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

Abstract

Given a smooth, bounded domain $Ω\subset\mathbb{R}^N$, 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 $Δ_λ=Δ_x+|x|^{2λ}Δ_y$ denotes the Grushin Laplacian Operator, $z=(x,y)\inΩ$, $N=n+m;\, n,\, m\geq 1$, $λ>0$, $0\leq r<1<s<2^*_λ-1$ and $μ$ is a positive parameter. The functions $g$ and $h$ may change sign and $2^*_λ=\frac{2Q}{Q-2}$ is the critical Sobolev exponent associated with the homogeneous dimension $Q=n+(1+λ)m$ of $Δ_λ$. In the critical case $s=2^*_λ-1$, we further show that the problem admits at least two non-trivial, non-negative solutions under the additional assumptions $g\geq 0$ and $h\equiv 1$.