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

Jacobo Pejsachowicz

DOI: http://dx.doi.org/10.12775/TMNA.2001.033

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.

Keywords


Bifurcation; index bundle; fredholm operators; Leray-Schauder degree; nonlinear elliptic BVP

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