Existence Of A Positive Ground State Solution For Quasilinear Asymptotically Periodic Schr(?)dinger Equation With Critical Exponent

In this master thesis, we pay an attention on the existence of solutions for the following quasilinear Schr???dinger equation with Sobolev critical exponent-?u + V?x?u - ??|u|²?u = g?x,u? + K?x?|u|^22*-2u,x?RN,where N ? 3,V, K and g are asymptotically periodic in x. Note that,22*= ?4N?/?N-2?corresponds to the critical exponent for the above critical problem.In chapter one, the background for quasilinear Schr???dinger equation are pre-sented. Then the paper's framework and main results are also summarized.In chapter two, some notations and preparatory results are stated.In chapter three, based on some existing results for quasilinear Schr???dinger equations, the existence of a solution for the above critical problem are studied. By using a change of variables, the quasilinear Schr???dinger equation is converted into a semilinear one. With the variational method, we establish the existence of a positive ground state solution for original problem.