On the existence of solutions of the Dirichlet problem for $p$-Laplacian on Riemannian manifolds

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