Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian

We consider the equation -Delta(p)u = f(u) in a smooth bounded domain of R-n, where Delta(p) is the p-Laplace operator. Explicit examples of unbounded stable energy solutions are known if n >= p + 4p/p-1 . Instead, when n < p + 4p/p-1 , stable solutions have been proved to be bounded only in the radial case or under strong assumptions on f. In this article we solve a long-standing open problem: we prove an interior C-alpha bound for stable solutions which holds for every nonnegative f epsilon C-1 whenever p >= 2 and the optimal condition n < p+4p/p-1 holds. When p epsilon (1,2) , we obtain the same result under the nonsharp assumption n < 5p . These interior estimates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when the domain is strictly convex. Our work extends to the p-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 .