Multiplicity of solutions to a class of degenerate elliptic equations in both sub-critical and critical cases

Given a smooth, bounded domain Ω⊂RN, we establish the existence of two non-trivial, non-negative solutions to the semilinear degenerate elliptic equation −Δλu=μg(z)|u|r−1u+h(z)|u|s−1uin Ωu∈H01,λ(Ω)} where Δλ=Δx+|x|2λΔy denotes the Grushin Laplacian Operator, z=(x,y)∈Ω, N=n+m; n, m≥1, λ>0, 0≤r<1<s<2λ∗−1 and μ is a positive parameter. The functions g and h may change sign and 2λ∗=2QQ−2 is the critical Sobolev exponent associated with the homogeneous dimension Q=n+(1+λ)m of Δλ. In the critical case s=2λ∗−1, we further show that the problem admits at least two non-trivial, non-negative solutions under the additional assumptions g≥ 0 and h≡1.