  We study the Dirichlet problem on a bounded convex domain of $\mathbb R^N$,
with zero boundary data, for truncated Laplacians ${\mathcal P}_k^\pm$, with
$k<N$. We establish a necessary and sufficient condition (Theorem 1) in terms
of the "flatness" of domains for existence of a solution for general
inhomogeneous term. This result, in particular, shows that the strict convexity
of the domain is sufficient for the solvability of the Dirichlet problem. The
result and related ideas are applied to the solvability of the Dirichlet
problem for the operator ${\mathcal P}_k^+$ with lower order term when the
domain is strictly convex and the existence of principal eigenfunctions for the
operator ${\mathcal P}_1^+$. An existence theorem is presented with regard to
the principal eigenvalue for the Dirichlet problem with zero-th order term for
the operator ${\mathcal P}_1^+$. A nonexistence result is established for the
operator ${\mathcal P}_k^+$ with first order term when the domain has a
boundary portion which is nearly flat. Furthermore, when the domain is a ball,
we study the Dirichlet problem, with a constant inhomogeneous term and a
possibly sign-changing first order term, and the associated eigenvalue problem.
