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