On the Existence of Solutions of the Dirichlet Problem for the $p$-Laplacian on Riemannian Manifolds

We obtain a criterion for the existence of solutions of the problem
$$ \Delta_p u=0 \quad\text{in}\quad M \setminus \partial M,\qquad u|_{\partial M}=h $$
with a bounded Dirichlet integral, where $M$ is an oriented complete Riemannian manifold with boundary and $h \in W_{p,\mathrm{loc}}^1 (M)$, $p > 1$.