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

We consider the equation $-{\mathrm{\Delta }}_{p}u=f(u)$ in a smooth bounded domain of ${\mathbb{R}}^n$, where ${\mathrm{\Delta }}_p$ is the $p$-Laplace operator. Explicit examples of unbounded stable energy solutions are known if $n\ge p+4p\mathrm{/}(p-1)$. Instead, when $n<p+4p\mathrm{/}(p-1)$, 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 $f\in C^1$whenever$p\geq2$andtheoptimalcondition$n<p+4p/(p-1)$holds\mathrm{.}When$p\in(1,2)$,weobtainthesameresultunderthenon-sharpassumption$n<5p$. 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 $n<10$.