INTRODUCTION

The object of this paper is to develop an additive, integer-valued

degree theory for quasilinear Fredholm mappings, and to use this theory to

study existence, multiplicity and bifurcation problems for solutions of

fully nonlinear elliptic partial differential equations with general

boundary conditions of Shapiro-Lopatinskij type.

Let X and Y be real Banach spaces, and X be another Banach space

in which X is embedded compactly. Let $n(X,Y) denote the subset of

L(X,Y) consisting of operators which are Fredholm of index zero. A mapping

f : X—Y is called quasilinear Fredholm provided that f may be

represented as f(x) = L(x)x + C(x) for x in X, where (1) C: X X

is compact and (2) L is the restriction to X of a continuous mapping

L: X $ (X,Y). Quasilinear Fredholm mappings were introduced by

Snirel'man [Sn] in his study of the nonlinear Riemann-Hilbert problem.

Another typical situation in which quasilinear Fredholm maps arise quite

naturally is in the study of the Dirichlet problem for quasilinear elliptic

equations. However, what is more interesting is that fully nonlinear

elliptic operators with general nonlinear elliptic boundary conditions

induce quasilinear Fredholm maps between appropriate function spaces,

provided that the "coefficients" are sufficiently smooth.

In [Sn], a rudimentary form of degree is defined for quasilinear

Fredholm mappings by approximating such mappings by vector bundle morphisms

and then reducing the definition to the intersection number of a propertly

defined section of this bundle with the zero-section. Owing to various

choices which are inherent to its construction, the degree in [Sn] is

*Research supported by the CNR (Italy) and a NATO Research Grant

ReceivedbytheeditorJune 11, 1990.

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