Existence of groundstates for a class of nonlinear Choquard equations

TL;DR

This paper proves the existence of groundstate solutions for a class of nonlinear Choquard equations involving Riesz potentials, under conditions on the nonlinearity, and establishes symmetry and sign properties of these solutions.

Contribution

It demonstrates the existence of groundstate solutions for nonlinear Choquard equations with minimal assumptions on the nonlinearity, extending previous results.

Findings

Existence of nontrivial solutions under general conditions.

Solutions are radially symmetric and of constant sign when nonlinearity is even and monotone.

Groundstates are proven to exist for a broad class of nonlinearities.

Abstract

We prove the existence of a nontrivial solution (u \in H^1 (\R^N)) to the nonlinear Choquard equation [- \Delta u + u = \bigl(I_\alpha \ast F (u)\bigr) F' (u) \quad \text{in (\R^N),}] where (I_\alpha) is a Riesz potential, under almost necessary conditions on the nonlinearity (F) in the spirit of Berestycki and Lions. This solution is a groundstate; if moreover (F) is even and monotone on ((0,\infty)), then (u) is of constant sign and radially symmetric.