A GROUND STATE SOLUTION TO THE CHERN-SIMONS-SCHRODINGER SYSTEM

In this paper,we consider the Chern-Simons-Schrodinger system{−Δu+[e^(2)|A|^(2)+(V(x)+2eA_(0))+2(1+κq/2)N]u+q|u|^(p−2)u=0,−ΔN+κ^(2)q^(2)N+q(1+κq2)u^(2)=0,κ(∂_(1)A_(2)−∂_(2)A_(1))=−eu^(2),∂_(1)A_(1)+∂_(2)A_(2)=0,κ∂_(1)A_(0)=e^(2)A_(2)u^(2),κ∂_(2)A_(0)=−e^(2)A_(1)u^(2),where u∈H^(1)(R^(2)),p∈(2,4),Aα:R^(2)→R are the components of the gauge potential(α=0,1,2),N:R^(2)→R is a neutral scalar field,V(x)is a potential function,the parametersκ,q>0 represent the Chern-Simons coupling constant and the Maxwell coupling constant,respectively,and e>0 is the coupling constant.In this paper,the truncation function is used to deal with a neutral scalar field and a gauge field in the Chern-Simons-Schrödinger problem.The ground state solution of the problem(P)is obtained by using the variational method.