Nonlinear scalar field equation with competing nonlocal terms

Abstract

We find radial and nonradial solutions to the following nonlocal problem $$-\Delta u+\omega u=(I_{\alpha }\ast F(u))f(u)-(I_{\beta }\ast G(u))g(u)\text{ in }{\mathbb{R}}^N$$ under general assumptions, in the spirit of Berestycki and Lions, imposed on $f$ and $g$, where $N\ge 3$, $0\le \beta \le \alpha <N$, $\omega \ge 0$, $f,g:\mathbb{R}\to \mathbb{R}$ are continuous functions with corresponding primitives $F,G$, and $I_{\alpha },I_{\beta }$ are the Riesz potentials. If $\beta >0$, then we deal with two competing nonlocal terms modelling attractive and repulsive interaction potentials.

18 pages, to appear in Nonlinearity