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

TL;DR

This paper analyzes the asymptotic behavior, uniqueness, and nondegeneracy of positive ground state solutions for fractional Schr"odinger equations with mixed nonlinearities as the parameter  approaches zero, covering subcritical and critical cases.

Contribution

It provides a detailed asymptotic analysis of ground states for fractional Schr"odinger equations with mixed nonlinearities, establishing their uniqueness and nondegeneracy for small parameter values.

Findings

Ground states asymptotically match solutions of simpler equations for small .

For subcritical p, solutions converge to the ground state of the base fractional Schrd6dinger equation.

For critical p, solutions exhibit a specific rescaled asymptotic profile.

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.