Nontrivial solution for Klein-Gordon equation coupled with Born-Infeld theory with critical growth

Abstract

In this article, we study the following system: −Δu+V(x)u−(2ω+ϕ)ϕu=λf(u)+∣u∣4u,inR3,Δϕ+βΔ4ϕ=4π(ω+ϕ)u2,inR3,\left\{\begin{array}{ll}-\Delta u+V\left(x)u-\left(2\omega +\phi )\phi u=\lambda f\left(u)+| u{| }^{4}u,& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{1em}{{\mathbb{R}}}^{3},\\ \Delta \phi +\beta {\Delta }_{4}\phi =4\pi \left(\omega +\phi ){u}^{2},& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{1em}{{\mathbb{R}}}^{3},\\ \end{array}\right. where f(u)f\left(u) is without any growth and Ambrosetti-Rabinowitz condition. We use a cut-off function and Moser iteration to obtain the existence of nontrivial solution. Finally, as a by-product of our approaches, we obtain the same result for the Klein-Gordon-Maxwell system.

Keywords

• klein-gordon-born-infeld system
• moser iteration
• mountain pass theorem
• 35a15
• 35b33
• 35j60