The existence of multiple solutions for the Klein–Gordon equation with concave and convex nonlinearities coupled with Born–Infeld theory on R3. (December 2017) - Chen, Shang-Jie; Song, Shu-Zhi
Abstract: In this paper, we prove the existence of multiple solutions for the following Klein–Gordon equation with concave and convex nonlinearities coupled with Born–Infeld theory { − Δ u + a ( x ) u − ( 2 ω + ϕ ) ϕ u = λ k ( x ) | u | q − 2 u + g ( x ) | u | p − 2 u, x ∈ R 3, Δ ϕ + β Δ 4 ϕ = 4 π ( ω + ϕ ) u 2, x ∈ R 3, where 1 < q < 2 < p < 6 . Under appropriate assumptions on a, k, g and λ, the existence of multiple nontrivial solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory.