Multiplicity results for critical growth problems involving the p-Grushin operator

We prove the existence of multiple nontrivial solutions for degenerate elliptic critical growth problems of the form $${aligned} {\{ {array}{ll}-Δ γ ,p u= λ |u|ʳ⁻²u+| u| ^{pγ*-2}u & in Ω \&\\ u=0 & on ∂ Ω , {array}. } {aligned}$$ where $$Δ γ , p$$ denotes the Grushin p -Laplace operator, $$Ω ⊂ RN$$ is a bounded domain intersecting the degeneration set of the operator, $$λ >0$$ and $$p , where p^*_γ is the Grushin critical exponent. We also investigate critical Grushin ( p , q )-Laplacian type problems, that is, Dirichlet critical growth problems governed by the operator L:= -Δ γ ,p- Δ γ ,q. In particular, we prove that the considered problems have arbitrarily many solutions for all sufficiently large λ . The results are based on the application of the abstract multiplicity theorems established by Perera in [27].