Semiclassical states of nonlinear Dirac equations with degenerate potential

  • Xu Zhang
  • Zhi-Qiang WangEmail author


In this paper, we study the following nonlinear Dirac equation
$$\begin{aligned} -\,i\varepsilon \alpha \cdot \nabla u+a\beta u+V(x)u=|u|^{p-2}u,\ x\in {\mathbb {R}}^3, \quad \mathrm{for}\ u\in H^1({\mathbb {R}}^3, {\mathbb {C}}^4), \end{aligned}$$
where \(p\in (2,3)\), \(a > 0\) is a constant, \(\alpha =(\alpha _1,\alpha _2,\alpha _3)\), \(\alpha _1,\alpha _2,\alpha _3\) and \(\beta \) are \(4\times 4\) Pauli–Dirac matrices. Our investigation focuses on the case in which |V(x)| may approach a as \(|x|\rightarrow \infty \). This is a degenerate case as most works in the literature assume a strict gap condition \(\sup _{x\in {\mathbb {R}}^3} |V(x)|< a\), which is a key condition used in setting up an infinitely dimensional topological linking structure as well as in dealing with the compactness issues of the variational formulation. Under the assumption that V has a local trapping potential well, for \(\varepsilon >0\) small, we construct bound state solutions concentrating around the local minimum points of V. As a consequence we construct an infinite sequence of localized bound state solutions as \(\varepsilon \rightarrow 0\).


Dirac equation Degenerate potential Semiclassical states Concentration 

Mathematics Subject Classification

35Q40 49J35 



Research was supported by the Specialized Fund for the Doctoral Program of Higher Education of China, NSFC 11771324 and a Simons Collaboration Grant.


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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaChina
  2. 2.School of Mathematics and InformaticsFujian Normal UniversityFuzhouChina
  3. 3.Department of Mathematics and StatisticsUtah State UniversityLoganUSA

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