Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity

TL;DR

This paper investigates multiple localized solutions for a nonlinear Choquard equation with a broad class of nonlinearities, employing penalization methods without relying on common growth conditions.

Contribution

It introduces a novel approach to find multi-peak solutions for the Choquard equation without the usual monotonicity or Ambrosetti-Rabinowitz assumptions.

Findings

Existence of multi-peak solutions concentrating at potential minima

Solutions are obtained without monotonicity of f(s)/s

No need for Ambrosetti-Rabinowitz condition

Abstract

In this paper, we study a class of nonlinear Choquard type equations involving a general nonlinearity. By using the method of penalization argument, we show that there exists a family of solutions having multiple concentration regions which concentrate at the minimum points of the potential $V$. Moreover, the monotonicity of $f(s)/s$ and the so-called Ambrosetti-Rabinowitz condition are not required.