Strong solutions for singular Dirichlet elliptic problems

IF 1.1 4区数学 Q1 MATHEMATICS

Electronic Journal of Qualitative Theory of Differential Equations Pub Date : 2022-01-01 DOI:10.14232/ejqtde.2022.1.40

T. Godoy

{"title":"Strong solutions for singular Dirichlet elliptic problems","authors":"T. Godoy","doi":"10.14232/ejqtde.2022.1.40","DOIUrl":null,"url":null,"abstract":"We prove an existence result for strong solutions <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>u</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant=\"normal\">Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of singular semilinear elliptic problems of the form <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>−</mml:mo> <mml:mi mathvariant=\"normal\">Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>⋅</mml:mo> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi mathvariant=\"normal\">Ω</mml:mi> <mml:mo>,</mml:mo> </mml:math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>τ</mml:mi> </mml:math> on <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi mathvariant=\"normal\">∂</mml:mi> <mml:mi mathvariant=\"normal\">Ω</mml:mi> <mml:mo>,</mml:mo> </mml:math> where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:mi>q</mml:mi> <mml:mo><</mml:mo> <mml:mi mathvariant=\"normal\">∞</mml:mi> <mml:mo>,</mml:mo> </mml:math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi mathvariant=\"normal\">Ω</mml:mi> </mml:math> is a bounded domain in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:math> with <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> boundary, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>2</mml:mn> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>q</mml:mi> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi mathvariant=\"normal\">∂</mml:mi> <mml:mi mathvariant=\"normal\">Ω</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:math> and with <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>g</mml:mi> <mml:mo>:</mml:mo> <mml:mi mathvariant=\"normal\">Ω</mml:mi> <mml:mo>×</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant=\"normal\">∞</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo stretchy=\"false\">→</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant=\"normal\">∞</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> belonging to a class of nonnegative Carathéodory functions, which may be singular at <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> and also at <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>S</mml:mi> </mml:math> for some suitable subsets <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>S</mml:mi> <mml:mo>⊂</mml:mo> <mml:mover> <mml:mi mathvariant=\"normal\">Ω</mml:mi> <mml:mo accent=\"false\">¯</mml:mo> </mml:mover> <mml:mo>.</mml:mo> </mml:math> In addition, we give results concerning the uniqueness and regularity of the solutions. A related problem on punctured domains is also considered.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Journal of Qualitative Theory of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.14232/ejqtde.2022.1.40","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

We prove an existence result for strong solutions u ∈ W 2 , q ( Ω ) of singular semilinear elliptic problems of the form − Δ u = g ( ⋅ , u ) in Ω , u = τ on ∂ Ω , where 1 < q < ∞ , Ω is a bounded domain in R n with C 2 boundary, 0 ≤ τ ∈ W 2 − 1 q , q ( ∂ Ω ) , and with g : Ω × ( 0 , ∞ ) → [ 0 , ∞ ) belonging to a class of nonnegative Carathéodory functions, which may be singular at s = 0 and also at x ∈ S for some suitable subsets S ⊂ Ω ¯ . In addition, we give results concerning the uniqueness and regularity of the solutions. A related problem on punctured domains is also considered.

查看原文本刊更多论文

奇异狄利克雷椭圆问题的强解

我们证明了奇异半线性椭圆型问题的强解u∈W 2, q (Ω)的存在性，其形式为- Δ u = g(⋅，u)在Ω中，u = τ在∂Ω上，其中1 q∞，Ω是R n中具有c2边界的有界域，0≤τ∈W 2−1 q，q(∂Ω)，并且与g: Ω x(0，∞)→[0，∞)属于一类非负的carathodory函数，对于某些合适的子集s∧Ω¯，该类函数在s = 0和x∈s处可能是奇异的。此外，我们给出了解的唯一性和正则性的结果。同时考虑了穿孔域上的一个相关问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Electronic Journal of Qualitative Theory of Differential Equations 数学-数学

CiteScore

1.40

自引率

9.10%

发文量

审稿时长

3 months

期刊介绍： The Electronic Journal of Qualitative Theory of Differential Equations (EJQTDE) is a completely open access journal dedicated to bringing you high quality papers on the qualitative theory of differential equations. Papers appearing in EJQTDE are available in PDF format that can be previewed, or downloaded to your computer. The EJQTDE is covered by the Mathematical Reviews, Zentralblatt and Scopus. It is also selected for coverage in Thomson Reuters products and custom information services, which means that its content is indexed in Science Citation Index, Current Contents and Journal Citation Reports. Our journal has an impact factor of 1.827, and the International Standard Serial Number HU ISSN 1417-3875. All topics related to the qualitative theory (stability, periodicity, boundedness, etc.) of differential equations (ODE''s, PDE''s, integral equations, functional differential equations, etc.) and their applications will be considered for publication. Research articles are refereed under the same standards as those used by any journal covered by the Mathematical Reviews or the Zentralblatt (blind peer review). Long papers and proceedings of conferences are accepted as monographs at the discretion of the editors.