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Nonlinear Dirac equations with Coulomb-type potentials and nonlinearities resonating at essential spectrums

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  15 April 2018

Shaowei Chen  and
Shuai Jiang
Shaowei Chen
Affiliation:
School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, People's Republic of China (swchen6@163.com; beijingren123@163.com)
Shuai Jiang
Affiliation:
School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, People's Republic of China (swchen6@163.com; beijingren123@163.com)
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Abstract

We consider the nonlinear Dirac equation

The potential function V satisfies the conditions that the essential spectrum of the Dirac operator is and this Dirac operator has infinitely many eigenvalues in (−1, 1) accumulating at 1. This potential function V may change sign in ℝ3 and contains the classical Coulomb potential V (x) = −γ/|x| with γ > 0 as a special case. The nonlinearity F satisfies the resonance-type condition lim. Under some additional conditions on V and F, we prove that this equation has infinitely many solutions.

Keywords

Dirac equationsresonant problemssign-changing potentialsCoulomb-type potentials

MSC classification

Secondary: 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 4 , August 2018 , pp. 713 - 730
Copyright
Copyright © Royal Society of Edinburgh 2018 

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