A nonlinear Picone’s identity and its applications

The purpose of this note is to show a generalization to Picone’s identity in a nonlinear framework. The classical Picone’s identity turns out to be a particular case of our result. We show, as an application of our results, that the Morse index of the zero solution to a semilinear elliptic boundary value problem is 0 and also establish a linear relationship between the components of the solution of a nonlinear elliptic system.