Nonlinear anisotropic Picone type identities for general vector fields and applications

The main goal of this paper is to prove two generalisations of nonlinear Picone type identities for anisotropic subelliptic p-Laplacian and p-biLaplacian in the context of the general vector fields. Furthermore, we present several applications which range from Hardy type and Rellich type inequalities to Liouville type and Sturmian type comparison principles, to first eigenvalue monotonicity and Morse index of positive solutions.