Journal of the
Korean Mathematical Society JKMS

We prove that for any continuous function $f$ on the $s$-harmonic $(1 < s < \infty)$ boundary of a complete Riemannian manifold $M$, there exists a solution, which is a limit of a sequence of bounded energy finite solutions in the sense of supremum norm, for a certain elliptic operator $\mathcal A$ on $M$ whose boundary value at each $s$-harmonic boundary point coincides with that of $f$.
If $E_1, E_2, \ldots, E_l$ are $s$-nonparabolic ends of $M$, then we also prove that there is a one to one correspondence between the set of bounded energy finite solutions for $\mathcal A$ on $M$ and the Cartesian product of the sets of bounded energy finite solutions for $\mathcal A$ on $E_i$ which vanish at the boundary $\partial E_i$ for $i =1,2,\ldots,l$.

Keywords: $s$-harmonic boundary, $\mathcal A$-harmonic function, end

MSC numbers: Primary 58J05, 31B05