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

Abstract

We obtain necessary and sufficient existence conditions for solutions of the boundary value problem $$ Δ_p u = f \quad \mbox{on } M, \quad \left. \left| \nabla u \right|^{p - 2} \frac{\partial u}{\partial ν} \right|_{ \partial M } = h, $$ 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 $\partial M$.