Multiplicity of solutions to a degenerate elliptic equation in the sub-critical and critical cases

Given a smooth and bounded domain $\mathrm{\Omega }(\subset {\mathbf{R}}^N)$, we prove the existence of two non-trivial, non-negative solutions for the semilinear degenerate elliptic equation \begin{align} \left. \begin{array}{l} -Δ_λu=μg(z)|u|^{r-1}u+h(z)|u|^{s-1}u ;$\text{in}$; Ωu\in H^{1,λ}_0(Ω) \end{array}\right} \end{align} where ${\mathrm{\Delta }}_{\lambda }={\mathrm{\Delta }}_x+\mathrm{\mid }x{\mathrm{\mid }}^{2\lambda }{\mathrm{\Delta }}_y$ is the Grushin Laplacian Operator, $z=(x,y)\in \mathrm{\Omega }$, $N=n+m;n,m\ge 1$, $\lambda >0$, $0\le r<1<s<2_{\lambda }^{\ast }-1$; the functions $g,h$ are of indefinite sign and $2_{\lambda }^{\ast }=\frac{2Q}{Q-2}$ is the critical Sobolev exponent, where $Q=n+(1+\lambda )m$ is the homogeneous dimension associated to the operator ${\mathrm{\Delta }}_{\lambda }$. As for the critical case $s=2_{\lambda }^{\ast }-1$, we prove the existence of at least two non-trivial, non-negative solutions provided $g\ge 0$ and $h\equiv 1$.