Existence Of Solutions For Klein-gordon-Born-infeld System And Schr?dinger-bopp-Podolsky System

In this article,we mainly use the mountain pass theorem,minimization method and other nonlinear analysis tools to study the existence of solutions for two kinds of Klein-Gordon-Born-Infeld system and Schr?dinger-Bopp-Podolsky system with strong physical background.Firstly,consider the following Klein-Gordon-Born-Infeld system:where 2<p<2*,ω>0 is a frequency parameter,m,μ is a real constant andβ>0,u,φ>:R3→R,Δ4φ=div(|▽φ|2▽φ).The mountain pass theorem is used to prove the existence of the ground state solution.Secondly,consider the following Klein-Gordon-Born-Infeld system without growth condition of nonlinear term:where ω>0 is a constant,u,φ:R3→R,V(x):R3→R is a coercive potential,f(u)without any growth and(AR)conditions.A new system obtained by cut-off function is used to prove the existence of a nontrivial solution to this system,and then Moser iteration method is used to prove that this nontrivial solution is still a nontrivial solution of Klein-Gordon-Born-Infeld system.Finally,consider the following Schrodinger-Bopp-Podolsky system:where ω>0 is a constant,p∈(2,10/3),u,φ:R3→R.The minimization sequence is used to prove that the system satisfies strong subadditivity conditions to overcome the compactness problem of the minimization sequence,and then the existence of normal solutions is proved.Later we prove the orbital stability for Schrodinger-Bopp-Podolsky system.