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

Abstract

There have been intensive studies for the system on existence/non-existence and classification 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 Schrödinger system. Then we generalize our results to Ncoupled Schrödinger 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 (d 1 w, d 2 w, • • • , d N w) 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.