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

Abstract

In this paper we study the Dirac equation with Coulomb potential $$\begin{array}{r}-\mathit{i}\mathit{\alpha }·\nabla \mathit{u}+\mathit{a}\mathit{\beta }\mathit{u}-{\displaystyle \frac{\mathit{\mu }}{|\mathit{x}|}}\mathit{u}=\mathit{f}\mathrm{(}\mathit{x}\mathrm{,}|\mathit{u}|\mathrm{)}\mathit{u}\mathrm{,}\mathit{x}\in {\mathbb{R}}^3\end{array}$$ where a is a positive constant, μ is a positive parameter, $\mathit{\alpha }=\mathrm{(}{\mathit{\alpha }}_1\mathrm{,}{\mathit{\alpha }}_2\mathrm{,}{\mathit{\alpha }}_3\mathrm{)}$ , ${\mathit{\alpha }}_{\mathit{i}}$ and β are $4\times 4$ 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 μ. Moreover, we are able to obtain the asymptotic property of ground state solution as $\mathit{\mu }\to 0^{+}$ , this result can characterize some relationship of the above problem between $\mathit{\mu }\mathrm{>}0$ and $\mathit{\mu }=0$ .