Topological methods for non-linear elliptic equations of arbitrary order
Felix E. Browder · Pacific Journal of Mathematics · 1966
Consider a strongly elliptic nonlinear partial differential equation (e): F(x,u,Du, ,D 2m u) = 0, of order 2ra on a bounded, smoothly bounded subset Ω of R n .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 Holder 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. Theorem 1 is our general result of Leray-Schauder type for the most general sort of strongly elliptic nonlinear equation. Its proof is