The Dirichlet problem for possibly singular elliptic equations with degenerate coercivity

TL;DR

This paper investigates the existence, uniqueness, and regularity of nonnegative solutions to a class of possibly singular elliptic equations with degenerate coercivity, extending understanding of such problems under various conditions.

Contribution

It establishes new results on the solvability and regularity of elliptic equations with singular and degenerate features, including cases where the nonlinearity may blow up at zero.

Findings

Proved existence of solutions under broad conditions.

Established regularity results for solutions.

Analyzed the impact of singularities and degeneracy on solution behavior.

Abstract

We deal with existence, uniqueness and regularity of nonnegative solutions to a Dirichlet problem for equations as \begin{equation*} \displaystyle -\operatorname{div}\left(\frac{|\nabla u|^{p-2}\nabla u}{(1+u)^{\theta(p-1)}}\right) = h(u)f \quad \text{in }\Omega, \end{equation*} where $\Omega$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge 2$), $p>1$, $\theta\ge 0$, $f\geq 0$ belongs to a suitable Lebesgue space and $h$ is a continuous, nonnegative function which may blow up at zero and it is bounded at infinity.