Solutions To The Chern-Simons-Schr(?)dinger System With External Potential

In this paper,we consider the existence of static solutions to the nonlinear Chern-Simons-Schr(?)dinger system (?)with an external potential V(x),where D0=(?)t+i?A0 and Dk=(?)xk-i?Ak,k=1,2,for(x1,x2,t)?R2,1 are covariant derivatives,? is the coupling number.In the first chapter,we introduce some associated physical background and preliminary knowl-edge.In the second chapter,we consider the case of p>4,Suppose that V(x)satisfies lim|x|?? V(x)=+? and x·?V(x)?0,we show problem(0.0.2)admits a mountain pass solution for all ?>0.In the third chapter we discuss the case of 2<p<4,Suppose that V(x)satisfies lim|x|?? V(x)=+?,we show that there exists ?*>0 such that for 0<?<?*,problem(0.0.2)has two nontrivial static solutions(??,A0?,A1?,A2?).Moreover,we show there also exists ?>0 such that if ?>?,problem(0.0.2)has no nontrivial solutions.