Semilinear degenerate elliptic equation in the presence of singular nonlinearity

Abstract

Given $\Omega (\subseteq R^{1+m})$, a smooth bounded domain and a nonnegative measurable function $f$ defined on $\Omega$ with suitable summability. In this paper, we will study the existence and regularity of solutions to the quasilinear degenerate elliptic equation with a singular nonlinearity given by: \begin{align} -Δ_λu&=\frac{f}{u^ν} \text{ in }Ω\nonumber &u>0 \text{ in } Ω\nonumber &u=0 \text{ on } \partialΩ\nonumber \end{align} where the operator ${\Delta }_{\lambda }$ is given by $${\Delta }_{\lambda }u=u_{xx}+|x{|}^{2\lambda }{\Delta }_{y}u;(x,y)\in R\times R^m$$ is known as the Grushin operator.