Research On The Existence Of Solutions For A Class Of Schrodinger Poisson System With Asymptotically Periodic Potential

This paper is dedicated to studying the following Schrodinger-Poisson system with critical Sobolev exponent{-△u + V(x)u + φu = K(x)f(u)+ |u|5,in R3,{-△φ=u2,in R3,where V(x)is asymptotically periodic potential,K(x)is a positive continuous potential.f is continuous function.We employ the Mountain Pass Theorem,concentration-compactness and Nehari manifold approach to obtain a ground s-tate solution in two cases:the periodic one and the asymptotically periodic case,by introducing a weaker condition that there exists θ0 ∈(0,1)such that K(x)[f(T)/T3-f(tT)/(tT)3]sign(1-t)+θ0V(x)|1-t2|/(tT)2≥0,(?)x∈R3,t>0,T ≠0 instead of the usual Nehari-type monotonic condition on f(t)/t|3.