A GENERALIZATION OF THE LERAY-SCHAUDER INDEX FORMULA AND AN EXISTENCE THEOREM FOR THE LICHNEROWICZ EQUATION.

摘要

My thesis consists of two parts; the first deals with the Leray-Schauder degree theory and provides the following generalization of their index formula:;deg (F,(OMEGA),0) = (chi)((zeta)) (1).;where (zeta) is the vector bundle with base M and fibre B/Range DF(x), X((zeta)) its Euler characteristic, and (kappa) the number of eigenvalues (lamda) < 0 of DF(x).;Corollary. If the eigenvalue (lamda) = 0 has geometric multiplicity equal to m, then (1) reads:;Let B be a Banach space F:(OMEGA) (L-HOOK) B (--->) B; (OMEGA) open, F of the form "identity plus a compact operator", and F (NOT=) 0 on (PAR-DIFF)(OMEGA). Assume that (OMEGA) (INTERSECT) F('-1)(0) = M('m), a manifold of dimension m (necessarily finite), and that the null space of DF(x) is exactly equal to TM('m) for all x(epsilon)M. Then.;deg (F,(OMEGA),0) = (-1)('(kappa)) (chi)(M).;The second part of my thesis deals with nonlinear elliptic equations in unbounded domains. Some partial results extend existence theorems of Chaljub-Simon and Choquet-Bruhat for the Lichnerowicz equation in IR('n):;8(DELTA)(phi) - R(phi) + a(phi)('-7) + b(phi)('-3) + c(phi)('5) = 0.;(phi)(TURN)1 near (INFIN).;These authors dealt with the case.;R,a,b,c, (LESSTHEQ) c/(1+(VBAR)x(VBAR)) ('2+(lamda)), (lamda) > 0.;My results replace these with weaker assumptions, asking rather that these functions can be written as linear combinations of derivatives of functions which decay as c/(1+(VBAR)x(VBAR))('1+(lamda)), (lamda) > 0 (the functions R,a,b,c may, in fact, grow provided that they oscillate rapidly near (INFIN)).