Semi-classical states for the Choquard equation
(2015) Calculus of Variations and Partial Differential Equations — Vol. 52, n° 1-2, p. 199-235 (2014)
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- Abstract
- We study the nonlocal equation −ε²Δuε + Vuε = ε−α (Iα∗∣uε∣p)∣uε∣p−2uε in 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,α+22)/(N−2)+, or p>2 and lim inf ∣x∣→∞ V(x)∣x∣2>0, or p=2 and inf x∈ℝN 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.
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Citations
Moroz, V., & Van Schaftingen, J. (2015). Semi-classical states for the Choquard equation. Calculus of Variations and Partial Differential Equations, 52(1-2), 199-235. https://doi.org/10.1007/s00526-014-0709-x (Original work published 2014)