The Existence Of Ground State Solutions For Two Classes Of Elliptic Partial Differential Equations Under Variational Framework

In this paper,we use variational method to study the existence of ground state solutions for a class of Choquard equations with general nonlinearities and a class of Gross-Pitaevskii equations in trapped dipolar quantum gases inR~2.We mainly use the mountain pass theorem,the general minimax principle and the concentration compactness principle to prove the main results.In Chapter 1,we introduce the background and the research status of the two types of equations studied in this paper.Moreover,we will give the main results,some required notation and structural arrangements.In Chapter 2,we study the existence of ground state solutions for a class of Choquard equations with general nonlinearities.We mainly use the general minimax principle to construct a special bounded(PS)sequence which is related to the Pohozaev’s identity to prove the existence of ground state solutions to this class of Choquard equations,where the nonlinear term is considered to be almost optimal.Our results generalize the existence results of ground state solutions for Choquard equations with subcritical growth to the case of critical growth.In Chapter 3,we consider the existence of ground state solutions to the Gross-Pitaevskii equations in trapped dipolar quantum gases inR~2.Under certain assumptions,we prove the existence of the ground state solution to this class of Gross-Pitaevskii equations.Our results generalize the existence of ground state solutions to the Gross-Pitaevskii equations in trapped dipolar quantum gases inR~3to the case inR~2.