Some qualitative questions on the equation −div(a(x,u,∇u))=f(x, u)

In this article, we establish several applications of Picone's identity for the operator of the form−div(a(x,u,∇u)), such as Hardy type inequality, Sturmian comparison theorem, monotonicity property of the first eigenvalue, nonexistence of positive supersolutions and Caccioppoli inequality under certain conditions on a.