Regularizing effect of absorption terms in singular and degenerate elliptic problems

TL;DR

This paper investigates how absorption terms influence the existence and regularity of solutions to singular and degenerate elliptic problems, showing that lower order absorption terms can have a regularizing effect.

Contribution

It demonstrates the regularizing effect of absorption terms on solutions to singular elliptic problems with degenerate coercivity, expanding understanding of solution regularity.

Findings

Absorption terms improve solution regularity.

Existence of solutions is established under certain conditions.

Lower order terms have a regularizing effect in degenerate elliptic problems.

Abstract

In this paper we study the existence and regularity of solutions to the following singular problem \begin{equation} \left\{ \begin{array}{lll} &-\displaystyle\mbox{div} \big(a(x,u)|\nabla u|^{p-2}|\nabla u|\big) + |u|^{s-1}u =\frac{f}{u^{\gamma}} &\mbox{ in } \Omega \\ &u>0 &\mbox{ in }\Omega \\ &u=0 &\mbox{ on } \delta\Omega \end{array} \right. \end{equation} proving that the lower order term $u|u|^{s-1}$ has some regularizing effects on the solutions in the case of an elliptic operator with degenerate coercivity.