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

Abstract

Abstract We consider the equation - Δ p ⁢ u = f ⁢ ( u ) {-\Delta_{p}u=f(u)} in a smooth bounded domain of ℝ n {\mathbb{R}^{n}} , where Δ p {\Delta_{p}} is the p-Laplace operator. Explicit examples of unbounded stable energy solutions are known if n ≥ p + 4 ⁢ p p - 1 {n\geq p+\frac{4p}{p-1}} . Instead, when n < p + 4 ⁢ p p - 1 {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 α {C^{\alpha}} bound for stable solutions which holds for every nonnegative f ∈ C 1 {f\in C^{1}} whenever p ≥ 2 {p\geq 2} and the optimal condition n < p + 4 ⁢ p p - 1 {n<p+\frac{4p}{p-1}} holds. When p ∈ ( 1 , 2 ) {p\in(1,2)} , we obtain the same result under the nonsharp assumption n < 5 ⁢ p {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 {p=2} in the optimal range n < 10 {n<10} .