  We prove the existence of viscosity solutions for fractional semilinear
elliptic PDEs on open balls with bounded exterior condition in dimension $d\geq
1$. Our approach relies on a tree-based probabilistic representation based on a
(2s)-stable branching processes for all $s\in (0,1)$, and our existence results
hold for sufficiently small exterior conditions and nonlinearity coefficients.
In comparison with existing approaches, we consider a wide class of polynomial
nonlinearities without imposing upper bounds on their maximal degree or number
of terms. Numerical illustrations are provided in large dimensions.
