Abstract
We consider the equation −Δpu=f(u) in a smooth bounded domain of Rn, where Δp is the p-Laplace operator. Explicit examples of unbounded stable energy solutions are known if n≥p+4pp−1. Instead, when nf. In this article we solve a long-standing open problem: we prove an interior Cα bound for stable solutions which holds for every nonnegative f∈C1 whenever p≥2 and the optimal condition np-Laplacian some of the recent results of Figalli, Ros-Oton, Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutions when p=2 in the optimal range n<10.