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

We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case, we study some cooperative elliptic systems involving critical nonlinearities in $ {\mathbb{R}}^n $.