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We provide the asymptotic behavior of solutions, at the singularity and at infinity, for a class of subelliptic Dirichlet problems with Hardy perturbation and critical nonlinearity of the type $$-\,{\mathcal {L}}_\alpha u -\mu \dfrac{\psi ^2}{d^2} u =K(z)|u|^{2^*-2}u$$-Lαu-μψ2d2u=K(z)|u|2∗-2u in $$\varOmega $$Ω, where $${\mathcal {L}}_\alpha = \varDelta _x + |x|^{2\alpha }\varDelta _y$$Lα=Δx+|x|2αΔy, $$\alpha >0$$α>0 is the so-called Grushin operator, $$\varOmega $$Ω is an open subset of $${\mathbb {R}}^N$$RN, $$0\in \varOmega $$0∈Ω, d is the gauge norm naturally associated with $${\mathcal {L}}_\alpha $$Lα, $$\psi :=|\nabla _\alpha d|$$ψ:=|∇αd|, where $$\nabla _\alpha $$∇α is the Grushin gradient, $$K\in L^\infty $$K∈L∞ and $$0\le \mu < {\overline{\mu }}$$0≤μ<μ¯, where $${\overline{\mu }}$$μ¯ is the best Hardy constant for $${\mathcal {L}}_\alpha $$Lα. Furthermore, we establish some Pohozaev-type nonexistence results.

Asymptotic estimates and nonexistence results for critical problems with Hardy term involving Grushin-type operators