Uniqueness and nondegeneracy for some nonlinear elliptic problems in a ball. (Q1419809)

This paper is devoted to the uniqueness of non-degeneracy of solutions of nonlinear problems of the following type \[ \begin{cases} \Delta_p u+ f(r,u)= 0\quad &\text{in }B,\\ u> 0\quad &\text{in }B,\\ u= 0\quad &\text{on }\partial B,\end{cases} \] where \(B\) is the unit ball in \(\mathbb R^N\) centered at the origin, \(N\geq 2\), \(r=| x|\) and \(\Delta_p\) denotes the \(p\)-Laplace operator \[ \Delta_p u= \text{div}(|\nabla u|^{p-2}\nabla u),\quad p> 1. \] To this end the authors use the maximum principle and an implicit function theorem that they derive in a suitable weighted space.