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 $$(-\Delta {)}^{s}u+\mu u=(I_{\alpha }\ast F(u))f(u)on{\mathbb{R}}^N$$ where $s\in (0,1)$, $N\ge 2$, $\alpha \in (0,N)$, $\mu >0$, $I_{\alpha }$ 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.

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• Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities