On The Solutions To A Class Of Quasilinear Elliptic Equations

This paper is concerned with the following type of quasilinear elliptic equations:-?pu+V(x)|u|p-2u-?p(|u|2)u=|u|q-2u,X?RN,where 1<p<N,p*=Np/N-p,the potential V(x)is a continuous function.When 2p<q<2p*,we prove that the equation has nontrival solutions by Mountain Pass Lemma.When q? 2p*,we prove that the equation has no nontrival solutions by the Pohozaev identity.Firstly,we introduce the background of these equations and the researching situation about Pohozaev identities.Then,we derive some important lemma,such as Pohozaev identity,the estimation of transform and so on.At last,in Section 3,we prove that the equation has nontrival solutions by Mountain Pass Lemma and the equation has no nontrival solutions by the Pohozaev identity.