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On semi-classical limits of ground states of a nonlinear Maxwell-Dirac system. (English) Zbl 1297.35197

Summary: We study the semi-classical ground states of the nonlinear Maxwell-Dirac system: \[ \begin{cases}\alpha\cdot(i\hslash\nabla+q(x)\mathbf A(x))w-a\beta w-\omega w-q(x)\phi(x)w=P(x)g(|w|) w\\-\Delta\phi =q(x)|w|^2\\ -\Delta A_k=q(x)(\alpha _kw)\cdot\bar w\quad k=1,2,3\end{cases} \] for \(x\in\mathbb R^3\), where \(\mathbf A\) is the magnetic field, \(\phi \) is the electron field and \(q\) describes the changing pointwise charge distribution. We develop a variational method to establish the existence of least energy solutions for \(\hslash\) small. We also describe the concentration behavior of the solutions as \(\hslash\to 0\).

MSC:

35Q40 PDEs in connection with quantum mechanics
49J35 Existence of solutions for minimax problems
35A15 Variational methods applied to PDEs
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