Nonexistence and existence of nontrivial solutions for Klein–Gordon–Maxwell systems with competing nonlinearities

Abstract

Abstract In this paper, we study the following nonlinear Klein–Gordon–Maxwell system: {−Δu+u−(2ω+ϕ)ϕu=λa(x)|u|r−2u−b(x)|u|q−2u,x∈R3,Δϕ=(ω+ϕ)u2,x∈R3,(P) $$\begin{aligned} \textstyle\begin{cases} -\Delta u+ u-(2\omega +\phi )\phi u =\lambda a(x) \vert u \vert ^{r-2}u-b(x) \vert u \vert ^{q-2}u, & x\in \mathbb{R}^{3},\\ \Delta \phi = (\omega +\phi )u^{2}, & x\in \mathbb{R}^{3}, \end{cases}\displaystyle \hspace{75pt}(\mathrm{P}) \end{aligned}$$ where ω is a positive constant, q>2 $q>2$, r∈(2,min{6,q}) $r\in (2,\min \{6,q\})$, a∈L66−r(R3) $a\in L^{\frac{6}{6-r}}(\mathbb{R}^{3})$ is a positive potential, b∈Lloc1(R3) $b\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{3})$ is also a positive potential. Under some integrability assumption on aq−2br−2 $\frac{a^{q-2}}{b^{r-2}}$, nonexistence and existence results are obtained depending on λ via variational methods.

Keywords

• Klein–Gordon–Maxwell system
• Competing nonlinearities
• Ekeland’s variational principle
• Mountain pass theorem
• Variational methods