Regularity of radial extremal solutions for some non local semilinear equations

Abstract

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