A proof of existence of particle-like solutions of Einstein Dirac equations

We prove existence of a ground state particle-like solution to the Einstein-Dirac equations: a b '= 1&parl0;r Ar&parr0; -wTr +m&parl0;A r&parr0; wT r-m&parl0; Ar&parr0; -1&parl0;rA r&parr0; ˙a b A'r= 1-Ar r-16pwr T2r a2r+b 2r T'r=T rA r-1 2rAr -16pwT3r &parl0;a2r +b2r&parr0; 2rAr +32pT2r ab2r2 Ar +16pmT2r a2r -b2r 2rAr By a ground state particle-like solution we mean a solution of the above equations that satisfies the constraints: limr→infinity r1-Ar <infinitylim r-infinityTr <infinity0infinity &parl0;a2r +b2r&parr0; TAdr<infinity .