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Ground state of \(N\) coupled nonlinear Schrödinger equations in \(\mathbb R^n\), \(n \leq 3\). (English) Zbl 1119.35087

Summary: We establish some general theorems for the existence and nonexistence of ground state solutions of steady-state \(N\) coupled nonlinear Schrödinger 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 \(N=3\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
47J30 Variational methods involving nonlinear operators
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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