Semi-classical states for the Choquard equation - Moroz, Vitaly; Van Schaftingen, Jean (Calculus of variations and partial differential equations, 2015)
We study the nonlocal equation - ε 2 Δ u ε + V u ε = ε - α ( I α ∗ | u ε | p ) | u ε | p - 2 u ε in R N , where N ≥ 1 , α ∈ ( 0 , N ) , I α ( x ) = A α / | x | N - α is the Riesz potential and ε > 0 is a small parameter. We show that if the external potential V ∈ C ( R N ; [ 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 , α + 2 2 ) / ( N - 2 ) + , or p > 2 and lim inf | x | → ∞ V ( x ) | x | 2 > 0 , or p = 2 and inf x ∈ R N V ( 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.