The Existence And Multiplicity Of Solutions For Chern-Simons-Schr(?)dinger Systems

In this paper,by using the variational method,we study the following Chern-Simons-Schrodinger systemwhere(?)for x=(x2,x2)? R2,Aj:R2?R(j=0,1,2)is the gauge field and the nonlinearity f? C(R2 × R,R).We assume the following conditions on nonlinearity f:(F1)f?C(R,R)satisfies limt?0f(t)/t=0(f2)there exists Ca for any ?>0,such that |f(t)?C?e?t2 for all t?0(f3)there exists ?>4 such that f(t)t??F(t)>0 for all t ? R,where F(t)=?0t f(s)ds,(f4)f(t)is odd.Firstly,we consider ground state solutions for Chern-Simons-Schrodinger sys-tems in H1(R2).We use Trudinger-Moser inequality and combine the Minimax principle to construct the Nehari-Pohozaev-Palais-Smale and so we get the bound-edness of sequence and the existence of nontrivial solutions of problem(0.0.2).By the standard method,we prove that problem(0.0.2)has at least a ground state solution.Secondly,based on the previous work,we study infinitely many high energy radial solutions for Chern-Simons-Schrodinger systems in Hr1(R2).If(f1)-(f3)hold,and we assume that the nonlinear term f is odd,by using Symmetric mountain pass,we obtain that problem(0.0.2)have infinitely many high energy radial solutions.