  In these notes we study the Dirichlet problem for critical points of a convex
functional of the form \[ F(u)=\int_{\Omega}\phi\left( \left\vert \nabla
u\right\vert \right) , \] where $\Omega$ is a bounded domain of a complete
Riemannian manifold $\mathcal{M}.$ We also study the asymptotic Dirichlet
problem when $\Omega=\mathcal{M}$ is a Cartan-Hadamard manifold. Our aim is to
present a unified approach to this problem which comprises the classical
examples of the $p-$Laplacian ($\phi(s)=s^{p}$, $p>1)$ and the minimal surface
equation ($\phi(s)=\sqrt{1+s^{2}}$). Our approach does not use the direct
method of the Calculus of Variations which seems to be common in the case of
the $p-$Laplacian. Instead, we use the classical method of a-priori $C^{1}$
estimates of smooth solutions of the Euler-Lagrange equation. These estimates
are obtained by a coordinate free calculus. Degenerate elliptic equations like
the $p-$Laplacian are dealt with by an approximation argument.
  These notes address mainly researchers and graduate students interested in
elliptic partial differential equations on Riemannian manifolds and may serve
as a material for corresponding courses and seminars.
