EXISTENCE THEOREMS FOR NONLINEAR

PARTIAL DIFFERENTIAL EQUATIONS

FELIX E. BROWDER

PART I

Introduction. It is my purpose in the present discussion to describe the applica-

tion of some recently developed general techniques in nonlinear functional

analysis to the study of the existence theory for solutions of various classes of

nonlinear partial differential equations, specifically nonlinear elliptic boundary

value problems, nonlinear elliptic eigenvalue problems, and nonlinear equations

of evolution of parabolic and wave equation type. Corresponding to these three

classes of problems, we shall consider three related abstract theories: the theory

of nonlinear operators of monotone type from a reflexive Banach space X to its

conjugate space X*, the application of the Lusternik-Schnirelman category in

infinite dimensional spaces, and the theory of semigroups of nonlinear non-

expansive operators in Hilbert and Banach spaces.

Our partial differential equations will be defined for the sake of simplicity on

an open subset G of an Euclidean n-dimensional space

Rn

rather than upon a

manifold and we shall speak for convenience of notation of a single equation

rather than a system of equations. We shall make no use, however, of the

maximum principle for second order scalar elliptic or parabolic equations or of

the Di Giorgi-Nash estimates or their consequences so that there is no conceptual

loss of generality in the restriction.

The points of the open set G will be denoted by x = (xx, ...,x„) and the

elementary differential operators by

Da

= flj= 1

(^/^xjTJ f°r a n

ordered n-tuple

a = (xl9..., a„) of nonnegative integers, with the order of the operator

Da

being

written as |a| = Yj=i

a

/ To write nonlinear partial differential operators in a

convenient form, we introduce the vector space

RSm

whose elements are

£ m

=

{£x |

|a|

^

m} anc*

divide each such £ into two parts £m = ((, rj) where

f

=

{fyl |y5| m — 1} e

RSmi

is the lower order part of £m, and ( = {£a| |a| = m}

is the part of £, corresponding to the mth derivatives. In this notation, a general

(nonlinear) partial differential operator of order m is defined by a mapping

F.G x

RSm

-+

R1

and defines an operator on functions u on G which assigns to

each such, w, another function v on G with

v(x) = F(x, £m(u)(x)), xeG,

where £»(* ) =

{Dau(x)\

|a| m}.

The systematic study of boundary value problems for nonlinear partial differen-

tial equations goes back as far at least as the work of Sergei Bernstein in the first

decade of the twentieth century who made a determined and difficult attack

upon second-order nonlinear elliptic equations in the plane using perturbation

l

http://dx.doi.org/10.1090/pspum/016/0269962