The Ground State Solution Of The Nonlinear Fractional Schrödinger Equation (set)

In this paper,we investigate the problems with fractional Laplacian.Systems involving fractional operators play an important role in the real world,and have a strong physical meaning that appears in many fields such as fractional quantum mechanics,finance,soft thin films and so on.Because of the appearance of the terms fractional Laplacian,which implies that problems are no longer a point-wise identity.This phenomenon provokes some mathematical difficulties,which makes the study of such problem particularly interesting.The first chapter is an introduction,which briefly introduce the background and the main results.In the second chapter,we investigate the nonlinear fractional Schr(?)dinger system with linear couplings as follows(?) where(-?)?,??(0,1)denotes the fractional Laplacian and ?>0 is the coupling parameter.Under some assumptions,we prove the existence of positive ground state solutions to the above system with the help of the method of Nehari manifold and concentration compactness Lemma.In the third chapter,we consider the existence of ground state solutions to the following fractional Schr(?)dinger-Poisson systems with a general potential(?) where(??)s and(-?)t denote the fractional Laplacian,0<s ?t<1 and 2s+ 2t>3,the potential V(x)is weakly differentiable and f?C(R,R).Under some assumptions on potential V(x)and f(u),a nontrivial ground state solutions of Nehari-Pohozaev type(u,?)is established through using a subtle approach developed by Jeanjean and global compactness Lemma.