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

Abstract

Given a smooth, bounded domain $\Omega \subset {\mathbb{R}}^N$, we establish the existence of two non-trivial, non-negative solutions to 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}\; Ω u\in H^{1,λ}_0(Ω) \end{array}\right\} \end{align*} where ${\Delta }_{\lambda }={\Delta }_x+|x{|}^{2\lambda }{\Delta }_y$ denotes the Grushin Laplacian Operator, $z=(x,y)\in \Omega$, $N=n+m;n,m\ge 1$, $\lambda >0$, $0\le r<1<s<2_{\lambda }^{\ast }-1$ and $\mu$ is a positive parameter. The functions $g$ and $h$ may change sign and $2_{\lambda }^{\ast }=\frac{2Q}{Q-2}$ is the critical Sobolev exponent associated with the homogeneous dimension $Q=n+(1+\lambda )m$ of ${\Delta }_{\lambda }$. In the critical case $s=2_{\lambda }^{\ast }-1$, we further show that the problem admits at least two non-trivial, non-negative solutions under the additional assumptions $g\ge 0$ and $h\equiv 1$.

28 pages