Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth

Abstract

Abstract This work is concerned with the following Klein-Gordon-Maxwell system: − Δ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = f ( u ) , x ∈ R 2 , Δ ϕ = ( ω + ϕ ) u 2 , x ∈ R 2 , \left\{\begin{array}{ll}-\Delta u+V\left(x)u-\left(2\omega +\phi )\phi u=f\left(u),\hspace{1.0em}& x\in {{\mathbb{R}}}^{2},\\ \Delta \phi =\left(\omega +\phi ){u}^{2},\hspace{1.0em}& x\in {{\mathbb{R}}}^{2},\end{array}\right. where ω > 0 \omega \gt 0 is a constant, u , ϕ : R 2 → R u,\phi :{{\mathbb{R}}}^{2}\to {\mathbb{R}} , V ∈ C ( R 2 , R ) V\in {\mathcal{C}}\left({{\mathbb{R}}}^{2},{\mathbb{R}}) , and f ∈ C ( R , R ) f\in {\mathcal{C}}\left({\mathbb{R}},{\mathbb{R}}) obeys exponential critical growth in the sense of the Trudinger-Moser inequality. We give some new sufficient conditions on f f , specifically related to exponential growth, to obtain the existence of nontrivial solutions. Our results improve and extend the previous results. In particular, we give a more precise estimation than the ones in the existing literature about the minimax level.