Semilinear degenerate elliptic equation in the presence of singular nonlinearity

TL;DR

This paper investigates the existence and regularity of solutions to a degenerate elliptic equation with singular nonlinearity involving the Grushin operator in a bounded domain.

Contribution

It provides new results on solutions to a quasilinear degenerate elliptic equation with singular nonlinearity involving the Grushin operator.

Findings

Existence of solutions under certain conditions.

Regularity properties of solutions established.

Analysis of the impact of singular nonlinearity on solution behavior.

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} -\Delta_\lambda u&=\frac{f}{u^{\nu}} \text{ in }\Omega\nonumber &u>0 \text{ in } \Omega\nonumber &u=0 \text{ on } \partial\Omega\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.