Vol. 17, No. 1, 1966

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Topological methods for non-linear elliptic equations of arbitrary order

Felix Earl Browder

Vol. 17 (1966), No. 1, 17–31

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.

Mathematical Subject Classification
Primary: 35.47
Received: 23 October 1964
Published: 1 April 1966
Felix Earl Browder