Existence Of Ground State Solutions For A Class Of Quasilinear Elliptic Equations In The Heisenberg Group

Heisenberg group plays an important role in representation theory,partial d-ifferential equation,number theory,harmonic analysis and quantum mechanics.In this article,we mainly use the variational method and the principle of concentrated compactness to study the existence of ground state solutions for a class of quasilinear elliptic equations.The thesis is divided into two sections according to contents.In chapter one,we introduce the research background and research status of the quasilinear elliptic equation on a class of equations involving asymptotic autonomy in the Heisenberg group.And the framework of Heisenberg group,such as group algorithm,norm,scaling map is given.The horizontal gradient,natural inner prod-uct,horizontal divergence,Kohn Spencer Laplace operator on Heisenberg group are also defined.In chapter two,we research the existence of the ground state solution of this kind of quasilinear elliptic autonomy equations-?H,pu+V(z)|u|p-2u=f(z,u),z?HN where f:R?R is continuous,V,f are periodic potential functions.We using the Ambrosetti-Rabinowitz and subcritical of the nonlinearity f to prove that the functional satisfies the geometric structure of mountain road theorem,then we use the principle of concentration compactness to deal with the problem of lack of com-pactness.Finally,the existence of ground state solution is proved.