Nontrivial Solutions Of Klein-Gordon-Maxwell Systems With Exponential Growth In R~2

The Klein-Gordon-Maxwell system is derived from the electromagnetic field model in quantum mechanics,which is used to describe the problem of solitary waves generated by the interaction between the nonlinear Klein-Gordon field and Maxwell field,and has a strong mathematical physics background.In recent years,this system has received extensive attention from scholars at home and abroad,but it is worth noting that in the existing literature,most of the research is on the case of N≥3,but there are very few studies on the two-dimensional case.In this paper,the variational method is used to study the existence of non-trivial solutions of Klein-Gordon-Maxwell system in two-dimensional space,and the related results in the existing literature are generalized and improved.The full text is divided into three chapters,the main contents are as follows:Firstly,it briefly introduces the research background and significance of Klein-GordonMaxwell system,the research status and the research problems of this paper,and gives the corresponding preliminary knowledge at the same time.Secondly,Next,consider the following Klein-Gordon-Maxwell system where ω>0 is a constant,u,φ,V,G:R2→R,f:R→R and the nonlinear term f has subcritical exponential growth.With the help of the arbitrariness of the parameter α in Trudinger-Moser-type subcritical exponential growth,the existence of non-trivial solutions of the system is obtained under weaker assumptions compared with previous work.Finally consider the following Klein-Gordon-Maxwell system where ω>0 is a constant,u,φ,V:R2→R,f:R→R,and the nonlinear term f with critical exponential growth,the existence of non-trivial solutions of the above system is proved under relatively loose conditions.In particular,in order to overcome the difficulty of missing compactness caused by critical growth,we establish some new inequalities,which give new estimates of the minimax level value of the energy functional.