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The concentration behavior of ground state solutions for nonlinear Dirac equation. (English) Zbl 1435.35325

Authors’ abstract: In this paper, we study the following nonlinear Dirac equation \[ -i \varepsilon \sum_{k=1}^3 \alpha_k \partial_k u +a \beta u + V(x)u=f(|u|)u, \text{ in } \mathbb{R}^3, \] where \(\varepsilon\) is a small parameter, \(a > 0\) is a constant, \(\alpha_1\), \(\alpha_2\), \(\alpha_3\) and \(\beta\) are \(4 \times 4\) Pauli-Dirac matrices, \(V\), \(K\) and \(f\) are continuous but are not necessarily of class \(\mathcal{C}^1\). We prove the existence of ground state solution by using variational methods, and we determine a concrete set related to the potentials \(V\) and \(K\) as the concentration position of these ground state solutions as \(\varepsilon \rightarrow 0\). Moreover, we consider some properties of these ground state solutions, such as convergence and exponential decay estimate. The result presented in this paper generalizes the result in [Y. Ding and X. Liu, J. Differ. Equations 252, No. 9, 4962–4987 (2012; Zbl 1236.35133)].

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
35Q40 PDEs in connection with quantum mechanics
49J35 Existence of solutions for minimax problems
81V10 Electromagnetic interaction; quantum electrodynamics
83C10 Equations of motion in general relativity and gravitational theory

Citations:

Zbl 1236.35133
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References:

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