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

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.

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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