Fractional Schr\"odinger equations with mixed nonlinearities: asymptotic profiles, uniqueness and nondegeneracy of ground states

Abstract

We study the fractional Schr\"odinger equations with a vanishing parameter: $$ (-\Delta)^s u+u =|u|^{p-2}u+\lambda|u|^{q-2}u \text{ in }\mathbb{R}^N,\quad u \in H^s(\mathbb{R}^N),$$ where $s\in(0,1)$, $N>2s$, $2<q<p\leq 2^*_s=\frac{2N}{N-2s}$ 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 $(-\Delta)^s u+u=u^p$, 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.