Existence and asymptotics of ground states to the nonlinear Dirac equation with Coulomb potential. (English) Zbl 1498.35459
Summary: In this paper we study the Dirac equation with Coulomb potential
\[
-i \alpha \cdot \nabla u + a \beta u - \frac{\mu}{| x |} u = f ( x , | u | ) u, \quad x \in \mathbb{R}^3
\]
where \(a\) is a positive constant, \( \mu\) is a positive parameter, \( \alpha = ( \alpha_1 , \alpha_2 , \alpha_3)\), \( \alpha_i\) and \(\beta\) are \(4 \times 4\) 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 \(\mu \). Moreover, we are able to obtain the asymptotic property of ground state solution as \(\mu \rightarrow 0^+\), this result can characterize some relationship of the above problem between \(\mu > 0\) and \(\mu = 0\).
MSC:
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
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