On multiplicity of solutions to nonlinear Dirac equation with local super-quadratic growth

Abstract

In this article, we study the following nonlinear Dirac equation: −iα⋅∇u+aβu+V(x)u=g(x,∣u∣)u,x∈R3.-i\alpha \hspace{0.33em}\cdot \hspace{0.33em}\nabla u+a\beta u+V\left(x)u=g\left(x,| u| )u,\hspace{1em}x\in {{\mathbb{R}}}^{3}. The Dirac operator is unbounded from below and above so the associated energy functional is strongly indefinite. Compared with some existing issues, the most interesting problem in this article is that we assume that the nonlinearity satisfies a local super-quadratic growth condition, which is weaker than the usual global version. This case allows the nonlinearity to be super-quadratic at some domains and asymptotically quadratic at other domains. Using variational tools together with deformation lemma, genus theory, and some special techniques, we prove the existence of ground-state solutions and infinitely many geometrically distinct solutions under local super-quadratic growth condition. Our results complement and extend the related results of Bartsch and Ding [Solutions of nonlinear Dirac equations, J. Differ. Equ. 226 (2006), 210–249] and of Chen et al. [Ground state solutions for the Dirac equation with periodic external force field, Sci. Sin. Math. (Chinese) 54 (2024), 747–764].

Keywords

• dirac equation
• local super-quadratic condition
• ground-state solutions; multiplicity
• 35j50
• 35q40
• 49j35