Existence of nontrivial solutions for the Klein-Gordon-Maxwell system with Berestycki-Lions conditions

Abstract

Abstract In this article, we study the following Klein-Gordon-Maxwell system: − Δ u − ( 2 ω + ϕ ) ϕ u = g ( u ) , in R 3 , Δ ϕ = ( ω + ϕ ) u 2 , in R 3 , \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}-\Delta u-\left(2\omega +\phi )\phi u=g\left(u),\hspace{1.0em}{\rm{in}}\hspace{1em}{{\mathbb{R}}}^{3},\hspace{1.0em}\\ \Delta \phi =\left(\omega +\phi ){u}^{2},\hspace{1.0em}{\rm{in}}\hspace{1em}{{\mathbb{R}}}^{3},\hspace{1.0em}\end{array}\right. where ω \omega is a constant that stands for the phase; u u and ϕ \phi are unknowns and g g satisfies the Berestycki-Lions condition [Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313–345; Nonlinear scalar field equations. II. Existence of infinitelymany solutions, Arch. Rational Mech. Anal. 82 (1983), 347–375]. The Klein-Gordon-Maxwell system is a model describing solitary waves for the nonlinear Klein-Gordon equation interacting with an electromagnetic field. By using variational methods and some analysis techniques, the existence of positive solution and multiple solutions can be obtained. Moreover, we study the properties of decay estimates and asymptotic behavior for the positive solution.