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 $u$.
Categories: math.AP