Stable solutions to fractional semilinear equations: uniqueness, classification, and approximation results

TL;DR

This paper investigates stable solutions to fractional semilinear equations, proving uniqueness, classification, and approximation results, and establishing interior regularity in low dimensions for the half-Laplacian.

Contribution

It provides new uniqueness and classification theorems for stable solutions and demonstrates approximation by regular solutions, extending regularity results to fractional equations.

Findings

Stable solutions are unique and classifiable under certain conditions.

Weak stable solutions can be approximated by bounded, regular solutions.

Interior regularity is established for solutions in dimensions 1 to 4.

Abstract

We study stable solutions to fractional semilinear equations $(-\Delta)^s u = f(u)$ in $\Omega \subset \mathbb{R}^n$, for convex nonlinearities $f$, and under the Dirichlet exterior condition $u=g$ in $\mathbb{R}^n \setminus \Omega$ with general $g$. We establish a uniqueness and a classification result, and we show that weak (energy) stable solutions can be approximated by a sequence of bounded (and hence regular) stable solutions to similar problems. As an application of our results, we establish the interior regularity of weak (energy) stable solutions to the problem for the half-Laplacian in dimensions $1 \leq n \leq 4$.