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On the concentration of semiclassical states for nonlinear Dirac equations

  • * Corresponding author: Xu Zhang

    * Corresponding author: Xu Zhang

The first author is supported by the China Postdoctoral Science Foundation 2017M611160

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  • In this paper, we study the following nonlinear Dirac equation

    $\begin{equation*}-i\varepsilonα·\nabla w+aβ w+V(x)w = g(|w|)w, \ x∈ \mathbb{R}^3, \ {\rm for}\ w∈ H^1(\mathbb R^3, \mathbb C^4), \end{equation*}$

    where $a > 0$ is a constant, $α = (α_1, α_2, α_3)$, $α_1, α_2, α_3$ and $β$ are $4×4$ Pauli-Dirac matrices. Under the assumptions that $V$ and $g$ are continuous but are not necessarily of class $C^1$, when $g$ is super-linear growth at infinity we obtain the existence of semiclassical solutions, which converge to the least energy solutions of its limit problem as $\varepsilon \to 0$.

    Mathematics Subject Classification: Primary: 35Q40; Secondary: 49J35.

    Citation:

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