Hits:

Indexed by:期刊论文

Date of Publication:2016-09-01

Journal:TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS

Included Journals:SCIE、Scopus

Volume:48

Issue:1

Page Number:213-233

ISSN No.:1230-3429

Key Words:Bifurcation; eigenvalue; Kirchhoff type equation; positive solutions

Abstract:In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem:
   {-(a + b integral(Omega) vertical bar del(u)vertical bar(2) dx) Delta u = lambda u + h(x, u, lambda) in Omega,
   u = 0 on Omega.
   Under some natural hypotheses on h, we show that (a lambda(1), 0) is a bifurcation point of the above problem. As an application of the above result, we shall determine the interval of lambda, in which there exist positive solutions for the above problem with h(x, u; lambda) = lambda f (x, u) - lambda u, where f is asymptotically linear at zero and asymptotically 3-linear at infinity. To study global structure of bifurcation branch, we also establish some properties of the first eigenvalue for a nonlocal eigenvalue problem. Moreover, we provide a positive answer to an open problem involving the case a = 0.