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 ε>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 uε∈H1(R2).Moreover,our results show that,as ε→0,the global maximum point xε of uε must concentrate at the global minimum point x0 of V.