Existence and asymptotics of ground states to the nonlinear Dirac equation with Coulomb potential

In this paper we study the Dirac equation with Coulomb potential -i alpha center dot del u + alpha beta u - mu/vertical bar x vertical bar u = integral(x, vertical bar u vertical bar)u, x is an element of R-3 where a is a positive constant, mu is a positive parameter, alpha = (alpha(1), alpha(2), alpha(3)), alpha i and beta are 4 x 4 Pauli-Dirac matrices. The Dirac operator is unbounded from below and above so the associate energy functional is strongly indefinite. Under some suitable conditions, we prove that the problem possesses a ground state solution which is exponentially decay, and the least energy has continuous dependence about mu. Moreover, we are able to obtain the asymptotic property of ground state solution as mu -> 0(+), this result can characterize some relationship of the above problem between mu > 0 and mu = 0.