We consider the boundary value problem involving the one dimensional pLaplacian, and establish the precise intervals of the parameter for the existence and nonexistence of solutions with prescribed numbers of zeros. Our argument is based on the shooting method together with the qualitative theory for half-linear differential equations.
We establish Hartman-Wintner type criteria for the half-linear second order differential equation r(t)Φ(x ′ ))′ + c(t)Φ(x) = 0, Φ(x) = |x| p−2 x, p > 1, where this equation is viewed as a perturbation of another equation of the same form.