MULTIPLICITY RESULTS FOR DIFFERENTIAL EQUATIONS WITH SYMMETRY (BORSUK-ULAM THEORY)

Motivated by the usefulness of both the classical ZZ('2) and S('1) versions of the Borsuk-Ulam theorem in establishing multiplicity results for certain differential equations with appropriate symmetry, we present a generalized ZZ('p) Borsuk-Ulam theorem. In its simplest form the theorem states that for (OMEGA) a bounded, open, ZZ('p) invariant neighborhood of the origin in (//C)('n), if f is a continuous equivariant map of (PAR-DIFF)(OMEGA) into (//C)('m), where m < n, then f must vanish somewhere. The proof offered is most tractable using only degree theory and the transversality lemma.;linear Dirichlet problem with a ZZ('p) symmetry for a system of func- tions u = (u('1),...,u('2N)) defined in a bounded domain (OMEGA), with smooth boundary, in (//R)('n). (UNFORMATTED TABLE FOLLOWS).;(DELTA)u + F(,u)(u) = 0 in (OMEGA).;u = 0 on (PAR-DIFF)(OMEGA).;This topological result is used with a min-max argument to give the next result in partial differential equations. Consider the non-.;(TABLE ENDS).;With appropriate growth conditions on F, this system has infinitely many solutions.;The generalized ZZ('p) Borsuk-Ulam theorem is essential in the development of the following result--a ZZ('p) index theory. Similar to the S('1) index of Benci, this index is a map from ZZ('p) invariant sets to the extended positive integers including zero. This is accomplished through another continuous map with a particular equivariant prop- erty mapping these ZZ('p) invariant sets into (//C)('n)(FDIAG) 0 ; the index being the dimension of the lowest dimensional range for which such a map exists. A special case of the ZZ('p) index is the Ljusternik-Schnirelmann category theory. A very natural application of the ZZ('p) index to the problem of finding subharmonic solutions with minimal period for a nonautonomous nonlinear Hamiltonian system is cited.;The final result uses the Mountain Pass Lemma to prove exist- ence of a positive solution for a nonlinear Laplace equation with.;boundary conditions of mixed type on a bounded domain (OMEGA), with smooth boundary, in (//R)('n). (UNFORMATTED TABLE FOLLOWS).;(DELTA)u + f(x,u) = 0 in (OMEGA).;(PAR-DIFF)u/(PAR-DIFF)(nu) + a(x)u = 0 on (PAR-DIFF)(OMEGA).;(TABLE ENDS).;Here a(x) is a positive continuous function and appropriate growth conditions are assumed for f.