Asymptotic decay of solutions for sublinear fractional Choquard equations

Abstract

Goal of this paper is to study the asymptotic behaviour of the solutions of the following doubly nonlocal equation ${(-\Delta )}^{s}u+\mu u=(I_{\alpha }\ast F\left(u\right))f\left(u\right)\text{on}{R}^N$where $s\in (0,1)$, $N\ge 2$, $\alpha \in (0,N)$, $\mu >0$, $I_{\alpha }$ denotes the Riesz potential and $F\left(t\right)={\int }_0^{t}f\left(\tau \right)d\tau$ is a general nonlinearity with a sublinear growth in the origin. The found decay is of polynomial type, with a rate possibly slower than $\sim \frac{1}{{\left|x\right|}^{N+2s}}$. The result is new even for homogeneous functions $f\left(u\right)={\left|u\right|}^{r-2}u$, $r\in [\frac{N+\alpha }{N},2)$, and it complements the decays obtained in the linear and superlinear cases in Cingolani et al. (2022); D’Avenia et al. (2015). Differently from the local case $s=1$ in Moroz and 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.