*preprint*

**Inserted:** 15 may 2024

**Year:** 2023

**Abstract:**

Goal of this paper is to study the asymptotic behaviour of the solutions of
the following doubly nonlocal equation $$(-\Delta)^{s} u + \mu u =
(I_{{\alpha}F}**(u))f(u) \quad \hbox{on $\mathbb{R}^N$}$$ where $s \in (0,1)$,
$N\geq 2$, $\alpha \in (0,N)$, $\mu>0$, $I_{\alpha}$ denotes the Riesz
potential and $F(t) = \int_0^t f(\tau) d \tau$ is a general nonlinearity with a
sublinear growth in the origin. The found decay is of polynomial type, with a
rate possibly slower than $\sim\frac{1}{x^{N+2s}}$. The result is new even
for homogeneous functions $f(u)=u^{r-2}u$, $r\in [\frac{N+\alpha}{N},2)$, and
it complements the decays obtained in the linear and superlinear cases in
D'Avenia, Siciliano, Squassina (2015) and Cingolani, Gallo, Tanaka (2022).
Differently from the local case $s=1$ in Moroz, Van Schaftingen (2013), new
phenomena arise connected to a new "$s$-sublinear" threshold that we detect on
the growth of $f$. To gain the result we in particular prove a Chain Rule type
inequality in the fractional setting, suitable for concave powers.
**