Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian

TL;DR

This paper proves boundedness and regularity of stable, radially decreasing solutions to fractional Laplacian equations in certain dimensions, extending known results for classical Laplacian to fractional cases.

Contribution

It establishes $L^ abla$ bounds for stable solutions of fractional Laplacian equations in specific dimensions, including all $s o(0,1)$, and applies to nonlinearities $f ext{ in } C^2$.

Findings

Boundedness of stable solutions in dimensions $2 leq n < 2(s+2+\sqrt{2(s+1)})$.

Regularity results for extremal solutions when $f$ is scaled by $\lambda > 0$.

Applicable to all nonlinearities $f ext{ in } C^2$.

Abstract

We study the regularity of stable solutions to the problem $$ \left\{ \begin{array}{rcll} (-\Delta)^s u &=& f(u) & \text{in} \quad B_1\,, u &\equiv&0 & \text{in} \quad \mathbb R^n\setminus B_1\,, \end{array} \right. $$ where $s\in(0,1)$. Our main result establishes an $L^\infty$ bound for stable and radially decreasing $H^s$ solutions to this problem in dimensions $2 \leq n < 2(s+2+\sqrt{2(s+1)})$. In particular, this estimate holds for all $s\in(0,1)$ in dimensions $2 \leq n\leq 6$. It applies to all nonlinearities $f\in C^2$. For such parameters $s$ and $n$, our result leads to the regularity of the extremal solution when $f$ is replaced by $\lambda f$ with $\lambda > 0$. This is a widely studied question for $s=1$, which is still largely open in the nonradial case both for $s=1$ and $s<1$.