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.