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

Abstract

We 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 $n\ge p+4p/(p-1)$. 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 $f\in C^1$ whenever $p\ge 2$ and the optimal condition $n<p+4p/(p-1)$ holds. When $p\in (1,2)$, we obtain the same result under the non-sharp 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 $n<10$.

Adv. Calc. Var. 2020 https://doi.org/10.1515/acv-2020-0055. Errors in Theorem 1.8, Proposition A.3, and Step 1-Case 1 of the Proof of Theorem 1.1 corrected and commented. They do not change the validity of our main results in the previous and printed versions of the article

References in corpus (1)

• Boundedness of stable solutions to nonlinear equations involving the $p$-Laplacian