Nodal solutions for the Choquard equation

TL;DR

This paper constructs minimal action nodal solutions for the Choquard equation using a new minimax principle and concentration-compactness techniques, highlighting differences from the nonlinear Schrödinger equation.

Contribution

It introduces a novel minimax method and concentration-compactness lemmas for finding nodal solutions to the Choquard equation, a nonlocal nonlinear PDE.

Findings

Constructed minimal action odd solutions for specific p ranges.

Developed new concentration-compactness lemmas for sign-changing sequences.

Highlighted the absence of such solutions in the nonlinear Schrödinger equation.

Abstract

We consider the general Choquard equations $$ -\Delta u + u = (I_\alpha \ast |u|^p) |u|^{p - 2} u $$ where $I_\alpha$ is a Riesz potential. We construct minimal action odd solutions for $p \in (\frac{N + \alpha}{N}, \frac{N + \alpha}{N - 2})$ and minimal action nodal solutions for $p \in (2,\frac{N + \alpha}{N - 2})$. We introduce a new minimax principle for least action nodal solutions and we develop new concentration-compactness lemmas for sign-changing Palais--Smale sequences. The nonlinear Schr\"odinger equation, which is the nonlocal counterpart of the Choquard equation, does not have such solutions.