Existence and decays of solutions for fractional Schr\"{o}dinger equations with decaying potentials

TL;DR

This paper investigates the existence and decay behavior of positive solutions for fractional Schrödinger equations with decaying potentials, establishing explicit thresholds for the nonlinearity exponent that determine solution existence.

Contribution

It introduces a uniform penalization method combined with comparison principles to identify explicit thresholds for solution existence based on potential decay rates.

Findings

Existence of solutions for p in (p_*, 2_s^*) under certain decay conditions.

Non-existence of solutions for p in (2, p_*) when p_* > 2.

Decay properties of solutions depend on the decay rate of the potential.

Abstract

We revisit the following fractional Schr\"{o}dinger equation \begin{align}\label{1a} \varepsilon^{2s}(-\Delta)^su +Vu=u^{p-1},\,\,\,u>0,\ \ \ \mathrm{in}\ \R^N, \end{align} where $\varepsilon>0$ is a small parameter, $(-\Delta)^s$ denotes the fractional Laplacian, $s\in(0,1)$, $p\in (2, 2_s^*)$, $2_s^*=\frac {2N}{N-2s}$, $N>2s$, $V\in C\big(\R^N, [0, +\infty)\big)$ is a potential. Under various decay assumptions on $V$, we introduce a uniform penalization argument combined with a comparison principle and iteration process to detect an explicit threshold value $p_*$, such that the above problem admits positive concentration solutions if $p\in (p_*, \,2_s^*)$, while it has no positive weak solutions for $p\in (2,\,p_*)$ if $p_*>2$, where the threshold $p_*\in [2, 2^*_s)$ can be characterized explicitly by \begin{equation*}\label{qdj111} p_*=\left\{\begin{array}{l} 2+\frac {2s}{N-2s} \ \ \ \text { if } \lim\limits_{|x| \to \infty} (1+|x|^{2s})V(x)=0,\vspace{1mm} 2+\frac {\omega}{N+2s-\omega} \text { if } 0<\inf (1+|x|^\omega)V(x)\le \sup (1+|x|^\omega)V(x)< \infty \text { for some } \omega \in [0, 2s],\vspace{1mm} 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text { if } \inf V(x)\log(e+|x|^2)>0. \end{array}\right. \end{equation*} Moreover, corresponding to the various decay assumptions of $V(x)$, we obtain the decay properties of the solutions at infinity.