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Removable Singularities for Quasilinear Elliptic Equations with Source Terms Involving the Solution and Its Gradient

Removable Singularities for Quasilinear Elliptic Equations with Source Terms Involving the... This paper establishes a removable singularity theorem for the quasilinear elliptic equations with source terms like -Δpu=a|u|q+b|∇u|s+c|u|σ|∇u|τ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} -\Delta _p u = a |u|^q + b|\nabla u|^s + c|u|^\sigma |\nabla u|^\tau \end{aligned}$$\end{document}with nonnegative bounded Borel measurable functions a, b, c and positive numbers q,s,σ,τ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$q,s,\sigma ,\tau $$\end{document}. In particular, we give upper bounds of exponents q,s,σ,τ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$q,s,\sigma ,\tau $$\end{document} and a sharp growth condition for nonnegative weak solutions in RN\E\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^N{\setminus } E$$\end{document} to be extended to the whole of RN\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^N$$\end{document} as solutions, when E is a compact set satisfying a uniform Minkowski condition. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Bulletin of the Brazilian Mathematical Society, New Series" Springer Journals

Removable Singularities for Quasilinear Elliptic Equations with Source Terms Involving the Solution and Its Gradient

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Publisher
Springer Journals
Copyright
Copyright © Sociedade Brasileira de Matemática 2022
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/s00574-022-00283-y
Publisher site
See Article on Publisher Site

Abstract

This paper establishes a removable singularity theorem for the quasilinear elliptic equations with source terms like -Δpu=a|u|q+b|∇u|s+c|u|σ|∇u|τ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} -\Delta _p u = a |u|^q + b|\nabla u|^s + c|u|^\sigma |\nabla u|^\tau \end{aligned}$$\end{document}with nonnegative bounded Borel measurable functions a, b, c and positive numbers q,s,σ,τ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$q,s,\sigma ,\tau $$\end{document}. In particular, we give upper bounds of exponents q,s,σ,τ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$q,s,\sigma ,\tau $$\end{document} and a sharp growth condition for nonnegative weak solutions in RN\E\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^N{\setminus } E$$\end{document} to be extended to the whole of RN\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^N$$\end{document} as solutions, when E is a compact set satisfying a uniform Minkowski condition.

Journal

"Bulletin of the Brazilian Mathematical Society, New Series"Springer Journals

Published: Jan 15, 2022

Keywords: Removable singularity; Quasilinear elliptic equation; Wolff potential; Primary 35J92 Secondary 31C45; 35B60

References