Semi-classical states for fractional Choquard equations with decaying potentials

This paper deals with the following fractional Choquard equation: [Formula: see text] where [Formula: see text] is a small parameter, [Formula: see text], [Formula: see text] is the fractional Laplacian, [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] is a Riesz potential, [Formula: see text] is an external potential. Under some assumptions on the decay rate of [Formula: see text] and the corresponding range of [Formula: see text], we prove that the problem has a family of solutions [Formula: see text] concentrating at a local minimum of [Formula: see text] as [Formula: see text]. Since the potential [Formula: see text] vanishes at infinity, we need to employ a type of penalized argument and implement delicate analysis on the double nonlocal terms to establish regularity, positivity and asymptotic behavior of [Formula: see text], which is totally different from the local case. As a contrast, we also develop some nonexistence results, which imply that the assumptions on [Formula: see text] and [Formula: see text] for the existence of [Formula: see text] are almost optimal. To prove our main results, a general strong maximum principle and comparison function for the weak solutions of fractional Laplacian equations are established.