TY - JOUR

T1 - Semi-classical states for the nonlinear Choquard equations

T2 - Existence, multiplicity and concentration at a potential well

AU - Cingolani, Silvia

AU - Tanaka, Kazunaga

N1 - Funding Information:
Authors started this research during the second author?s visit to Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari in 2016 and the first author?s visit to Department of Mathematics, Waseda University in 2017. They would like to thank Politecnico di Bari and Waseda University for their kind hospitality. The first author is partially supported by INdAM-GNAMPA Project 2017 ?Metodi matematici per lo studio di fenomeni fisici nonlineari?, and by PRIN 2017JPCAPN ?Qualitative and quantitative aspects of nonlinear PDEs?. The second author is partially supported by JSPS KAKENHI Grants No. JP25287025, JP17H02855, JP16K13771, JP26247014, JP18KK0073, JP19H00644, and NSFC-JSPS bilateral joint research project ?Variational study of nonlinear PDEs?.
Publisher Copyright:
© European Mathematical Society.

PY - 2019

Y1 - 2019

N2 - We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation −ε2Δv + V (x) v = ε 1 α (Iα ∗ F(v))f(v) in RN , where N ≥ 3, α ∈ (0, N), Iα(x) = Aα/|x|N-α is the Riesz potential, F ∈ C1(R, R), F´(s) = f(s) and ε > 0 is a small parameter. We develop a new variational approach and we show the existence of a family of solutions concentrating, as ε → 0, to a local minima of V (x) under general conditions on F(s). Our result is new also for f(s) = |s|p-2s and applicable for p ∈ (N N +α, N N + - α 2 ). Especially, we can give the existence result for locally sublinear case p ∈ (N N +α , 2), which gives a positive answer to an open problem arisen in recent works of Moroz and Van Schaftingen. We also study the multiplicity of positive single-peak solutions and we show the existence of at least cupl(K)+1 solutions concentrating around K as ε → 0, where K ⊂ Ω is the set of minima of V (x) in a bounded potential well Ω, that is, m0 ≡ infxεΩ V (x) < infxε∂Ω V (x) and K = {x ∈ Ω; V (x) = m0}.

AB - We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation −ε2Δv + V (x) v = ε 1 α (Iα ∗ F(v))f(v) in RN , where N ≥ 3, α ∈ (0, N), Iα(x) = Aα/|x|N-α is the Riesz potential, F ∈ C1(R, R), F´(s) = f(s) and ε > 0 is a small parameter. We develop a new variational approach and we show the existence of a family of solutions concentrating, as ε → 0, to a local minima of V (x) under general conditions on F(s). Our result is new also for f(s) = |s|p-2s and applicable for p ∈ (N N +α, N N + - α 2 ). Especially, we can give the existence result for locally sublinear case p ∈ (N N +α , 2), which gives a positive answer to an open problem arisen in recent works of Moroz and Van Schaftingen. We also study the multiplicity of positive single-peak solutions and we show the existence of at least cupl(K)+1 solutions concentrating around K as ε → 0, where K ⊂ Ω is the set of minima of V (x) in a bounded potential well Ω, that is, m0 ≡ infxεΩ V (x) < infxε∂Ω V (x) and K = {x ∈ Ω; V (x) = m0}.

KW - Non-local nonlinearities

KW - Nonlinear Choquard equation

KW - Positive solutions

KW - Potential well

KW - Relative cup-length

KW - Semiclassical states

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U2 - 10.4171/rmi/1105

DO - 10.4171/rmi/1105

M3 - Article

AN - SCOPUS:85074890458

VL - 35

SP - 1885

EP - 1924

JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

SN - 0213-2230

IS - 6

ER -