Ground State Solutions And Multiple Solutions Of Chern-simons-schr(?)dinger System

In this paper,by using the variational method,we study the following Chern-Simons-Schrodinger system where(?)for x=(x1,x2)?R2,Aj:R2?R isthe gauge field for j=0,1,2,??R,V(x)?C1(R2,R)and f?C(R,R).Firstly,we consider system(0.0.1)with ?=1,V(x)=0 in radial space Hr1(R2),namely considering the solutions to system(0.0.1)of the form where u,k,h are real value functions depending only |x|.Then u satisfies where h(s)=1/2?0 s ru2(r)dr,f?C(R,R).When f satisfies asymptotically 3-linear at infinity,we obtain that system(0.0.2)exists ground state solutions by using general minimax principle.Moreover,if f is odd,we obtain the multiplicity of solutions by symmetric mountain pass approach.Secondly,we consider the normalized solutions of system(0.0.1)with non-constant potential V(x)and general 3-superlinear nonlinearity f.Motivated by Jeanjean[L.Jeanjean,Nonlinear Anal.1997],we construct a Nehari-Pohozaev-Palais-Smale sequence to prove the boundedness of sequence.Under the conditions of V(x),we obtain that the energy functional of system(0.0.1)satisfies Palais-Smale condition.Thus,we obtain the existence of normalized solutions.Finally,we prove that system(0.0.1)exists ground state normalized solutions by taking minimizing sequence.