Existence And Concentration Of Semi-classical Ground State Solutions For Chern-simons-schr(?)dinger System

In this paper,we study the following Chern-Simons-Schr(?)dinger system where ?>0 is a small parameter,V is the external potential,(?)for(x1,x2)?R2,Ai(i=0,1,2)is the gauge field and f is a superlinear nonlinear term.V and f satisfy the following conditions(V)V?C1(R2,R),0<V0=infx?R2 V(x)<V?=lim inf|x|?? V(x),(f'1)lim s?0 f(s)/s=0 and there exist constant C>0 and q?(2,+?)such that|f(s)|?C(1+|s|q-1),for all s?R,(f'2)there exists p?(5,+?)such that lim|s|?? f(s)/s=+?,(f'3)f?C(R,R)is odd and the function s(?)(fs)/s5 is nondecreasing on(0,?).When u?Hr1(R2),the above equation becomes where h(s)=?0s1/2u2(l)dl,V and f satisfy the following conditions(V1)V?C1(R2,R),0<V0?V(|x|)?V(x),(V2)lim inf|x|?? V(x)=V??(0,+?),(f1)f?C(R2,R)is odd and there exist constants C>0 and q?(2,+?)such that|f(s)|?C(1+|s|q-1),for all s?R,(f2)(?)f(s)/s=0 and(?)F(s)/s6=+?,where F(s)=?0s f(t)dt,(f3)s(?)f(s)/s5 is nondecreasing on(0,?).Firstly,we prove that there is a positive ground solution for the Chern-Simons-Schr(?)dinger system in Hr1(R2)by the Nehari manifold.Then,by using variational methods and analytic technique,we prove that this system possesses a positive ground state solution u??H1(R2).Moreover,our results show that,as ??0,the global maximum point x? of u? must concentrate at the global minimum point x0 of V.