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Nodal solutions for a fourth-order two-point boundary value problem. (English) Zbl 1085.34015

Summary: We consider boundary value problems of fourth-order differential equations of the form

u '''' +βu '' -αu=μh(x)f(u),0<x<r,
u(0)=u(r)=u '' (0)=u '' (r)=0,

where μ is a parameter, β(-,), α[0,) are constants with

r 2 β π 2 +r 4 α π 4 <1,

hC(0,r],[0,)) with h¬0 on any subinterval of [0,r], fC(,) satisfies f(u)u>0 for all u0, and

lim u- f(u) u=0,lim u+ f(u) u=f + ,lim u0 f(u) u=f 0 ,

for some f + , f 0 (0,). We use bifurcation techniques to establish existence and multiplicity results on nodal solutions to the problem.


MSC:
34B15Nonlinear boundary value problems for ODE
34C23Bifurcation (ODE)