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 α>0,such that |f(t)≤Cαeαt2 for all t≥0(f3)there exists μ>4 such that f(t)t≥μF(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.