We survey old and recent results dealing with the existence and properties of
solutions to the Choquard type equations −Δu+V(x)u=(∣x∣−(N−α)∗∣u∣p)∣u∣p−2uin RN, and some of its variants and extensions.Comment: 39 page
We study the nonlocal equation −ε2Δuε+Vuε=ε−α(Iα∗∣uε∣p)∣uε∣p−2uεin RN, where N≥1, α∈(0,N), Iα(x)=Aα/∣x∣N−α is the Riesz
potential and ε>0 is a small parameter. We show that if the
external potential V∈C(RN;[0,∞)) has a local minimum
and p∈[2,(N+α)/(N−2)+) then for all small ε>0
the problem has a family of solutions concentrating to the local minimum of V
provided that: either p>1+max(α,2α+2)/(N−2)+,
or p>2 and liminf∣x∣→∞V(x)∣x∣2>0, or p=2 and infx∈RNV(x)(1+∣x∣N−α)>0. Our assumptions on the decay of V and admissible range of
p≥2 are optimal. The proof uses variational methods and a novel nonlocal
penalization technique that we develop in this work.Comment: 28 pages, updated bibliograph