Standing waves with a critical frequency for nonlinear Choquard equations

TL;DR

This paper investigates the existence and concentration of standing wave solutions for a class of nonlinear nonlocal Choquard equations with a critical frequency, analyzing how solutions behave as a small parameter approaches zero.

Contribution

It establishes the existence of groundstate solutions for small parameters and describes their concentration behavior in the presence of a potential that vanishes or is bounded away from zero.

Findings

Existence of groundstate solutions for small epsilon

Concentration behavior as epsilon approaches zero

Solutions adapt to potential's vanishing or bounded nature

Abstract

In this paper, we study the nonlocal Choquard equation $$ -\varepsilon^2 \Delta u_\varepsilon + V u_\varepsilon= (I_\alpha * |u_\varepsilon|^p)|u_\varepsilon|^{p-2}u_\varepsilon $$ where $N\geq 1$, $I_\alpha$ is the Riesz potential of order $\alpha \in (0, N)$ and $\varepsilon>0$ is a parameter. When the nonnegative potential $V\in C (\mathbb{R}^N)$ achieves $0$ with a homogeneous behaviour or on the closure of an open set but remains bounded away from $0$ at infinity, we show the existence of groundstate solutions for small $\varepsilon>0$ and exhibit the concentration behaviour as $\varepsilon\to 0$.