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On singular $p$-Laplacian problems with discontinuous convection terms

Umberto Guarnotta, Salvatore A. Marano

Abstract

The existence of positive strong solutions to a homogeneous Dirichlet $p$-Laplacian problem, with reaction sum of a both singular at zero and highly discontinuous nonlinearity and of a discontinuous convection term, is established. Locality arguments, based on suitable measure theoretical results (see Section 3), are employed.

On singular $p$-Laplacian problems with discontinuous convection terms

Abstract

The existence of positive strong solutions to a homogeneous Dirichlet -Laplacian problem, with reaction sum of a both singular at zero and highly discontinuous nonlinearity and of a discontinuous convection term, is established. Locality arguments, based on suitable measure theoretical results (see Section 3), are employed.
Paper Structure (4 sections, 15 theorems, 122 equations)

This paper contains 4 sections, 15 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and auxiliary results
  3. Measure-theoretical results
  4. Proof of the main result

Key Result

Theorem 1

Let $({\rm H}_f)$, $({\rm H}_g)$, hypatzero, and hypatinfty be satisfied. Then prob admits a strong solution $u\in C^{1,\alpha}_0(\overline{\Omega})$ for some $\alpha\in(0,1)$.

Theorems & Definitions (31)

  • Theorem 1
  • Proposition 2: see GM, Proposition 2.1
  • Proposition 3: cf. OK, Theorem 21.3
  • Proposition 4: see GM, Proposition 2.3
  • Theorem 5
  • proof
  • Corollary 6
  • Example 7
  • Remark 8
  • Remark 9
  • ...and 21 more