Regularity of radial extremal solutions for some non local semilinear equations

TL;DR

This paper studies the regularity and existence of stable solutions to fractional Laplacian equations with superlinear nonlinearities in a bounded domain, focusing on how solutions behave depending on the parameter lambda.

Contribution

It provides new insights into the regularity of radial extremal solutions for nonlocal semilinear equations involving the fractional Laplacian.

Findings

Regularity results for stable solutions depending on lambda

Existence criteria for solutions based on parameters

Analysis of extremal solutions in fractional elliptic equations

Abstract

We investigate stable solutions of elliptic equations of the type \begin{equation*} \left \{ \begin{aligned} (-\Delta)^s u&=\lambda f(u) \qquad {\mbox{ in $B_1 \subset \R^{n}$}} \\ u&= 0 \qquad{\mbox{ on $\partial B_1$,}}\end{aligned}\right . \end{equation*} where $n\ge2$, $s \in (0,1)$, $\lambda \geq 0$ and $f$ is any smooth positive superlinear function. The operator $(-\Delta)^s$ stands for the fractional Laplacian, a pseudo-differential operator of order $2s$. According to the value of $\lambda$, we study the existence and regularity of weak solutions $u$.