Sharp conditions for the existence of sign-changing solutions to equations involving the one-dimensional pp-Laplacian

We consider the boundary value problem involving the one-dimensional pp-Laplacian (|u′|p−2u′)′+a(x)f(u)=0,01p>1. We establish sharp conditions for the existence of solutions with prescribed numbers of zeros in terms of the ratio f(s)/sp−1f(s)/sp−1 at infinity and zero. Our argument is based on the shooting method together with the qualitative theory for half-linear differential equations.