Ground State Solutions For Schr?dinger-Poisson System With Various Potentials

In this paper,by using the variational method,we study the following Schr?dinger-Poisson system(?)where V?C(R3,R)and g?C(R,R)and V(x)is potential function.Firstly,we consider problem(0.0.1)with V(x)=1,namely(?)where g is a continuous function and superlinear at zero,and g(t)t>3G(t)>0 with G(t)=?0t g(s)ds,t ? R.We establish the existence of ground state solutions of(0.0.2),by using a condition lim|t|??g(t)/|t|2=v with(?)<v<+?,instead of the usual 2-superlinear condition lim|t|??G(t)/|t|3=+?.Secondly,we consider a class of critical growth problem(0.0.1)with steep po-tential well,that is(?)where q ?(3,6)and ?,? are two positive constants,and ?a(x)represents a po-tential well whose depth is controlled by ?.?a(x)is called a steep potential well if ? large.Combining the theory of minimax structure to construct the Nehari-Poho(?)aev-Palais-Smale and Poho(?)aev identity,we derive the boundedness of se-quence.and then we use the characteristics of the steep potential to establish the existence of nontrivial solutions of(0.0.3).We prove that problem(0.0.3)has at least a ground state solution.Moreover,the concentration behavior of the ground state solution is also described as ???,i.e.,as ???,the ground state solution-s of(0.0.3)weakly converge to the solutions of the following Schr?dinger-Poisson system(?)...