Article Contents
Article Contents

# A complete classification of ground-states for a coupled nonlinear Schrödinger system

• In this paper, we establish the existence of nontrivial ground-state solutions for a coupled nonlinear Schrödinger system

$-\Delta u_j+ u_j=\sum\limits_{i=1}^mb_{ij}u_i^2u_j, \quad\text{in}\ \mathbb{R}^n,\\ u_j(x)\to 0\ \text{as}\ |x|\ \to \infty, \quad j=1,2,\cdots, m,$

where $n=1, 2, 3, m\geq 2$ and $b_{ij}$ are positive constants satisfying $b_{ij}=b_{ji}.$ By nontrivial we mean a solution that has all components non-zero. Due to possible systems collapsing it is important to classify ground state solutions. For $m=3$, we get a complete picture that describes whether nontrivial ground-state solutions exist or not for all possible cases according to some algebraic conditions of the matrix $B = (b_{ij})$. In particular, there is a nontrivial ground-state solution provided that all coupling constants $b_{ij}, i\neq j$ are sufficiently large as opposed to cases in which any ground-state solution has at least a zero component when $b_{ij}, i\neq j$ are all sufficiently small. Moreover, we prove that any ground-state solution is synchronized when matrix $B=(b_{ij})$ is positive semi-definite.

Mathematics Subject Classification: Primary: 35J20, 35J47; Secondary: 35J50.

 Citation:

• Table 1.  The number of non-zero components of ground-state solutions

 case condition 1 condition 2 type 1 $det(B) > 0$ $\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3 > \sqrt{\mu_1\mu_2}$ $p= 3$ 2 $det(B) > 0$ $\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3 < \sqrt{\mu_1\mu_2}$ $p=1$ 3 $det(B) < 0$ $p=1, 2$ 4 $rank(B)=1$ $p=1, 2, 3$ 5 $rank(B)=2$ $\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3=\sqrt{\mu_1\mu_2}$ $p=2, 3$ 6 $rank(B)=2$ $\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3=\sqrt{\mu_1\mu_2}$ $p=1, 2$ 7 $rank(B)=2$ $\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3 > \sqrt{\mu_1\mu_2}$ $p=1, 2, 3$ 8 $rank(B)=2$ $\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3 < \sqrt{\mu_1\mu_2}$ $p=1$
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