{"title":"Semilinear problems with right-hand sides singular at u = 0 which change sign","authors":"Juan Casado-Díaz , François Murat","doi":"10.1016/j.anihpc.2020.09.001","DOIUrl":null,"url":null,"abstract":"<div><p>The present paper is devoted to the study of the existence of a solution <em>u</em><span> for a quasilinear second order differential equation with homogeneous Dirichlet conditions, where the right-hand side tends to infinity at </span><span><math><mi>u</mi><mo>=</mo><mn>0</mn></math></span><span>. The problem has been considered by several authors since the 70's. Mainly, nonnegative right-hand sides were considered and thus only nonnegative solutions were possible. Here we consider the case where the right-hand side can change sign but is non negative (finite or infinite) at </span><span><math><mi>u</mi><mo>=</mo><mn>0</mn></math></span>, while no restriction on its growth at <span><math><mi>u</mi><mo>=</mo><mn>0</mn></math></span> is assumed on its positive part. We show that there exists a nonnegative solution in a sense introduced in the paper; moreover, this solution is stable with respect to the right-hand side and is unique if the right-hand side is nonincreasing in <em>u</em>. We also show that if the right-hand side goes to infinity at zero faster than <span><math><mn>1</mn><mo>/</mo><mo>|</mo><mi>u</mi><mo>|</mo></math></span>, then only nonnegative solutions are possible. We finally prove by means of the study of a one-dimensional example that nonnegative solutions and even many solutions which change sign can exist if the growth of the right-hand side is <span><math><mn>1</mn><mo>/</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>γ</mi></mrow></msup></math></span> with <span><math><mn>0</mn><mo><</mo><mi>γ</mi><mo><</mo><mn>1</mn></math></span>.</p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":"38 3","pages":"Pages 877-909"},"PeriodicalIF":1.8000,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.09.001","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0294144920300858","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 2
Abstract
The present paper is devoted to the study of the existence of a solution u for a quasilinear second order differential equation with homogeneous Dirichlet conditions, where the right-hand side tends to infinity at . The problem has been considered by several authors since the 70's. Mainly, nonnegative right-hand sides were considered and thus only nonnegative solutions were possible. Here we consider the case where the right-hand side can change sign but is non negative (finite or infinite) at , while no restriction on its growth at is assumed on its positive part. We show that there exists a nonnegative solution in a sense introduced in the paper; moreover, this solution is stable with respect to the right-hand side and is unique if the right-hand side is nonincreasing in u. We also show that if the right-hand side goes to infinity at zero faster than , then only nonnegative solutions are possible. We finally prove by means of the study of a one-dimensional example that nonnegative solutions and even many solutions which change sign can exist if the growth of the right-hand side is with .
期刊介绍:
The Nonlinear Analysis section of the Annales de l''Institut Henri Poincaré is an international journal created in 1983 which publishes original and high quality research articles. It concentrates on all domains concerned with nonlinear analysis, specially applicable to PDE, mechanics, physics, economy, without overlooking the numerical aspects.