Consider a strongly elliptic
nonlinear partial differential equation (e): F(x,u,Du,⋯,D2mu) = 0, of order 2m on a
bounded, smoothly bounded subset Ω of Rn. For second-order operators, Leray
and Schauder, using the theory of the topological degree for completely
continuous displacements of a Banach space, showed that the existence of
solutions of the Dirichlet problem for (e) could be proved under the assumption
of suitable a-priori bounds for solutions of the type of (e). In the present
paper, using precise results on the solutions of linear elliptic differential
operators with Hölder continuous coefficients as well as a variant of the
Leray-Schauder method, we extend this result to equations of arbitrary even order.
We also obtain results on uniqueness in the large under hypotheses of local
uniqueness.
