Asymptotic behaviour of solutions to the anisotropic doubly critical equation

The aim of this paper is to deal with the anisotropic doubly critical equation \(\begin{aligned} -\Delta _p^H u - \frac{\gamma }{[H^\circ (x)]^p} u^{p-1} = u^{p^*-1} \qquad \text {in } {\mathbb {R}}^N, \end{aligned}\) $$\begin{array}{c}\hfill -{\mathrm{\Delta }}_p^{H}u-\frac{\gamma }{{[H^{\circ }(x)]}^p}{u}^{p-1}=u^{p^{\ast }-1}\text{in}{\mathbb{R}}^N,\end{array}$$ where H is in some cases called Finsler norm, \(H^\circ \) $H^{\circ }$ is the dual norm, \(1<p<N\) $1<p<N$ , \(0 \le \gamma < \left( (N-p)/p\right) ^p\) $0\le \gamma <{\left((,N,-,p,),/,p\right)}^p$ and \(p^*=Np/(N-p)\) $p^{\ast }=Np/(N-p)$ . In particular, we provide a complete asymptotic analysis of \(u \in \mathcal {D}^{1,p}({\mathbb {R}}^N)\) $u\in {\mathcal{D}}^{1,p}({\mathbb{R}}^N)$ 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.