Multiplicity of Positive Solutions for a Fourth-Order Quasilinear Singular Differential Equation

This paper is concerned with the multiplicity of positive solutions of boundary value problem for the fourth-order quasilinear singular differential equation (broken vertical bar u '''broken vertical bar(p-2)u '')'' = lambda g(t)f(u), 0 < t < 1, where p > 1, lambda > 0. We apply the fixed point index theory and the upper and lower solutions method to investigate the multiplicity of positive solutions. We have found a threshold lambda* < +infinity, such that if 0 < lambda <= lambda*, then the problem admits at least one positive solution; while if lambda > lambda*, then the problem has no positive solution. In particular, there exist at least two positive solutions for 0 < lambda < lambda*.