  In this paper we provide the classification of positive solutions to the
critical $p-$Laplace equation on $\mathbb{R}^n$, for $1<p<n$, possibly having
infinite energy. If $n=2$, or if $n=3$ and $\frac 32<p<2$ we prove rigidity
without any further assumptions. In the remaining cases we obtain the
classification under energy growth conditions or suitable control of the
solutions at infinity. Our assumptions are much weaker than those already
appearing in the literature. We also discuss the extension of the results to
the Riemannian setting.
