ORIENTATION AND DEGREE FOR NONLINEAR BOUNDARY VALUE PROBLEMS 11

~ . . .- k 2k+s-m.+3/2

•:

irK s

z9(n)

- HS

^(n)

x n

H

*

(an), (1.7)

i=l

which is quasilinear Fredholm provided that the Fredholm index of the first

variation of (BVP)is zero. If f€C ' and each g.€C ij , and

0 y 1, we formulate (BVP)as the zero of the mapping

2k+2 r 2 if

k 2 + 2 k

"

m

i^

i=l

which also is quasilinear Fredholm, if its first variation has Fredholm

index 0.

Once the solutions of (BVP) have been formulated as the zeroes of

(1.7) or of (1.8), the degree may be used to study existence and multi-

plicity questions for (BVP), and bifurcation and continuation questions

for parametrized families of such (BVP)*s.

Before outlining the contents of the sections, we make a few brief

general comments on nonlinear functional analytic methods in the treatment

of nonlinear elliptic boundary value problems. The Leray-Schauder degree

is defined for compact vector fields which are nonvanishing on the boundary

of an open bounded set. Already in Sections IV and V of [Le-Sc], Leray

and Schauder explicitly considered the formulation of partial differential

equations as solutions of an operator equation G(x,x) = 0, where for fixed

x € X, the equation G(x,») = 0 has a unique solution C(x) and themap

C is compact. Moreover, in [Le-Sc] this diagonal representation is used

to consider not only the Dirichlet problem for second-order quasilinear

elliptic problems, but also fully nonlinear elliptic second-order