Standing waves with a critical frequency for nonlinear Choquard equations

We study the nonlocal Choquard equation −ε2Δuε+Vuε=(Iα∗|uε|p)|uε|p−2uε in RNwhere N≥1, Iα is the Riesz potential of order α∈(0,N) and ε>0 is a parameter. When the nonnegative potential V∈C(RN) achieves 0 with a homogeneous behavior or on the closure of an open set but remains bounded away from 0 at infinity, we show the existence of groundstate solutions for small ε>0 and exhibit the concentration behavior as ε→0.