Release Time:2019-03-12  Hits:

Indexed by: Journal Article

Date of Publication: 2018-11-05

Journal: ELECTRONIC JOURNAL OF DIFFERENTIAL EQUATIONS

Included Journals: Scopus、SCIE

Volume: 2018

ISSN: 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.