### Abstract

We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation −ε^{2}Δv + V (x) v = _{ε} ^{1} _{α} (Iα ∗ F(v))f(v) in R^{N} , where N ≥ 3, α ∈ (0, N), Iα(x) = Aα/|x|^{N}-^{α} is the Riesz potential, F ∈ C^{1}(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}-^{2}s 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) < inf_{x}ε∂_{Ω} V (x) and K = {x ∈ Ω; V (x) = m0}.

Original language | English |
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Pages (from-to) | 1885-1924 |

Number of pages | 40 |

Journal | Revista Matematica Iberoamericana |

Volume | 35 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2019 |

### Keywords

- Non-local nonlinearities
- Nonlinear Choquard equation
- Positive solutions
- Potential well
- Relative cup-length
- Semiclassical states

### ASJC Scopus subject areas

- Mathematics(all)