Ground State of N Coupled Nonlinear Schr�dinger Equations in Rn,n?3

Abstract

We establish some general theorems for the existence and nonexistence of ground state solutions of steady-state N coupled nonlinear Schrodinger equations. The sign of coupling constants βij’s is crucial for the existence of ground state solutions. When all βij’s are positive and the matrix Σ is positively definite, there exists a ground state solution which is radially symmetric. However, if all βij’s are negative, or one of βij’s is negative and the matrix Σ is positively definite, there is no ground state solution. Furthermore, we find a bound state solution which is non-radially symmetric when N=3.