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

TL;DR

This paper studies the existence and multiplicity of positive solutions to a nonlinear PDE involving the Grushin operator with a singular term, analyzing how solutions vary with the exponent p relative to a critical Sobolev exponent.

Contribution

It provides new results on positive solutions for a critical Grushin problem with a singular nonlinearity, considering subcritical, critical, and supercritical cases.

Findings

Existence of solutions varies with the exponent p.

Multiple solutions can occur depending on parameters.

The analysis covers subcritical, critical, and supercritical regimes.

Abstract

We investigate the existence and multiplicity of positive solutions to the problem \begin{equation} \begin{cases} \begin{aligned} - \Delta_{\gamma} u &= \lambda u^{p} + u^{-\delta} &\quad \text{in } \Omega, \quad u &= 0 &\quad \text{on } \partial \Omega, \end{aligned} \end{cases} \end{equation} where $\Delta_{\gamma}$ denotes the Grushin operator defined by \begin{equation} \Delta_{\gamma} := \Delta_x + (1+\gamma)^2 |x|^{2\gamma}\Delta_y, \end{equation} with $\gamma>0$, $z=(x,y)\in \mathbb{R}^N$, $N=n+m$, $n \geq 1$, $m\geq 1$, $\Omega \subset \mathbb{R}^N$ 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, $p<2_\gamma^*-1$, $p=2_\gamma^*-1$, or $p>2_\gamma^*-1$, respectively, where $2_\gamma^*=\frac{2Q}{Q-2}$ is the critical Sobolev exponent associated with the Grushin operator, and $Q=m+(1+\gamma)n$ is the corresponding homogeneous dimension.