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??R,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 ??R,2 ? p ? 6,|u|⁴u and ?|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 ??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?.