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

Abstract

In this work we study the problem -div (|∇u|p-2 ∇u) = λƒ(u) in the unit ball of ℝN, with u = 0 on the boundary, where p > 2, and λ is a real parameter. We assume that the nonlinearity ƒ has a zero ū0 > 0 of order k ≥ p-1. Our main contribution is showing that there exists a unique positive solution of this problem for large enough λ and maximum close to ū0. 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.