In the paper titled "On the existence and stability properties of positive solution for some p-Laplacian Dirichlet problems", World Journal of Modelling and Simulation, Vol. 3 (2007) No. 1, pp. 21-26,
The author claimed that.: if u be any nonnegative solution of $-\Delta _{p}u=g(x,u) \ in \ \Omega, u=0\ on \ \partial \Omega $, then the linearized equation about u is
$$-(p-1)\ div[\nabla u|^{p-2}\nabla v]-g_{u}(x,u)v=\lambda \ v \ in \ \Omega, v=0\ on \ \partial \Omega$$ where $g_{u}(x,u)$ denotes the partial derivative of $g(x,u)$ with respect to $u$. The above equation was obtained from the formal derivative of the operator $\Delta _{p}u$.
My question is: How we can derive this linearized equation? Aslo, what is the formal derivative of the operator $\Delta _{p}u$?