Positive Solutions Of Critical Schr(?)dinger-Poisson Systems

In this paper,we use the variational methods to study the following Schr（?）dinger-Poisson system:where 0≠λR,f:R³ × R→R,the functions K（x）and M（u）= ∫⁰_u m（s）ds are nonnegative and nonzero,and for any given u ∈ D^1,2（R³）,the function K（x）M（u）∈L^6/5（R³）.Firstly,we consider one kind of Schr（?）dinger-Poisson system with two critical exponents,where λ∈R,2 ≤ p ≤ 6,|u|⁴u and φ|u|³u are local and nonlocal Sobolev critical terms respectively.By combining the variational methods,Mountain Pass theo-rem,strong maximun principle and Pohoz（?）ev identity,we prove the existence and nonexistence results for nontrivial solutions of system（SP₁）.Secondly,we study the following Schr（?）dinger-Poisson system which involves two critical exponents but no perturbation terms,where μ∈R.By using the variational methods,minimization techniques,Lagrange multiplier rule and Pohoz（?）ev identity,we prove the existence and nonexistence re-sults for nontrivial solutions of system（P₁）.At last,we study the following modified critical Schr（?）dinger-Poisson system:where V,K,g are asymptotically periodic functions of x and g ∈C（R³ × R,R）.By combining the variational methods,Mountain Pass theorem,the Concentration-compactness principle and the strong maximum principle,we prove the existence of a positive ground state solution of system（SP₂）.