Existence And Orbital Stability Of Standing Waves For A Class Of Fractional Schr?dinger Equations

This paper mainly discusses the existence of the standing waves for a class of nonlinear fractional Schr ¨odinger equations and its orbit stability problems.Throughout the paper,the main tool we use is the variational method.This paper is mainly composed of five parts.In the first chapter,We describe the background and the main results of this paper.In the second chapter,we give some basic knowledge and necessary symbolic descriptions.In the third chapter,we consider the following nonlinear fractional Schr ¨odinger equationsin the case of μ= 0.Where 0<α<1,i=（-1）^1/2,N ≥ 1,（H₀）:μ₁,μ₂,β>0,2<p₁,p₂<2+4α/N,r₁,r₂>1,r₁+r₂<2+4α/N.We prove that in the case of μ = 0,the corresponding energy functionals have lower bounds under constraints,for this we introduce a global minimization problem,and this very small element is the solution of this equations.Here we use the concentrationcompactness principle,which proves that any minimization sequence is tight,and the very small element is orbitally stable.In the fourth chapter,we prove the nature of the solution in the case of 0<μ<（?）=（（N-2）/2）².We have studied that the corresponding energy functionals have lower bounds under constraints,for concentration-compactness principle,we established that the compactness of any minimization sequence and the orbital stability of its minimal elements.In the fifth chapter,the conclusions and problems of this paper are given.