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 (-Delta)(s)u + mu u = (I-alpha * F(u))f(u) on R-N where s is an element of (0, 1), N >= 2, alpha is an element of (0, N), mu > 0, I-alpha denotes the Riesz potential and F(t) = integral(t)(0) 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 similar to 1/vertical bar x vertical bar(N+2s). The result is new even for homogeneous functions f(u) = vertical bar u vertical bar(r-2) u, r is an element of[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.