The family index theorem and bifurcation of solutions of nonlinear elliptic bvp

Details The family index theorem and bifurcation of solutions of nonlinear elliptic bvp The family index theorem and bifurcation of solutions of nonlinear elliptic bvp

2011-07-08

arxiv.org/abs/1107.1727v3

Mathematics

Analysis of PDEs

20 pages, Corrections in sections 2,4 and references. Added Section 5 (example) and an Appendix. To appear on JDE

Scientific paper

We obtain some new bifurcation criteria for solutions of general boundary value problems for nonlinear elliptic systems of partial differential equations. The results are of different nature from the ones that can be obtained via the traditional Lyapunov-Schmidt reduction. Our sufficient conditions for bifurcation are derived from the Atiyah-Singer family index theorem and therefore they depend only on the coefficients of derivatives of leading order of the linearized differential operators. They are computed explicitly from the coefficients without the need of solving the linearized equations. Moreover, they are stable under lower order perturbations.

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