AbstractWe consider the equation {-\Delta_{p}u=f(u)} in a smooth bounded domain of {\mathbb{R}^{n}}, where {\Delta_{p}} is the p-Laplace operator. Explicit examples of unbounded stable energy solutions are known if {n\geq p+\frac{4p}{p-1}}.
Instead, when {n<p+\frac{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\in C^{1}} whenever {p\geq 2} and the optimal condition {n<p+\frac{4p}{p-1}} holds.
When {p\in(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}.
Cover's open problem: “The Capacity of the Relay Channel”
