Ground states and non-existence results for Choquard type equations with lower critical exponent and indefinite potentials. (August 2020) - Wu, Qingfang; Qin, Dongdong; Chen, Jing
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.