Ground states and non-existence results for Choquard type equations with lower critical exponent and indefinite potentials. (August 2020)

• Record Type: Journal Article
• Authors: Wu, Qingfang; Qin, Dongdong; Chen, Jing
• Is Part Of: Nonlinear analysis. Volume 197(2020)
• Publication Date: 2020-08
• Subjects: 35J61 -- 35J20 -- 35B33; Choquard type equations -- Pohožaev identity -- Critical exponent -- Characterization of ground states -- Non-existence; 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.2020.111863
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we study following Choquard equation with lower critical exponent: − Δ u + V ( x ) u = ( I α ∗ | u | α N + 1 ) | u | α N − 1 u + f ( x, u ), x ∈ R N, u ∈ H 1 ( R N ), where N ≥ 1, I α is the Riesz potential, and V : R N → R allows to be sign-changing. Under some mild assumptions imposed on the nonlinearity f, we prove that above equation has a ground state solution for the periodic case and asymptotically periodic case, respectively. The characterization of the ground states is also investigated by a direct approach that are constrained to the Nehari manifold. Moreover, we show a non-existence result for the equation via a generalized Pohožaev identity established for the non-autonomous nonlinearity f . The results extend and improve some recent ones in the literature.