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

Abstract In this article, we study the following system: − Δ u + V ( x ) 2 ω ϕ = λ f ∣ 4 , in width="1em" mathvariant="double-struck">R 3 β π columnalign="left" \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 ){u}^{2},& \end{array}\right. where xmlns:m="http://www.w3.org/1998/Math/MathML"> f\left(u) is without any growth and Ambrosetti-Rabinowitz condition. We use a cut-off function Moser iteration to obtain existence of nontrivial solution. Finally, as by-product our approaches, same result for Klein-Gordon-Maxwell system.