,
and for $2\le N<q_0(p,a,b)$ in the second equation.
Here $q_0$ and $q_c$ are exponents, which are always larger than the
classical critical ones and depend on the parameters a,b.
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