Existence of Solutions to the Second Boundary-Value Problem for the $$p$$-Laplacian on Riemannian Manifolds
Vadim Vadimovich Brovkin, Andrej Alexandrovich Kon'kov · Mathematical Notes · 2021
We obtain necessary and sufficient conditions for the existence of solutions to the boundary-value problem $$ \Delta_p u=f\quad\text{on}\quad M,\qquad | abla u|^{p-2}\,\frac {\partial u}{\partial u}\bigg|_{\partial M}=h, $$ where $$p > 1$$ is a real number, $$M$$ is a connected oriented complete Riemannian manifold with boundary, and $$ u$$ is the outer normal vector to $$\partial M$$ .