Ground state of N coupled nonlinear Schrodinger equations in Rn, n ≤ 3

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 beta(ij)'s is crucial for the existence of ground state solutions. When all beta(ij)'s are positive and the matrix Sigma is positively definite, there exists a ground state solution which is radially symmetric. However, if all beta(ij)'s are negative, or one of beta(ij)'s is negative and the matrix Sigma is positively definite, there is no ground state solution. Furthermore, we find a bound state solution which is non-radially symmetric when N = 3.