problem and the lack of radial symmetry.

As a more general motivation, the solutions of problems of type with $f{\left(}x,u{\right)}=\left|u\right|^{p^*-2}u$ turn out to play a central role in the blow-up theories of critical equations in general domains. Possible references in book form on this subject and its applications in the isotropic regime are Druet–Hebey–Robert [@DruHebRob], Ghoussoub [@Gho], and Struwe [@Str2]. A first step in the direction of a blow-up theory in the anisotropic regime was taken in El Hamidi–Vétois [@ElHVet] where we extended the bubble tree decompositions of Struwe [@Str1]. Now, if one wants to go further and investigate a pointwise blow-up theory, then it is essential to know the asymptotic behavior of the solutions of with $f{\left(}x,u{\right)}=\left|u\right|^{p^*-2}u$. The results in this paper can be seen as a crucial step in this direction.

Anisotropic equations of type have received much attention in recent years. In addition to the above cited references [@ElHRak; @ElHVet] and without pretending to be exhaustive, we mention for instance the works by Cianchi [@Cia2] on symmetrization properties, Cîrstea–Vétois [@CirVet] on the fundamental solutions, Cupini–Marcellini–Mascolo [@CupMarMas] on the local boundedness of solutions, Fragalà–Gazzola–Kawohl [@FraGazKaw] on the existence and non-existence of solutions in bounded domains, Lieberman [@Lie] on 