Properties of the minimizers for a constrained minimization problem arising in fractional NLS system

Abstract

In this paper, we study a fractional NLS system with trapping potentials in ${\mathbb {R}}^{2}$. By constructing a constrained minimization problem, we show that minimizers exist for the minimization problem if and only if the attractive interaction strength $a_{i}<a^{*}{:}{=}\Vert Q\Vert _{2}^{2s}$, where $i=1, 2$ and Q is the unique positive radial solution of $(-\Delta )^{s}u+su-|u|^{2s}u=0$ in ${\mathbb {R}}^{2}$, $s\in (0, 1)$. Moreover, by analyzing some precise energy estimates, we obtain the concentration and blow-up behavior for the minimizers of the minimization problem as $(a_{1}, a_{2})\nearrow (a^{*}, a^{*})$. Comparing to the NLS system and fractional NLS equation, we encounter some new difficulties because of the nonlocal nature of the fractional Laplace. One of the main difficulties is that the energy functional is changed, we have to develop a suitable trial function to do some precise integral computation for the energy of minimization problem. Another difficulty is given by the fact that the Q(x) is polynomially decay at infinity, which is in contrast to the fact that the ground state exponentially decays at infinity in $s=1$, we need to give a more detailed proof to establish the best estimate of the trial function. The last major difficulty lies in the decay estimates of the sequences of solution to the nonlocal problem at infinity are different from those in the case of the classical local problem, we must build decay estimates for nonlocal operators.