Degenerate elliptic problem with a singular nonlinearity

TL;DR

This paper establishes existence and regularity of solutions for a class of nonlinear elliptic equations involving degenerate operators and singular nonlinearities, expanding understanding of such complex boundary value problems.

Contribution

It provides new existence and regularity results for elliptic problems with degenerate coercive operators and singular right-hand sides, a topic with limited prior analysis.

Findings

Existence of solutions under specified conditions

Regularity results for solutions

Handling of singular nonlinearities in elliptic equations

Abstract

In this paper, we prove existence and regularity results for solutions of some nonlinear Dirichlet problems for an elliptic equation defined by a degenerate coercive operator and a singular right hand side. \begin{equation}\label{01} \left\{ \begin{array}{lll} -\displaystyle\mbox{div}( a(x,u,\nabla u))&=\displaystyle\frac{f}{u^{\gamma}} & \mbox{ in } \Omega \\ u&>0 &\mbox{ in }\Omega \\ u&=0 &\mbox{ on } \delta\Omega \end{array} \right. \end{equation} where $\Omega $ is bounded open subset of $I\!\!R^{N}(N\geq2),$ $\gamma>0$ and $ f$ is a nonnegative function that belongs to some Lebesgue space.