Existence Of Solutions For Elliptic System In The Whole Space

Elliptic equations and systems,as an important research object and theoretical tool,have been widely used in research of mathematica,physics,engineering,etc.In fact,there are many problem in the natural science in relation to elliptic equations and systems,such as geometry,elasticity,fluid mechanics,variational methods,etc.Therefore,many experts and scholars have conducted in-depth research on elliptic equations and elliptic systems.In this paper,we study the existence of solutions of two classes of elliptic system.First of all,we study the existence of solutions of the following nonlinear Schrodinger system with unbounded potential where the constants N?3,?>1,?>1,?+??(2,2*),where 2*denotes the critical Sobolev exponent,that is 2*=2N/(N-2).We assume that q(x):R~N?R is continuous and satisfies (?).By using the method of minimization of Nehari manifolds,we can prove that the above system have nontrivial solutions.Secondly,we study the existence of solutions of the following nonlinear Schrodinger system,where the constants N?3,?>1,?>1,?+??(2,2*),where 2*denotes the critical Sobolev exponent,that is 2*=2N/(N-2).We assume that Q(x)?C(R~N),satisfies Q(x)(?)1 and (?),By using the mountain pass theorem,we can prove that the above system have nontrivial solutions.