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

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