  The aim of this paper is to deal with the anisotropic doubly critical
equation $$-\Delta_p^H u - \frac{\gamma}{[H^\circ(x)]^p} u^{p-1} = u^{p^*-1}
\qquad \text{in } \R^N,$$ where $H$ is in some cases called Finsler norm,
$H^\circ$ is the dual norm, $1<p<N$, $0 \leq \gamma< \left((N-p)/p\right)^p$
and $p^*=Np/(N-p)$. In particular, we provide a complete
  asymptotic analysis of $u \in \mathcal{D}^{1,p}(\R^N)$ near the origin and at
infinity, showing that this solution has the same features of its euclidean
counterpart. Some of the techniques used in the proofs are new even in the
Euclidean framework.
