Fractional Schrödinger equations with mixed nonlinearities: asymptotic profiles, uniqueness and nondegeneracy of ground states

Abstract

We study the fractional Schrödinger equations with a vanishing parameter: $$(-\Delta {)}^{s}u+u=|u{|}^{p-2}u+\lambda |u{|}^{q-2}u\text{ in }{\mathbb{R}}^N,u\in H^s({\mathbb{R}}^N),$$ where $s\in (0,1)$, $N>2s$, $2<q<p\le 2_s^{\ast }=\frac{2N}{N-2s}$ 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 $2_s^{\ast }$. For $p<2_s^{\ast }$, the ground state solution asymptotically coincides with unique positive ground state solution of $(-\Delta {)}^{s}u+u=u^p$, whereas for $p=2_s^{\ast }$ 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.

37 pages