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In this paper, we study the following Klein–Gordon–Maxwell system − Δ u + ( λ a ( x ) + 1 ) u − ( 2 ω + ϕ ) ϕ u = f ( x , u ) , in R 3 , − Δ ϕ = − ( ω + ϕ ) u 2 , in R 3 . Using variational methods, we obtain the existence of ground state solutions under some appropriate assumptions on a and f .

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.

In this paper, we obtained the existence of infinitely many large energy solutions for the Schrodinger–Poisson system with concave and convex nonlinearities by using Z 2 − Mountain Pass Theorem in critical point theory. Some recent results from the literature are sharply improved and extended.

In this paper we study the following nonhomogeneous Klein–Gordon–Maxwell equations { − △ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = f ( x , u ) + h ( x ) , in R 3 , − △ ϕ + ϕ u 2 = − ω u 2 , in R 3 . Under appropriate assumptions on V , f and h , the existence of multiple solutions is proved via Variational methods.

In this paper, we consider the higher-order Wong–Zakai approximations of the non-autonomous stochastic reaction–diffusion equation driven by additive/multiplicative white noises. The solutions between the approximation equation and stochastic reaction–diffusion equation are compared in higher-order spaces, in terms of the initial data. Based on these results and the known L 2 -upper semi-continuity, we prove that the random attractor of the approximation random system converges to that of the non-autonomous stochastic reaction–diffusion equation with additive/multiplicative white noises in L p ( R N ) ∩ H 1 ( R N ) when the size of the approximation shrinks to zero.

In this paper we study the following nonhomogeneous Kirchhoff equation − ( a + b ∫ R N | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( x , u ) + h ( x ) , in R N , where f satisfies the Ambrosetti–Rabinowitz type condition. Under appropriate assumptions on V , f and h , the existence of multiple solutions is proved by using the Ekeland’s variational principle and the Mountain Pass Theorem in critical point theory.

In this paper we prove the existence of multiple solutions for the following nonhomogeneous Klein–Gordon equation coupled with Born–Infeld theory, { − Δ u + [ m 2 − ( ω + ϕ ) 2 ] u = | u | q − 2 u + h ( x ) , in R 3 , Δ ϕ + β Δ 4 ϕ = 4 π ( ω + ϕ ) u 2 , in R 3 , when 2 q 6 .

In this paper we study the existence of infinitely many large energy solutions for the superlinear Schrodinger–Maxwell equations { − Δ u + V ( x ) u + ϕ u = f ( x , u ) , in R 3 , − Δ ϕ = u 2 , in R 3 , via the Fountain Theorem in critical point theory.

Periodic solutions and infinitely distinct subharmonic solutions are obtained for a class of nonconvex and nonautonomous superquadratic Hamiltonian systems z = J H z (z,t) by using the minimax methods in critical point theory.