Asymptotic decay of solutions for sublinear fractional Choquard equations

Goal of this paper is to study the asymptotic behaviour of the solutions of the following doubly nonlocal equation (−Δ)su+μu=(Iα∗F(u))f(u)onRNwhere s∈(0,1), N≥2, α∈(0,N), μ>0, Iα denotes the Riesz potential and F(t)=∫0tf(τ)dτ is a general nonlinearity with a sublinear growth in the origin. The found decay is of polynomial type, with a rate possibly slower than ∼1|x|N+2s. The result is new even for homogeneous functions f(u)=|u|r−2u, r∈[N+αN,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.