{"title":"Nodal solutions for a nonlocal fourth order equation of Kirchhoff type","authors":"","doi":"10.1016/j.aml.2024.109292","DOIUrl":null,"url":null,"abstract":"<div><p>We study the bifurcation behavior of nodal solutions for the Kirchhoff type beam equation <span><math><mfenced><mrow><mtable><mtr><mtd></mtd><mtd><msup><mrow><mi>u</mi></mrow><mrow><mo>′</mo><mo>′</mo><mo>′</mo><mo>′</mo></mrow></msup><mo>−</mo><mi>M</mi><mrow><mo>(</mo><mrow><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup><msup><mrow><mrow><mo>|</mo><msup><mrow><mi>u</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mi>x</mi></mrow><mo>)</mo></mrow><msup><mrow><mi>u</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mo>=</mo><mi>λ</mi><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mi>x</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi>u</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mi>u</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span> where <span><math><mrow><mi>λ</mi><mo>∈</mo><mi>R</mi></mrow></math></span> is a parameter, <span><math><mrow><mi>M</mi><mo>:</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>→</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>f</mi><mo>:</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mo>×</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow></math></span> are smooth functions. We obtain the existence of nodal solutions under some suitable conditions. The proof of our main result is based upon bifurcation techniques.</p></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":2.9000,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics Letters","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0893965924003124","RegionNum":2,"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: 0
Abstract
We study the bifurcation behavior of nodal solutions for the Kirchhoff type beam equation where is a parameter, and are smooth functions. We obtain the existence of nodal solutions under some suitable conditions. The proof of our main result is based upon bifurcation techniques.
期刊介绍:
The purpose of Applied Mathematics Letters is to provide a means of rapid publication for important but brief applied mathematical papers. The brief descriptions of any work involving a novel application or utilization of mathematics, or a development in the methodology of applied mathematics is a potential contribution for this journal. This journal''s focus is on applied mathematics topics based on differential equations and linear algebra. Priority will be given to submissions that are likely to appeal to a wide audience.