p-Laplacian problem in a Riemannian manifold

This paper is divided into two parts. First, we will prove the existence of solutions of the p-Laplacian equation in the Riemannian manifold in the space \({\mathcal {H}}^{\alpha ,p}_{loc}({\mathcal {N}})\). On the other hand, we will give a criterion to obtain a positive lower bound for \(\lambda _{1,p}(\Omega )\), where is a bounded domain \(\Omega \subset {\mathcal {N}}\). In the first result, we do not consider a bounded subset on the Riemannian manifold \({\mathcal {N}}\).