Existence and asymptotics of ground states to the nonlinear Dirac equation with Coulomb potential

Article type: Research Article

Authors: Wang, Tianfang^a | Zhang, Wen^{b; c; d} | Zhang, Jian^{b; c; d; *}

Affiliations: [a] School of Educational Science, Baotou Teachers’ College, Baotou, 014030 Inner Mongolia, P.R. China. E-mail: [email protected] | [b] College of Science, Hunan University of Technology and Business, Changsha, 410205 Hunan, P.R. China. E-mails: [email protected], [email protected] | [c] Key Laboratory of Hunan Province for Statistical Learning and Intelligent Computation, Hunan University of Technology and Business, Changsha, 410205 Hunan, P.R. China | [d] Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, 200585 Craiova, Romania

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: In this paper we study the Dirac equation with Coulomb potential −iα·∇u+aβu−μ|x|u=f(x,|u|)u,x∈R3 where a is a positive constant, μ is a positive parameter, α=(α1,α2,α3), αi and β are 4×4 Pauli–Dirac matrices. The Dirac operator is unbounded from below and above so the associate energy functional is strongly indefinite. Under some suitable conditions, we prove that the problem possesses a ground state solution which is exponentially decay, and the least energy has continuous dependence about μ. Moreover, we are able to obtain the asymptotic property of ground state solution as μ→0+, this result can characterize some relationship of the above problem between μ>0 and μ=0.

Keywords: Dirac equation, Coulomb potential, asymptotic property, ground state solutions

DOI: 10.3233/ASY-211748

Journal: Asymptotic Analysis, vol. 130, no. 1-2, pp. 187-212, 2022