Semi-classical states for the Choquard equation

TL;DR

This paper investigates semi-classical states for a nonlocal Choquard equation, establishing existence and concentration of solutions near potential minima using variational methods and a new nonlocal penalization technique.

Contribution

The authors develop a novel nonlocal penalization method and prove the existence of concentrating solutions for the Choquard equation under optimal decay and parameter conditions.

Findings

Solutions concentrate near local minima of the potential V.

The method applies for a wide range of p and decay conditions on V.

The nonlocal penalization technique is new and effective for such problems.

Abstract

We study the nonlocal equation $$-\varepsilon^2 \Delta u_\varepsilon + V u_\varepsilon = \varepsilon^{-\alpha} \bigl(I_\alpha \ast \lvert u_\varepsilon\rvert^p\bigr) \lvert u_\varepsilon \rvert^{p - 2} u_\varepsilon\quad\text{in \(\mathbf{R}^N\)}, $$ where $N \ge 1$, $\alpha \in (0, N)$, $I_\alpha (x) = A_\alpha/\lvert x \rvert^{N - \alpha}$ is the Riesz potential and $\varepsilon > 0$ is a small parameter. We show that if the external potential $V \in C (\mathbb{R}^N; [0, \infty))$ has a local minimum and $p \in [2, (N + \alpha)/(N - 2)_+)$ then for all small $\varepsilon > 0$ the problem has a family of solutions concentrating to the local minimum of $V$ provided that: either $p > 1 + \max (\alpha, \frac{\alpha + 2}{2})/(N - 2)_+$, or $p > 2$ and $\liminf_{\lvert x\rvert \to \infty} V (x) \lvert x \rvert^2 > 0$, or $p = 2$ and $\inf_{x \in \mathbb{R}^N} V (x) (1 + \lvert x \rvert^{N - \alpha}) > 0$. Our assumptions on the decay of $V$ and admissible range of $p\ge 2$ are optimal. The proof uses variational methods and a novel nonlocal penalization technique that we develop in this work.