Nonlinear elliptic boundary value problems. II
Felix E. Browder · Transactions of the American Mathematical Society · 1965
In a preceding paper on nonlinear elliptic boundary value problems [7], the writer established an existence theorem for variational solutions of nonlinear elliptic boundary value problems for systems of the form Au= 2 D*Ax(x,u,-,Dmu) MS"with Ax having at most polynomial growth.This theorem was derived from an abstract theorem concerning the solvability of a class of nonlinear functional equations in reflexive Banach spaces.Our result in [7] extended and generalized earlier results anonunced by M. I. Vishik [20], [21], [22] and obtained by more concrete-analytic arguments.Very recently Vishik has published in [23] a detailed account of his methods and obtained more precise results than those announced in his Notes listed above.The one feature of the results of [23] which goes beyond the framework of the methods given in [7] (and one on which Vishik has laid great emphasis) is that the monotonicity or strong ellipticity hypotheses imposed on the system A involve essentially only the variation of the Ax with respect to Dmu and not with respect to the lower-order derivatives of u.It is our object in the present discussion to give an extension of our methods which allows us to obtain results under weaker hypotheses of this type.As in [7], our approach is based on a general theorem on nonlinear functional equations in Banach spaces.In §1, we formulate our main results on the solvability of nonlinear elliptic boundary value problems and the corresponding abstract theorem.In §2, we prove the abstract theorem.In §3, we prove our main theorem on the existence of solutions of boundary value problems.In §4, we consider extensions and specializations of this theorem.In §5, we turn back to the abstract theory and analyze the general method applied in [7] and here in the general context of locally convex linear spaces.1. Let Í2 be a bounded and smoothly bounded open subset of R".The general