Positive Solutions For Schr(?)dinger-Poisson System With Critical Exponent

First, we consider one kind of Schr（?）dinger-Poisson equations system: where u∈H¹（R³）,1< q< 2,λ>0, and k≠0. Assume that h（x） satisfies the following condition:In the critical growth conditions, by the Mountain Pass Theorem and variational methods, we can obtain the following result:Theorem 1 Suppose that 1<q<2,λ>0, k≠0 and （H₀） holds. Then there exists λ^*> 0 such that problem （SP） has at least one positive solutions （u,φ_u） ∈H¹（R³） x D^1,2（R³） for λ ∈（0,λ^*）.Second, we consider the other kind of Schr（?）dinger-Poisson system: where 1<q<2 and λ> 0. We will make the following assumptions on h and l:In the critical growth conditions, by the Mountain Pass Theorem, variation-al methods and concentration compactness principle, we can obtain the following result:Theorem 2 Assume 1< q< 2, the hypotheses （H₁） and （H₂） hold. Then there exists λ^*> 0 such that problem （P） has at least two positive solutions （u,φ_u）∈ D¹,2（R³）×D^1,2（R³） for λ ∈ （0,λ^*）.