Nonlinear scalar field equation with competing nonlocal terms * *d’Avenia and Pomponio are members of GNAMPA of INdAM and are partially supported by PRIN 2017JPCAPN Qualitative and quantitative aspects of nonlinear PDEs. d’Avenia is also supported by FRA2019 of Politecnico di Bari. Mederski was partially supported by the National Science Centre, Poland, Grant No. 2017/26/E/ST1/00817 and by the Deutsche Forschungsgemeinschaft

Abstract

We find radial and nonradial solutions to the following nonlocal problem under general assumptions, in the spirit of Berestycki and Lions, imposed on f and g, where N ⩾ 3, 0 ⩽ β ⩽ α < N, ω ⩾ 0, are continuous functions with corresponding primitives F, G, and I α , I β are the Riesz potentials. If β > 0, then we deal with two competing nonlocal terms modelling attractive and repulsive interaction potentials.