Existence And Multiplicity Results For The Choquard Equation In Exterior Domain

In this paper,we show that the following Choquard equation:(?)has at least catQ(?)positive solutions under certain condition on 2N-?/N<q<2N-?/N-2 if ?>?0 for some ?0=?0(q)>0,where ?(?)RN,N?3,is an unbounded domain with the boundary(?)??0,and the set RN\? is bounded.This thesis is divided into four chapters.In chapter 1,we introduce the background of the problem.In chapter 2,we give some preliminaries and obtain the infimum of the energy functional associated with(0.0.2)are equal in ? and RN.In chapter 3,we derive from Brezis-Lieb lemma and the vanish lemma the global compactness result.In chapter 4,we use the barycenter map to establish the relationship between the level set on the Nehari manifold and the region ?,and then,using the Lusternik-Schnirelman theorem we prove that the(0.0.2)has at least cat?(?)positive solutions.