Ground State Solutions For Non-autonomous Schr?dinger-Poisson Systems And Critical Schr?dinger Equations

In the present paper,we study non-autonomous Schr?dinger-Poisson systems and a class of critical Schr?dinger equations.Using variational principle and some analysis tricksjwe prove the existence of ground state solutions for relevant problems.Firstly,we investigate the following Schr?dinger-Poisson system (?) where 3?p?4.K(x),Q(x)are nonnegative functions satisfying some asymptotic property and some other assumptions.Applying Nehari manifold and Pohozaev iden-tity,a bounded Nehari-Pohozaev-Palais-Smale sequence is constructed.We obtain that system(P)admits at least a ground state solution.Secondly,we consider the system(P)again,where 4<p<6.K(x)is a nonneg-ative function.limsup/|x|?? Q(x)may not be equal to lim inf|x|?? Q(x)such that lim/|x|?? Q(x)may not exist.Q(x)may be indefinite in sign and Q satisfies some other conditions.By using Nehari manifold,Ekeland's variational principle and Lagrange multiplier rule,the existence of negative ground state solution for system(P)is proved.Finally,we study Schr?dinger equation with critical Sobolev exponent:-?u+u=?|u|p-2u+L(|x|)|u|2*-2u,X?RN,where N?>3,2<p<2*=2N/N-2,? is a positive real number.We don't need to assume that L(x)is periodic or asymptotically periodic.Applying Nehari manifold,Ekeland s variational principle and Lagrange multiplier rule,we obtain that the equation admits at least a positive ground state radial solution.