ON SEMICLASSICAL GROUND STATES OF A NONLINEAR DIRAC EQUATION

This paper is concerned with existence and concentration phenomena of semiclassical solutions of the following nonlinear Dirac equation -i (h) over bar Sigma(3)(k=1) alpha(k)partial derivative(k)u + a beta u + V(x)u - Sigma(J)(j=1) W-j(x)vertical bar u vertical bar(pj-2)u for x is an element of R-3 where p(j) is an element of ( 2, 3) are subcritical. Under some conditions, we show that there are two families of semiclassical solutions, for (h) over bar > 0 small, with the least energy, one concentrating on the set of minimal points of V and another on the set of maximal points of W-j. Both are exponentially decay as vertical bar x vertical bar -> infinity.