Existence and multiplicity of solutions for a critical Grushin problem with a singular nonlinearity

Abstract

We investigate the existence and multiplicity of positive solutions to the problem \begin{equation} \begin{cases} \begin{aligned} - Δ_γ u &= λu^{p} + u^{-δ} &\quad \text{in } Ω, \quad u &= 0 &\quad \text{on } \partial Ω, \end{aligned} \end{cases} \end{equation} where ${\Delta }_{\gamma }$ denotes the Grushin operator defined by \begin{equation} Δ_γ := Δ_x + (1+γ)^2 |x|^{2γ}Δ_y, \end{equation} with $\gamma >0$, $z=(x,y)\in {\mathbb{R}}^N$, $N=n+m$, $n\ge 1$, $m\ge 1$, $\Omega \subset {\mathbb{R}}^N$ a smooth bounded domain, $\lambda >0$, $1<p<\infty$, and $\delta >0$. The analysis depends on the exponent $p$, which may be subcritical, critical, or supercritical, that is, $p<2_{\gamma }^{\ast }-1$, $p=2_{\gamma }^{\ast }-1$, or $p>2_{\gamma }^{\ast }-1$, respectively, where $2_{\gamma }^{\ast }=\frac{2Q}{Q-2}$ is the critical Sobolev exponent associated with the Grushin operator, and $Q=m+(1+\gamma )n$ is the corresponding homogeneous dimension.