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

## TL;DR

This paper establishes a criterion for the existence of solutions to the p-Laplacian Dirichlet problem on complete Riemannian manifolds with boundary, focusing on solutions with bounded Dirichlet integral.

## Contribution

It provides a new criterion for the existence of solutions to the p-Laplacian Dirichlet problem on Riemannian manifolds with boundary, extending previous results.

## Key findings

- Derived a criterion for solution existence
- Applied to manifolds with boundary
- Focused on solutions with bounded Dirichlet integral

## Abstract

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 $h \in W_{p, loc}^1 (M)$, $p > 1$.