Uniqueness and nondegeneracy for some nonlinear elliptic problems in a ball

In this paper we study the uniqueness and nondegeneracy of positive solutions of nonlinear problems of the type Δpu+f(r,u)=0 in the unit ball B, u=0 on ∂B. Here Δp denotes the p Laplace operator Δp=div( up−2 u), p>1. The main ideas rely on the Maximum Principle and an implicit function theorem that we derive in a suitable weighted space. This space is essential to deal with the case p≠2.