Nonlinear scalar field equation with competing nonlocal terms

TL;DR

This paper establishes the existence of radial and nonradial solutions for a nonlinear scalar field equation involving competing nonlocal terms, modeling attractive and repulsive interactions, under broad assumptions.

Contribution

It introduces new existence results for solutions to a nonlocal scalar field equation with competing terms, extending previous work to more general conditions.

Findings

Existence of radial solutions.

Existence of nonradial solutions.

Modeling of attractive and repulsive interactions.

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.