Classification of Positive Ground State Solutions with Different Morse Indices for Nonlinear N-Coupled Schrödinger System

In this paper,we study the following N-coupled nonlinear Schrodinger system■,wheren≤3,N≥3,uj>0,βi,j=βj,i>0 are constants andβj,j=μj,j=1,...,N.There have been intensive studies for the system on existence/non-existence and clas-sification of ground state solutions when N=2.However fewer results about the classification of ground state solution are available for N≥3.In this paper,we first give a complete classification result on ground state solutions with Morse indices 1,2 or 3 for three-coupled Schrodinger system.Then we generalize our results to N-coupled Schrodinger system for ground state solutions with Morse indices 1 and N.We show that any positive ground state solutions with Morse index 1 or Morse index N must be the form of(d1w,d2w,...,dNw)under suitable conditions,where w is the unique positive ground state solution of certain equation.Finally,we generalize our results to fractional N-coupled Schrödinger system.