We study the fractional Schrödinger equations with a vanishing parameter: where \(s\in (0,1)\) , \(N>2s\) , \(2<q<p\le 2^*_s=\frac{2N}{N-2s}\) are fixed parameters and \(\lambda >0\) is a vanishing parameter. We investigate the asymptotic behaviour of positive ground state solutions for \(\lambda \) small, when p is subcritical, or critical Sobolev exponent \(2^*_s\) . For \(p<2_s^*\) , the ground state solution asymptotically coincides with unique positive ground state solution of \((-\Delta )^s u+u=|u|^{p-2}u\) , whereas for \(p=2_s^*\) the asymptotic behaviour of the solutions, after a rescaling, is given by the unique positive solution of the nonlocal critical Emden-Fowler type equation. Additionally, for \(\lambda >0\) small, we show the uniqueness and nondegeneracy of the positive ground state solution using these asymptotic profiles of solutions.
