A Brézis–Nirenberg Type Problem for a Class of Degenerate Elliptic Problems Involving the Grushin Operator

Motivated by the seminal paper due to Brézis–Nirenberg (in: Commun Pure Appl Math 36:437–477, 1983), we will establish the existence of solutions for the following class of degenerate elliptic equations with critical nonlinearity: {-Δγv=λ|v|q-2v+|v|2γ∗-2vinΩ,v=0on∂Ω, where Δ γ: = Δ x+ (1 + γ) 2| x| 2γΔ y is the Grushin operator, z: = (x, y) ∈ RN , N= m+ n, m, n≥ 1 , Ω ⊂ RN is a smooth bounded domain, λ> 0 , q∈[2,2γ∗) and 2γ∗=2NγNγ-2 is the critical Sobolev exponent in this context, where Nγ= m+ (1 + γ) n is the so-called homogeneous dimension attached to the Grushin operator Δ γ . In order to prove our main results it was necessary to do a careful study involving the best constant Sγ(m, n) of the Sobolev embedding for the spaces associated with Δ γ . In order to do that, we prove a version of the Lions’ Concentration-Compactness Principle for the Grushin operator. We also provide existence results for a critical problem involving the Grushin operator on the whole space RN .