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

We obtain some new criteria for bifurcation of solutions of general boundary value problems for nonlinear systems of elliptic partial differential equations which are rather different 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. In fact they are computed explicitly from the coefficients without any need of solving the linearized equations. Moreover, as opposite to the local bifurcation invariants they are stable under lower order perturbations.