Nonlinear scalar field equation with competing nonlocal terms

Abstract

We find radial and nonradial solutions to the following nonlocal problem $$-\Delta u +\omega u= \big(I_\alpha\ast F(u)\big)f(u)-\big(I_\beta\ast G(u)\big)g(u) \text{ in } \mathbb{R}^N$$ under general assumptions, in the spirit of Berestycki and Lions, imposed on $f$ and $g$, where $N\geq 3$, $0\leq \beta \leq \alpha<N$, $\omega\geq 0$, $f,g:\mathbb{R}\to \mathbb{R}$ 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.