Index bundle, Leray-Schauder reduction and bifurcation of solutions of nonlinear elliptic boundary value problems

Abstract

We show that a family $F_p;\ p\in P$ of nonlinear elliptic boundary value problems of index $0$ parametrized by a compact manifold admits a reduction to a family of compact vector fields parametrized by $P$ if and only if its index bundle $\text{\rm Ind}F$ vanishes. Our second conclusion is that, in the presence of bounds for the solutions of the boundary value problem, the non vanishing of the image of the index bundle under generalized $J$-homomorphism produces restrictions on the possible values of the degree of $F_p$. The most striking manifestation of this arises when the first Stiefel-Whitney class of the index bundle is nontrivial. In this case, the degree of $F_p$ must vanish! From this we obtain a number of corollaries about bifurcation from infinity for solutions of nonlinear elliptic boundary value problems.