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Indexed by:期刊论文

Date of Publication:2018-11-05

Journal:ELECTRONIC JOURNAL OF DIFFERENTIAL EQUATIONS

Included Journals:SCIE、Scopus

Volume:2018

ISSN No.:1072-6691

Key Words:Bifurcation; spectrum; nonlocal problem; nodal solution

Abstract:In this article, we consider a Dancer-type unilateral global bifurcation for the Kirchhoff-type problem
   -(a+b integral(1)(0) vertical bar u'vertical bar(2) dx)u '' =lambda u+h(x,u,lambda) in (0,1),
   u(0) = u(1) = 0.
   Under natural hypotheses on h, we show that (a lambda(k), 0) is a bifurcation point of the above problem. As applications we determine the interval of lambda, in which there exist nodal solutions for the Kirchhoff-type problem
   -(a+b integral(1)(0) vertical bar u'vertical bar(2) dx)u '' =lambda f(x,u) in (0,1),
   u(0) = u(1) = 0.
   where f is asymptotically linear at zero and is asymptotically 3-linear at infinity. To do this, we also establish a complete characterization of the spectrum of a nonlocal eigenvalue problem.