A probabilistic approach tо a nonlinear differential equation on a Riemannian manifold

We investigate the minimal solution of the problem
\begin{gather*} Lu=u^\alpha в D, u=f на O, \end{gather*}
where $1\le\alpha\le2$, $D$ is an open subset f a Riemannian manifold, O is a regular relatively, open subset of $\partial D$, and $f$ is a mapping from $\partial D$ to $[0,\infty]$ which is continuous on $O$ and vanishes on $\partial D\setminus O$. An explicit formula for such a solution is given in terms of the $(L,\alpha)$-superdiffusion.