Standing waves with a critical frequency for nonlinear Choquard equations

Abstract

In this paper, we study the nonlocal Choquard equation $$-{\epsilon }^2\Delta u_{\epsilon }+V{u}_{\epsilon }=(I_{\alpha }\ast |u_{\epsilon }{|}^p)|u_{\epsilon }{|}^{p-2}{u}_{\epsilon }$$ where $N\ge 1$, $I_{\alpha }$ is the Riesz potential of order $\alpha \in (0,N)$ and $\epsilon >0$ 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 $\epsilon >0$ and exhibit the concentration behaviour as $\epsilon \to 0$.

22 pages

References in corpus (3)

• A guide to the Choquard equation
• Gravitation and quantummechanical localization of macroobjects
• Choquard equations under confining external potentials

Cited by in corpus (1)

• Semiclassical states for Choquard type equations with critical growth: critical frequency case