Existence And Concentration Of Solutions For Chern-Simons-Schr?dinger System

In this paper,the existence of non-radial solutions and the existence and con-centration of radial sign-changing solutions for Chern-Simons-Schrodinger system in H~1(R~2)or H_r~1(R~2)are studied by using the variational method and some analytical techniques.Firstly,we study the following Chern-Simons-Schrodinger system in Hr1(R~2(?)(0.0.1)where the parameters ?,?>0,hu(s)=1/2integral from n=0 to S ru~2(r)dr and f is the nonlinearity.In Chapter 2 and Chapter 3,we mainly consider the existence and concentration of the ground state radial sign-changing solution of the system(0.0.1).In Chapter 2,we assume that f satisfies asymptotically 5-linear growth and certain monotonic-ity conditions.By using the variational method,combining constrained minimiza-tion arguments and the quantitative deformation lemma,we prove the existence of ground state radial sign-changing solution for system(0.0.1),which changes sign ex-actly once.Further,we obtain that,as ??0,0,the ground state radial sign-changing solution of system(0.0.1)strongly converges to the ground state radial sign-changing solution of the following Schrodinger equation-?u+?u=f(u)in R~2.(0.0.2)In Chapter 3,we consider that.f satisfies a class of critical exponential growth in R~2.By using the constrained minimization method and Trudinger-Moser inequality,we obtain the existence and concentration of ground state radial sign-changing solution for system(0.0.1).Secondly,we investigate the following Chern-Simons-Schr?dinger system in H~1(R~2)(?)(0.0.3)where V is coercive potential and may be sign-changing.In Chapter 4,the existence of nontrivial solutions for system(0.0.3)are obtained via the local link theorem when f satisfies 5-superlinear growth and generalized subcritical growth conditions.