Strong solutions to singular discontinuous $p$-Laplacian problems

TL;DR

This paper proves the existence of positive strong solutions for a class of Dirichlet $p$-Laplacian problems with singular and highly discontinuous reactions, where discontinuities are limited to measure-zero sets.

Contribution

It establishes existence results for solutions under minimal discontinuity assumptions, advancing understanding of singular $p$-Laplacian problems with discontinuous reactions.

Findings

Existence of positive strong solutions proven.

Solutions exist despite reaction discontinuities on measure-zero sets.

Applicable to problems with singular behavior at zero.

Abstract

In this paper, the existence of positive strong solutions to a Dirichlet $p$-Laplacian problem with reaction both singular at zero and highly discontinuous is investigated. In particular, it is only required that the set of discontinuity points has Lebesgue measure zero.