Ground states and non-existence results for Choquard type equations with lower critical exponent and indefinite potentials

Abstract

In this paper, we study following Choquard equation with lower critical exponent: −Δu+V(x)u=(Iα∗|u|αN+1)|u|αN−1u+f(x,u),x∈RN,u∈H1(RN),where N≥1, Iα is the Riesz potential, and V:RN→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.