Boundary Value Problems
Volume 2010 (2010), Article ID 570932, 11 pages
doi:10.1155/2010/570932
Research Article

Nodal Solutions for a Class of Fourth-Order Two-Point Boundary Value Problems

Jia Xu1,2 and XiaoLing Han1

1Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
2College of Physical Education, Northwest Normal University, Lanzhou 730070, China

Received 18 February 2010; Accepted 27 April 2010

Academic Editor: Irena Rachůnková

Copyright © 2010 Jia Xu and XiaoLing Han. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the fourth-order two-point boundary value problem u′′′′+Mu=λh(t)f(u), 0<t<1, u(0)=u(1)=u′(0)=u′(1)=0, where λ∈ℝ is a parameter, M∈(-π4,π4/64) is given constant, h∈C([0,1],[0,∞)) with h(t)≢0 on any subinterval of [0,1], f∈C(ℝ,ℝ) satisfies f(u)u>0 for all u≠0, and lim⁡u→-∞f(u)/u=0, lim⁡u→+∞f(u)/u=f+∞, lim⁡u→0f(u)/u=f0 for some f+∞,f0∈(0,+∞). By using disconjugate operator theory and bifurcation techniques, we establish existence and multiplicity results of nodal solutions for the above problem.