Infinitely many bound states for Choquard equations with local nonlinearities. (December 2019)

• Record Type: Journal Article
• Authors: Li, Xinfu; Liu, Xiaonan; Ma, Shiwang
• Is Part Of: Nonlinear analysis. Volume 189(2019)
• Publication Date: 2019-12
• Subjects: 35J20 -- 35J60; Multiplicity -- Choquard equations -- Bound states; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111583
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we consider the following Choquard equation ( CH ) − Δ u + u = ( I α ∗ | u | p ) | u | p − 2 u + V ( x ) | u | q − 2 u in R N, u ∈ H 1 ( R N ), where N ≥ 3, α ∈ ( ( N − 4 ) +, N ), p ∈ [ 2, N + α N − 2 ), q ∈ ( 2, 2 N N − 2 ) ∩ ( 1 + ( p − 1 ) ( N − 2 ) α N 2, 1 + 2 N ( N − α ) + N 2 ( p − 1 ) ( N − 2 ) α ) and I α is the Riesz potential. Under some suitable decay assumptions but without any symmetry property on V ( x ), we prove that the problem has infinitely many solutions, whose energy can be arbitrarily large.