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.