Semi-classical limits of ground states of a nonlinear Dirac equation

We study the semi-classical states of the following nonlinear Dirac equation−iℏ∑k=13αk∂kw+aβw+V(x)w=W(x)g(|w|)w for x∈R3x∈R3 where the nonlinearity is of superlinear and subcritical growth as |w|→∞|w|→∞. The Dirac operator is unbounded from below and above so the associate energy functional is strongly indefinite. We develop an argument to establish the existence of least energy solutions for ℏ   small. We also describe the concentration phenomena of the solutions as ℏ→0ℏ→0.