Uniqueness for degenerate elliptic sublinear problems in the absence of dead cores

In this work we study the problem $$ -mathop{ m div}(| abla u|^{p-2} abla u)=lambda f(u) $$ in the unit ball of $mathbb{R}^N$, with $u=0$ on the boundary, where $p>2$, and $lambda$ is a real parameter. We assume that the nonlinearity $f$ has a zero $ar{u}_0>0$ of order $kge p-1$. Our main contribution is showing that there exists a unique positive solution of this problem for large enough $lambda$ and maximum close to $ar{u}_0$. This will be achieved by means of a linearization technique, and we also prove the new result that the inverse of the $p$-Laplacian is differentiable around positive solutions.