Hardy inequality and properties of the quasilinear Sturm–Liouville problem

Abstract

We present necessary and sufficient condition for discreteness of the set of all eigenvalues (having the usual Sturm–Liouville properties) of a quasilinear Sturm–Liouville-type problem with weights on an infinite interval. We point out that the same condition is necessary and sufficient for the compact embedding of certain weighted Sobolev and Lebesgue spaces. Our result completes those from the linear theory.