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

We consider boundary value problems of fourth-order differential equations of the formu⁗+βu″−αu=μh(x)f(u),00f(u)u>0 for all u≠0u≠0, andlimu→−∞f(u)u=0,limu→+∞f(u)u=f+∞,limu→0f(u)u=f0 for some f+∞,f0∈(0,∞)f+∞,f0∈(0,∞). We use bifurcation techniques to establish existence and multiplicity results of nodal solutions to the problem.