Solvability of Linear and Quasilinear Elliptic Boundary Value Problems via the A-Proper Mapping Theory
W. V. Petryshyn · Numerical Functional Analysis and Optimization · 1980
In this paper we use the theory of A-proper mappings and their uniform limits to obtain in a simple way general variational approximation-solvability and/or existence theorems for quasilinear elliptic BVProblems in divergence form of order 2m. In Section 1 we first show how our simple approach is used to obtain constructive solvability of linear elliptic equations under a somewhat different definition of strong ellipticity condition which allows us to dispense with the usage of the Garding inequality. In view of this, the same approach is also used in Section 2 to establish the constructive solvability of nonlinear BVProblems when the leading part satisfies a nonlinear version of the strong ellipticity condition used here. This result is then used to establish a new existence result (see Theorem 2.3) for a not necessarily coercive quasilinear elliptic BVProblem which properly includes earlier results of Višik, Browder, Pohožayev, Leray and Lions and others. In Section 3 some special cases of our abstract results are stated. These are applicable to ODE's and PDE's which are not in divergence form.