On the concentration of semiclassical states for nonlinear Dirac equations

Xu Zhang

doi:10.3934/dcds.2018238

Article Contents

2018, Volume 38, Issue 11: 5389-5413. Doi: 10.3934/dcds.2018238

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On the concentration of semiclassical states for nonlinear Dirac equations

Xu Zhang^,

School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China

^* Corresponding author: Xu Zhang
^* Corresponding author: Xu Zhang

Received: June 30, 2017

Revised: November 30, 2017

Published: August 2018

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

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Abstract

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 $\varepsilon \to 0$.

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

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