Nodal solutions for a fourth-order two-point boundary value problem

We consider boundary value problems of fourth-order differential equations of the form u'''' + beta u'' - alpha u = mu h(x)f (u), 0 < x < r, u(0) = u(r) = u"(0) = u"(r) = 0, where mu is a parameter, beta is an element of (-infinity, infinity), alpha is an element of [0, infinity) are constants with r(2)beta/pi(2) + r(4)alpha/pi(4) < 1, h is an element of C([0, r], [0, infinity)) with h not equivalent to 0 on any subinterval of [0, r], f is an element of C(R, R) satisfies f(u)u > 0 for all u not equal 0, and lim(u ->-infinity) f(u)/u = 0 lim(u ->+infinity) f(u)/u = f(+infinity) lim(u -> 0) f(u)/u = f(0) for some f(+infinity), f(0) is an element of (0, infinity). We use bifurcation techniques to establish existence and multiplicity results of nodal solutions to the problem. (c) 2005 Elsevier Inc. All rights reserved.