Multiple-peak Semi-clasical States For The Choquard Equation

We study the nonlocal equation where N≥1,α∈(0.N),p∈[2,(N+α)/(N-2)+),Iα(x)=Aα/｜x｜N-α is the Riesz potential,V∈C(RN;[0.∞))is the external potential which has K(≥1)lo-cal mimimums and ε>0 is a small parameter. We show that the problem has a family of solutions concentrating to the K mimimums of V provided that:either p>1+max(α.α+2/2)/(N-2)+,or p>2 and lim inf｜x｜â†'∞V(x)｜x｜2>0.or p=2 and infx∈RN V(x)(1+｜x｜N-α)>0.The proof uses variational methods and a nonlocal penalization technique developed by Vitaly Moroz and Jean Van Schaftingen in[8].