Strong solutions to singular discontinuous $p$-Laplacian problems

TL;DR

This work establishes the existence of a positive strong solution to the Dirichlet problem $-\\Delta_p u=f(u)$ in a bounded domain when the reaction $f$ is singular at $0$ and highly discontinuous on a measure-zero set. The authors combine sub-solution construction with a truncation-regularization scheme, solve auxiliary problems via a variational energy $J_\\varepsilon$, and pass to a limit to obtain a differential inclusion $-\\Delta_p u\in\\partial F(u)$; a strong locality argument then upgrades this to $-\\Delta_p u=f(u)$ a.e., yielding $u\in C^{1,\\alpha}(\\overline{\\Omega})$. The approach accommodates discontinuities in $f$ (with $|\\mathcal{D}_f|=0$) and singular behavior near zero, and it extends to non-homogeneous $(p,q)$-Laplacians under suitable conditions. Key technical tools include Hardy-Sobolev inequalities, sub-/super-solution constructions, and Clarke sub-differential techniques for non-smooth reactions.

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.

Theorems & Definitions (27)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Definition 1.4
• Example 1.5
• Proposition 2.1
• proof
• Proposition 2.2: Hardy-Sobolev inequality
• Proposition 2.3
• proof