On critical Klein–Gordon–Maxwell systems with super-linear nonlinearities. (July 2020) - Tang, Xianhua; Wen, Lixi; Chen, Sitong
Abstract: This paper is concerned with two classes of critical Klein–Gordon–Maxwell systems as follows − Δ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = μ f ( u ) + u 5, x ∈ R 3, Δ ϕ = ( ω + ϕ ) u 2, x ∈ R 3 and − ε 2 Δ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = g ( x, u ) + K ( x ) u 5, x ∈ R 3, Δ ϕ = ( ω + ϕ ) u 2, x ∈ R 3, where μ > 0, ε > 0, V, K ∈ C ( R 3, R + ), f ∈ C ( R, R ) and g ∈ C ( R 3 × R, R ) are super-linear at infinity. Instead of the expression " μ sufficiently large" in the existing works, we give a certain range μ ≥ μ 0 under which the first system admits a ground state solution when V is positive and periodic. Under some mild conditions on V, K and g, we show that the second system has at least one nontrivial solution provided ε ∈ ( 0, ε 0 ], where the bound ε 0 > 0 is expressed explicitly in terms of V, K and g .