On the existence of solutions of the second boundary value problem for $p$-Laplacian on Riemannian manifolds

TL;DR

This paper establishes necessary and sufficient conditions for the existence of solutions to a second boundary value problem involving the p-Laplacian on Riemannian manifolds, extending the understanding of nonlinear PDEs in geometric contexts.

Contribution

It provides a comprehensive characterization of solution existence for the p-Laplacian boundary value problem on Riemannian manifolds, a significant extension of classical PDE theory.

Findings

Derived necessary and sufficient conditions for solutions.

Extended existence theory to Riemannian manifolds.

Addressed boundary conditions involving the p-Laplacian.

Abstract

We obtain necessary and sufficient existence conditions for solutions of the boundary value problem $$ \Delta_p u = f \quad \mbox{on } M, \quad \left. \left| \nabla u \right|^{p - 2} \frac{\partial u}{\partial \nu} \right|_{ \partial M } = h, $$ where $p > 1$ is a real number, $M$ is a connected oriented complete Riemannian manifold with boundary, and $\nu$ is the external normal vector to $\partial M$.