Existence And Limit Behavior Of Nonlocal Minimization Problems With Steep Potential

We consider the following non-local Hartree type Minimization Problemswhere the energy functional E?(u)is defined by?>0 is a parameter.Under some suitable assumptions on potential h(x)we prove that there exists a threshold N*>17,independent of ?h(x),such that if N?N*,then e?(N)has no minimizers for any ?>0;if 0<N<N*there exists a constant ?*(N),such that e?(N)still has no minimizers for 0<?<?*(N),while e?(N)admits minimizers for ?>?*(N).For any given 0<N<N*,we also investigate limit behavior of non negative minimizers of e?(N)as ???.