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+iλA0 and Dk=(?)xk-iλAk,k=1,2,for(x1,x2,t)∈R2,1 are covariant derivatives,λ 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 λ>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 λ*>0 such that for 0<λ<λ*,problem(0.0.2)has two nontrivial static solutions(Ψλ,A0λ,A1λ,A2λ).Moreover,we show there also exists λ>0 such that if λ>λ,problem(0.0.2)has no nontrivial solutions.